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SCUOLA NORMALE SUPERIORE - PISA
CLASSE DI SCIENZE
Tesi di perfezionamento in fisica
String Cosmology
Candidato:
Riccardo Sturani
Relatore:
Prof. Michele Maggiore
Aprile 2002
Ai miei genitori
Contents
Introduction
viii
Notations
xi
1 Elements of cosmology
1.1 The observed Universe . . . . . . . . . .
1.2 Standard cosmology . . . . . . . . . . .
1.3 Shortcomings of the standard cosmology
1.4 Inflation . . . . . . . . . . . . . . . . . .
1.4.1 Problems of inflationary models .
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2 Elements of string theory
2.1 The bosonic string . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Type II superstrings . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Heterotic string . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Type I superstring . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Conformal Killing groups for zero genus surfaces . . . . . . . . . .
2.6.1 The disk conformal Killing group . . . . . . . . . . . . . . .
2.6.2 The projective plane conformal Killing group . . . . . . . .
2.7 Low energy effective action . . . . . . . . . . . . . . . . . . . . . .
2.8 Compactification and T-duality . . . . . . . . . . . . . . . . . . . .
2.9 Some non-perturbative aspects . . . . . . . . . . . . . . . . . . . .
2.10 Extended objects’ tension and charge from tree level computations
2.10.1 D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10.2 Orientifold planes . . . . . . . . . . . . . . . . . . . . . . .
3 The
3.1
3.2
3.3
3.4
3.5
3.6
pre-big bang model
The model . . . . . . . . . . . . .
Phenomenological consequences .
Effect of α0 corrections . . . . . .
The issue of the initial conditions
Effects of a “stringy” phase . . .
Summary . . . . . . . . . . . . .
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vi
4 Supersymmetric vacuum configurations in string cosmology
4.1 The supergravity action . . . . . . . . . . . . . . . . . . . . . .
4.2 The supersymmetry conditions . . . . . . . . . . . . . . . . . .
4.3 Unbroken supersymmetry by fermion condensate . . . . . . . .
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Loop corrections and graceful exit
5.1 Supersymmetric action in four dimensions . .
5.2 Necessary condition for a graceful exit . . . .
5.3 The effective action with loop corrections . .
5.3.1 Terms with two derivatives . . . . . .
5.3.2 Terms with four derivatives . . . . . .
5.4 The cosmological evolution . . . . . . . . . .
5.4.1 The evolution without loop corrections
5.4.2 The effect of the loop-corrected Kähler
5.4.3 The effect of threshold corrections . .
5.5 Transition to a D-brane dominated regime . .
5.6 Conclusions . . . . . . . . . . . . . . . . . . .
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. 132
. 136
6 The
6.1
6.2
6.3
6.4
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potential
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generalized second law of thermodynamics in string cosmology
Entropy bounds and geometric entropy . . . . . . . . . . . . . . . . . . .
Geometric and quantum entropy . . . . . . . . . . . . . . . . . . . . . .
The generalized second law . . . . . . . . . . . . . . . . . . . . . . . . .
Application to the pre-big bang scenario . . . . . . . . . . . . . . . . . .
7 Higgs-graviscalar mixing in type I string theory
7.1 The large extra dimension scenario . . . . . . . .
7.2 Branons’ effective action . . . . . . . . . . . . . .
7.3 Non-Abelian generalization . . . . . . . . . . . .
7.3.1 One open-one closed string on the disk . .
7.3.2 Two open-one closed string amplitude . .
7.4 Conformal invariance . . . . . . . . . . . . . . . .
7.5 Higgs-graviscalar mixing . . . . . . . . . . . . . .
7.6 Two open string cylinder amplitude . . . . . . .
7.7 Higgs on branes intersection . . . . . . . . . . . .
7.8 Conclusions . . . . . . . . . . . . . . . . . . . . .
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8 Conclusions
137
Acknowledgments
139
A Perturbations in inflationary cosmology
A.1 The Bogolubov coefficients . . . . . . . .
A.2 Density perturbations . . . . . . . . . .
A.3 Particle perturbations . . . . . . . . . .
A.3.1 Quantum description . . . . . . .
A.3.2 Classical description . . . . . . .
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vii
B The moduli problem
152
C Superstring
155
D Theta functions
157
E Supersymmetry transformation on PBB solutions
159
Bibliography
161
Introduction
In the present work we summarize our research activity in the field of string phenomenology. In particular we focus our attention on the pre-big bang scenario, first proposed by
Veneziano and Gasperini about ten years ago. 1
A problem of standard cosmology is to explain the observed high degree of isotropy
of the Universe, which can be dynamically realized if the Universe underwent a period
of inflation, i.e. accelerated expansion, during its early stage of evolution. Inflation
is naturally incorporated in the pre-big bang scenario as this model predicts that the
standard cosmological evolution is dual to an inflating one, which takes place before the
big bang. The big bang itself is interpreted as the natural outcome of the Universe starting
from generic initial conditions of the pre-big bang type, i.e. with asimptotically vanishing
curvature and coupling, eventually evolving towards higher curvature and coupling. This
pre-big bang phase can be mapped by a duality transformation into an almost standard
Friedmann-Robertson-Walker expansion, which could then be the natural outcome of the
primordially inflating Universe.
Actually a matching between the pre-big bang phase and the classical FriedmannRobertson-Walker expansion has not been achieved and what separates the two phases is
nothing but the big bang singularity. Our study points to the direction of realizing this
matching, being our goal to exhibit a fully viable cosmological model wich will have several
attracting theoretical and phenomenological features, like for instance the possibility of
naturally incorporating inflation, without ad hoc potentials and fields.
The approach we propose here is to tackle the issue of the big bang singularity by studying the low energy effective action of the string massless modes. In our first investigation
about the smoothing of the singularity we consider the effect of imposing supersymmetry on the classical background field configuration describing the pre-big bang Universe.
Supersymmetry is preserved if the variations of the fields under a supersymmetric transformation vanish and we found a particular supersymmetry preserving solution to the cosmological equations of motion which is nonsingular, by admitting that a dilatino-gravitino
condensate appears.
After this encouraging result we turned to a sistematic study of the string-derived effective actions, which get two kinds of corrections to the lowest level form: higher derivative
terms, which take account of the massive modes of the string, controlled by the string
constant α0 , and quantum loop corrections, controlled by the dilaton expectation value.
In general only the first terms of the perturbative expansion of both α 0 and loop corrections are known, but supersymmetry can improve the situation, allowing to know the
1
An update collection of
http://www.ba.infn.it/˜gasperin.
papers
on
the
viii
pre-big
bang
scenario
is
available
at
ix
correct form of the effective action to all order in the loop counting paramater, even if
still at lowest order in α0 . The result we shall show is that corrections to the low energy
effective action go in the right direction towards the regularization of the cosmological
solution, but they also drive the parameters towards the strong coupling regime, where
new non-perturbative degrees of freedom may be relevant and play a key role in ensuring
a fully viable cosmological model.
From the analysis of the low energy effective action we get a sensible picture of the
cosmological evolution but not free of difficulties, then in a work with Ram Brustein we
devoted our attention to general arguments relying on thermodynamics considerations.
We started from the idea that that it is possible to associate an entropy, defined geometric
because it is associated to the structure of space-time itself, to a general spacetime with
horizons, related to the fundamental lack of knowledge of the physics beyond the horizon.
We found that any “would be” singular cosmological solution violates the generalized
second law of thermodynamics before reaching the singularity, and adding to the geometric
entropy a second source of entropy represented by quantum field fluctuactions, which
turns out to be decreasing during an inflating phase, consistency with the second law of
thermodynamics requires the pre-big bang inflating solution to be driven into a phase with
decreasing curvature when the coupling becomes of order one.
Then we turn towards a more field theoretical issue, still in the context of string theory,
where the required existence of more than four dimensions makes natural the appearance
of lower dimensional objects, D-branes, living in a higher dimensional background. Out of
the plethora of new interesting phenomena this might lead to, we focused in the work we
developed with Ignatios Antoniadis on the effect resulting by admitting that the Standard
Model Higgs is described by open strings attached to a D-brane, while particles interacting
only gravitationally are free to move in the full space. This effect has been considered
within the framework of the large extra dimensions scenario proposed by Antoniadis,
Arkani-Hamed, Dimopoulos and Dvali. We shall exhibit a string setup for computing the
Higgs-graviscalar mixing, i.e. the mixing amplitude between a standard model particle and
a particle with purely gravitational interaction, that may lead to a phenomenologically
interesting invisible width of the Higgs.
The work is organized as follows. In ch. 1 we discuss the basics of cosmology, including inflation, exposing the main observational data and introducing the theoretical tools
for discussing cosmological models. In ch. 2 we give a short introduction to perturbative
superstring theory with mention to non-perturbative aspects. Beside summarizing material well known in literature, we also show how to compute string scattering amplitudes
with zero, one or two vertex operators, which are relevant for the determination of the
brane and orientifold tension in a different way than through the celebreted Polchinski’s
cylinder amplitude. In ch. 3 the pre-big bang model is described, with emphasis on its
phenomenological aspects. In ch. 4 we start to explain in detail how we tried to match
the pre-big bang phase with a standard post-big bang expansion examining the effect
that the preservation of supersymmetry has on low energy solutions. In ch. 5 the quest
for a regularization of the big bang singularity will lead us to the analysis of quantum
corrections to the low energy effective action from realistic string compactifications. In
ch. 6 the part of this work devoted to the pre-big bang scenario is concluded by examining
the consequences of the implementation of thermodynamics arguments and the concept of
holography in a cosmological context. Finally in chap. 7 we investigate a possible role of
x
D-branes in particle physics by showing that a mixing between open (representing Standard Model particles) and closed string excitations (i.e. gravitationally interacting only
particles) can be important even for its experimental signatures in a scenario with the
fundamental string scale at the TeV.
Notations
We use natural units
~ = c = kBoltzmann = 1 .
The Newton constant is denoted by GN and it is often traded with κ according to
8πGN ≡ κ2 .
We recall that
r
~c
= 1.221 · 1019 GeV .
GN
The metric is written in the “mostly plus” convention: η µν =diag(−, +, +, . . . , +). Given
the metric gµν we denote by g the absolute value of its determinant.
The conventions for the Riemann and Ricci tensors are
2
MP l c =
Rµνρσ = Γµνσ,ρ + Γµαρ Γανσ − ρ ↔ σ ,
Rνσ = Rµνρσ δµρ .
Capitol Latin letters will stand for 10-dimensional indices, the first part of the Greek
alphabet (α . . . δ) for spinor indices and second part of the Greek alphabet (µ . . . ω) for
lower dimensional space-time indices.
The gamma matrix satisfying the Clifford algerba in D = 10
M N
Γ Γ
= 2η M N
can be decomposed into the following form
Γ
M
=
γ Mα β̇
M β
γ α̇
0
0
!
,
thus having one index of positive (undotted) and one of negative (dotted) chirality, the
chirality matrix being
Γ11 = Γ0 Γ1 · · · Γ9 .
The charge conjugation matrix in D = 10 is
0 C αβ̇
,
C=
C α̇β 0
with the property
C αγ̇ Cγ̇β = δβα .
xi
1 Elements of Cosmology
“Mi pare strano che l’universo
sia nato da un’esplosione,
mi pare strano che si tratti invece
del formicolio di una stagnazione.”
E. Montale, Quaderno di quattro anni
In this chapter we briefly recall some notions of general relativity and cosmology emphasizing what are the main observational data on the Universe, the successes and the
shortcomings of the standard cosmological model and we finally sketch the main ideas of
inflation.
1.1
The observed Universe
Since the discovery of an isotropic extraterrestial electromagnetic radiation in the microwave region (from few mm to few cm), the cosmic microwave background radiation
(CMBR) [1], it is commonly believed that the Universe evolved from a hot dense state
expanding and relaxing towards the present cold and “almost empty” Universe.
As the CMBR is endowed with a Planck spectrum, it must be originated in a Universe
much hotter and denser than the present, where interactions between radiation and matter were frequent enough to create thermal equilibrium. Then, because of the expansion
and the subsequent cooling, interaction stopped and the Universe became optically thin,
allowing the CMBR to “relax” as it was and to reach us today with little change apart
from the red-shift due to the expansion.
COBE observation [2] gives a black body spectrum with a temperature T γ0 = 2.726 ±
0.010K (which corresponds to a photon number density n γ = 422cm−3 ) and a quadrupole
anisotropy amplitude ∆Tl=2 = 11 ± 3µK. The dipole anisotropy can be ascribed to the
local motion of our system with respect with the “cosmic rest frame” and it corresponds
to v = 365 ± 18 km/s.
The cosmological expansion is also witnessed by the relation between the (luminosity)
distance dL (dL ≡ L/4πF , being L the object’s luminosity and F the measured flux) and
the red-shift z (1 + z ≡ λr /λe , where λr is the wavelength of the received wave, made
longer than the emitted one λe by the cosmological expansion) of a galaxy, which can be
expressed (up to second order in z) as
1
H0 dL = z + (1 − q0 )z 2 + . . .
2
1
(1.1)
2
1 Cosmology
where the Hubble constant H0 is the present expansion rate of the Universe, H 0 ≡
ȧ(t0 )/a(t0 ), a(t) is the cosmic scale factor and in the second order term the deceleration factor q0 ≡ −äa/ȧ2 |t0 appears (t0 stands for the present time). The experimental
value for the Hubble constant [3] is H 0 = 100h km sec−1 Mpc−1 which can be traded for
a length scale or a time scale H0−1 = 3000h−1 Mpc = 9.3 × 1027 h−1 cm = 9.8 × 109 h−1
years, where the experimental uncertainty is in the adimensional parameter h in the range
0.4 . h . 1.0.
Another keypoint in the standard comological model is the analysis of big bang nucleosynthesis, which is the theory explaining the origin of the light element isotopes
(D, 3 He, 4 He and 7 Li). Nuclear reactions took place in the early Universe from t ' 0.01
to 100 sec after the big bang (corresponding to a temperature T ' 10 MeV to 0.1 MeV).
The comparison between predicted and observed abundances of light element isotopes
provides a test of the standard cosmology. Concordance is observed for a value of the
free parameter η (defined as the number of baryon to photon ratio, which is constant as
both scale as a−3 (t)) given by η ≡ nB /nγ = 4 − 7 × 10−10 , corresponding to a density of
baryons ΩB (density normalized to units of critical mass density, see sec. 1.2) constrained
by 0.015 ≤ ΩB h2 ≤ 0.026 [4]. This leads to the conclusion that if we consider a Universe
with critical energy density, Ω = 1, most of it must be supplied by non-baryonic matter.
Big-bang nucleosynthesis is also a probe of the early Universe and particle physics,
as it constrains the existence of additional hypothetical light (≤ MeV) particle species,
which would, if present, affect the energy density and rate of expansion of primordial
Universe, so affecting the predicted abundances of light element isotopes. The present
most conservative bound is Nν < 4.3 [4] for the number of equivalent neutrino species N ν .
After the epoch of thermal equilibrium, during which the Universe was comprised of a
soup of elementary particles with short free path, the expansion made interactions lesser
and lesser frequent, eventually leading to the decoupling between photons and baryons and
electrons. The isotropy of the CMBR bounds the fluctuations in the mass distribution
of baryons averaged over a Hubble length δ B ≡ δρB /ρB ∼ 10−4 at the time of matterradiation decoupling, being the decoupling epoch roughly at z ∗ ' 103 . Matter coupled
to photons can start gravitational collapse only after decoupling, whereas dark matter,
not charged under electromagnetic (and strong) interactions, can collapse already after
the time of matter radiation equality, z eq ∼ 2 × 104 h2 , and at that epoch the required
anysotropy needed to give rise to present structures is at least δ DM ≡ δρDM /ρ ∼ 10−3 .
The presence of a cosmological constant willl make this estimate bigger as perturbations
stop growing when the cosmological constant starts driving the expansion.
−1 z
The red-shifted Hubble length at the time of matter-radiation equality is λ eq ' Heq
eq
−1
∼ 10h Mpc, setting the characteristic length scale of the relevant density perturbations
for gravitational collapse2 .
Our view of a homogeneous and isotropic Universe holds when mass distribution is
averaged over scales larger than 100 Mpc, corresponding to a red-shift z ∼ 3 × 10 −2 ,
but gravitational instabilities give rise to structures which are the main features that we
presently observe: planets, stars, galaxies, groups and cluster of galaxies, superclusters,
2
The gravitational collapse can start only after the perturbation wavelength become sub-Hubble length
sized, see app. A, for longer wavelength this happens later thus having less time to collpase and shorter
wavelengths anyway do not start growing until teq . The short wavelength cutoff in the primordial spectrum
of density perturbation is model-dependent.
1.1 The observed Universe
3
Figure 1.1: CMBR spectrum of anisotropies from Boomerang data [10].
voids . . . A tipical galaxy, like ours, is made out of about 10 11 stars or so (MJ = 2 × 1033
g) which are distributed over a region whose typical size is 10 23 cm ' 30kpc. A tipical
value for the galaxy correlation length is ξ GG ' 5h−1 Mpc and the analogous quantity for
cluster of galaxies is ξCC ' 25h−1 Mpc [5].
Dark matter is required also by astrophysical observations as the comparison between
the luminosity profiles and rotation curves for spiral galaxies [6] indicate. From the study
of rotational velocity v of matter placed outside the point where the light of galaxies
effectively ceases one should expect, if light faithfully traced mass, v ∝ r −1/2 (denoting by
r the distance from the center of the galaxy) according to Newtonian gravity, whereas it
is observed that v ∼ const. This indicates that the mass in the outer part is dominated
by low-luminosity material, a dark halo.
Of outstanding cosmological interest are the recent data concerning the CMBR anisotropies at small scales from the Boomerang [7] and MAXIMA [8] balloon experiments and
DASI [9] interferometer. The anisotropy dependence on the sky position is decomposed
in sperical harmonics obtaining the multipole coefficients C l showed in fig. 1.1.
The oscillatory pattern is due to the acoustic oscillations in the primordial plasma
which are set by the density perturbations that “reenter” the Hubble scale, i.e. whose
wavelength gets smaller than the Hubble length (perturbations with wavelength bigger
than the Hubble length do not oscillate but are “frozen”, see app. A). Normal modes,
labelled by their comoving wave vectors k c (which is constant throughout the expansions
and that is related to the physical one k by k c /a = k), evolve independently
and the
Rt
phase kc η∗ of their oscillations is frozen in at last scattering, being η ∗ = ∗ 1/a dt the
conformal time (see sec. 1.2) markingR the decoupling epoch. The relevant length scale at
η
that time is the sound horizon s∗ = ∗ cs dη which is of the order of the inverse Hubble
scale at decoupling, H∗−1 . Perturbations with wavelength longer then s ∗ are still outside
the Hubble scale and hence they have not started yet to set oscillations in the plasma at
t∗ , whereas shorter wavelengths have reentered earlier and the decoupling freezes them at
4
1 Cosmology
1.7
1.2
∆Τ/Τ
0.7
0.2
−0.3
−0.8
−1.3
−1.8
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9
kη
Figure 1.2: Driven oscillation in the adiabatic (continuos line) and isocurvature
(dashed line) case of the primordial photon-baryon plasma. Peaks correspond to compression of the plasma, wells to rarefaction.
different phases (they also undergo some damping because of the diffusion in the plasma).
The normal modes roughly oscillate as a combination of cosine and sine functions according
to the initial conditions set in by the perturbations. As a function of k c there will be a
serie of temperature fluctuation peaks and wells at k c m = mπ/(2s∗ ). The k dependence
of the phase of the oscillation can be converted to the corresponding length scale, which
is projected on our sky over an angle θ related to the multipole index l by l ∝ 1/θ.
Thus the positions of the peaks depend essentially on the type of perturbations and the
sound horizon s∗ . The oscillation patterns triggered by purely adiabatic and isothermal
density fluctuations are displayed in fig.1.2: they drive respectively the cosine and the sine
oscillation, thus producing different peak positions.
Adiabatic pertubations correspond to perturbations in baryons and photons balanced
so that δ(nB /nγ ) = 0, whereas isocurvature ones are such that δρ B = −δργ . A general
perturbation in the primordial plasma can be expressed as a combination of this two
fundamental types. Data strongly suggest that perturbations are adiabatic [11, 12] (and
gaussian) and constrain the spectrum of initial density perturbations k 3 |δk |2 ∝ k n−1 to
have n = 1 within ten percent [7], where δ k ≡ δρk /ρ. The relevant wavelength λa in
determining the acoustic oscillations correspond to k a ∼ lH∗ /z∗ where l ∼ 200 − 1000
denotes the multipole position of the acoustic peaks and z ∗ has been inserted to take
account of the redshift, giving ka ∼ (0.1 − 0.5 Mpc)−1 h.
The interpeak distance depends only on the Hubble scale at decoupling and the geometry of the Universe so it is a measure of the spatial curvature of the Universe (see sec. 1.2).
The result is that a Universe with flat (i.e. Ω k = 0, see sec. 1.2) spatial sections is favoured
by data or differently stated that Ωm + ΩX ∼ 1 with an accuracy of less then ten percent
[13, 14, 7], where we denoted by Ωm and ΩX respectively the normalized energy density
in non relativistic matter and in any form of relativstic dark matter.
5
1.2 Standard cosmology
Other recent, even if not conclusive, data indicate that the Universe is now accelerating,
based on distance vs. red-shift measurments from high-z supernovae Ia (SNe Ia) [15,
16], which give an experimental estimate of q 0 , that roughly speaking depends on the
combination ρ + 3p, see eq. (1.8b).
Finally study on the average mass and light profiles of galaxy clusters give an independent determination of the contribution Ω m to the energy density of the Universe [17]. The
three possible contribution to energy density content of the Universe (matter, curvature
and relativistic dark energy) are fixed by, see sec. 1.2,
Ωm + Ω X + Ω k = 1 .
(1.2)
The three above mentioned observations (CMBR, distance vs. redshift, mass and light
profiles) constrain thus two indipendent quantities, and at the moment data give indications for Ωm ∼ 0.3, ΩX ∼ 0.7 and Ωk ∼ 0 [18] for ΩX being made of a no better understood
dark energy with very negative presure (p X < −0.6ρX ). Even if this data are not conclusive we remark that we have three indipendent observational pieces of data which depend
(theoretichally) upon two parameters (Ω m and ΩΛ , say) so to give an overconstrained
system with a unique solution.
An obvious candidate for ΩX is a cosmological constant, but from a field theoretical
point of view the cosmological constant represents a dramatic hierarchy problem as its
order of magnitude is fixed by the Hubble constant, which is
Λ ∼ H02 ' (2 · 10−42 GeV )2 ,
(1.3)
which corresponds to a “vacuum energy” (ρ Λ )1/4 ≡ (3Λ/(8πGN ))1/4 ' 3 × 10−3 eV which
is, say, 14 order of magnitude less than the electroweak mass scale. At the moment there
is no satisfactory explanation of this mismatch in scales.
1.2
Standard cosmology
Now we pass to consider the theoretical framework which enables to interpret the cosmological data. The starting point is the cosmological principle which states that [19] it is
possible to define in space-time a family of space-like sections such that on top of all of
them the Universe has the same physical properties in each point and in every direction.
We consider a 4-dimensional space-time compatible with the cosmological principle:
the most general metric for such a space having homogeneous and isotropic spatial sections
(i.e. which has a maximally symmetric three-dimensional subspace) can be parametrized
by the following line element
dr 2
2 2
2
2
2
+ r dθ + r sin θdφ ,
ds = −dt + a (t)
1 − kr 2
2
2
2
(1.4)
where a(t) can be interpreted as the cosmic scale factor and the parameter k can be chosen
to be 1, 0, −1 corresponding respectively to spherical, euclidean and hyperbolic spatial
sections. Another popular coordinate choice is obtained by substituting in previous (1.4)
dt = a(t(η))dη. η is often called conf ormal time, whereas t is the cosmic time.
6
1 Cosmology
The Einstein equations can be derived from the action S = S EH + Sm where
Z
X Z
√
1
D √
SEH = 2
d x g(R − 2Λ) ,
Sm =
dD x −gLf ields ,
2κ
(1.5)
f ields
where we included the cosmological constant Λ and we denoted by S M the ‘matter’ action,
giving the equations
Gµν + Λgµν = 8πGN Tµν ,
(1.6)
where we introduced the Einstein tensor G µν defined as Gµν ≡ Rµν − 1/2Rgµν and we
used the definition Tµν ≡ √2−g δSm /δg µν .
By using the Bianchi identity Gµν ;ν = 0 we obtain for consistency the continuity equation
for the sources
Tµν ;ν = 0
(1.7)
plus the condition Λ,µ = 0, which ensures that the cosmological constant is actually
constant. Inserting the ansätz (1.4) into (1.6) we get
8πGN
Λ
k
=
ρ+ ,
a2
3
3
ä
4πG
Λ
N
Ḣ + H 2 = = −
ρ(1 + 3p/ρ) + .
a
3
3
H2 +
(1.8a)
(1.8b)
The cosmological principle fixes the form of the stress-energy tensor to
Tµν = a2 diag(ρ, p, p, p)
(1.9)
so that eq. (1.7) can be rewritten in a non covariant form as
ρ̇ + 3H(ρ + p) = 0 ,
(1.10)
where an overdot stands for derivative with respect to the cosmic time t defined by eq. (1.4).
The cosmological constant can be treated on equal footing with any other form of energy
provided ρΛ = −pΛ = Λ/(8πGN ) is set. The previous relation (1.10) implies that assuming
a barotropic equation of state for the cosmological fluid (p = wρ) we obtain
ρ ∝ a−3(w+1) .
(1.11)
To keep contact with standard notation we introduce
Ωm ≡
8πGN ρ
,
3H 2
ΩΛ ≡
Λ
,
3H 2
Ωk ≡ −
k
,
(aH)2
(1.12)
whose sum must equal unity according to eq. (1.8a). The critical energy density ρ c (i.e.
Ω = 1) corresponds to
ρc ≡
3H02
= 1.88h2 × 10−29 g · cm3 = 8.10h2 × 10−47 GeV4 .
8πGN
(1.13)
7
1.2 Standard cosmology
Using eqs. (1.8) we can now relate the cosmological parameter q 0 with the energy
parameter Ωi0 according to
Ωm0
3p
q0 =
1+
− ΩΛ 0 .
(1.14)
2
ρ
We can solve the eqs. (1.8), or alternatively any of the eqs. (1.8) and eq. (1.10), to
find the qualitative evolution of the cosmic scale factor once p(ρ) is known. Any stable
energy component with negative pressure is of little importance in the early stage of
the cosmological evolution: at early times it must have been a minor fraction of the
cosmological source because of (1.11). In fig. 1.3 the evolution of the scale factor for
w = 1/3 is displayed. The main feature of this example is that the time coordinate cannot
2.0
k=−1
k=0
1.5
a
k=1
1.0
0.5
0.0
0.0
0.5
t
1.0
1.5
2.0
Figure 1.3: Evolution of the cosmic scale factor in a pure radiation Universe (w = 1/3)
for the three possible cases of spatial curvature k = 0, ±1.
be extended beyond a critical value in the past, conventionally set to t = 0, where a
singularity is met almost unavoidably if the dominant energy source has p/ρ > 0. Clearly
our analysis based on classical general relativity breaks down at t ∼ t P l . The singularity is
not an artifact of our particular solution, but within the framework of general relativity it
has been demonstrated by Hawking and Penrose [20, 21] under well defined but reasonable
hypotheses. They demonstrated that a spacetime M necessarily contains incomplete,
inextendible timelike or null geodesics (i.e. a singularity) under the following hypotheses:
1. M contains no closed timelike curves (unavoidable causality requirement).
2. At each point in M and for each unit timelike vector with component u α the energy
momentum tensor satisfies (null energy condition)
1
α
Tµν − gµν Tα uµ uν ≥ 0
2
8
1 Cosmology
3. The manifold is not too highly symmetric so that for at least one point the curvature
is not lined up with the tangent through the point, or in formula
u[µ Rν]ρσ[τ uυ] uρ uσ 6= 0
at some point on the geodesic
4. M does not contain any trapped surface.
All conditions except the last one, which is rather technical, are completely reasonable for
a realistic spacetime. In particular to violate the first hypothesis, in term of quantities
appearing in the ansätz (1.9), 3p < −ρ should hold.
We finally give numerical values for some of the relevant physical quantities. As the
contribution of matter and that of the cosmological constant to the present energy content
of the Universe is comparable, in the past matter, and particularly relativistic matter,
must have dominated, since ρm ∼ a−3(w+1) , ρΛ ∼ const, ρk ∼ a−2 and w > 0 for ordinary
sources.
Conversion formula between time and red-shift is given by the integration of eq. (1.8a)
t(z) = H0−1
Z
(1+z)−1
0
dx
(Ωm0 x−1−3w + ΩΛ0 x2 + Ωk0 )1/2
(1.15)
where we assumed that matter is dominated by a single energy source with p/ρ = w. The
previous equation reduces, in the case in which radiation is the dominant energy source
(w = 1/3, i.e. in the radiation dominated epoch)
t∼
1
−1/2
(1 + z)−2 H0−1 Ωm0 .
2
(1.16)
The epoch of nucleosynthesis corresponds to z ∼ 4 × 10 6 .
As the Universe expands a particle species can be in thermal equilibrium provided
that the rate Γ of the interaction mantaining equilibrium is greater than the rate of expansion, namely Γ > H. For example neutrinos decoupled when T ∼ 1 MeV, (t ∼ 1
sec, z ∼ 106 ), after decoupling their temperature decreases as a −1 . Photons decoupled from electrons when the electrons began to recombine with nuclei to form atoms at
−2
,
Trec = Tγ0 (1 + zrec ) ' 0.25 eV = 3000 K, zrec ∼ 1100 − 1200, trec ' 6.6 × 102 Ωm0 h2
see [3]. As the present amount of energy in radiation and non relativistic matter is known,
we can infer the time when matter and radiation density made equal contribution to the
−1/2
energy content of the Universe teq ' 0.5H0−1 Ω0 (1 + zeq )−3/2 , with zeq ∼ 2 × 104 Ωm0 h2
−2
years.
or Teq = T0 (1 + zeq ) ∼ 5.5 Ω0 h2 eV, one has teq ' 1.4 × 103 Ωm0 h2
At later times the energy density ρm in non relativistic matter (characterized by a negligible pressure) dominates the energy content of the Universe up to the present time during
which we now know that ρm and ρX are roughly of the same order.
1.3
Shortcomings of the standard cosmology
Because of its finite age the Universe must have developed horizons, which means, roughly
speaking, that not all the regions of the Universe that are accessible to our observation
have had a chance to be in casual contact, having thus no reason to develop common
1.3 Shortcomings of the standard cosmology
9
features. As the Universe emerged at, say, t P l we cannot expect that it was in a highly
homogeneous state unless we admit that an unknown mechanism has conspired in prePlanck time to create very peculiar initial conditions, but unless we are ready to deal with
quantum gravity we cannot go beyond the Planck epoch. Let us explain more quantitavely
how this comes at odds with homogeneity and isotropy.
A light ray which leaves at the Planck epoch can travel till a time t up to a distance
given by the particle horizon dp
Z t
dτ
.
(1.17)
dp (t) = a(t)
tP l a(τ )
This formula shows that for the particle horizon being finite the behaviuor of a(t) for
small t is crucial. In standard cosmology a necessary condition for the integral in (1.17)
to converge is that for any kind of energy sources w > −1/3, as it can be derived from
eq. (1.8b).
Starting with generic initial conditions on a three-surface at t = t P l , we expect that
after a time t we should have isotropy on a region of space of linear dimension roughly
given by dp (t) which is roughly, assuming radiation domination,
dp (t(z)) '
2
∼ H −1 (t) ∼ t .
H0 (1 + z)−2
(1.18)
When we look back in time towards, say, the cosmic microwave background radiation,
we are looking at z∗ ∼ 103 , so we should expect to observe isotropy only across a region
corresponding to a Hubble length of the decoupling epoch. Let us estimate which is the
angle δ(H −1 ) subtended in our sky by an object of linear dimension H −1 (z∗ ), observed
from a distance dA . δ(D) = D/dA and if the light ray travels from r = r 1 to r = 0 we have
1/2
dA = a0 r1 (z)/(1 + z). But in a purely matter dominated Universe r 1 (z) = 1/(2H0 a0 Ωm0 )
(for z 1) and after substitution we get
z −1/2
∗
1/2
1/2
,
(1.19)
δ(H∗−1 ) ' 2Ωm0 H0 z∗ H∗−1 ' 0.860 Ωm0
1100
that is equivalent to say that the standard cosmology predicts isotropy in the CMBR on
an arc of sky whose angular size is less than one degree.
Another problem arises from the analysys of the temporal evolution of Ω m . Starting
with
Ωm =
8πGN ρ
8πGN ρa3
=
,
3 H2
3 aȧ2
(1.20)
taking the derivative with respect to t, using the continuity equation R −3 d(ρR3 )/dt =
−3pH and eq. (1.8b) to eliminate ä we have
Ω̇m
p
= H(Ωm − 1) 1 + 3
,
(1.21)
Ωm
ρ
where we did not consider the cosmological constant whose contribution is negligible in
the primordial Universe. After the substitution Ω̇m with ȧ(dΩm /da) we get
dΩm
1
p
1
1+3
.
(1.22)
=
Ωm (Ωm − 1) da
a
ρ
10
1 Cosmology
Admitting a single source with p = wρ the previous equation has the solution
1
∝ a1+3w
(1.23)
1−
Ωm
which shows that if today Ω is closed to 1 then in the primordial Universe it must have
been fantastically closed to one, raising a problem of fine tuning.
Another issue of fundamental importance in cosmology is how the structure we observe
today have originated or what was the origin of the small primeval fluctuations around
the homogenoeus background, which triggered the gravitational collapse of the galaxies,
but the standard cosmological scenario simply does not address this issue.
Finally we mention another way to look at the horizon problem, suggested by Penrose
[22]. Considering the CMBR, the entropy in our Universe at the big bang can be estimated
to be ∼ H0−3 T03 ∼ 1090 . If black holes are present today in the galaxies far bigger numbers
can be obtained for the present entropy (we remind that the entropy of a black hole is
given by its horizon area, see sec. 6.1). To make the computation easy, we can estimate
the entropy of the Universe by considering the entropy of a black hole with the mass
equivalent to the total mass in our Universe, which gives S ∼ 10 120 , compared to which
the initial entropy of the Universe appears ridicously small. Thus it is clear that we must
understand why the entropy in the big bang was so small compared with what it might
have been. Anthropic arguments will not help here as Universes endowed with an entropy
smaller than 1090 might have also given rise to planetary sistems potentially able to host
life.
1.4
Inflation
We now sketch how inflation [23, 24, 25] works in solving some of the problems of standard
cosmology. By inflation it is meant a stage of exponential expansion or more generally
a period characterized by ä > 0 and massive entropy production. A successful model of
inflation must satisfy:
• Sufficient inflation to solve the horizon and flatness problem. This is usally obtained
by admitting a source of energy in the cosmological equation with negative pressure so
that by (1.8b) ä ∝ −(ρ + 3p) > 0 which requires w < −1/3. This condition makes the
spatial curvature term k/a2 in (1.8a) to become irrelevant as
H
d
= 2|k|−1 ȧä .
(1.24)
dt |k|/a2
Then during inflation Ωm grows compared to Ωk , as it can be checked also by (1.11), and
the Universe will emerge from the inflationary phase in a radiation dominated phase with
temperature Trh . In order to solve the flatness problem, that is in order to have Ω m ∼ Ωk
now, the amount of inflation, conveniently measured by
ae He
N ≡ ln
,
(1.25)
ab Hb
where indices “e” and “b” stand respectively for end and begining of inflationary era, must
satisfy
Trh 1 Teq
Trh
N & ln
,
(1.26)
+ ln
∼ 60 + ln
Teq
2
T0
1016 GeV
11
1.4 Inflation
where instantaneous transition between the inflationary and the radiation phase has been
assumed.3 Inflation also widens the size of the particle horizon d p so that at the end of
inflation
dp ∼ eN H −1 H −1 ,
(1.27)
if H is the Hubble scale of inflation, making homogeneity possible on scale much bigger
of that Hubble scale.
• Suffciently high reheat temperature. Inflation is usually triggered by the potential
energy of a scalar field, the inflaton χ, whose equation of motion will generally be
χ̈ + 3H χ̇ + V 0 (χ) = 0 ,
(1.28)
being V its potential and
8πGN
H =
3
2
1 2
χ̇ + V (χ)
2
(1.29)
the relevant equation for the expansion of the Universe. In this setup inflation happens
when the inflaton is not initially at the minimum of the potential and it slowly rolls down
the potential so that its kinetic energy is negligible and its potential energy sustains an
accelerated expansion, as in this case
1
ρ = χ̇2 + V ,
2
p=
1 2
χ̇ − V (χ) ,
2
ρ ' −p ' V (χ) ,
ρ + p ' χ̇2 ρ ,
(1.30)
and then (1.28) reduces to
3H χ̇ + V 0 = 0 .
(1.31)
The condition to be matched to realize a slow roll are
|V 00 (χ)| 9H 2 ' 24π
V (χ)
,
MP2 l
MP l |V 0 /V | (48π)1/2 ,
(1.32)
which can be derived respectively by imposing the consistency of the approximation of
neglecting χ̈ in (1.28) to the time derivative of (1.31) and by making the kinetic energy
much smaller then the potential energy. These conditions can be matched by a variety of
potentials, three of which are shown in app. A.
During inflation the Universe will expand and overcool until eventually the inflationary
regime will break down as the inflaton will settle at the minimum of its potential, oscillating and decaying into ordinary matter and radiation, hence reheating the Universe and
releasing a huge amount of entropy. The highest cosmological scale that can be probed
by observation in this case is set by the reheating temperature T rh , which must be bigger
then the nucleosynthesys scale (∼ 10M eV ) and also high enough to allow baryogenesis
(at least Trh & TeV).
• The abundance of unwanted relics must be very small. If this is not the case a supermassive object may become non-relativistic well before the Universe is matter dominated,
3
Eq. (1.26) is derived by considering that during inflation Ωm gains over Ωk a factor e2N and during
2
2
/Teq
× Teq /T0 .
ordinary FRW expansion from Trh down to T0 Ωk gains over Ωm a factor Trh
12
1 Cosmology
roughly from zeq ∼ 104 on, and its energy density, scaling as a −3 compared to a−4 of radiation, will earn a factor a with respect to radiation and eventually dominate the Universe
well before it is required by the standard cosmological model and observations (for sure
a non-relativistic species cannot dominate the Universe at the epoch of nucleosynthesis).
Thus the abundance of any non relativistic species Y is constrained to be
Teq
ρY
(Tnr ) <
' 10−11
ρc
Tnr
Tnr
1TeV
−1
,
(1.33)
where Tnr is the photon bakground temperature at the epoch in which the species Y is
becomes non relativistic. The problem of relics is present also in standard cosmology and
inflation can solve it as it dilutes any preexisting density of relics, provided that they are
not overproduced in the reheating process.
• Adiabatic and almost scale-invariant spectrum of density perturbations. During inflation, or generally during a phase of accelerated expansion (or contraction) fluctuations
in any field get amplified. Quantum fluctuations of given mode k originate when they are
sub-Hubble scale sized4 , they are stretched outside the Hubble scale as k is redshifted,
they “freeze out”, i.e. they stay constant without evolving apart form the red-shift of
their wavelengths while they are super-Hubble sized and finally they become again subHubble sized in the radiation or matter dominated phase. This happens for perturbations
whose wavelength is long enough, i.e. such that k < k max = H, being H the (maximum)
curvature scale reached during the inflating phase. Physical lengths tend to become big˙ −1 = ä
ger(smaller) than the Hubble scale during inflation(decelerated expansion) as a/H
is positive during inflation and negative during ordinary FRW-like phase.
There exist a simple formula to relate the amount of perturbations in each k mode at
the time they become sub-Hubble sized (epoch “2”) in terms of the time they became
super-Hubble sized (epoch “1”) and it is [26]
1 δρk 1 δρk =
,
(1.34)
1 + w ρ 1,k=H
1 + w ρ 2,k=H
where ρk is the k mode energy density perturbation per logarithmic interval of momentum.
Applying this to a post-inflationary Universe
δρk 4 V δρk H 2 =
∼
. 10−5 ,
(1.35)
ρ 2,k=H
3 χ̇2 ρ 1,k=H
χ̇ 1,k=H
where the numerical bound comes from the isotropy of the CMBR at scale k = H(z ∗ ).
From the theoretical point of view this bound strictly constrains the parameters appearing
in the potential, but not the energy scale of inflation.
The density perturbations turn out to be adiabatic, as δρ 6= 0 is the initial condition in
the radiation phase, gaussian, as δρ k is linear in the inflaton fluctuation δχ k which is a
random variable with gaussian distribution, and almost scale invariant as the quantity
H 2 /χ̇ has little variation during inflation.
Inflation has the double effect of realizing large scale smoothness and small scale fluctuations.
4
Often in literature “horizon” is used as a synonym of “Hubble length”.
1.4 Inflation
1.4.1
13
Problems of inflationary models
In the inflationary scenario it is not automatic that everything works in detail. For instance:
• To start inflation the inflaton must be homogenous in a large enough region so
that the spatial gradients in χ give negligible contributions compared to potential energy,
(∇χ)2 V , otherwise inflation will not start.
In the chaotic version of the inflationary scenario it happens that some very tiny part of an
initially chaotic state may be in such a special state of spatial homogeneity which allows
the onset of an inflationary expansion, giving rise to the observable Universe. Anyway this
idea is still exposed to the same kind of criticism addressed to the anthropic considerations
(see end of sec. 1.3).
• Slow roll conditions in the potential must be matched and the initial position of the
inflaton must be far enough from the minimum of the potential to obtain a long period
of inflation. Moreover in order of the quantum fluctuation of the inflaton field not to
invalidate the classical analysis ∆χ q < ∆χcl which, using ∆χq ∼ H and ∆χcl ∼ χ̇/H,
holds once (1.35) is fulfilled. This leads to a very strong bound on the parameter of the
potential which has sometime been addressed as a fine tuning. Anyway this fine tuning
problem can be solved by considering a slow roll scenario characterized by a quadratic
potential and incorporating radiative corrections within the context of supergravity [27],
see also sec. A.2.
• In general production of massive stable particles can take place in inflationary scenarios, so that inflation might produce the same unwanted relics that it dilutes.
• Among the possible inflationary scenarios, none is a compelling part of a sensible
particle physics model.
• The trans-Planckian problem: scales of astrophysical interest, as for instance the
length corresponding to the present Hubble scale H 0−1 ∼ 3000Mpc have crossed the Hubble
length during the inflationary period when the scale factor was e N smaller than at the end
of inflation, with
2
1
ln(HdS /Heq ) + ln(Heq /H0 ) ∼ 60 + 1/2 ln(HdS /1013 GeV) ,
(1.36)
2
3
where HdS is the Hubble parameter during inflation, here assumed to be constant and H eq
the Hubble parameter at radiation-matter equality. This implies that more than about 60
e-folds before the end of inflation5 all present astrophysical scales were sub-Planckian, if
it is assumed that length scales keep redshifting linearly with the scale factor even below
the Planck length. This may represent a problem as the quantum vacuum state used
in the computation of the primordial spectrum of perturbations, see sec. A.2, might be
different from the one which is selected by the unknown fundamental theory at work at
the (sub-)Planck scale.
• Anyway a fundamental question remains unanswered: what is the fate of the initial
singularity?
In the following chapters we propose a tentative explanation of how inflation can be
incorporated in a sensible theory of gauge and gravity, string theory, and how it is possible
to address the problem of the cosmological singularity in a string theory context.
N=
5
The inflationary phase has lasted more than 60 e-folds in the case of the standard inflationary models,
see (A.27) and (A.30).
2 Elements of string theory
“You’re so narrow-minded: you think
in such three-dimensional terms.”
The Borg Queen to Data,
Star Trek: First contact
We expose in this chapter some elements of string theory, recalling briefly both perturbative and non perturbative aspects. The reference guides are the book by Green, Schwarz
and Witten [28] and the one by Polchinski [29]. For conformal field theory techniques in
string theory [30] is the original paper, and also the review [31] has been useful. We first
describe the usual perturbative formulations of string theory and in sec. 2.6 we show how
to compute amplitudes whose relative Conformal Killing Group is completely or partly
unfixed by the vertex operators. After mentioning the issues of compactification and Dbranes, the result obtained in sec. 2.6 will be applied in sec. 2.10 to compute the brane
tension from a disk amplitude.
2.1
The bosonic string
Strings are fundamental objects with one-dimensional extension; ordinary particles will
emerge from the theory as different excitations of a single string: in particular a massless
mode with spin 2 will be present, which can be identified as the graviton, making gravity
the first “phenomenological prediction” of string theory.
String theory needs the introduction of a new dimensionful fundamental constant α 0
which can be traded for a length λs (the string length), for a tension T (the string tension)
or a mass Ms (the string mass) via the definition
λ2s ≡
1
≡ Ms−2 ≡ α0 .
2πT
We start describing the bosonic string by using the Polyakov action
Z
1
SP = −
dτ dσ γ 1/2 γ ab ∂a X M ∂b XM ,
4πα0 Σ
(2.1)
(2.2)
where Σ is the world-sheet, i.e. the surface spanned by the string in its temporal evolution
parametrized by coordinate τ (time-like) and σ (space-like), γ ab is the the world-sheet
metric and the X M ’s are the coordinates of the D-dimensional Minkowskian target space
which embeds the world-sheet. Action (2.2) reduces to the geometric area of the surface
Σ once the value of γab obtained by its equations of motion is substituted in it.
14
15
2.1 The bosonic string
Beside D−dimensional Poincaré invariance and reparametrization of the world sheet
coordinates τ and σ, the action is invariant under two-dimensional Weyl rescaling, which
transforms the two-dimensional metric according to
0
γab → γab
= exp (2ω(τ, σ)) γab
(2.3)
for any ω(τ, σ), leaving unaltered the X M and the world-sheet coordinates, thus defining
a two dimensional conformal field theory. We can use the symmetries of the action to fix
the two-dimensional metric
γab = diag(−1, 1)
(2.4)
so that the gauge fixed action SPgf takes the form
Z
1
dτ dσ ∂ α X M ∂α XM ,
SPgf = −
4πα0 Σ
(2.5)
which leads to the equations of motions
gf
4π δS
1
− 1/2 Pab = Tab = 0
δγ
α
γ
M
∂a X ∂b XM
1
− γab ∂ a X M ∂b XM
2
δSPgf
∼ XM = 0 .
δX M
= 0,
(2.6)
(2.7)
The gauge choice (2.4) makes the X M equations of motion linear, which is essential for
quantization. The Eulero-Lagrange equations of motion must be supplemented by appropriate boundary conditions
X M (τ, σ) = X M (τ, σ + 2π)
M
∂σ X |σ=0,π = 0
NN
∂τ
DD
XM |
σ=0,π
=0
closed
(2.8)
open
(2.9)
where eq. (2.8) defines a closed string and eqs. (2.9) correspond to the open string case
with either Neumann-Neumann, or NN for short (implying no momentum flow at string
ends), or Dirichlet-Dirichlet (DD, implying constant end positions) boundary conditions.
Also mixed ND boundary conditions are allowed, however it should be remembered that
a Dirichlet boundary condition breaks D-dimensional Poincaré invariance. We have conventionally chosen the range of σ to be [0, 2π] for closed strings and [0, π] for open ones.
It is useful to introduce coordinates w, w̄
w ≡ iτ 0 − σ ,
w̄ ≡ iτ 0 + σ ,
(2.10)
where τ 0 = iτ is the Wick rotated world-sheet time. The domain of w is a strip in the
complex plane, with edges periodically identified for closed strings and with boundaries in
the open string case. The mostly used coordinates in our computations are
z = exp(−iw) ,
z̄ = exp(iw̄) ,
(2.11)
16
2 String theory
whose domain is the full complex plane for closed strings and the upper half plane (Im
z > 0) for open ones. In the z (half-)plane Euclidean time runs radially and equal time
contours are circles. In the z coordinates the gauge fixed metric (2.4) reads γ zz = γz̄z̄ = 0,
γz z̄ = γz̄z = 1/2 and eqs. (2.6) and (2.7) reduce to
α0 Tzz = ∂z X M ∂z XM = 0 ,
∂z ∂z̄ X
M
α0 Tz̄ z̄ = ∂z̄ X M ∂z̄ XM = 0 ,
(2.12)
=0
(2.13)
and Tz z̄ vanishes identically for the gauge fixed metric (2.4). We shall write from now on
simply T for Tzz , T̃ for Tz̄ z̄ , ∂ and ∂¯ for ∂z and ∂z̄ .
The conformal transformation (2.3) is in terms of z an arbitrary holomorphic change
of variables z → z 0 = f (z) for any purely holomorphic function f (z). Given any operator
φ(z), it is defined to have a conformal dimension J if under an arbitrary holomorphic
reparametrization it transforms to
φ(z) → φ0 (z 0 ) = (∂f )J φ(z) .
(2.14)
For open strings conditions (2.9) become
¯ M |z=z̄ = 0
∂X M − ∂X
¯ M |z=z̄ = 0
∂X M + ∂X
NN
(2.15)
DD ,
which are also written as ∂n X|∂Σ = 0 (∂t X|∂Σ = 0) for the NN (DD) case as the normal
(tangent) derivative to the world-sheet boundary is involved.
The general solution X M (τ, σ) for closed strings can be decomposed as a sum of left- and
right-moving modes
M
(z̄) ,
X M (τ, σ) → X M (z, z̄) = XLM (z) + XR
(2.16)
where
XLM (z)
α0
xM
ln z + i
− i pM
=
2
2 L
α0
2
1/2 X
n6=0
1 M −n
α z
n n
(2.17)
and analogously for the right movers provided the substitutions L ↔ R, α n ↔ α̃n , z ↔ z̄
are made. For open strings conditions (2.15) compel right and left movers to combine
according to
0 1/2 X
α
1 M −n
M
M
0 M
2
NN , (2.18a)
X (z, z̄) = x − iα p ln |z| + i
αn z + z̄ −n
2
n
n6=0
0 1/2 X
α
1 M −n
M
M
0 M
αn z − z̄ −n DD . (2.18b)
X (z, z̄) = x − iα p ln(z/z̄) + i
2
n
n6=0
p
p
M
M
In the previous mode expansions √
we implicitly identified p M
2/α0 , pM
2/α0
L = α0
R = α̃0
M
M
for closed string and p = α0 / 2α0 for open ones. This is justified by the explicit form
of the conserved current under spacetime translations
jM =
i
∂X M ,
α0
(2.19)
17
2.1 The bosonic string
which gives the momentum in terms of the α oscillators according to
Z
I
pM + p M
1 2π
1
αM + α̃M
M
M
0
R
p = 0
dσ∂τ X =
= 0√
j M dz − j̄ M dz̄ = L
α 0
2πi
2
2α0
(2.20)
for closed string. In the case of an ordinary, infinite coordinate p L = pR for the corresponding momenta. For open strings the integration over σ is traded for an integration
over a circle on a complex plane: this can be done as the open string world sheet, the
half plane, with both holomorphic and antiholomorphic fields can be traded for the full
complex plane with holomorphic fields only, provided we identify, for the generic pair of
holomorphic and antiholomorphic fields φ(z), φ̃(z̄),
φ(z̄) = φ̃(z̄)
for Im(z) > 0 ,
(2.21)
thus obtaining for the momentum operator
I
Z
1
1 π
αM
dσ∂τ X M =
pM = 0
jzM dz = √ 0 .
α 0
2πi
2α0
(2.22)
Using the mode expansion (2.17) and (2.18) the constraints (2.12) turn out to have Fourier
components Lm , the Virasoro generators
Lm
∞
1 X µ
=
α
αµn ,
2 n=−∞ m−n
(2.23)
which must be supplemented by their right-moving partners in the closed string case.
The theory is quantized by imposing the canonical quantization condition
[X µ (τ, σ1 ), T ∂τ X ν (τ, σ2 )] = iδ(σ1 − σ2 )η µν ,
(2.24)
which can be translated in terms of the modes of the expansions (2.17) and (2.18) into
M N
(2.25a)
x , p = iη M N ,
M N M N
MN
(2.25b)
αm , αn = α̃m , α̃n = mδm,−n η
(the other commutator vanishing) or with the singular part of the Operator Product
Expansion (OPE)
α0 M N
η
ln(z1 − z2 )
2
α0
M
N
XR
(z̄1 )XR
(z̄2 ) ∼ − η M N ln(z̄1 − z̄2 )
2

0
N
XLM (z1 )XR
(z̄2 ) ∼
α0 M N
∓ η
ln(z1 − z̄2 )
2
XLM (z1 )XLN (z2 ) ∼ −
(2.26a)
(2.26b)
closed
NN(DD)
open
.
(2.26c)
When the product of operators occurs, ∼ will stand for “singular part” of it.
Having introduced the oscillators α m the Fock vacuum |0, pM = 0i can be defined by
the condition
N
αM
n |0, p = 0i = 0 for n ≥ 0 .
(2.27)
18
2 String theory
Acting on the Fock vacuum, the oscillators α n with n < 0, together with eikX operator, will
generate the perturbative spectrum which is then characterized by the number operator
NX
X
NX ≡
αM
(2.28)
−n αM n
n>0
and analogously for ÑX in terms of α̃n . The spectrum of physical states is smaller than
the one spanned by the action of the α’s over the Fock vacuum, as only the equivalence
class of BRST closed states modded by the exact ones will be physical 1 . The result is that
for our gauge choice (2.4) the condition for a state |ψi to be physical is
(Ln − aδn,0 ) |ψi = 0
n ≥ 0,
(2.29)
(which must be supplemented by the analog relation for L̃m for closed strings) where the
constant a is due to a normal ordering effect and it is given by
a=−
D−2
.
24
(2.30)
In D = 26 (see below) a = −1 and the physical condition (2.29) for n = 0 gives the mass
formula
α0 m2cl = −pM pM = 4(NX − 1) = 4(ÑX − 1) ,
α
0
m2op
M
= −p pM = NX − 1 .
(2.31a)
(2.31b)
Strictly speaking the previus formula for open strings applies to the case of fully NN boundary conditions, in sec. 2.9 it will be qualified in the presence of DD boundary conditions.
We note moreover the presence of a tachyon in the spectrum of physical states.
In the BRST construction the X conformal field theory must be supplemented by the
b, c ghost theory, defined by
Z
1
¯ .
Sg =
d2 z b∂c
(2.32)
2πα0
The mode expansion is
b(z) =
X bn
,
z n+2
n
c(z) =
X cn
,
z n−1
n
(2.33)
and the energy-momentum tensor is
T g =: (∂b)c : −2∂ (: bc :) ,
(2.34)
which gives conformal weights Jb = 2 and Jc = −1 (colons stand for normal ordering).
Like the Fadeev-Popov ghost in gauge field theory the b, c ghosts are (worldsheet) bosons
which anticommute. They can be quantized by assigning the OPE 2
b(z1 )c(z2 ) ∼
1
1
.
z1 − z 2
(2.35)
The actual expression of the BRST operator for the bosonic string is not needed here.
The ghost theory impose further condition, beside (2.27), on the ground state, and they will be dealt
together with the conditions from superghosts in sec. 2.2.
2
19
2.2 Type II superstrings
The Virasoro generators (2.23) in the quantum interpretation satisfy a modified Virasoro algebra:
[Lm , Ln ] = (m − n)Lm+n +
cX 3
(m − m)δm,−n ,
12
(2.36)
where the central charge cX can be calculated to be cX = D. The ghost Virasoro generators
satisfy an analogous algebra with c g = −26, thus only in 26 dimensions the theory is Weyl
invariant at the quantum level. The presence of a non vanishing central charge means
that the Weyl symmetry is anomalous as indicated by the non vanishing of the trace of
the energy momentum tensor, which is given by
Taa = −4Tz z̄ = −
c (2)
R ,
12
(2.37)
where R(2) is the world-sheet Ricci scalar. Thus the gauge choice (2.4) is quantum mechanically consistent only if c ≡ cX + cg = 0 i.e. D = 26.
As far as two-dimensional diffeomorphism invariance, this symmetry is non anomalous
if the theory is right-left symmetric or at least if the central charge c from the rightmoving degrees of freedom equals the central charge c̃ arising from the left-moving degrees
of freedom.
2.2
Type II superstrings
Other string theories can be obtained by imposing supersymmetry on the world-sheet,
in particular the type II theories involve closed string only and their low energy limit is
N = 2 supergravity in D = 10.
Introducing a D−plet of Majorana (world-sheet) fermions which transform in the vector representation of the target space Lorentz group SO(D − 1, 1) we are lead to consider
the action
Z
1
1
M a
M α
2
∂a X ∂ XM + ψ̄ ρ ∂α ψM ,
(2.38)
d σ
SSP = −
4π
α0
where we introduced the two dimensional gamma matrix ρ a . Beside the symmetries already possessed by the bosonic string action, (2.38) is also invariant under the supersymmetry transformation (world-sheet supersymmetry)
√
δX M = α0 ¯ψ M ,
(2.39a)
√ a
M
M
0
(2.39b)
δψ = −1/ α ρ ∂a X ,
whose conserved Noether current Ja is the supercurrent given by
√
Ja = 1/ α0 ρb ρa ψ M ∂b XM .
(2.40)
In terms of the coordinates z, z̄ and introducing the right- and left-moving fermions ψ and
ψ̃ we can write the equations of motion as
∂ ψ̃ µ = 0 ,
¯ µ=0 ,
∂ψ
¯ µ=0 .
∂ ∂X
(2.41a)
(2.41b)
20
2 String theory
The super-Virasoro currents in the chiral basis are
r
2 M
J=
ψ ∂XM ,
α0
1
1
T = 0 ∂X M ∂XM + ψ M ∂ψM
α
2
(2.42a)
(2.42b)
and analogously from the antiholomorphic side. Physical states are annihilated by the
positive frequency modes of both J and T .
To quantize the fermions the canonical anticommutation relations have to be imposed
M
ψα (τ, σ1 ), ψβN (τ, σ2 ) = 2πδ(σ1 − σ2 )δαβ η M N ,
(2.43)
which give in terms of the OPE
ψ(z1 )M ψ(z2 )N ∼ ±η M N
1
z1 − z 2
NN (DD) .
(2.44)
The introduction of fermions opens the possibility to make the boundary term
2π
δψψ|2π
0 − δ ψ̃ ψ̃|0 = 0
(2.45)
vanish in two different ways
ψ(τ, σ = 0) = ψ(τ, σ = 2π)
Ramond ,
(2.46a)
ψ(τ, σ = 0) = −ψ(τ, σ = 2π)
Neveu-Schwarz ,
(2.46b)
and analogously for the anti-holomorphic part. The Ramond sector, R for short, has the
same periodicity as the X, the Neveu Schwarz (NS) the opposite one, which implies integer
mode expansion for the R sector and half integer modes for the NS sector:
ψM =
ψ
M
=
1
z 1/2
1
z 1/2
X
−n
dM
n z
R,
(2.47a)
NS ,
(2.47b)
n∈Z
X
−r
bM
r z
r∈Z+1/2
where the prefactor z −1/2 is due to the change of variable σ, τ → z, z̄ as J ψ = 1/2. The
commutation relations that the oscillator operators inherit are
M N
M N
dm , dn = η M N δm,−n R ,
br , bs = η M N δr,−s NS .
(2.48)
The number operators are
Nd =
∞
X
n=1
ndM
−n
· dM n ,
Nb =
∞
X
r=1/2
rbM
−r · bM r .
(2.49)
In addition to condition (2.27) the ground states must satisfy the conditions
bM
r |0iN S = 0 for r ≥ 1/2 ,
dM
n |αiR = 0 for n ≥ 1 .
(2.50)
21
2.2 Type II superstrings
The NS ground state is a scalar, whereas the Ramond ground state furnishes a representation of dM
0 , which commute with the energy operator and satisfy a rescaled version of the
Clifford algebra relations as it follows from (2.48), the Ramond ground state is then a Majorana fermion which can be splitted in two Majorana-Weyl spinors of opposite chirality,
as in ten dimensions Majorana and Weyl conditions are compatible with each other.
Also the ghost are supplemented by superpartners, the commuting world-sheet fermions β and γ whose action is
Z
¯ .
Ssg =
dτ dσ β ∂γ
(2.51)
Σ
The supercurrents and energy momentum tensor are
3
Jsg = ∂βc + β∂c − 2bγ ,
2
3
Tsg =: (∂β) γ : − ∂ (: βγ :)
2
(2.52a)
(2.52b)
and the quantization condition is
β(z1 )γ(z2 ) ∼
1
.
z1 − z 2
(2.53)
The β, γ superghosts can be traded for the world-sheet bosons ϕ, ξ, η, through bosonization,
according to the rule
βγ ∼ ∂ϕ
β ∼ e−ϕ ∂ξ
γ ∼ eϕ η .
(2.54)
The singular parts of the OPE’s are
ϕ(z1 )ϕ(z2 ) ∼ − ln z12
η(z1 )ξ(z2 ) ∼
1
z12
(2.55)
and as usual analogous relations hold for the antiholomorphic part. The superghost mode
expansion is
β(z) =
X
n∈Z(+1/2)
βn
n+3/2
z
,
γ(z) =
X
n∈Z(+1/2)
γn
n−1/2
z
,
(2.56)
where the index n is integer in the R case and half integer in the NS case. The ghost part
of the vacuum is defined by
βn |0iN S = 0 , n ≥ 1/2 ,
βn |αiR = 0 , n ≥ 0 ,
bn |0iN S = bn |αiR = 0 , n ≥ 0 ,
γn |0iN S = 0 , n ≥ 1/2 ,
γn |αiR = 0 , n ≥ 1 ,
cn |0iN S = cn |αiR = 0 , n ≥ 1 .
(2.57a)
(2.57b)
(2.57c)
Expansion (2.56) allows the identifications involving the superghost part of the vacua and
the operators
|0iN S ∼ e−ϕ ,
|αiR ∼ e−ϕ/2 ,
(2.58)
22
2 String theory
as β(z)e−ϕ ∼ 1/z, γ(z)e−ϕ ∼ z, β(z)e−ϕ/2 ∼ z −1/2 and γ(z)e−ϕ/2 ∼ z 1/2 . The conformal
weight of elϕ is −l2 /2 − l as it can be checked by the explicit form of the superghost
energy momentum tensor (2.52b). We shall always consider the ground states with these
superghost charges.
We have now to deal with the super-Virasoro algebra (see app. C) which has an anomaly
in a generic number D of target-space dimensions. The full central charge responsible
for the anomaly is the sum of the X, ψ, ghosts and superghosts contributions given
respectively by cX = D, cψ = D/2, cg = −26 and csg = 11. The condition for the
cancellation of the anomaly is then
D+
D
− 26 + 11 = 0 ⇒ D = 10 .
2
(2.59)
A state |ψi is physical when it is annihilated by the Fourier modes with strictly positive
index of both the energy momentum tensor and of the supercurrent. For n = 0 the analog
of the condition (2.29) becomes
NX + Nd − aR |ψi = 0
NX + Nb − aN S |ψi = 0 .
(2.60)
The normal ordering costant for a receives a contribution −1/24 from each transverse
X degree of freedom, +1/24 from each periodic fermion and each antiperiodic fermion
contributes for −1/48 (the ghost and superghost contributions cancel the contribution
from longitudinal X and ψ fields), thus giving
aN S = −
aR = 0 ,
1
.
2
(2.61)
The masses allowed in the spectrum are then
α
0
m2II
=4×
NX + N d
NX + N b −
1
2
=4×
ÑX + Ñd
ÑX + Ñb −
1
2
R
,
NS
(2.62)
where the left and right moving modes can be combined in the 4 possible ways to give the
NS-NS, R-R, R-NS and NS-R sectors. The NS ground state is a scalar tachyon and the R
one is a massless spinor.
The perturbative spectrum has to be truncated to eliminate the tachyon, which is
eliminated by the Gliozzi-Scherk-Olive (GSO) projection that is built out introducing the
world-sheet spinor number F under which the world-sheet scalar X, b, c are even and the
world-sheet spinors ψ, β, γ are odd. To define the GSO action on perturbative states its
action over the total, matter plus ghost ground state, is needed
(−1)F |0iN S = −|0iN S ,
F
(−1) |αiR = |βiR Γ11 βα ,
(2.63a)
(2.63b)
where on the fermionic Ramond ground state (−1) F equals the chirality operator Γ11 .
In the NS sector the GSO projection (1 + (−1) F )/2 is forced by the requirement of
eliminating the tachyon, in the R sector the additonal choice (1 − (−1) F )/2 is available.
The two choices in the R sector correspond to the choice of which chirality to project out
23
2.3 Heterotic string
and which to keep in the spectrum and they are related by a spacetime reflection on a
single axis and so they are equivalent. 3
Finally the right-moving string states have to be tensored with an equivalent set for
left movers. Two choices are available for the R sector: either to pick up one chirality for
the right-moving and the other one for the left-moving modes, obtaining the non-chiral
type IIA theory, or to use the same GSO projection for the R sector for both right- and
left-moving modes, that gives type IIB theory, which is chiral.
The bosonic excitations of type II theories come from both the NS-NS sector and the
R-R one. The NS-NS spectrum is the same as the bosonic string one and it comprises the
graviton, an antisymmetric tensor and the dilaton
GM N , B M N , Φ ,
(2.64)
which amount to 35+28+1=64 physical degrees of freedom. In the R-R sector, where the
tensor product of two spinors give antisymmetric tensors of different ranks, we have the
potentials
CM , C M N P
type IIA ,
(2.65a)
C, CM N , CM N P Q
type IIB ,
(2.65b)
which give 8+56=1+28+35=64 physical degrees of freedom as the field strength of the
type IIB 4-form is subject to the self-duality constraint dC = ∗dC.
The fermions arise from the NS-R and R-NS sector and they are the two MajoranaWeyl gravitinos
ψαM , ψβ̇M
type IIA ,
(2.66a)
ψαM , ψβM
type IIB ,
(2.66b)
which are the superpartners of the bosonic fields in the N = 2 supermultiplet and have
2 × 8 × 23 = 128 physical degrees of freedom.
2.3
Heterotic string
Right and left-moving degrees of freedom in closed strings are decoupled, thus it is conceivable a string theory where the left-moving part of the bosonic string theory is tensored with
the-right moving part of the supersymmetric type II. Let’s consider then the world-sheet
action
!
Z
26
X
1
1
1
¯ A + ψ̃ M ∂ ψ̃M
¯ M+
∂X A ∂X
(2.67)
∂X M ∂X
d2 z
S=
4π
α0
α0
A=11
with the constraint
¯ A=0.
∂X
3
(2.68)
The requirements that the spectrum is tachyon free and that the partition function of the theory is
1-loop consistent (i.e. it is invariant under the modular group of the torus, which is the relevant surface
at 1-loop level, see sec. 2.8) fix the GSO projection to be the one we have just described.
24
2 String theory
This is a constrained system which should be quantized by Dirac brackets but the result is
simple in terms of conformal correlators: the left part can be quantizied as in the bosonic
string case, see correlator (2.26a), the right part as in the type II case, see correlators
(2.26b) and (2.44).
No spacetime interpretation can be given to the sixteen purely left-moving X’s and
we compactify them into a torus T 16 whose size and shape will be determined by the
consistency requirement of the theory, which will maximize the resulting gauge symmetry
of the target space theory.
Let us consider the X A theory. Writing the expansion (2.17) in terms of the coordinate
σ, τ and of the rescaled momentum kL ≡ (α0 /2)1/2 pL we have
0 1/2 X
0 1/2
α
α
1 A in(τ +σ)
xA
A
A
kL (τ + σ) +
+
α e
,
(2.69)
XL =
2
2
2
n n
n6=0
from which the heterotic string mass formula is
1 A
ÑX + Ñd
0 2
0 µ
,
α mhet = −α p pµ = 4 NX + kL kLA − 1 = 4 ×
ÑX + Ñb − 1/2
2
(2.70)
where NX is the bosonic string number operator (2.28) which involves 26 sets of oscillator
whereas ÑX involves only 10. The right-moving part of the spectrum is exactly the same
as the right part of type II string theory. The left-moving tachyonic mode which appears
for NX = kL = 0 is harmless as it does not belong to the spectrum because the last
equality in (2.70) cannot be matched in that case. The lightest excitations are then the
massless ones, among which we have
(
b̃µ−1/2 |0̃iN S
,
(2.71)
αA
−1 |pL = 0i ⊗
|α̃iR
corresponding to the Kaluza-Klein vector gauge fields associated with the U (1) 16 isometry
of the torus tensored to a massless Lorentz vector and its gaugino superpartner. Additional
massless modes are given by NX = 0 and kL2 = 2 and the corresponding states are
(
p
b̃µ−1/2 |0̃iN S
.
(2.72)
exp i 2/α0 kLA XA |0i ⊗
|α̃iR
We thus have the states (2.71) associated to the ∂X A currents corresponding to the Amomenta and the states (2.72) defined by momenta k LA ’s which are the charges under the
former currents, the kLA ’s are then the weights of the representation of the group the states
(2.72) form. This symmetry group turns out to be a target space gauge group symmetry
as it can be checked by explicit scattering amplitude computations.
The allowed momenta kLA lie on a lattice Γ which must be even, as k L2 = 2n for some
integer n in order to satisfy (2.70), and self dual for 1-loop consistency (see sec. 2.8). Only
two possibilities satisfying these two requirements are available: the lattice Γ 16 generated
by
kSO(32) = (±1, ±1, 0, . . . , 0)
| {z }
14×0
(2.73)
25
2.3 Heterotic string
and permutations, and the lattice Γ 8 × Γ8 , each of which Γ8 is generated by combinations
of the vectors
kE8 = (±1, ±1, |{z}
. . . ) ⊕ (±1/2, . . . , ±1/2) ,
|
{z
}
6×(0)
(2.74)
8×(±1/2)
where the components of the first vector can appear in any permutation and the components of the second one have an even number of plus and minus signs. The 480 vectors
(2.73) are the roots of SO(32) and the 240 vectors defined in (2.74) are the roots of E 8
(2.74), which in the way they have been written they are splitted between the the 112
roots of SO(16) (integer components) and the 128 spinorial weigths of SO(16) (half-integer
components).
p
The resulting torus over which the X’s are compactified
is α0 /2 times the the fun√
damental cell of Γ, thus its basis vectors have length α0 .
This construction can be described equivalently in terms of fermions if the 16 bosonic
currents ∂X A are traded, or fermionized, for 32 spinors λ B according to
√ 0 A
p
√
2/α0 ∂X A ∼ λA λA+16 ,
(2.75)
ei 2/α X ∼ λA + iλA+16 / 2 ,
and following the rule that integer coordinates for the compactification lattice points correspond to fermionic antiperiodic boundary conditions, half integer coordinates correspond
to periodic identifications on the fermionic side.
The 32 spinors are naturally endowed with a SO(32) global symmetry as they enter
¯ A . In terms of
the world-sheet lagrangian as λA ∂λ
0
NX
=
9 X
X
M =0 n
αM
−n αM n ,
Nλ =
32 X
X
B=1 n
the left part of the mass formula (2.70) becomes
0
NX + N λ − 1
0 2
0 µ
α mhet = −α p pµ = 4 ×
0 +N +1
NX
λ
nλB
−n λBn ,
NS
R
(2.76)
for periodic (R) or antiperiodic (NS) λ’s. We thus have in the NS sector the 32×31/2 = 496
massless states
B
λA
−1/2 λ−1/2 |0i ,
(2.77)
which are in the adjoint representation of SO(32). The GSO projection acts taking out of
the spectrum the λA
−1/2 |0i states, as here the vacuum is GSO even because there is no (−1)
contribution from the superghosts. The NS vacuum is then projected out not because of
the GSO projection but because it has no match in the right sector. In the R sector there
are no massless excitations.
If the set of 32 fermions is broken into two subsets of 16 fermions each and consequently
the SO(32) symmetry to SO(16) × SO(16), separate NS and R sectors with independent
GSO projections are available in the two subsets. The mass formula becomes
 0
 NX + Nλ − 1 NS-NS
0 2
0 µ
0 +N
R-NS, NS-R ,
(2.78)
α mhet = −α p pµ = 4 × NX
λ
 0
NX + Nλ + 1 R-R
26
2 String theory
where the R sectors furnish spinorial representations of SO(16). In the NS-NS sector,
after the two GSO projections are applied, we are left with two sets of massless states, the
first(second) of which is in the 120-dimensional adjoint representation of the first(second)
SO(16) and in the trivial representation of the second (first) SO(16). Additional massless
states come from the R-NS and the NS-R sectors and they are respectively in the (128,1)
and in the (1,128) of SO(16) × SO(16) where the 128 is the spinorial representation of
SO(16) of definite chirality, as out of the two possible chiralities one is GSO projected
out. Full agreement is then found between the bosonic and the fermionic formulation of
the heterotic string.
Summarizing the bosonic spectrum is made of the graviton, an antisymmetric potential
2 tensor, the dilaton and a gauge field with gauge group SO(32) or E 8 × E8 :
GM N , BM N , Φ, AM
(2.79)
for a total of 35+28+1=64 bosonic physical degrees of freedom in the gravitational multiplet and 8 × 496 bosonic physical degrees of freedom in the gauge multiplet.
The fermionic part is made of one gravitino in the N = 1 graviton multiplet with
8 × 23 = 64 physical degrees of freedom and one gaugino in the N = 1 gauge multiplet
with 23 × 496 physical degrees of freedom
ψαM , λα .
2.4
(2.80)
Type I superstring
We now consider the theory in (2.38) for open strings. In order to fulfil the boundary
conditions from the Eulero-Lagrange equations
(δψψ − δ ψ̃ ψ̃)|π0 = 0
(2.81)
we can impose the following
ψ(τ, σ = 0) = ±ψ̃(τ, σ = 0)
R,NS
(2.82)
and we can always set ψ = +ψ̃ at σ = π. As for the bosonic coordinate X also for ψ we
can trade holomorphic plus antiholomorphic fields on the upper half complex plane for
purely holomorphic fields on the whole complex plane via the identification ψ(τ, σ + π) =
ψ̃(τ, π − σ). The mass formula is
α0 m2I = NX + Nb −
1
2
α0 m2I = NX + Nd
NS ,
(2.83a)
R,
(2.83b)
thus having at the massless level a vector b M
−1/2 |0iN S and a fermion |αiR , which form the
N = 1 supersymmetric multiplet in 10 dimensions.
This is not quite the whole story as open strings have special points, their ends, which
can be endowed with new, non dynamical degrees of freedom that can be in any of n
states. Thus any perturbative state is labelled by a n × n complex Chan Paton matrix t ij ,
that is normalized according to
Tr(ta tb ) = δ ab .
(2.84)
27
2.4 Type I superstring
Open string amplitudes turn out to be invariant under a U (N ) rotation of the Chan-Paton
indices of the type t → AtA† where A is a generic U (N ) matrix, thus each string state of
the oriented theory transforms in the adjoint representation of the symmetry group. The
massless vector is the gauge field as the actual interactions testify, thus converting the
global U (N ) world-sheet symmetry into a spacetime gauge symmetry.
Closed strings must be allowed in the theory as open string ends can merge. As
described in sec. 2.2 type II closed strigs have N = 2 supersymmetry, whereas open
strings have N = 1 supersymmetry and there is no consistent way to couple a N = 1
gauge multiplet in 10 dimensions to a N = 2 gravity multiplet in 10 dimensions. The way
out of this problem is to perform an orientation projection that leads to the unoriented
open plus closed string theory, to achieve which the world-sheet parity operator Ω whose
action takes σ → π − σ or equivalently z → −z̄, has to be introduced. In terms of mode
expansions this means for open strings
n M
αM
n → ±(−1) αn ,
n M
dM
n → ±(−1) dn ,
r M
bM
r → ±(−1) br ,
(2.85)
where the ambiguity in sign depends on the choice of Neumann (+) or Dirichlet (−)
boundary condition. For closed strings we have
M
αM
n ↔ α̃n ,
n ˜M
dM
n ↔ (−1) dn ,
r M
bM
r ↔ (−1) b̃r ,
(2.86)
and
xM → x M ,
pM → pM ,
(2.87)
hold for both open and closed strings. The NS and R vacua satisfy
Ω|0iN S = −i|0iN S ,
Ω|αiR = −|αiR .
(2.88)
The closed string vacua undergo the transformations
|0iN S ⊗ |0̃iN S → |0̃iN S ⊗ |0iN S = −|0iN S ⊗ |0̃iN S ,
(2.89)
|0iR ⊗ |0̃iR → |0̃iR ⊗ |0iR = −|0iR ⊗ |0̃iR .
As type IIB superstrings have the same chiralities on both sides, a world-sheet parity
symmetry Ω can be imposed on it. Performing the orientation projection (1 + Ω)/2 on
type IIB massless spectrum we are left with the closed string massless spetrum of type I
in 10 dimensions, whose NS-NS sector is made of
H
GM N , B
H
MN , Φ
(2.90)
and in the RR sector we have
Z
C,
Z
PP
NP
CM N , C
M
PQ .
(2.91)
28
2 String theory
The NS-R and R-NS sector combine so that only one combination of them survive,
leaving one gravitino ψαM .
For open string Ω projects out the massless states A M and λα in absence of ChanPaton factors. Ω reverses the open string end points and therefore allowing a non trivial
gauge rotation in the Ω action over the Chan-Paton factors we have that the Chan Paton
factors of the surviving states must fulfil
Ω|a, iji = γjj 0 |Ω̂a, j 0 i0 iγi−1
0i ,
(2.92)
where Ω̂ is the part of Ω which acts on the fields. The requirement that Ω square to the
identity forces γ T = ±γ which modulo a change in the Chan-Paton basis is equivalent to
0 11
γ = 11 or γ = M ≡ i
.
(2.93)
−11 0
Due to the way we defined the orientation action over states in (2.85) and (2.88) we have
0
2
λT = −(−1)α mI λ
or
0
2
M λT M = −(−1)α mI λ ,
(2.94)
thus impling that the massless vectors in the spectrum are in the adjoint of SO(N ) or in
the adjoint of Sp(N ). We shall see in sec. (2.9) that only the choice of SO(32) is 1-loop
consistent.
The counting of the degrees of freedom at the massless level is the same as in the
heterotic string case.
2.5
Interactions
To keep account of interactions we must consider the possibility of string merging and
splitting. The resulting world-sheets are Riemann surfaces which can be topologically
classified according to their Euler number χ.
The relevance of the Euler number for world-sheets is due to the fact that the Polyakov
action can be added of a term sharing its same symmetries
Z
Z
λ
λ
1/2 (2)
dτ dσγ R +
ds k ,
(2.95)
Sχ =
4π Σ
2π ∂Σ
where R(2) is the two dimensional Ricci scalar and the last term involve the extrinsic
curvature of the world-sheet boundary. S χ does not affect the equations of motion and
it does not alter the quantization procedure of free strings as it is a total derivative in 2
dimensions: it depends only on the topology of the manifold. Performing the integrations
in (2.95) we obtain Sχ = λχ, where χ counts a combination of the number of holes g (the
genus), of boundaries b and of crosscaps c of the world-sheet according to 4
χ = 2 − 2g − b − c .
(2.96)
A crosscap is a hole with opposite points identified. Cutting a crosscap on a world-sheet
does not introduce boundaries. Adding a handle to a closed string world-sheet decreases
4
To eq. (2.96) nc /2 must be added in the case the world-sheet has nc corners.
29
2.5 Interactions
(a)
(b)
(c)
(d)
Figure 2.1: Closed string scattering at tree level (a) and one-loop level (b) and open
string scattering at tree level (c) and one-loop level (d).
χ by two and corresponds to the emission and reabsorption of a closed string, whereas the
addition of a boundary to an open string world-sheet corresponds either to absorption and
emission of an open string or to emission or absorption of a closed string, thus leading,
for the closed (open) string coupling parameter g c (go ), to
gc ∼ go2 ∼ eλ .
(2.97)
In other words a diagram with g handles will be a closed string g−loop diagram and it will
be weighted in the path integral by a factor g c2g ∝ e−λχ . Thus the perturbative expansion
in string theory is an expansion in powers of e 2λ for closed string and eλ for open ones: the
analog of the loop expansion in point particle quantum field theory is then an expansion
in the world-sheet genus. We shall see moreover in sec. (2.7) that λ is not a free parameter
but its value is set by the vacuum expectaction value of the dilaton.
The tree level amplitudes correspond to surfaces with the highest possible Euler number. There are three Riemann surfaces of positive Euler number: the sphere, S 2 which
is the world-sheet spanned by the temporal evolution of oriented closed strings, the real
projective plane RP2 which is the world-sheet of unoriented closed strings (involving the
presence of orientifold planes, that will be defined in sec. 2.9) and the disk D 2 , the open
string world-sheet. The real projective plane is topologically equivalent to a sphere with
the insertion of a crosscap at a point.
At each genus the external (incoming and outgoing) strings that correspond to physical
on-shell mass eigenstates are accounted for by inserting in the path integral vertex operators
at given points. Vertex operators carry the quantum numbers of the excitations and they
are mapped to the boundary of the world-sheet in the case they represent open string
excitations, to internal points if they describe closed string modes.
To evaluate scattering amplitudes for a generic process involving particles of type
i1 , . . . , in with momenta k1 , . . . kn we have to perform the path-integral of the (1+1)−dimensional quantum field theory that rules the propagation of a string, with the insertion of
30
2 String theory
operators Vin (ki ) and summing over compact world-sheets of different topologies
(2.98)
Si1 ...in (k1 , . . . , kn ) =
n Z
X Z [dX] [dψ] [dγ]
Y
exp (−SSP − λχ)
gn
d2 σj γ 1/2 (σj )Vij (kj )
V
S(diff×Weyl)
world−
j=1
sheets
where we divided by the infinite volume of the gauge group made of 2-dimensional supersymmetrized diffeomorphisms and Weyl rescaling and a factor g for each vertex operator
has been added. The Faddeev-Popov method can be used to factorize in the integrand the
volume of the gauge group so to simplify the functional integration. The presence of the
b, c ghosts is related to the existence of parameters in the metric that cannot be removed
by symmetries, the moduli, and symmetries that are not fixed by the choice of the metric,
which make up the conformal Killing group (CKG). The result that will not be derived
but only shown here is that the Fadeev-Popov determinant is expressed in terms of ghost
path integral with insertions, one b insertion for each metric modulus and one c insertion
for each independent confomal Killing vectors (CKV). For each c insertion the coordinate
of one vertex operator is not integrated over but it is fixed, thus cancelling part of the
CKG volume at the denominator of (2.98). The number of moduli µ and CKVs κ are
constrained on general grounds by the Riemann-Roch theorem
µ − κ = −3χ .
(2.99)
In particular the sphere has 3 complex CKVs, the projective plane and the disk three
real ones and none of them has any modulus. At 1-loop the relevant surfaces are the
torus for oriented closed strings, which has one complex modulus and one complex CKV,
the cylinder for oriented open strings, the Möbius strip (or equivalently the cylinder with
one boundary replaced by a crosscap) for unoriented open strings and the Klein bottle
(equivalent to a cylinder with boundaries replaced by crosscaps) for unoriented closed
strings. The cylinder, the Möbius strip and the Klein bottle all have one real modulus
and one real CKV.
Superghost are dealt with by ensuring that the total ϕ charge of vertex operators
involved in the amplitude equals the number of superconformal Killing vectors (SCKV) and
including as many picture changing operators (PCO)’s as supermoduli. The PCO operator
P (z) is defined in term of the BRST operator Q BRST and the ξ from the bosonization of
the superghost, according to
1/2
2
eϕ ηψM ∂X M ξ .
(2.100)
P ≡ QBRST ξ =
α0
The commutator of the PCO operator with a vertex operator of a pysical states gives a
vertex operator for the same physical states but with a ϕ charge or picture increased by
one. The obtained vertex operator is not BRST trivial as the commutator involves the
zero mode of ξ which does not enter the βγ path integral. Thus a string amplitude for
the disk, for instance, looks like
Z
Z
1
2 1
a
a
n
dXdψ exp −
Si1 ...in (k1 , . . . , kn ) = go
d σ 0 ∂a X∂ X + ψρ ∂a ψ ×
4π
α
(2.101)
Qn
Q
3
2
(2)
j=1 d σj
k=1 δ (σk − σ̄k ) Vij (kj )JB ,
2.6 Conformal Killing groups for zero genus surfaces
31
where B takes account of the insertion of the b-ghost and J involves the the c−ghost path
integral. J can be interpreted as the Jacobian relative to the change of variables from the
vertex operator positions which are fixed to the CKG parameters as we are going to show
in the next section.
To ensure that the amplitude is a scalar under all the symmetry of the action we
started with, the vertex operator must have conformal weight J V = 1 for open strings and
JV = J˜V = 1 for closed ones, beside being invariant under world-sheet diffeomorphisms
and D−dimensional Poincaré transformations.
2.6
Conformal Killing groups for zero genus surfaces
We now wish to describe an amusing (at least to the author) issue which is usually not
covered in standard expositions on string theory: how to deal with amplitudes involving a
partly or completely unfixed CKG. The application of this will be described in sec. 2.10,
where the brane tension will be computed through a disk computation rather through the
well-known cylinder amplitude.
When dealing with string amplitudes the conformal Killing groups of the relevant
surfaces have to be considered. In particular for amplitudes involving a number of vertex
operators which is less than the number of CKV’s for the relative surface the recipe given
in the previuos section does not apply and the factor J in (2.101) from the ghost path
integral must be substituded by an explicit evaluation of the volume V CKG of the CKG.
For the sphere the CKG is SL(2, C)/Z2 and it acts on the complex plane coordinate z
according to
αz + β
,
z → z0 =
γz + δ
with α, β, γ, δ ∈ C and αδ − βγ = 1. The three generators of the algebra sl(2, C) generate
the following transformations (we also report their infinitesimal form)
e λ0 L 0 : z → z 0 = e λ0 z ∼ z + λ 0 z ,
z
e λ1 L 1 : z → z 0 =
∼ z − λ1 z 2 ,
1 + λ1 z
eλ−1 L−1 : z → z 0 = z + λ−1 ,
(2.102a)
(2.102b)
(2.102c)
with λi ∈ C.5
2.6.1
The disk conformal Killing group
The disk is obtained from the plane by identifying points z and z 0 = 1/z̄, considering the
region |z| ≤ 1 as the fundamental one, with border given by |z| = 1. The CKG of the
sphere which survives the previous identification and which maps the disk to itself is the
SU (1, 1)/Z2 subgroup of SL(2, C)/Z2 , acting on the coordinate w of the disk according
to
αw + β
w → w0 =
,
β̄w + ᾱ
5
Note the difference between (2.102b) here and the first of (7.1.33) in [28]. Our sign choice is forced
by the requirement that being L1 + L−1 an hermitean operator eL1 +L−1 should generate a non-compact
subgroup. We shall reward with a fine bottle of Chianti who can explain this discrepancy.
32
2 String theory
with the constraint |α|2 − |β|2 = 1 (α, β ∈ C). The su(1, 1) algebra is generated in terms
of the λ’s of the parent sl(2, C) by λ0 ∈ I and λ1 = λ̄−1 . The disk with coordinate w can
be mapped into the upper half plane with coordinate z and vice-versa according to
z = −i
w+1
,
w−1
w=
z−i
.
z+i
(2.103)
In the half plane representation of the open string world sheet the CKG turns out to be
SL(2, R)/Z2 (isomorphic to SU (1, 1)/Z2 ) who has the following action on the coordinates
z → z0 =
az + b
,
cz + d
(2.104)
with a, b, c, d ∈ R and ad − bc = 1 (thus generated by λ 0,±1 ∈ R).
SL(2, R)/Z2 (SU (1, 1)Z2 ) is a noncompact group, whose volume is infinite. 6 We may
expect amplitudes which leave unfixed a non compact subgroup of SL(2, R)/Z 2 to vanish,
due to the suppressing factor of VCKG at the denominator, but we shall qualify this
statement as divergent quantities need regularization.
Expanding the generic generator of the CKG over purely hermitean or anti-hermitean
basic generators the generic element of the algebra can be parametrized as follows
Lsu(1,1) = i2θL0 + b+ (L−1 + L1 ) + ib− (L−1 − L1 ) ,
Lsl(2,R) = θ(L−1 − L1 ) − 2b+ L0 + b− (L−1 + L1 ) .
(2.105a)
(2.105b)
The subgroups generated by each of the above three basic generators are
w → w0 = e2iθ w
0≤θ≤π ,
z cos θ + sin θ
z → z0 =
−z sin θ + cos θ
w
cosh b+ + sinh b+
w → w0 =
w sinh b+ + cosh b+
b+ ∈ R ,
z→
z0
=
(2.106b)
e−2b+ z
w cosh b− + i sinh b−
−iw sinh b− + cosh b−
z cosh b− + sinh b−
=
z sinh b− + cosh b−
w → w0 =
z → z0
(2.106a)
b− ∈ R ,
(2.106c)
and the group volume VSU/Z2 can be conveniently written in terms of the parameters θ
q
and b ≡ b2+ + b2− [32] as
VSU/Z2 =
Z
π
dθ
0
Z
∞
0
db 2πb
sinh 2b
.
2b
(2.107)
The overall normalization is chosen so that the volume element at the origin is dθdb + db− .
We note that as 2L0 , (L−1 + L1 ) and i(L−1 − L1 ) are hermitian they generate noncompact subgroup, whereas the antihermitian 2iL 0 , (L−1 −L1 ) generate compact subgroup
6
Non compactness does not necessarily implies infinite volume as the size of the volume depends on the
metric. We shall show later that SL(2, R) has indeed infinite volume.
2.6 Conformal Killing groups for zero genus surfaces
33
according to (2.106). The full CKG volume is infinite, but to make sense of it we put a cut
off so that the maximum square “distance” on the CKG is 1/ which is equivalent to put
a cutoff on the coordinate bmax = −1/2 ln . Taking this upper limit the volume becomes
[32]
cutof f
VSU/Z
=
2
π2 π2
−
.
4
2
(2.108)
Indeed only logarithmic divergences are really infinite, as for instance
Z 1
dx −β
1
x =
for β > 0
x
β
0
can be defined by analitical continuation for any β < 0 leaving only the logarithmic
divergence from β = 0 as the real infinite. So discarding the 1/ pole term in (2.108) a
renormalized volume can be assigned to the CKG
ren
VSU/Z
=−
2
π2
.
2
(2.109)
To determine the volume in terms of the usual literature parameters λ 0,±1 of SL(2, R) the
Jacobian has to be used


−2 0 0
∂ (λ0 , λ−1 , λ1 ) 
(2.110)
=
0 1 1 =4
∂ (b+ , b− , θ)
0 1 −1
then obtaining
ren
= −2π 2 .
VSL/Z
2
(2.111)
The change of variables in the parametrization of the algebra between λ i and b± , θ is
defined by
λ0 L0 + λ−1 L−1 + λ1 L1 = −2b+ L0 + θ(L−1 − L1 ) + b− (L−1 + L1 ) ,
(2.112)
which implies −2b+ = λ0 , 2b− = λ−1 + λ1 and 2θ = λ−1 − λ1 , from which (2.110) follows.
Trusting the renormalized value of the CKG volume we are forced to admit that the zero
point amplitude on the disk is nonvanishing, being proportional to
2
AD
0 ∝−
C D2
1
= − 2 0 2 6= 0
2
2π
2π α go
(2.113)
where the coefficient of proportionality involves the volume of the fermionic part of the
CKG VSCKG and we used [29]
C D2 =
1
α0 go2
(2.114)
which holds in the case the group is parametrized by the λ’s given in (2.112).
Being convinced now (?) that the full CKG volume of the disk is finite we may run
into trouble as non vanishing corrections to one and two-point open string amplitudes may
34
2 String theory
appear, meaning open string tadpoles and mass and possibly wave function renormalization. However this is not the case as we now show.
The 1-open string amplitude leaves unfixed the subgroup of the CKG generated by (2.106b)
and a non compact combination of (2.106a) and (2.106c) (this is more easily checked in
the case the point is fixed at w = 1 or w = −1 on the unit disk or z = 0 or z = ∞ on the
half plane) which has volume
Z ∞
sinh 2b
π
π
1pt
db 2πb
=
− ,
(2.115)
VSU/Z2 =
2b
4 2
0
which again does not have a logarithmic singularity, so consistently with our previous
1pt
reasoning the assignment VSU/Z
= ∞ is not allowed. But in the case of one open string
2
the correlator of a single vertex operator trivially vanishes, so indeed the one-point open
string amplitude is vanishing.
In the case of two open string, fixed say at w 1 = 1 (z = 0) and w2 = −1 (z2 = ∞) the
unfixed CKG turns out to be the one parametrized in (2.106b) which has volume
2pt
VSU
Z
=
e2bmax
4
0
1
1
∼ − ln(2)
db+ q
2
1 + 4b2+
(2.116)
which has indeed a “true” logarithmic divergence, thus making the two-open string amplitude on the disk vanishing. Fixing instead one closed string vertex operator on the disk
leads to a finite residual VCKG as we shall see in sec. 2.10.1.
To check the full consistency of the renormalization of the CKG volume we also have to
show that troubles are not encountered with closed string amplitudes on the sphere. The
sphere CKG is SL(2, C)/Z2 , which is generated by
Lsl(2,C) = 2 (b0 + ia0 ) L0 + (b+ + ia+ ) (L−1 + L1 ) + (ib− − a− ) (L−1 − L1 ) . (2.117)
Its volume, normalized so that the volume element near the origin is d 3 ad3 b, is [32]
Z π/2
2
2 Z ∞
π3
π3
2 sin a
2 sinh 2b
da 4πa
VS 2 =
db
4πb
=
+
ln .
(2.118)
a2
4b2
162
4
0
0
The full volume is infinite as it has a logarithmic divergence. As again the correlator for
a single vertex operator is vanishing, only the unfixed CKG in the two points case is left
to be checked. Fixing two points on the sphere at z 1 = 0 and z2 = ∞, the unfixed CKG
is parametrized by
L2pt
SL(2,C) = 2(b0 + ia0 )L0 ,
whose volume
2pt
Vsl(2,C)
=
Z
0
π
da0
Z
e2bmax
4
0
1
π
db0 p
= − ln(2)
2
2
1 + 4b0
is truly infinite having a logarithmically divergent part. Summarizing the full picture is
consistent with the fact that a cosmological constant and a tadpole term for closed string
fields can appear on the disk but not on the sphere. We also see that an open string
tadpole is forbidden as well as a renormalization of two point amplitudes for open string
on the disk and for closed strings on the sphere.
35
2.7 Low energy effective action
2.6.2
The projective plane conformal Killing group
After studying the case of the disk we now turn to consider the projective plane. To
compute the coupling of closed strings to orientifold we have to consider a world-sheet
with the topology of the projective plane. It can be obtained from the sphere after the
identification z = −1/z̄, thus obtaining the unit disk as a fundamental region with points
at |z| = 1 diametrically identified. The resulting space has no border and open strings do
not couple to it. The CKG is the subgroup of SL(2, C)/Z 2 which respects the previous
identification, that is SU (2)/Z2 = SO(3). It acts on the points of the projective plane
according to
w → w0 =
αw + β
,
−β̄w + ᾱ
(2.119)
provided that |α|2 + |β|2 = 1, α, β ∈ C. The generic generator of SU (2) can be written as
Lsu(2) = 2ia0 L0 + a1 (L−1 − L1 ) + ia2 (L−1 + L1 ) .
(2.120)
In terms of the parent sl(2, C) parameters λ’s, see (2.102), su(2) is the subalgebra spanned
by λ0 = 2ia0 and λ−1 = −λ̄1 = (a1 + ia2 ). Its volume is genuinely finite
Z π/2
sin2 a
(2.121)
da 4πa2 2 = π 2 .
a
0
The subgroups generated by each of the generators appearing in (2.120) are
w0 = e2ia0 w
w cos a1 + sin a1
w0 =
−w sin a1 + cos a1
w cos a2 + i sin a2
w0 =
,
iw sin a2 + cos a2
(2.122a)
(2.122b)
(2.122c)
with a0,1,2 ∈ [0, π].
2.7
Low energy effective action
From the calculation of string amplitudes it is possible to derive the target space effective
action which reproduces the string mode scattering amplitudes. The low energy effective
action exhibits a double expansion, in
• α0 , as at high curvature, namely α0 R > 1 the non-point-like nature of the string will
emerge and massive modes will also be important.
• e2Φ , as we identified the λ in (2.95) with the dilaton expectation value. At strong
coupling, Φ ∼ 0, quantum effects, or in the string language, world-sheets with non trivial
topology contributions will not be suppressed in the path integral summation.
We now show the basic low energy effective action, restricting to the perturbative
regime in both the string length and the coupling. For the bosonic degrees of freedom of
type II string theory we have, neglecting Chern-Simons interactions,
Z
√ −2Φ
1
1
10
2
N S2
d x Ge
(2.123)
R + 4∂Φ∂Φ − (dB)
SII = 2
2
2κ10
36
2 String theory
in the NS-NS sector and
2
R
SII
=
1
4κ210
Z


X
√
d10 x G 
(dC (p+1) )2 
(2.124)
−1≤p≤4
in the R-R sector, where C (p+1) are (p + 1)-antsymmetric potential and only even(odd)
values of p are allowed for type IIA(B). For p = 3 the relative 5-form field strength is self
dual, i.e. dC (4) = ∗dC (4) .
In the heterotic string case the action turns out to be
Z
√
α0
1
1
d10 x Ge−2Φ R + 4∂Φ∂Φ − (dB)2 − TrF 2 ,
(2.125)
Shet = 2
2
8
2κ10
where the trace in the last term is understood to be over gauge degrees of freedom. The
heterotic string relation between gauge coupling and Planck mass is then in 10 dimensions
2
g10
=4
κ210
,
α0
(2.126)
which is valid even after compactification over an internal dimension volume V with p + 1
non compact dimensions as
2
gp+1
=
2
g10
,
V
κ2p+1 =
κ210
,
V
(2.127)
thus implying that the string scale is close to the Planck scale for reasonable values of the
gauge coupling.
For type I the low energy effective action is
−2Φ Z
√
e
1
e−Φ
10
(2) 2
2
SI = d x G
R + 4∂Φ∂Φ − (dC ) − 2 TrF
.
(2.128)
2
2κ210
2g10
In this case the gauge coupling and the Planck mass in 10 and lower dimensions are related
through
2
g10
∝ κ10 α0 ,
2
gp+1
∝
κp+1 V 1/2
α0
7−p
2
= V̄ 1/2
κp+1
α
p−1
0 4
α0
p−3
2
,
(2.129)
thus leaving open the interesting possibility that for small extra-dimensions, i.e. V̄ ≡
V /α0 (9−p)/2 < 1 the string energy scale can be lower than the Planck scale even for
gp+1 ∼ 1, possibility which will be dealt with in ch. 7. The proportionality coefficient
in (2.129) is given in (7.88).
These actions are written in the so-called string frame which displays explicitly the
genus of the world-sheet to which they are associated. The type II, heterotic and the
gravitational part of type I low energy effective actions correspond to sphere amplitudes,
the gauge part of type I to disk amplitudes.
The implicit identification of the λ parameter of (2.97) with the dilaton expectation
value can be understood considering the world-sheet action for the propagation of a string
in a non trivial background, that is a space where some of the massless modes are excited
Z
h
i
1
2
1/2
ab
ab
M
N
0 (2)
Scur = −
d
σγ
γ
G
+
i
B
∂
X
∂
X
+
α
R
Φ
, (2.130)
M
N
M
N
a
b
4πα0 Σ
37
2.8 Compactification and T-duality
where only the bosonic coordinate X has been displayed and we restricted non trivial
background value to the NS-NS fields only. Actually, to allow strings to propagate in
curved space, world-sheet conformal invariance (T = 0) should be checked first. From
the point of view of the two-dimensional world-sheet theory the above G M N , BM N , Φ are
field dependent coupling, which get renormalized by higher order interactions, being α 0
the coupling parameter, thus giving
T =
i B ab
1 Φ (2)
1 G ab
M
N
βM N γ ∂a X M ∂b X N − 0 βM
,
N ∂a X ∂b X − β R
0
2α
2α
2
(2.131)
where the β i are the beta-functionals for our interacting theory. The equation β i = 0 are
the equation of motions derived from previous actions for the NS-NS fields. The power
expansion in α0 means that the dimensionless expansion parameter is α 0 R, where R is the
characteristic curvature of the target space, thus the actions displayed acquires corrections
to all order of α0 , being trustworthy only in the limit α 0 R 1.7
The actions can be rescaled so to give an Einstein term canonically normalized via the
transformation
GM N → gM N = e−2Φ/(D−2) GM N = e−Φ/4 GM N .
(2.132)
One then obtains the low energy effective action in the so called Einstein frame which is
for the terms involving the graviton and the dilaton
Z
1
1
10 √
µ
d x g R − ∂µ Φ∂ Φ ,
S=−
(2.133)
16πGN
2
whereas for other fields it is the same as before a part from a multiplicative factor of the
type eaΦ for some constant a.
2.8
Compactification and T-duality
Closed strings are defined by the periodicity condition X(τ, σ) = X(τ, σ + 2π) that implies
pL = pR unless identification X ∼ X + 2πR is assumed for some R. The previous
identification means that the dimension is compactified on a circle S 1 of radius R.
As the spacetime momentum is given by the combination (p L + pR )/2 according to
eq. (2.20), the combination
pµL − pµR
wR
= 0
2
α
(2.134)
is the winding around the compact dimension, being w the winding number. Compactification of some of the coordiantes is a relevant issue as string theory predicts a number of dimensions higher than 4 and the simplest way to achieve an effective 4−dimensional theory
7
One could expect that the contribution of massive particles should be included as higher order α 0
corrections are considered, which are equivalent to higher loop contributions in the 2-D field theory, since
they are associated with non-renormalizable interactions and all terms can be generated in the α 0 expansion.
However the massive modes play no role as the associated operators do not mix with the massless sector;
more generally to renormalize background fields up to a given mass level only fields to the same or lower
mass level are needed [33].
38
2 String theory
is to reduce the 6 extra-dimensions into a compact manifold (or even not a smooth manifold, as we shall see) leaving only 4 non-compact or at least huge dimensions. The physics
at distances much longer than the compactification radius is essentially 4-dimensional.
The simplest way to reduce the dimensions of spacetime is to compactify a space
coordinate into a 6-dimensional torus T 6 by allowing the periodic identification
X m ∼ X m + 2πRm ,
(2.135)
where the Rm are the radii of the flat T 6 . The mass formula for closed strings is modified
with respect to the uncompactified case (2.62) or (2.70) to
0 m
α0 m2II = 4(NX + Nd ) + α0 pm
L pLm = 4(ÑX + Ñd ) + α pR pRm
0 m
α0 m2het = 4 (NX − 1) + α0 pm
L pLm = 4 ÑX + Ñd + α pR pRm
(2.136a)
(2.136b)
m
m
m m
0
where pm
L,R = (n /R ± w R /α ), n, w are the integer momentum and winding number
around the compact dimensions and only the R sector has been considered for the sake of
brevity. The mass formulas (2.136) are unaltered under the transformation
Rm ↔
α0
,
Rm
n↔w
(2.137)
which is the simplest example of T −duality symmetry. This simmetry holds at any order
of perturbation theory√and it can be extended to more general compactification patterns.
It allows to consider α0 as a minimal length in string theory as shorter scale can be
traded for bigger ones. The T-duality action over fields can be summarized by
XL → X L
XR → −XR
ψ → ψ,
ψ̃ → −ψ̃ .
(2.138)
Denoting by m̄ the dimension over which the T-duality transformation is performed, it
affects the closed string spacetime fields according to
1/2
1/2
Φ → Φ0 = Φ − ln R/α0
= Φ + ln R0 /α0
,
(2.139a)
1
,
Gm̄m̄
(
(p)
Cµ0 ...µp−1 m ∈ {µi }
.
=
(p+2)
Cµ0 ...µp m m ∈
/ {µi }
0
Gm̄m̄ → Gm̄
m̄ =
Cµ(p+1)
→ Cµ0 0 ...
0 ...µp
(2.139b)
(2.139c)
Thus the dilaton change so that the effective coupling in the dimensionally reduced theory
is not altered and the R-R potential looses or acquires the dimension over which the
T-duality is performed.
In the case of compactification of a single dimension we have no constraints over the size
R, but in more dimensions a condition analog to the heterotic string case must be matched.
This can be checked in the explicit form of the one-loop partition function, which is the
path integral computed on the torus world-sheet with no insertions of external particles.
The torus has one CKV and one complex modulus τ = τ 1 +iτ2 , thus the partition function
involves an integration over τ . The torus partition function can be obtained by taking
39
2.8 Compactification and T-duality
θ34 /η 4
NS
θ44 /η 4
(−1)F NS
θ24 /η 4
R
θ14 /η 4
(−1)F R
Table 2.1: Relation between θ functions and spin structures.
the world sheet field theory on a circle, evolving for time 2πτ 2 , traslating the spatial
world-sheet coordinate σ by 2πτ1 and then identifying the ends, leading to
ZT 2 (τ ) = Tr [exp (2πiτ1 P − 2πτ2 H)] ,
(2.140)
where P is the momentum and H the Hamiltonian. Expressing P and H in terms of
physical modes of the string, making explicit the integration over the modulus domain
F given by −1/2 < τ1 < 1/2, |τ | > 1, dividing by VCKG = (2π)2 τ2 as there are no
vertex operator to fix, considering the b ghost insertion which introduces a factor π 2 (an
additional factor 1/2 comes from the GSO projector), considering n compact dimensions
and 10 − n infinite ones, and finally integrating over τ the integrated partition function is
obtained
Z 2 Z
X
0 2
0 2
0 2
dp+1 k
d τ
q α pL /4 q̄ α pR /4 ×
(q q̄)α k /4
ZT 2 = V10−n
p+1
(2π)
F 8τ2
9−p
(2.141)
4
4 n,w∈Z
∗
4
4
4
4
1 θ3
θ4
θ2
θ1
θ4
θ24
θ14
1 θ3
,
−
−
−
−
−
±
η8 η4 η4 η4 η4
η8 η4 η4 η4 η4
where q ≡ exp(2πiτ ). The ghost path integral, beside adding numerical factors, cancels
the contributions from non-physical longitudinal states.
The functions η(τ ), θ1..4 (ν, τ ) are defined in app. D and they gather the summation over
the particle content of the teory. Their argument are understood to be τ for η and 0 and τ
for the θ’s. The first factor η 8 takes account of the X excitations, the θ i /η 4 of the various
spin structures which the torus can be endowed of as summerized in tab.2.1. The sector
label NS or R defines the σ periodicity and the presence or the absence of the factor (−1) F
the time periodicity on the world-sheet (periodic condition in the time direction if (−1) F
is present).
The first square bracket correspond to the left moving side oscillators, the second one to
the right moving side, where the ± sign allows to choose between type IIA (+) or type
IIB (−). In heterotic string the first square bracket is substituted by the contribution of
26 X, that is
4
X
1 θ3
θ44
θ24
θ14
2
1
−
−
+
→
(2.142)
q kl /2 24 .
η8 η4 η4 η4 η4
η
n
Γ
kl ∈
16
Γ8 Γ8
The tori with modulus τ , τ + 1 or −1/τ are equivalent, so must be their partition
function. This requirement forces the compactification lattice to be even and self dual of
Lorentzian signature (j, k), where j is the number of left dimensions and k the number
40
2 String theory
of right dimensions. The heterotic string in the bosonic formulation described in sec. 2.3
involves an even self dual lattice of the type (16, 0).
We have been cavalier with the superghost insertions. Indeed the NS, (−1) F NS and
R sector have no superghost zero mode, but the (−1) F R (the odd spin structure) has.
Anyway we shall not dwell on this detail, but we just mention that θ 1 (0, τ ) ≡ 0 and this
spin structure does not play any role in our analysis (even if it does play a role in other
interesting effects), so the superghost complications can be safely forgotten here.
Let us now turn our attention to the action of a T-duality transformation on open
strings, which do not have T-duality symmetry, as they have Kaluza-Klein momenta or
winding modes depending on their boundary condition being NN or DD. Thus the type I
mass formula in the open string R sector is
m 2
n
0 2
0 µ
0
NN ,
(2.143a)
α mop = −α p pµ = NX + Nd + α
Rm
α0 m2op = −α0 pµ pµ = NX + Nd +
(wm Rm )2
α0
DD ,
(2.143b)
where the µ indices are along the noncompact dimensions and the boundary conditions
over compact dimensions are NN in the first case and DD in the second. For open strings
a Neumann boundary condition is turned into a Dirichlet one by T-duality, as it is clear by
comparing (2.15) with (2.138). A Dirichlet boundary condition breaks Poincaré invariance
as the string endpoint is fixed to a lower dimensional hyperplane or Dp-brane where p is
the number of its spatial coordinates. Thus strings with DD boundary conditions over
nDD coordinates are attached to Dp-branes with codimension n DD or equivalently with
p = 9 − nDD .
Also mixed ND boundary conditions are possible, corresponding to an open string with
one end on a brane an the other end on a brane of different dimension or not parallel to
the first one. The mode expansion of an ND string is
0 1/2 X
α
1 µ −r
m
µ
X (z, z̄) = x + i
αr z ± z̄ −r
ND(DN) ,
(2.144)
2
r
r∈Z+1/2

−(n+1/2)
 dm
X
R
1
n+1/2 z
m
.
(2.145)
ψ (z) = 1/2
 bm z −n
z
NS
n∈Z
n
Contrarily to NN and DD cases, here the X coordinate is antiperiodic as well as the
fermionic R sector, whereas the NS sector is periodic. This is the same moding of a
twisted state under a Z2 orbifold (see below). The mass formula for a string with n N D
coordinates is
NX + N d
R
0 2
,
(2.146)
α mN D =
NX + Nb − 1/2 + nN D /8 NS
thus the R ground state is a massless scalar and the NS one is a fermion (referring to the
spacetime symmetry group of the coordinate with ND conditions) which is tachyonic for
nN D < 4, massless for nN D = 4 and massive for nN D > 4. However only the configurations
with nN D = 0, 4, 8 can be supersymmetric and they turn out to be the only stable ones.
41
2.8 Compactification and T-duality
Compactifications can also be realized so to leave no more than N = 1 supersymmetry
in D = 4, corresponding to 4 supercharges, whereas the N = 1 (N = 2) theory in D = 10
has 16 (32) supercharges, whose straightforward truncation will lead to N = 4(8) in
D = 4. A supersymmetry algebra larger than N = 1 in D = 4 forbids the existence of
chiral gauge coupling (we shall not deal with the issue of complete supersymmetry breaking
necessary for a completely realistic scenario). Compactification mechanisms that break
some of the supersymmetries and possibly part of the large gauge group which emerges in
the 10−dimensional theory to smaller and phenomenologically more interesting ones are
available.
Adding to the periodic identification (2.135) the identification under a discrete group
G whose action on T is not free, i.e. it has fixed points, the compactification manifold is
no more a smooth one, as at fixed points singularities develop, but the resulting space is
called an orbifold. Nevertheless the strings propagation on such orbifolds makes perfect
sense provided the spectrum is truncated in a proper way. For instance by taking the
double identification, for simplicity over a pair of coordinates
X i,i+1 ∼ X i,i+1 + 2πR ,
X i,i+1 → −X i,i+1 ,
(2.147)
the orbifold T 2 /Z2 is defined. Now the states which are not invariant under G = Z 2 have
to be projected out. The G action on a generic untwisted state is, for instance in the NS
sector,
|NX , Nb , n, wiN S → (−1)NX +2Nb |NX , Nb , −n, −wi .
(2.148)
The truncation to G−invariant states spoils the consitency of the theory unless a new
sector is added, the twisted sector, in which the compact coordinate satisfies
X(τ, σ = 0) = −X(τ, σ = (2)π) .
(2.149)
Under a Z2 orbifold the moding of the X and ψ is the same as the one described in
(2.144,2.145) for the ND string.
A non trivial action of G on the four spacetime dimensions is not allowed as it would break
Poincaré invariance, but it can be allowed on the heterotic gauge indices, thus breaking
part of the huge gauge invariance present in the theory.
For realistic compactifications the usual pattern is
M10 → M4 × R6 → M4 × T 6 → M4 × T 6 /G .
(2.150)
The 10-dimensional Lorentz group SO(1, 9) has a subgroup SO(1, 3) × SO(6) where
SO(1, 3) is the four-dimensional Lorentz group and SO(6) ∼ SU (4) is the subgroup of
SO(1, 9) that commutes with SO(1, 3). The 4 spinors of N = 4 in four dimension transform as a 4 of SU (4). The amount of supercharges in the compactified theory will be
given by the number of supercharges left invariant by G. If G for instance is chosen to
belong to a SU (3) subgroup of SU (4) then one element of the 4 of SU (4) is invariant
under SU (3) and that will lead to 4 unbroken supercharges (corresponding to the state of
SU (4) which is a singlet under its SU (3) subgroup action) or N = 1 supersymmetry in
D = 4. If G ∈ SU (2) then there will be 8 unbroken supercharges and N = 2 in D = 4.
42
2 String theory
A similar reasoning is at the base of Calabi-Yau construction. The gravitino supersymmetry transformation with parameter η is
δΨM = ∇M η .
(2.151)
Acting on the gravitino twice with exchanged order of derivatives and equating to zero to
find the condition for supersymmetry to hold we obtain
[∇M , ∇N ] η = RAB
M N σAB η ,
(2.152)
and the previous equation can be satisfied in the case of a smooth manifold if the holonomy
group is not the whole SU (4) but some proper subgroup of it. In particular if it is SU (3)
we have 4 unbroken supercharges (N = 1 in 4-D). Calabi-Yau manifolds with complex
dimension n are defined to be manifold endowed with a Kähler structure (i.e. manifold
with a complex structure) whose holonomy takes value in the SU (N ) subgroup of U (N ).
Thus compactification on M4 × CY 3 preserve N = 1 supersymmetry in D = 4.
2.9
Some non-perturbative aspects
In the previous section D-branes emerged from T-duality cosiderations. They are nonperturbative objects, their mass being inversely proportional to the coupling parameter
as we shall see, their presence and stability in the spectrum at low coupling is ensured by
their BPS nature.
As D-branes are open string related object, lowest order amplitudes in which they
are involved are computed on a world-sheet with boundary, the disk and the open string
boundary conditions are invariant under only D=10, N = 1 supersymmetry, implying that
out of the two supersymmetry charges Q and Q̃ only their combination Q + Q̃ is conserved
in the presence of D9-branes. Performing a T-duality over a direction m̄ the right moving
charge undergoes the transformation
Q̃ → β m̄ Q̃ ,
(2.153)
where β m̄ = Γm̄ Γ is the parity transformation on spinor, as required by (2.138). Moreover
Dp-branes carry charge µp under the RR (p + 1)-form potential
Z
µp = C (p+1)
(2.154)
which is vital to ensure the BPS relation. For a D9-brane the related potential is a
non dynamical (and as such not present in the perturbative spectrum) 10-form, but by
T-duality forms of any rank can be reached.
The full supersymmetry algebra in the presence of a brane can be written as
{Qα , Q̄β } = −2PL M ΓM
αβ ,
M
˜ } = −2P
{Q̃α , Q̄
R M Γαβ ,
β
M1
˜ } = − 2µp QR
· · · β Mp αβ ,
{Qα , Q̄
β
M1 ...Mp β
p!
(2.155a)
(2.155b)
(2.155c)
43
2.9 Some non-perturbative aspects
where QR is the volume form of the D-brane.
The Dp-branes are characterized by a tension τ p = µp , relation which ensures that
they are BPS, as τp enters the relations (2.155a,b). The brane tension can be computed
by the cylinder vacuum or, as we shall derive in sec. 2.10 by computing the gravitational
tadpole on the disk in the presence of a brane.
Let us first compute the cylinder partition function, i.e. the open string one-loop
vacuum amplitude with NN boundary conditions over all the coordinates and gauge theory
SO(n) or Sp(n). The cylinder has one real modulus t and one CKV, thus an integration
over the modulus range (0 < t < ∞) is needed and as no vertex operator is inserted we
cyl
have to divide by the volume of the CKG V CKG
= 4πt. Keeping then account of a factor
1/2 from the GSO projection, another 1/2 from the Ω projection and of a 2π from the b
ghost insertion, summing over the n Chan-Paton degrees of freedom and assuming all of
the 10 dimensions to be non-compact we finally have:
NS
ZC = iV10 n
2
Z
0
∞
R
dt −12 4
η
θ3 − θ24 − θ44 − θ14
8t
N S−N S
Z
d10 k α0 k2
q
,
(2π)10
(2.156)
R−R
where now q = e−2πt and the argument of η and θ functions are respectively it and 0, it.
The cylinder partition function has a dual interpretation as a propagation of a closed
string. Looking at fig. 2.2, if time flows vertically the diagram corresponds at any given
time to a loop of open string, if we let time flow horizontally a closed string is obtained
by slicing at fixed times. In (2.156) it has also been explicitly displayed which terms
contribute to the open NS and R sectors and which to the closed NS-NS and R-R sectors.
In the open string interpretation the cylinder has length π and circumference 2πt,
whereas in the closed string representation the circumference has to be normalized to 2π,
the closed string length, which then propagates for a distance s = π/t. Performing the
change of variable in the integral (2.156) t → s = π/t and using the modular transformation (D.4) of the theta functions we end up with the two representations of the cylinder
partition function
Z ∞
dt
(8π 2 α0 t)−5 η −12 (it) θ34 (0, it) − θ44 (0, it) − θ24 (0, it) − θ14 (0, it)
ZC = iV10 n2
8t
0
Z ∞
(2.157)
iV10 n2
dt −2πt
16 + O(e
) ,
= (1 − 1) × 18 10 0 5
t6
2 π α
0
−5
Z ∞
4
ds 8π 2 α0
2
ZC = iV10 n
θ3 (0, is/π) − θ44 (0, is/π) − θ24 (0, is/π) − θ14 (0, is/π)
−12
8π η (is/π)
0
(2.158)
Z ∞
iV10 n2
−2s
ds 16 + O(e ) ,
= (1 − 1) × 18 11 0 5
2 π α
0
where in the second lines the θ functions have been developed and only the contributions
of the massless modes has been displayed explicitly. The amplitude is vanishing, because
of mutually cancelling contributions from NS and R or NS-NS and R-R sectors. Analyzing
the divergences of separately the NS and R sectors we see from (2.157) that the amplitude
is clearly convergent in the t → ∞ limit (IR limit of the gauge theory, see (2.159)); in
44
2 String theory
σ2
2πτ
σ1
Torus
0
2π
2π t
Cylinder
0
π
4π t
2πt
Klein
bottle
0
2π
0
π
0
π /2
4πt
2π t
Möbius
strip
0
π
Figure 2.2: One loop diagrams. The torus is the fundamental region of the plane
under the identification w ∼ w + 2π ∼ w + 2πτ . In the second figure the identifications
are made explicit. The cylinder can be obtained from the torus with modulus τ = t
by identifying under the involution w 0 = −w̄ which leaves the boundaries w = 0, π.
Taking from the plane the identifications w ∼ w + 2π ∼ −w̄ + 2πit the Klein bottle
is obtained. Cutting it vertically at σ1 = π and gluing at σ2 = 2πt the cylinder
with boundaries replaced by crosscaps is obtained. Identifying the torus under the
two involutions w 0 = −w̄ and w0 = w + π(2it + 1) the Möbius strip (which has one
boundary) is obtained. Cutting it at σ1 = π/2 and gluing at σ2 = 2πt the cylinder
with one boundary replaced by a crosscap is obtained.
45
2.9 Some non-perturbative aspects
(2.158) both NS-NS and R-R contributions are divergent in the s → ∞ limit (IR limit of the
gravitational theory, see (2.160)). The cylinder partition function in the t representation
is (as far as the massless level is concerned) just the sum of the vacuum energy of the
particles as (in the open string sector)
2
Zvac (m ) = VD
Z
dD k
(2π)D
Z
0
∞
dt −2α0 (k2 +m2 )t
VD
e
∼ D/2
2t
α0
Z
0
∞
dt
t1+D/2
0
2
e−2α m t , (2.159)
where Zvac is the connected part of the vacuum amplitude. Thus the Z C divergence at
t → 0 is the related to the ordinary UV divergence of field theory. But in string theory
things are more interesting as this divergence is the same as the s → ∞ divergence of
(2.158) which is an IR, tree level, effect in the gravitational sector
Z ∞
2
1
ds e−m s ,
(2.160)
=
2
2
k + m k2 =0
0
due to the propagation of a massless particle. We will see in ch. 7 another application of
this UV-IR duality of string theory.
Let us consider now the Möbius amplitude. We have seen that the T-duality action is
0 = −X X 0 → −X 0 , implying
XL ↔ XR or in terms of T-dual coordinates XL0 = XL , XR
R
that in the T-dual picture the Ω symmetry turns out to be a space-time reflection, thus like
in the orbifold case the twisted sector has to be considered. If we suppose to take T-duality
along direction xm̄,n̄ then the states wave function is determined by the Ω projection at
xm to be the same at −xm modulo a sign, as for instance
Gµν = Gµν ,
Gµm̄ = −Gµm̄ ,
Gm̄,n̄ = Gm̄n̄ ,
Bµν = −Bµν ,
Bµm̄ = Bµm̄ ,
(2.161)
Bm̄n̄ = −Bm̄n̄ ,
where the first members have argument (x µ , xm ) and the second members (xµ , −xm ). In
other words in the presence of an orientifold plane coordinates perpendcular to it are
identified under a Z2 reflection just as in the orbifold construction. In particular we
see that in the case of compact dimensions, the orientifold projection involves also the
momentum and winding modes. The Möbius amplitude can then be interpreted as the
cylinder stretched between a brane and an orientifold fixed plane, see fig.2.2, or between
a brane and its mirror image under orientifold reflection, and in the case of fully NN the
Möbius partition function is
4
Z ∞
θ2 (0, 2it)θ44 (0, 2it) θ44 (0, 2it)θ24 (0, 2it)
dt
2 0 −5 −12
−
8π α t
η (2it)
ZM = ±inV10
8t
θ34 (0, 2it)
θ34 (0, 2it)
0
Z ∞
V10
dt
= ±(1 − 1)in 18 10 0 5
(16 + O(e−πt )) ,
(2.162)
t6
2 π α
0
−5 Z ∞
ds 8π 2 α0
θ24 (0, 2is/π)θ44 (0, 2is/π) θ44 (0, 2is/π)θ24 (0, 2is/π)
5
ZM = ±2in2 V10
−
8π η 12 (2is/π)
θ34 (0, 2is/π)
θ34 (0, 2is/π)
0
Z
∞
25 V10
= ±(1 − 1)2in 18 11 0 5
ds(16 + O(e−2s )) ,
(2.163)
2 π α
0
46
2 String theory
where the change of variables t → s = π/(4t) has been made and relations (D.4a,b,c) have
been used. Again an interpretation in terms of both open and closed strings is possible.
The sign ambiguity is related to the gauge group choice, as the positive sign gives the
massless open string modes in the symmetric (n(n + 1)/2 states) adjoint representation
of Sp(n) and the negative sign gives the antisymmetric (n(n − 1)/2 states) adjoint representation of SO(n). In the closed string interpretation the sign is the relative sign of
orientifold plane charge and tension with respect to the brane one.
Finally let us add the Klein bottle amplitude, which has no open string interpretation
as the Klein bottle as no boundary. We now understand this amplitude to describe the
vacuum graph relevant when dealing with orientifold planes only, that in the case of
9−orientifold is
Z ∞
dt
ZK = iV10
(4πα0 t)−5 η −12 (2it) θ34 (0, 2it) − θ44 (0, 2it) − θ24 (0, 2it) − θ14 (0, 2it)
8t
0
Z ∞
ds (4π 2 α0 )−5 4
θ3 (0, is/π) − θ24 (0, is/π) − θ44 (0, is/π) − θ14 (0, is/π)
= i25 V10
8π η(is/π)
0
Z ∞
10
2 V10
ds 16 + O(e−2s ) ,
(2.164)
= i 18 11 0 5
2 π α
0
where the rescaling t → s = π/(2t) has been used. The overall divergences of these 3
world-sheets add up to
Z ∞
V10
5 2
ZC + ZM + ZK ∼ i(n ± 2 ) 14 11 0 5
ds ,
(2.165)
2 π α
0
which means that only in the case of the gauge group SO(32) the theory is non-singular.
Type I string theory can be identified as type IIB/Ω with 16 Dp-branes, as one brane
on the top of an orientifold plane is endowed with the gauge group SO(2) because of the
presence of its mirror brane. Applying T-duality in the direction 9̂, say, we shall have 16
D8−branes and two orientifold fixed planes, one at each end of the coordinates. The branes
can be separated freely, as the net force they exert on each other and on the orientifold is
vanishing: the fields parametrizing their position, and then also their fluctuations in the
directions orthogonal to their world-volume, which we name branons, are flat directions
in the theory and separating the branes correspond to reduce the gauge group by making
some of the gauge bosons massive, giving a geometric description of the Higgs mechanism.
An interbrane separation is equivalent to a Wilson line and in the field theory language
it corresponds to giving an expectation value to a branon field. More on branons will be
said in chap. 7.
Moreover once the branes are no more on the top of the orientifold, which is possible
only for Dp-branes with p < 9, the Ω projection act on them by correlating the degrees of
freedom at x9̂ to the ones at −x9̂ thus giving no local constraint. Away from the orientifold
plane then the theory is locally oriented and unitary gauge groups are allowed, the low
energy theory is D = 10 N = 2 supergravity.
All the ingredients have been gathered now to compute the Dp-brane tension and RR charge by one loop-computations. The amplitude to be considered is the cylinder one
(2.158) with p+1 non compact dimensions, with n = 1, with a factor exp[−2πty 2 /(4π 2 α0 )]
as the open string stretches between two parallel Dp-branes at distance y and the mass
47
2.9 Some non-perturbative aspects
spectrum of the stretched string is shifted according to (2.143b). From the relevant amplitude in the closed string channel the massless mode contribution is to be picked up,
which is, in the oriented, type II case,
Z ∞
4
ty 2
dt
−12
4
2 0 − p+1
N S2
2
η
(it)
θ
(0,
it)
−
θ
(0,
it)
(8π tα )
Abb = iVp+1
exp −
3
2
4t
2πα0
0
Z ∞
5−p
4Vp+1
−ty 2 −2π/t
2 exp
=i
dt
t
1
+
O(e
)
(2.166)
2πα0
(8π 2 α0 )(p+1)/2 0
3−p
7−p
2
0
−1 p−7
Γ
∼ iVp+1 2 π 2 4π α
y p−7 ,
2
where the relation (valid for x, a ∈ R + )
Z ∞
dt ta−1 e−tx = Γ(a)x−a
(2.167)
0
has been used. In the case of unoriented type I branes the amplitude is half of it because
of the Ω projection.
Analogously the closed string exchange between a brane and an orientifold at distance
y is related to the Möbius amplitude, whose closed string channel can be interpreted as a
closed string emitted from a Dp-brane and absorbed by its image behind the orientifold
plane. The relevant amplitude is then obtained by (2.163) (for n = 1), adding an exponential factor exp[−2y 2 t/(πα0 )] and choosing the orthogonal projection for the massless
modes to give
Z ∞
2ty 2
dt
θ24 (0, 2it)θ44 (0, 2it)
2 0 − p+1
N S2
−12
(8π tα ) 2 exp −
Abo = −iVp+1
η
(2it)
8t
πα0
θ34 (0, 2it)
0
Z
5−p
2Vp+1
−2ty 2 −π/(2t)
2 exp
dt
t
= −i
1
+
O(e
)
(2.168)
πα0
(8π 2 α0 )(p+1)/2
p−7
7−p
y p−7 .
∼ −iVp+1 2p−5 π 2 (4π 2 α0 )3−p Γ
2
To identify the tension the comparison to the analog field theory amplitude is to be
considered. The low energy effective brane action is given by the Born-Infeld and ChernSimons actions, whose relevant parts are
Z
Z
√
√
p+1
−Φ
SBI = −τp d x e
G = −τp dp+1 x e(p−3)Φ/4 g ,
(2.169)
Z
SCS = µp dp+1 x C (p+1) .
(2.170)
From the bulk action (2.133) expanded at quadratic order in the field around a trivial
background
gM N = η M N + h M N ,
Φ = φ,
C (p+1) = c(p+1)
(2.171)
2
2
N
and adding the gauge-fixing term −(∂ N hN
M − ∂M hN /2) /(4κ ) the propagator for the
graviton in the deDonder gauge and the dilaton propagator are obtained
16πGN
1
hM N hRS =
gM R gN S + gM S gN R − gM N gRS ,
(2.172a)
k2
4
16πGN
φφ =
.
(2.172b)
k2
48
2 String theory
The linearized version of the Born-Infeld (in the Einstein frame) and Chern-Simons actions
is
Z
1 µ
3−p
p+1
(p+1)
φ − hµ + µ p C
Slin ∼ d x τp
(2.173)
4
2
and an analogous reasoning for the R-R form lead to the field theory amplitude
(
(O)
8πGN
τp2 , 2 × 2τp τp
Af t = i2 2 ×
(O) ,
k
µ2p , 2 × 2µp µp
(2.174)
where the additional factors of 2 in the presence of the orientifold are required by the
halving of the spacetime volume and by the summing of the orientifold-brane amplitude
with the symmetric brane-orientifold one. Making use of the explicit form of the massless
scalar Green function in d dimensions
Z
y 2−d
dd k eik·y
Γ(d/2 − 1)
(2.175)
=
Gd =
d k2
4π d/2
Rd (2π)
the tension and R-R charge of brane and orientifold are obtained
p
3−p
π/2
4π 2 α0 2 ,
τp = µ p =
κ10
(O)
τp = µ(O)
= −2p−5 τp .
p
(2.176a)
(2.176b)
(O)
For p = 9 we have τ9 = 16τ9 and we recover the previous result that the √
theory has
SO(32) gauge group. In the oriented case the Dp-brane tension τ pII is τpII = 2τp . The
orientifold has negative tension, which is not inconsistent as open strings do not attach to
an orientifold plane, thus it cannot fluctuate: its fluctuations would have ghost-like kinetic
term.
About the divergence in the cylinder graph at t → 0, in the presence of lower dimensional branes it appears only for p ≥ 7. For p = 7, 8 it can be iterpreted as the usual
Green function divergence in 1 and 2 dimensions (for p = 7 a ln y will appear in (2.166)
and (2.168) instead of y p−7 ). Allowing the space to be noncompact models with arbitrary
numbers of branes are consistent, the flux line of NS-NS and R-R field can go to infinity
and does not need to be absorbed by sinks as in the case of compact dimensions.
Actually for the NS-NS field even in the compact case the net amount of charge does
not need to vanish, giving rise in that case to a gloablly curved spacetime, which however
may give some problem as generally it is not known how to quantize string theory on
curved backgrounds. Differently, R-R must always add up to zero in compact spaces
as otherwise the R-R gauge symmetry will be spoiled and the theory inconsistent. Non
supersymmetric model in which orientifold planes with reversed sign of the tension and/or
charge and anti-Dp-branes (i.e. Dp-branes with reversed sign of the R-R charge with
respect to Dp-branes) are conceivable [34].
2.10
Extended objects’ tension and charge from tree level
computations
We now apply the methods developed in sec. 2.6 to compute the brane tension from the
disk amplitude and to dicuss the projective plane amplitude related to the orientifold
2.10 Extended objects’ tension and charge from tree level computations
49
tension. Computations of analog quantities where first performed in a different formalism
in [35] for the bosonic string and in [36] for the type I superstring.
2.10.1
D-branes
The Dp-brane tension and R-R charge can also be derived through tree level computations,
as they are related to the NS-NS and R-R tadpoles, whose computation involves a closed
string on the disk.
Amplitudes on the disk involve the correlator of 3 c ghost insertions or, which is the
same, the ratio between the integration over the positions of 3 open string vertex operators
and the CKG volume with the appropriate Jacobian for the change of variables. For 3
open string vertex operators placed at x 1 , x2 , x3 , the Jacobian is


R
x1 x2 x3 dx1 dx2 dx3
∂ (x1 , x2 , x3 )
1
1  =
J3 =
=
=  1
(2.177)
VCKG
∂ (λ0 , λ−1 , λ1 ) −x2 −x2 −x2 1
2
3
|(x1 − x2 )(x2 − x3 )(x1 − x3 )| .
In the case the position of a closed vertex operator is fixed, which is given by a complex
rather than a real number, its real and its imaginary part can be considered as independent
components and apply the same reasoning. Fixing the vertex to w = 0 on the unit disk
(z = i on the half plane), fixes the CKG except for its compact subgroup of rotations
(2.106a) which has a genuinely finite volume
Z π
Vθ =
dθ = π .
(2.178)
0
The Jacobian between the position of the vertex operators and the group parameters is
R
R
∂(x1 , x2 , θ) 2
dzdz̄dθ
dzdz̄
1
2
R
=
=R
= ,
(2.179)
J2 =
VCKG
∂(λ0 , λ−1 , λ1 ) Vθ
π
dθ dλ0 dλ−1 dλ1
where
|dzdz̄| = 2dx1 dx2
as
z ≡ x1 + ix2
x1,2 ∈ R ,
has been used together with
∂(x1 , x2 , θ) ∂(x1 , x2 , θ) ∂(b+ , b− , θ) =
∂(λ0 , λ−1 , λ1 ) ∂(b+ , b− , θ) ∂(λ0 , λ−1 , λ1 ) =


2x1
2x2 0
− det  1 − x21 + x22 −2x1 x2 0  41 = 1 ,
1 + x21 − x22 2x1 x2 1
(2.180)
and to obtain the final number we used (2.110) and the assignment x 1 = 0, x2 = 1 has
been made. On the disk vertex operators must have a total (left plus right) ghost ϕ
supercharge equal to −2 and this is realized in the NS-NS sector by (dependence on the
world sheet coordinates is understood)
(−1,−1)
VN S−N S (ζ, k) = gc ζM N e−ϕ e−ϕ̃ ψ M ψ̃ N eikXL eikXR ,
(2.181)
50
2 String theory
where from now on kL = kR = k as winding modes will not be relevant. The polarization
tensor ζM N properties define the particle which is dealt with:
M = kM ζ
ζM
M N = 0 graviton ,
ζM N = ζ N M ≡ h M N ,
k M ζM N = 0
√
− kM lN − kN lM ) / 8 , l2 = 0, kl = 1
ζM N = −ζN M ≡ bM N ,
antis. tensor ,
ζM N = φ (ηM N
dilaton.
(2.182)
Using the first correlator of (2.55) and
N
XLM (z1 )XR
(z̄2 ) = −
α0 M N
D
ln(z1 − z̄2 ) ,
2
ψ M (z1 )ψ̃ N (z̄2 ) =
DM N
,
z1 − z 2
(2.183)
where
DM N ≡ ηµν ⊗ (−δmn ) ,
(2.184)
where p + 1 NN boundary conditions are on µ, ν indices and 9 − p DD conditions on m, n
indices, the amplitude turns out to be
AN S 2 = iCD2 gc J2 he−ϕ e−ϕ̃ ihψ M ψ̃ N iheikX eikX̃ i =

hµµ /2
igc 2 1
−2igc 
MN
D
ζ
=
M
N
α0 go2 π (2i)2
πα0 go2  φ(p − 3)/4√2
(2.185)
and vanishing amplitude for the antisymmetric tensor. 8
The R-R vertex operator involves the spin field S α , which describes the R vacuum and
whose insertion in the fermionic path integral introduces a branch cut in ψ µ . The spin
field correlators involve the introduction of a matrix which is the spinor analog of (2.184)
±Γ0 Γ1 . . . Γp
p even
,
(2.186)
M≡
±Γ0 Γ1 . . . Γp Γ11 p odd
where the two possible signs correspond to the two R-R charges of the brane. The relevant
correlators for the amplitude computation are
hSα (z1 )S̃β (z̄2 )i =
hSα (z1 )S̃β̇ (z̄2 )i =
Cαγ̇ Mβγ̇
(z1 − z̄2 )5/4
Cαγ̇ Mβ̇γ̇
(z1 − z̄2 )5/4
p even ,
(2.187a)
p odd ,
(2.187b)
where spinor of equal(opposite) chirality in type IIA(B) are allowed as they occurs for
R-R vertex operators in a non symmetric picture and the charge conjugation matrix C
has been used.
8
Strictly speaking in the case of one particle the on-shell momentum is vanishing and thus there is
no transverse space to define the dilaton polarization, the result (2.185) is obtained by considering a
momentum k with no components parallel to the brane world volume and taking the limit k → 0.
2.10 Extended objects’ tension and charge from tree level computations
51
The p + 1 form R-R potential in the (−1/2, −3/2) picture is [37]
gc
(C (p+1) , kL = kR = k) = 1/2 e−ϕ/2 e−3ϕ̃/2 eikX eikX̃ ×
α0 i
 h αβ

(p+1)
(p+3)
− C C
+C
S̃β p even
Sα
h iαβ̇

 C C (p+1) + C (p+3)
S̃β̇ p odd
−1/2,−3/2
V R2
(2.188)
where contraction of Lorentz indices of the potentials with gamma product ones is understood. The on-shell conditions (obtained as usual imposing BRST invariance of the vertex
operator) are
d ∗ C (p+1) = 0 ,
dC (p+1) + ∗d ∗ C (p+3) = 0 ,
dC (p+3) = 0 ,
k2 = 0 ,
(2.189)
thus showing that C (p+3) is a pure gauge field which is necessary to make the field strength
of C (p+1) non vanishing9 . The GSO projection is consistent as the world-sheet fermion
number of e−ϕ/2 is the opposite of the one of e−3ϕ/2 . This vertex operator is equivalent
to the better known one in the (−1/2, −1/2) picture
(−1/2,−1/2)
VR−R
−1/2,−3/2
˜
= [Q̃BRST(, ξV
R−R
gc FM0 ...Mp+2 e−ϕ/2 e−ϕ̃/2 eikX eikX̃ Sα
]=
(CΓM0 ...Mp+2 )αβ̇ S̃β̇ p even
(CΓM0 ...Mp+2 )αβ S̃β p odd
(2.190)
where we introduced
FM0 ...Mp+2 = (p + 2)ik[M0 CM1 ...Mp+1 ] .
(2.191)
Neglecting the p + 3 form which gives vanishing contribution, the R-R amplitude in a
Dp-brane background is
iCD2 gc
J2 he−ϕ/2 e−3ϕ̃/2 iheikX eikX̃ ihS(CΓµ0 ...µp )S̃iCµ0 ...µp =
A R2 =
α0 1/2
−igc
8gc
Tr[CΓµ0 ...µp (C −1 M T )T ]Cµ0 ...µp = ±i
Cµ0 ...µp ,
3/2 2
0
2πα go
πα0 3/2 go2
9
The vertex operator in the (−3/2, −1/2) picture
−3/2,−1/2
VR−R
gc
(C (p+1) , kL = kR = k) = − 0 1/2 e−3ϕ/2 e−ϕ̃/2 eikX eikX̃ ×
α
8h “
”iαβ
>
< C C (p+1) + C (p+3)
S̃β p even
Sα h “
”iαβ̇
>
: C C (p+1) + C (p+3)
S̃β̇ p odd
could equally well have been written and still
−1/2,−1/2
VR−R
h
i
−3/2,−1/2
= QBRST , ξVR−R
,
with on-shell conditions
d ∗ C (p+1) = 0 ,
dC (p+1) − ∗d ∗ C (p+3) = 0 ,
dC (p+3) = 0 ,
k2 = 0 .
(2.192)
52
2 String theory
where the trace is over positive chirality indices only. The previous amplitude is nonvanishing only in the presence of a R − R form of the right degree to match the number
of γ’s in M . Choosing the positive sign in the definition of M (2.186) gives a sign in front
of the amplitude according to the rule
p
0
1
2
3
4
5
6
7
8
9
sign
+
-
-
+
+
-
-
+
+
-
(2.193)
Under the rescaling
hM N → hM N /(2κ) ,
√
φ → 2φ/(2κ) ,
Cµ(p+1)
→
0 ...µp
(2.194a)
(2.194b)
0 1/2
α
C (p+1) ,
8κ µ0 ...µp
(2.194c)
and using the relation [29]
gc = κ10 /(2π) ,
(2.195)
the previous amplitudes leads to action (2.173) provided the tension τ p and charge µp of
the brane are given by the expression
τp = µ p =
1
,
2π 2 α0 go2
(2.196)
where it is understood that go = go (p), as it can be checked using T-duality, according to
go (p = 9) = go (p)(2πα0
1/2 (9−p)/2
)
.
(2.197)
Relation (2.196) is in agreement with (2.176) provided a proper relation between g o and κ
holds. Such relation will be obtained in sec. 7.6 by unitarity cutting open in two different
ways the cylinder amplitude with insertion at its borders.
The result (2.185) then requires a non vanishing 0-point amlitude for the disk, in
contradiction with the naive analysis that the CKG of the disk has infinite volume, but in
agreement with analysis performed in sec. 2.6. 10
About the volume of the fermionic part of the CKG V f CKG by comparing (2.113)
and (2.185) Vf KCG = 1 is found. If the disk is parametrized using the w coordinates
0
= CD2 /2 is obtained and also a different volume for
the different normalization CD
2
the bosonic part of the CKG because of the Jacobian (2.110). Consequently from the
analog of (2.113) in the w case the volume V f0CKG of the fermionic part of the CKG is
Vf0CKG = 2Vf CKG = 2.
10
The same procedure can be applied to a fractional brane, for instance, and its tension can be obtained
in terms of the coupling parameter g of the twisted NS-NS vertex operator. A relation between g and
physical quantity like κ can then be obtained by using unitarity arguments on scattering amplitudes
involving twisted and untwisted states
2.10 Extended objects’ tension and charge from tree level computations
2.10.2
53
Orientifold planes
The orientifold plane tension might be computed analogously by evaluating the dilaton,
graviton and R-R form tadpole on the projective plane if the projective plane normalization was known. The normalization factor C RP2 can be obtained by unitarity, which
in this case will require a one loop computation as the propagation of an internal closed
string particle is a one loop process, so we think that the best thing to do in this case is
to compare the one point amplitude for a closed string in an orientifold background to the
one loop vacuum computation performed in sec. 2.9 in order to obtain the normalization
factor itself.
The following correlation function on RP 2 will be needed
α0 M N
D
ln |1 + z1 z̄2 | ,
2
DM N
,
hψ M (z1 )ψ̃ N (z̄2 )i =
1 + z1 z̄2
hea1 ϕ(z1 ) ea2 ϕ̃(z̄2 ) i = (1 + z1 z̄2 )−a1 a2 .
N
(z̄2 )i = −
hXLM (z1 )X̃R
(2.198)
(2.199)
(2.200)
In the case of the projective plane the zero point amplitude is
2
ARP
∝
0
CRP2
CRP2
=
,
VCKG
π2
(2.201)
where again knowledge of the volume of the fermionic part of the CKG V f CKG is needed.
Fixing the coordinate of a closed string vertex operator at w = 0, say, on the projective
plane fixes the CKG apart from the subgroup generated by 2L 0
w → w0 =
a
w = e2iθ w ,
ā
whose volume is Vθ = π. The Jacobian needed in the amplitude computation is
1 ∂(w1 , w̄1 , a0 ) 1 ∂(x1 , x2 , a0 ) J2 =
=
Vθ ∂(a1 , a2 , a0 ) =
 Vθ 2∂(a0 ,2a1 , a2 )

(1 + x1 − x2 )
2x1 x2
−2x2
1
1
1
det 
2x1 x2
(1 − x21 + x22 ) 2x1  × =
.
π
2
2π
0
0
1
(2.202)
(2.203)
where we used w = x1 + ix2 = 0, |dwdw̄| = dx1 dx2 and the extra 1/2 factor is due to the
fact that not the full CKG spanned by (2.122b,c) is required to map the point w 1 = 0 into
the full disk, but as |β|2 < 1/2 is enough to do the job, see (2.119), only half of the group
volume spanned by a1,2 is needed.
The one-closed string amplitude is then
2
ARP
= iCRP2 J2 he−ϕ/2 e−ϕ̃/2 ihψ ψ̃iheikX eikX̃ i =
N S2

hµµ /2
iCRP2 gc M N
CRP2 gc 
,
D
ζM N = 2i
 φ(p − 3)/4√2
2π
π
and again the amplitude is vanishing for the antisymmetric tensor.
(2.204)
54
2 String theory
This has the opposite sign with respect to brane coupling (provided C RP2 is positive).
The Ramond-Ramond amplitude is (we again suppress the auxiliar (p + 2)-form)
RP2
AR
= iCRP2
2
8CRP2 gc
gc J2 −ϕ/2 −3ϕ̃/2
µ0 ...µp
Cµ0 ...µp , (2.205)
he
e
ihSCΓ
S̃iC
=
±i
µ
...µ
p
0
α0 1/2 V3
πα0 1/2
and the choice of the sign is just the opposite one with respect to (2.193). After the
2
2
rescalings (2.194) both ARP
and ARP
can be derived by the action (2.173) provided
N S2
R2
(O)
(O)
τp , µp are now substituded by τp , µp
and these are given by
τp(O) = µ(O)
= CRP2 /(2π 2 ) .
p
(2.206)
Using the one-loop result (2.176b) and relation (2.197) we can infer
CRP2 =
24 π 9−p
2p−5
= (p−7)/2
.
0
2
α go (p)
α0
go2 (p = 9)
(2.207)
If we now add the information that the the volume of the fermionic part of the CKG
equals the one for the disk in the w coordinates representation [36], so that according to
2
the discussion at the end of sec. 2.10.1 we can assume V fRP
CKG = 2, the relation
2
ARP
=i
0
is obtained, consistently with (2.206).
CRP2
CRP2
=i
2VCKG
2π 2
(2.208)
3 The pre-big bang model
“The inflaton should spring forth some
grander theory and not vice-versa.”
E. Kolb and M.S. Turner,
The early Universe
This chapter will discuss the pre-big bang model [38, 39, 40, 41]. We define it by taking
as the starting point the low energy effective action for the graviton and the dilaton and
we discuss the main phenomenological consequences of pre-big bang inflation (which does
not require ad hoc field or potentials), namely the spectrum of density perturbations and
the background gravitational waves. We also show how the pre-big bang model decouples
the problem of the initial conditions from the one of the big bang singularity and the issue
of the initial conditions is discussed. Finally we start investigating the problem of the
singularity by considering the regularization mechanism that α 0 corrections to the lowest
energy action provide, even if for a full development of the analysis of the cosmological
singularity we refer to the following three chapters.
3.1
The model
The string massless modes comprise for any string theory the graviton and the dilaton.
The low energy action involving these two fundamental excitations is
S=
1
2κ2d+1
Z
dd+1 x
i
√ −φ h
Ge
R + (∂φ)2 ,
(3.1)
where the number of dimensions are left generic, even if perturbative string theory requires
d = 10 and the action has been now expressed in terms of the dilaton φ ≡ 2Φ. 1 The
consequent equations of motion are
R−G
RM N + ∇ M ∇N φ = 0 ,
MN
∂M φ∂N φ + 2φ = 0 .
1
(3.2a)
(3.2b)
The φ used here and in the following chapters up to the sixth one is not the same as the φ defined in
(2.171) and used in ch. 7, we hope this will not create confusion but we ran out of φ-like characters.
55
56
3 The pre-big bang model
The Einstein frame version of action (3.1) corresponds to a scalar field, the dilaton, minimally coupled to gravity and the relative equations of motion are
1
∂M φ∂N φ ,
8
φ = 0 .
(E)
RM N =
(3.3a)
(3.3b)
The degrees of freedom connected to the dimensions in excess can be conveniently parametrized for instance by the simple metric ansätz
µ
ds2 = gµν dxµ dxν + e2βm (x ) dy m dym ,
(3.4)
where Greek indices are tangent to our (p+1)-dimensional world (we leave for the moment
p arbitrary) and Latin ones are tangent to the extra, or internal, dimensions. From the
previous ansätz the following action is obtained
!
Z
X
1
1
(p+1)
p+1 √ −φp+1
2
2
R
+ (∂φp+1 ) −
d x ge
,
(3.5)
(∂βi )
S=
2κp+1
2
i
where the p + 1-dimensional dilaton is related to the higher dimensional one by
φp+1 = φ − (d − p)β .
(3.6)
Further specializing the metric to the (p + 1)-homogeneous, isotropic and spatially flat
ansätz
ds2 = −N 2 (t)dt2 + a2 (t)(dr 2 + r 2 dΩ2p−1 ) + e2βm (t) dy m dym ,
(3.7)
the action
1
S= 2
2κp+1
Z
#
" X
ȧ 2
e−φ̄p+1
2
2
˙
β̇i
dtd x
− φ̄p+1 +
p
N
a
p
(3.8)
i
is obtained, where analogously to (3.6) the shifted dilaton
φ̄p+1 ≡ φp+1 − p ln a
(3.9)
has been introduced. Action (3.8) displays the symmetry
a(t) →
1
,
a(−t)
φ̄p+1 (t) → φ̄p+1 (−t) ,
(3.10)
called scale factor duality (SFD) [38] which can be generalized to the presence of the
antisymmetric 2-tensor of the heterotic string and anisotropic backgrounds, i.e. to general
homonogeneous cosmological backgrounds [39], provided the spacetime has an Abelian
group of isometries [42]2 .
Defining the Hubble parameter H ≡ ȧ/a, the SFD symmetry can relate a FRW expansion (H > 0) with decreasing curvature ( Ḣ < 0) and constant dilaton (as in standard
cosmology gauge and gravitational coupling must be constant in time) at t > 0 to a
super-inflationary phase with H, Ḣ > 0 and running dilaton at t < 0 as it is sketched
57
3.1 The model
H
+ branch
− branch
Pre−big bang
Post−big bang
0
0
t
Figure 3.1: Duality between a decelerating expanding Universe and a superinflating
one, which are the solutions (3.11). The instant t = 0 is the epoch of maximum
curvature H, identified with the big bang.
in fig. (3.1). The solutions to the equations of motions obtained from action (3.8) are
(making the choice N = 1 and dropping integration constants)
a(t) = |t|δ ,
φ̄p+1 (t) = − ln |t| ,
βi (t) = ζi ln |t| ,
(3.11)
provided that
pδ 2 +
X
ζi2 = 1 .
(3.12)
i
We denoted by t the time coordinate defined in the string frame by (3.7) and condition
N = 1, which does not equal to the analog t E defined by the same metric ansätz with
N = 1 in the Einstein frame.3 With the metric ansätz
(3.13)
ds2 = a2 (η) −N 2 (η)dη 2 + dr 2 + r 2 dΩ2p−1 + e2βi (t) dy m dym ,
the time coordinate, also called conformal time, to distinghuish it from the cosmic time
t, is the same in both frames and the solutions (3.11) become (againg we drop integration
constants to give explicitly only the dependence on time)
δ
a(η) = |η| 1−δ
φ̄p+1 (η) = −
1
ln |η|
1−δ
βi (η) =
ζi
ln |η| ,
1−δ
(3.14)
with the same Kasner constraint (3.12). To reproduce an expanding (H > 0) Universe at
late time δ > 0 for t > 0 must be chosen and the dual solution has δ < 0, H > 0 for t < 0.
2
For instance spatially curved models in D = 4, those denoted by k = ±1 in eq. (1.4), have non-Abelian
Killing vectors and then SFD does not apply to them.
3
In fact ln |tE | = p(1−δ)
ln |t|
p−1
58
3 The pre-big bang model
The solutions satisfy
1
φ̄˙ = − H
δ
(3.15)
so φ̄˙ and H have the same (opposite) sign, or the solution is on the “+” (“-”) branch, for
negative (positive) δ. The two branches are related by the SFD and both of them reach a
singularity at finite time tsing , conventionally chosen tsing = 0 by adjusting the integration
constants. But as the curvature scale H approaches α 0 −1/2 and the dilaton φ → 0 or more,
the original low energy effective action is no longer trustworthy as higher order terms in
α0 and eφ are no longer negligible. The dilaton is always growing on the “+” branch for
H > 0 (and also on the “-” branch for internal dimensions static or varying slowly enough)
as from (3.15)
1
φ̇p+1 = H p −
.
(3.16)
δ
Next section and chapters will describe attempts in regularizing the cosmological solutions and if some stringy mechanism is able to smoothen the transition between the two
branches this would be a non-singular cosmological scenario which naturally incorporates
inflation.
The solution (3.11) for erly times suggest that the Universe emerged from a state of
low curvature (H → 0) and small coupling (φ → −∞) and thus we can say that the
cosmological principle is replaced in string cosmology by the basic postulate of pre-big
bang cosmology [43]:
The Universe started its evolution from the most simple and natural state conceivable
in string theory, its perturbative vacuum, i.e. it started its evolution almost
empty, cold and flat,
as opposed to the standard cosmological scenario where the Universe started dense, hot
and highly curved, situation that in this scenario is dynamically generated out of the
simple Universe at the beginning. We note moreover that as
lP l = λs eφ4 /2 ,
(3.17)
the Planck length is much smaller the string scale for t → −∞ so that the Universe is
initially eminently classic. Afterwords the growth of the dilaton is faster than that of the
scale factor: i.e. the Universe evolve naturally from a classical to a quantum regime (in
the “+” branch). This can be restated by considering the Einstein frame, in which the
Planck length is constant. The (p + 1)-dimensional Einstein frame has metric g µν defined
by
gµν = Gµν e−2φp+1 /(p−1) ,
(3.18)
and the consequent Einstein frame scale factor is
aE (η) = ae−
φp+1
p−1
1
(η) = |η| p−1 ,
(3.19)
59
3.2 Phenomenological consequences
irrespectively of the value of δ. In the Einstein frame the Planck length is fixed and the
string length is time dependent with relation (3.17) still holding, the pre-big bang branch
thus corresponds to an accelareted contraction, the gravitational collapse is triggered by
the dilaton energy-momentum tensor. The horizon and flatness problem can equally well
be solved by a period of accelerated contraction, rather than accelerated expansion, as
the spatial curvature becomes negligible compared to the energy density in the dilaton
as aE → 0 and moreover the particle horizon grows much faster then the Hubble length,
which is actually shrinking during accelerated contraction 4 .
Of course the problem of mathcing the two branches, the graceful exit problem, is
highly non trivial and equivalent in the two frames, and it will be the main subject of the
next three chapters.
3.2
Phenomenological consequences
In this section, assuming that a regularization of the solutions is indeed possible, we shall
expose the phenomenological consequences of the PBB model, focusing mainly on those
which are regularization mechanism-independent 5 .
PBB inflation, like any accelerated cosmological phase of expansion, provides a natural
mechanism to amplificate primordial vacuum fluctuations of different kind of fields whose
detailed treatment can be found in app. A. Here we just state the more important results.
Let us start by considering gravitational waves [44]. An example of a gravitational
wave spectrum is displayed in fig. 3.2, where the relevant phenomenological constraints
are also shown. Introducing Ωgw (f ), the normalized energy density in gravitational waves
per unit of logarithmic interval of frequency f = k/(2π) analogously to (A.55)
Ωgw (f ) =
1 dρgw (f )
,
ρc d ln f
(3.20)
the main observational bounds can be quantified as [45]:
• The high degree of isotropy of the CMBR at angular scales of order of one tenth
of radiant (which would be spoiled via Sachs-Wolfe effect if gravitational waves are too
abundantly produced in the early Universe) compels Ω gw (f ) to be smaller than
2
2
−11 H0
h0 Ωgw < 7 × 10
for H0 < f < 30H0 ,
(3.21)
f
H0 being the Hubble constant.
• The regularity in the millisecond signal coming from binary pulsar, which after a
several years-long observation gives
h20 Ωgw (f = 10−8 Hz) < 10−8 .
(3.22)
4
Strictly speaking the particle horizon is infinite when computed over the background (3.14). We assume
that the pre-big bang inflation sets in at some finite time t0 , which is also the lower limit of the integral
defining the particle horizon as defined in (1.17).
5
If the smoothing of the singularity is achieved by physical effects introduced by new degrees of freedom
which become light at strong coupling, we may expect that the phenomenological consequences predicted
on the basis of perturbative physics should be consistently altered as the light degrees of freedom in the
strong coupling region are generally some complicated non local function of the weak coupling ones. Thus
we assume that the regularization mechanism can be investigaterd by means of perturbative physics.
60
3 The pre-big bang model
• The success of nucleosynthesis to explain the observed cosmological abundance of
light elements would be spoiled if too many gravitational waves were present at the epoch
of nuclei formation, thus providing the constraint over the integrated spectrum
Z
d(ln f )Ωgw (f ) ' 6.3 × 10−6 .
(3.23)
The gravitational wave spectrum predicted by inflation [46] is much lower because it is
almost flat at all scales (as the Hubble parameter during inflation is almost costant) and
the CMBR bound compels the spectrum to be rather low at large scale and thus at all
scales. For the PBB case the spectrum has a positive tilt, thus the COBE bound is easily
evaded and the stringest bound is given by the nucleosynthesis one.
The spectrum shown in fig. 3.2 is calculated for a specific regularization mechanism,
the one described in sec. 3.3, but the low frequency part, the f 3 raise for low f , the
maximum height and the upper frequency cutoff are rather general and regularization
mechanism-independent [44].
Defining Hs as the (square root of the) maximum curvature achieved during pre-big
bang inflation, Hs must be of the order of the inverse string length λ −1
and using the
s
2
relation λ2s ∼ (2/αGU T )lP2 L ∼ 40lP2 l , where αGU T ≡ gGU
/4π
∼
1/20,
the (properly
T
red-shifted) maximum amplified frequency turns out to be
fmax
Hs
'
2π
Heq
Hs
1/2
aeq
' 4 × 1010 Hz
a0
Hs
0.15MP l
1/2
.
(3.24)
The redshift factor is
a(teq )
a(ts )
a(ts )
=
×
∼
a0
a(teq )
a0
Heq
Hs
1/2
×
1
,
zeq
(3.25)
as a ∝ H −1/2 ∝ t1/2 during the radiation dominated epoch, z eq ' 2 × 104 is used to give
the numerical estimate and < 1 is a pure number (see app. A). For higher frequencies
the relative wavelength will be always sub Hubble length-sized and thus the mechanism
discussed in app. A is not active. This frequency corresponds to the production of one
graviton per mode per polarization, so that the energy density in gravitational waves is
ρgw (fmax ) =
(2πfmax )4
,
π2
(3.26)
and at higher frequency the spectrum has a sharp cutoff [47]. Thus for a positively tilted
spectrum the maximum height in Ωgw can be estimated to be [48]
Ωmax
gw '
4 )
3 (2πfmax
H2
∼ 10−4 2 2s ,
2
2
3
8π H0 MP l
MP l
(3.27)
where Hs is the maximum curvature scale reached in the cosmological evolution.
As far as the density perturbation are concerned we see that because of the slope of
the spectrum, the gravitational fluctuations have little power at the Mpc scale, 6 which,
6
We remind that 1 Mpc ' 3 × 1024 cm ' 1014 sec.
61
3.2 Phenomenological consequences
0
10
f
3−|3−2β |
−10
10
−20
h0 Ωog
10
f
−30
3
2
10
−40
10
Pulsar
COBE
−50
10
−60
10
Nucleosintesi
fs
−18
10
−13
10
−8
10
f(Hz)
−3
10
2
10
Figure 3.2: Spectrum of gravitational waves in the pre-big bang model as computed
in [49]. The cosmological model involve a PBB phase, a string phase with constant
curvature and linearly running dilaton as described in sec. 3.3 and a FRW radiation
dominated phase. Here the free parameter fs is arbitrarily chosen to be fs = 10Hz.
The experimental bounds discussed in the text are also displayed.
whithin few order of magnitudes, is the length scale involved in structure formation and
the CMB anisotropies, thus showing that this kind of perturbations are not relevant for
those issues.
Electromagnetic perturbations are also produced and they may provide the seeds for
the birth of the extragalactic magnetic fields which are observed 7 [50]. Differently from
standard cosmology, where the usual Yang-Mills action is conformally invariant in D =
4 and the mechanism of amplification of photon vacuum fluctuations is not operative,
in stringy physics electromagnetic fluctuations are amplified because of their non trivial
coupling to the dilaton, as shown in tab. A.1.
The most important phenomenological test of the model is if PBB cosmology can provide density perturbations with the right features to fit the temperature anisotropy of the
CMBR and to set the right initial conditions for structure formation, at least as well as
inflation does. Other sources of inhomogeneities than the gravitons and dilatons have to
be investigated and the PBB scenario provides several of them (see again tab. A.1)[51, 52].
For instance it was first noticed in [53] that a flat spectrum of perturbations can be provided by the axion, which is defined as the Poincaré dual of the antisymmetric tensor in
(A.63). The axion perturbation spectrum depends on the dynamics of the internal dimen7
There exist actually a mechanism wich allows to explain how a magnetic field can be amplified by a
rotating galaxy, the dinamo mechanism, provided that a seed of the field is present at the onset of the
gravitational collapse.
62
3 The pre-big bang model
sions and it is flat for δ = 1/3, case which include isotropic contraction in the 6 internal
dimensions with βi = −1/3, see (3.12). The axions are not part of the homogeneous
gravi-dilaton background and therefore the density perturbation they trigger are set in
the δρ = 0 initial condition, which leads to isocurvature perturbations, that on general
grounds have problems in giving a correct fit to CMB anisotropies, as mentioned in sec. 1.1
and studied for instance in [12]. Moreover, the density perturbation will be proportional to
the perturbed axion energy-momentum tensor, which is quadratic in the axion fluctuation
and since the axion perturbation δσ is gaussian, δρ ∝ (δσ) 2 will have a χ2 distribution,
which is not in agreement with fits from galaxy distribution [54]. Finally it is not a strictly
flat spectrum the best fit of isocurvature axion perturbations to CMBR, which has been
computed in [55]. Indeed a small positive slope (d ln Ω/d ln k ' 0.33) is required and then
in order not to have too much power at small length scales a kink making the spectrum
flat at high frequency is needed, thus requiring a transition in the comsological evolution.
PBB scenario also suffer of the problem of dangerous relics, which is explained in
detail in app. B. For instance scalar fields with gravitational interaction, like the fields
parametrizing the degrees of freedom of the internal dimensions, which we shall generically
denote as moduli, or χ, are produced gravitationally, with typical relative abundances
Yχ ∼ 10−3 − 10−4 , being Yχ the ratio between the density of the species χ and the entropy
of the Universe; Yχ is constant throughout the expansion.
In the post-big bang the dilaton and the moduli must become massive as otherwise
they will mediate long-range non-universal gravitational interaction [56]. Giving a mass
mχ to the moduli the range of the non-universal interaction they mediate can be limited so
to become harmless provided mχ > 10−4 eV [57, 58]. Moreover the 4-dimensional dilaton
must be stabilized as otherwise the gauge field coupling, whose value is set by the dilaton
expectation value, will “slide” in time, whereas experimental bounds constrain the change
rate of the fine structure constant α̇/α < 10 −15 ys−1 [59]. Once the dilaton is stabilized
the string and Einstein-frame are equivalent, being related by a constant metric rescaling.
However a potential problem arise if the moduli are endowed with masses m χ in the
range 100 MeV< mχ < 10 TeV. In fact if this is the case they will decay after nucleosynthesis and before the present epoch, the photons produced in their decay may break deuterium thus spoiling the success of nucleosynthesis. This can be avoided if the abundance
of moduli is preventively diluted by a huge production of entropy so to make Y χ < 10−13
[60, 61].
The best candidate process for such a massive entropy production is the decay of a
non-relativistic particle which takes place after it has come to dominate the energy density
of the Universe. Its decay into relativistic particles will release enough entropy to dilute
the abundance of other, stable or unstable, moduli to acceptable densities. The needed
amount of entropy can be achieved for m χ < 103 TeV, which is still compatible with bound
mχ > 10 TeV needed to let the Universe reheat to a temperature higher than the MeV
scale, so that nucleosynthesis can take place after the modulus decay [62].
The benefic effect of a modulus decay are not restricted to solving the moduli problem
as it may be exploited to give an interesting spectrum of adiabatic, scale invariant and
gaussian density perturbations [63]. This can be achieved in a string cosmological context,
as shown in [64], where axions present in the 4-dimensional low energy effective action
are considered. As already mentioned the axion σ can be endowed with an almost flat
spectrum of perturbations, provided the internal dimensions are not static in the PBB
3.3 Effect of α0 corrections
63
phase. Then, by acquiring a potential in the radiation dominated phase, the axion will
become non-relativistic and will be able to dominate the Universe energy density. Finally
by its decay into photons it can reheat the Universe solving the moduli problem and
imprints into the decay products its perturbation spectrum, which will give almost scale
invariant, adiabatic (as δρ 6= 0) and gaussian (as δρ ∝ δσ) density perturbation.
The potential the axion can be endowed with is usually a periodic one. For a potential
1/2
with a periodicity σ0 ∼ Ms and amplitude V0 the axion mass will be mσ ∼ V0 /σ0 ∼
103 TeV for V0 ∼ (1012 GeV)4 , which is good for solving the moduli problem. In the case
the spectral tilt is positive the density perturbations are gaussian and they have the simple
form
γ
k
δρk
∼
' 10−4
(3.28)
ρ
k1
(10−4 is the experimental input) being k1 ∼ Ms , thus requiring that the axion spectral
tilt γ ∼ 0.14 for k/k1 ∼ 10−28 and a smaller γ can be obtained by lowering the amount of
perturbation by a multiplicative factor. In the negative tilted case the periodicity nature
of the potential is crucial to damp the resulting huge fluctuations at big length scales, thus
ensuring the right power at the astrophysical scale k ∼ (10 −2 Mpc)−1 and still a value for
γ close to zero. A problem may indeed arise if the fluctuations are bigger than the field
value itself, leading to δρk ∝ (δσ)2 and thus not gaussian fluctuations, which is not a good
phenomenological prediction [54].
In this model isocurvature fluctactions from other axionic fields which never dominate
the Universe are not exluded, but they can give sub-dominant isocurvature contributions
to density perturbations consistently with observations [12].
3.3
Effect of α0 corrections
Let us consider the regularization of singular solutions (3.11) as a result of stringy α 0
corrections, at the lowest order in the string coupling parameter e φ . We thus add to
action (3.1), with frozen internal dimensions, the first order corrections in α 0
Z
√
α0
1
dD x G e−φ4
Rµνρσ Rµνρσ .
(3.29)
S α0 =
2
8
As we the exact conformal field theory of the string on general background is not known
(and we do not know the scattering amplitude for n massless particles, with generic n) we
must be content with the first terms of a perturbative expansion in power of α 0 , even if
when the curvature reaches the string scale higher order corrections in α 0 will be equally
important as the one given by (3.29).
Moreover there is a source of ambiguity in the use of low energy effective action derived
from a n-particle
scattering amplitudes. For instance the contribution of the terms
√
√
GRµν Rµν and GR2 to the 4-graviton scattering vanishes because of a cancellation
between the contact graph and the one particle reducible exchange graph [65], thus these
terms could be added to the low energy effective action (3.29) with any coefficient in front
of them. This cancellation would not hold off-shell, but an off-shell formulation of string
scattering amplitude is still lacking. The coefficient of the (R µνρσ )2 term is instead fixed
by the 4-point scattering amplitude.
64
3 The pre-big bang model
The ambiguity can be understood in terms of the two dimensional world-sheet field
theory as its conformal invariance is equivalent to the vanishing of the beta functionals
defined in (2.131), but the pertubative coefficients of the beta functional depends on the
renormalization scheme from two loops on, i.e. starting from the terms R 2 .
Due to these sources of ambiguities, in [66] the α 0 corrections
Z
1
d4 x g 1/2 e−φ R2GB − (∂φ)2
(3.30)
S=
2
to the low nergy effective action are used, instead of the previous (3.29), where R 2GB
denotes the Gauss-Bonnet invariant combination
R2GB ≡ Rµνρσ Rµνρσ − 4Rµν Rµν + R2 ,
(3.31)
which has the virtue of not having derivatives of the metric higher than the second.
The resulting equation of motions are then ordinary second order differential equations
whith the remarkable property to possess fixed points solutions for constant H and φ̇
and numerical integrations show that solutions with pre-big bang-like initial conditions
are attracted into the fixed points. The solutions are displayed in fig. 3.3 and in the
asymptotic region, where H and φ̇ are constant they can be parametrized as
a(η) = −
1
,
Hs η
φ4 (η) = φs − 2β ln |η/ηs | ,
for ηs < η < η1 < 0 ,
(3.32)
which leads to constant H ≡ ȧ/a and φ̇4 , where we assumed that the above parametrization
well describes the actual solutions for the limited amount of time ranging from η s , before
which (3.11) holds, to η1 , after which a graceful exit to a FRW phase is conjectured. We
shall christen stringy phase the epoch characterized by the above (3.32).
This has been the first example of an explicit non-singular pre-big bang model. Nevertheless it has not been achieved a complete exit as solutions evolve from the +branch
towards a fixed point, rather than a FRW background. More general α 0 corrections will
be studied in ch. 6, where we shall deal with holography and a generalized second law of
thermodynamics in cosmology.
It is understood in this model that α 0 corrections becomes important earlier that
quantum corrections, i.e. higher order terms in α 0 , which is consistent with the PBB
postulate. As the dilaton is growing on this fixed point solution quantum corrections are
expected to become important at some point, possibly being able to trigger a transition
to a post-bif bang phase.
The post-big bang solutions turn out not to be attracted by the fixed points. and we
note that the SFD (3.10) is not a symmetry of the action (3.29) nor (3.30). Indeed SFD
symmetry can be used to constrain the “allowed” α 0 corrections, by requiring that they
fulfil SFD, which is possible only with isotropic metric asätze, but this does not help in
finding solutions interpolating between pre and post-big bang [67]. With a general metric
ansätz a modification to order α0 of (3.10) is required to obtain a SFD invariant action
[68, 69] for a specific combination of α 0 correction terms, but it still does not allow to go
from pre- to post-big bang.
The antisymmetric tensor Bµν can also be introduced in the comsological equation of
motion, with the general result that even starting from an homogeneous ansätz, solutions
65
3.4 The issue of the initial conditions
2.0
H, dφ/dt
1.5
k=0
1.0
0.5
0.0
−15.0
−5.0
t
5.0
15.0
Figure 3.3: The solutions for H and φ̇ for a spatially flat Universe (k = 0) to the eqs.
derived from action (3.30) which is the low energy effective action with first order α 0
corrections included, from [66].
are driven to anisotropic ones, which may separate the evolution of 3 spatial dimension
from the internal ones, even if the problem of the big bang singularity remains unsolved
[70, 71].
In closing this section we also mention to the problem raised in [72] about the instability
that PBB solutions if potential of any rank are included in the background, which seems
to be natural in a string derived model. According to the analysis made in [73], when
potential forms of any rank are included in the Einstein action, cosmological solution will
exhibit near the cosmological singularity an oscillatory behaviour rather than a Kasnerlike (3.12) one, which may set in even at lower scale than the string energy. Anyway this
issue deserves further investigation.
3.4
The issue of the initial conditions
So far a perfectly homogeneous and isotropic model has been considered, and one might
wonder if spatial gradients may prevent the onset of PBB inflation, as it happens for
standard inflation. The analysis of classical inhomogeneities in pre-big bang cosmology has
been worked out in [74, 75]. In the PBB model the situation is different than in standard
inflation, as the initial state of the Universe is asympotically trivial, consistently with the
PBB postulate implying that spatial gradients as well as time derivatives are naturally
tiny in string units. Actually the condition for an accelerated evolution to start refers to
gravitational collapse in the Einstein frame, which can be translated in the string frame
to a chaotic version of PBB inflation. In the Universe patches where the dilatonic kinetic
66
3 The pre-big bang model
energy is a non-negligible fraction of the total energy density PBB inflation sets in. The
onset of gravitational collapse is a perfectly classical phenomenon, which is coherent with
Hawking and Penrose’s theorem on singularity in general relativity, however the amount of
inflation these pathces undergo will depend on the epoch they start inflating (or collapsing
to black hole singularities in the Einstein frame), i.e. on the initial conditions, as PBB
inflation should be stopped as soon as the curvature reaches the stringy scale (H ∼ λ −1
s )
or the strong coupling regime of the theory (e φ ∼ 1) is met. The total amount of inflation
between two generic values of time t i and tf can be conveniently measured by the factor
Z defined as
Z=
H(tf )a(tf )
,
H(ti )a(ti )
(3.33)
where in our case ti and tf denote respectively the time of the onset of inflation and
the time when higher order corrections become relevant. To have a sufficient amount of
inflation to solve the horizon and flatness problem, see (1.26), the intial values of the fields
(we can set tf ∼ λs ) must be constrained according to [51]
eφ(ti ) < 10−26 ,
R3 (ti ) < 10−38 λ−2
s ,
(3.34)
where R3 (ti ) is the initial 3-curvature of the inflating patch. It has been argued that the
previous initial conditions represent a fine tuning [76, 77] but it should be remembered
that PBB initial conditons require an almost flat and decoupled Universe, thus at its origin
the Universe is classic, it has no knowledge of any string or Planck length.
Moreover the classical action (3.1) has the symmetry for a costant shift in the dilaton
and a costant rescaling of the metric
φ → φ + cost ,
gµν → ec gµν ,
(3.35)
that means that the quantities constrained by eq. (3.34) are classically not even defined,
as they can be shifted at will.8
3.5
Effects of a “stringy” phase
Let us anyway study if the subsequent phase characterized by constant H and φ̇ may
help in providing an additional phase of exponential expansion [78]. The stringy phase
suggested by the solution obtained from action (3.30) is given by constant Hubble parameter and linearly running dilaton, see fig. 3.3, thus we can consider its contribution to
the solution of the homogeneity/flatness problem. The condition e φ(ti ) 1 ensures the
existence of a long stringy phase, as curvature H may become of the order of λ s when the
weak coupling condition eφ < 1 is still fulfilled. From (3.32)
1
Hs
=
,
2β
φ̇
(3.36)
in the stringy phase where both H and φ̇ are constant. The total amount of inflation
during this deSitter-like phase (as ȧ/a is constant) can be found by fixing the end of the
8
We also want to remark that to us the fine tuning problem is more an issue of taste than a scientific
well-posed problem.
67
3.5 Effects of a “stringy” phase
inflation at time tf from the condition φ(tf ) = 0 (as the stringy deSitter phase must start
for negative values of the dilaton):
|φs |
ZdS = exp (Hs (tf − ts )) = exp
,
(3.37)
2β
and it is very large if at the beginning of the string phase we are in the weak coupling
regime, |φs | 1 or if β 1 (dilaton almost costant). The amount of inflation during the
string phase alone, given by eq. (3.37), is sufficient to solve the cosmological problems if
|φs | & 120β .
(3.38)
In general, we can expect that a long inflationary phase at the string scale will produce a large density of stochastic gravitational waves, and we should ask whether the
condition (3.38) is consistent with the experimental bounds on the gravitational wave
spectrum. Neglecting finer details of the spectrum, which are discussed in ref. [49], in the
range fs < f < f1 the spectrum is approximately given by
3−|3−2β|
f
,
(3.39)
h20 Ωgw (f ) ' 3 × 10−7
f1
where fs is a free parameter of the model that can be traded for Z dS , as fs = f1 /ZdS ,
being f1 the end-point of the spectrum, estimated in (3.24). For f < f s the spectrum
varies as f 3 ln2 f .
In the following, for definiteness, we use f 1 = 10GHz and h20 Ωgw (f1 ) = 3 · 10−7 , as in
[49]. Our results can be easily rescaled using different values for these quantities.
Since f1 is fixed, the spectrum depends only on two free parameters β, and f s , or
equivalently β and ln ZdS . We now study the range of values of these parameter allowed
by the three observational constraint discussed above.
In order to compute the integral in the bound (3.23) the explicit form of the spectrum
has to be inserted, combining it with the COBE and pulsar bounds we obtain the results
presented in fig. 3.4. The shaded area is the region of parameter space forbidden by
these observational constraints. We observe that requiring ln Z dS > 60 implies β > 0.12 in
order to evade the COBE bound. This happens because for ln Z dS > 60 the frequency fs =
f1 /ZdS becomes smaller than the maximum frequency explored by COBE f ' 10 −16 Hz,
and we cannot take advantage of the ∼ f 3 behavior of Ωgw for f < fs in order to lower
the value at COBE frequencies in comparison to the value at f = f 1 . Rather, from 10−16
Hz up to f = f1 ∼ 10 GHz the spectrum varies as f 3−|3−2β| (i.e. as f 2β for β ≤ 3/2) and,
because of this, β cannot be too close to zero.
We can also consider a situation in which the amount of inflation ln Z dS > 60 is given
partly by the deSitter phase and partly by the super-inflationary phase. This reduces the
requirement on ZdS alone and therefore on β. A value of β as close as possible to zero
is the most favorable situation for the observation of the gravitational wave spectrum at
LIGO/Virgo frequencies, f = 6Hz−1kHz. Therefore, asking that the required amount of
inflation is provided uniquely by the string phase, the maximum value of the spectrum
at, say, 1kHz, is lowered. We present in the figure the lines in the parameter space that
correspond to a value of h20 Ωgw (1 kHz) equal to 10−8 and 10−7 .
In the range 40 < ln ZdS < 56 the stronger limit on β is given by the pulsar-timing
68
3 The pre-big bang model
constraint, and is β > 0.04. Finally for 21 < ln Z dS < 40 the primordial nucleosynthesis
constraint is the strongest one and we find the final smooth branch of the curve. For
ln ZdS < 21 we have no more restrictions on the value of β.
80
−8
70
Ω =10
60
Ln ZdS
50
40
−7
30
Ω =10
20
10
0.00
0.05
0.10
2β
0.15
0.20
Figure 3.4: The forbidden region in the parameter space lnZdS vs. 2β (defined in
(3.33) and (3.36)) is the shaded area. Along the dot-dashed line Ω ≡ h20 Ωgw (1 kHz) =
10−7 and Ω = 10−8 along the dashed line. From [78]
Let us is also useful to discuss our results in terms of the original parameters of the
model φs and β, rather than ln ZdS and β. A large value of ln ZdS = −φs /(2β) can be
obtained as a combination of two limiting cases:
• if β is very close to zero, so that for any reasonable value of the initial curvature
eq. (3.38) is satisfied, even without requiring an especially large value of |φ i | and hence of
|φs |;
• if φi is very large and negative.
Concerning the condition over β, we see from fig. 3.4 that β cannot be chosen to
be arbitrarily small because of the various observational constraints; still, we can reach
moderately small values β ' 0.12. This condition is analogous to the slow roll conditions in
standard implementations of the inflationary scenario, see eq. (1.32). It is a requirement
on the dynamics of the theory, not on the initial conditions, and it ensures that φ̇ is
sufficiently small so that the mechanism that terminates inflation, and that presumably
takes place when eφ = O(1), is sufficiently delayed.
String cosmology, however, also has the second option, namely −φ i 1. In this
case the inflationary phase is long not because the field φ obeys a slow-roll condition,
but rather because its initial value φ i is such that eφi 1 is very far from the point
where inflation terminates, eφ ∼ 1. Making a comparison with chaotic inflation [25], see
sec. A.2, we see that there the “natural” initial value of the inflaton field φ i is fixed by the
3.6 Summary
69
condition V (ϕi ) ∼ MP4 l , where V is the potential that triggers inflation, and this fixes the
dimensionful field φi in terms of the Planck mass and of the dimensionless parameters of
the potential.
In our case, instead, φ is a dimensionless field and g 02 = eφi is the initial value of the
gauge coupling. The initial condition g 02 1 means that the evolution starts deeply into
the perturbative regime and as such, it is possibly the most natural initial condition in
this context rather than a fine-tuned one.
From the phenomenological point of view we note that the nucleosynthesis constraint
is an integral bound and for a given shape of the spectrum fixes the heigth of the peak.
For ∼ 1 and Hs ∼ 0.15MP l the nucleosynthesis bound is saturated and it might be
interesting even for direct detection in the near future by the interferometer experiments
LIGO and Virgo, whose sensitivity is (optimistically) 10 −6 in the window 1Hz< f <1kHz
[79, 80].
Thus the model allows a long inflationary phase at the string scale, while at the same
time the existing observational bounds on the production of relic gravitational waves are
respected. This stringy phase can be long enough to solve the horizon/flatness problems,
or it can be combined with the superinflationary phase to provide the required amount
of inflation. In the former case, the value of the intensity of the relic gravitational wave
spectrum to be expected at ground based interferometers is of order Ω gw ∼ a few ×10−8
while in the latter case it can reach a maximum value Ω gw ∼ a few ×10−7 .
We note that as the spectral slope is 2β for f s < f < f1 , the stringy phase may
provide a flat adiabatic spectrum of density perturbations, provided the astrophysical
relevant scale fA for structure formation and CMB anisotropies lies in this range f s <
fA < f1 . As 2πfA ∼ (10 − 0.1)Mpc−1 , fs < fA will require at least ZdS ∼ O(60), that
is the perturbation must saturate the COBE bound and provide the small anisotropies
δρkA /ρ ∼ 10−4 of CMBR at the time of Hubble scale crossing at the decoupling epoch,
thus leading to a spectral slope 2β ∼ 0.12 − 0.13, which is slightly above the observational
bound, which requires, parametrizing the perturbtion spectrum relevant for the CMBR as
Pper ∝ k n−1 , n ' 1 within ten per cent accuracy.
3.6
Summary
Concluding this introductory chapter about the pre-big bang string cosmological model
we summarize the pluses and minuses of it:
The Goodies
• It provides a natural mechanism for realizing inflation, based on a fundamental
theory of physics rather than introducing ad hoc fields or potentials. The underlying
SFD symmetry suggests the existence of an inflating branch as the counterpart of a
standard decelerating one.
• Initial conditions are natural as the Universe starts simple and decoupled and naturally evolves into a rich and complex one by means of a gravitational instability.
The problem of the initial conditions is separated from that of the singularity.
• A post-inflationary hot big bang is a natural outcome and no fine-tuning is necessary
to overcome the problems of homogeneity and flatness.
70
3 The pre-big bang model
• Perturbations in any fundamental field are amplified and can play a role in seeding
the cosmic magnetic fields and the primeval density perturbation. An interesting
background of gravitational waves is produced.
The Baddies
• A scale invariant spectrum of adiabatic density perurbations is all but automatic,
even if not impossible.
• The graceful exit problem is mainly unsolved: the conjecture that a hot big bang
model is the outcome of the high curvature/coupling phase is attractive but it still
has to be proven.
4 Supersymmetric vacuum
configurations in string cosmology
Pre-big bang cosmology has been initially developed using the lowest-order effective action
of the bosonic string. This allowed to understand the basic features of this cosmological
model: the Universe starts at weak coupling and low curvature, follows a superinflationary evolution and enters a large curvature phase. Also the generality of initial conditions
and the phenomenological consequences of the model can be investigated within the approximation of the lowest order effective action, but in this framework the cosmological
evolution unavoidably reaches large curvatures and strong coupling and finally runs into a
singularity. The pre-big bang model cannot be considered consistent as far as the matching between pre- and post-big bang branches will not be completed: making this matching
will require to smoothen the big bang singulatity.
The α0 corrections analysed previously go in the right direction but do not complete
the job. Here we try a different approach based on supersymmetry [81]: we look for cosmological solutions of the pre-big bang type which are left unaltered by a supersymmetry
transformation and we exhibit one example which has the right inflating behaviour well
before the big bang and which does not run into a singularity because of to the formation
of a fermion condensate. This example thus shows a new mechanism for avoiding the
singularity within the context of a low energy effective action, even if it does not display
a full exit.
4.1
The supergravity action
Given a classical background field configuration |Ωi, it preserves supersymmetry if its
supersymmetry variation vanishes, and the condition for unbroken supersymmetry will be
also a sufficient condition for the fields to satisfy the equations of motion. Thus, being
Q the supersymmetry charge, the condition for Ω to represent a supersymmetric field
configuration is
Q|Ωi = 0 ,
(4.1)
or equivalently for any operator A we have
hΩ| {A, Q} |Ωi = hΩ|δA|Ωi = 0 ,
(4.2)
as the anticommutator {Q, A} is the supersymmetry variation of A, δA. Bosonic field
variation will be automatically satisfied as no fermion can acquire a vacuum expectation
71
72
4 Supersymmetric vacuums configuration
value without breaking Lorentz or SO(3) invariance, so only the fermionic field variation
has to be checked: to find a vacuum endowed with unbroken supersymmetry (at tree
level) is equivalent to find a field configuration which is annihilated by the variation under
a supersymmetry transformation.
We want to check if the requirement that a cosmological background be supersymmetric
may enforce the solution to avoid singularities. Let us write down for this purpose the
bosonic part of the action of N = 1 supergravity in D = 10 in the string frame:
Z
1
1
10 √
−φ
d x ge
S=
R + (∂φ)2 − HM N P H M N P − ψ̄M ΓM N P DN ψP
2
12
1
−λ̄ΓM DM λ − √ (∂N φ)ψ̄M ΓN ΓM λ + (∂N φ)ψ̄ N ΓM ψM
(4.3)
2 2
1
1 M
1
− (λ̄ΓABC λ)
ψ̄ ΓABC ψM + ψ̄ M ΓM AB ψC − ψ̄A ΓB ψC + . . . .
32
12
2
Our notations are as follows: the gravitino ψ M is a left-handed Weyl-Majorana spinor, the
dilatino λ is a right-handed Weyl-Majorana spinor, H = dB and the covariant derivative
DM is with respect to the spin connection ω(e), which is independent of the fermionic
fields [82, 83]. Indices A, B, M, N take values 0, . . . , 9, Γ ABC... denotes the antisymmetrized
product of ten-dimensional gamma matrices, with weight one. We also set κ 10 = 1. The
dots in eq. (4.3) stands for terms of the type (H M N P × fermion bilinears), terms with three
gravitino fields and one dilatino, and terms with four gravitinos. Their explicit form is
not needed below and can be obtained from ref. [83]. According to the previous discussion
to find a vacuum endowed with unbroken supersymmetry we must impose the conditions
hδλi = hδψM i = 0.
4.2
The supersymmetry conditions
We consider first the case in which the expectation values of all bilinears in the Fermi fields
are set to zero. This corresponds to solutions of the equations of motions of the bosonic
part of the action (4.3), and therefore to the pre-big bang cosmology. The supersymmetry
variations of the dilatino and gravitino field can be found, e.g., in ref. [83]. Writing
them in the string frame, and setting the fermion condensates to zero, the equations
hδλi = hδψM i = 0 give
1
MNP
M
η=0,
(4.4a)
Γ ∂M φ − HM N P Γ
6
1
(4.4b)
DM η − HM η = 0 ,
8
where η is the parameter of the supersymmetry transformation and H M ≡ HM N P ΓN P .
Note that in the string frame (where the Ricci term in the action is not canonically
normalized as it appears with a pre-factor e −φ ) eq. (4.4b) is independent of the dilaton
field, contrarily to what happens in the Einstein frame [84]. This simplifies considerably
the analysis of the equations. Writing D̂M ≡ DM − (1/8)HM , eq. (4.4b) implies the
integrability conditions [D̂M , D̂N ]η = 0, which gives
2RM N P Q ΓP Q + (DN HM ) − (DM HN ) − HM R Q HN RS ΓQS η = 0 ,
(4.5)
4.3 Unbroken supersymmetry by fermion condensate
73
which is therefore a necessary (but not sufficient) condition for supersymmetry. One
can now see by inspection that the solutions (3.11) used in homogeneous pre-big bang
cosmology do not satisfy equations (4.4a) and (4.5). This is obvious for the solutions with
vanishing HM N P , since in this case eq. (4.4a) requires a constant dilaton.
In fact, we tried a rather general ansätz compatible with a maximally symmetric 3dimensional space
ds2 = −dt2 + a2 (t)d~x2 + gmn (t, ~y )dy m dy n ,
(4.6)
in which the 3-space, with coordinates ~x, is isotropic and has a scale factor independent
of the internal coordinates ~y , while the metric in the six-dimensional internal space is
independent of the xi but otherwise arbitrary.
For HM N P we made the ansatz Hijk = const · ijk for i, j, k = 1, 2, 3, H0ij = 0,
HM N P vanishes also if indices of the three-space and indices of the internal space apppear
simultaneously, and HM N P is arbitrary if all the indices M N P take values 0, 4, . . . 9. We
also considered the case of spatially curved sections of the three-space, see app. E for
explicit computations. Even with this ansatz, which is the most general compatible with
maximal symmetry of the three-space when the metric of the three-space is independent
of the internal coordinates, it is straightforward to show that eqs. (4.4,4.5) do not admit
non-trivial cosmological solutions.
4.3
Unbroken supersymmetry by fermion condensate
The super-inflationary pre-big bang solutions are therefore rotated by supersymmetry
transformations into different classical solutions of the action (4.3). Since each classical
solution of the equations of motion corresponds to a string vacuum, this means that
selecting such a vacuum corresponds to a spontaneous breaking of supersymmetry. If we
want to preserve the advantages of supersymmetry for low-energy physics, for instance
for the hierarchy problem, supersymmetry should not be broken already in the pre-big
bang era. Therefore we now look for vacuum states with unbroken supersymmetry. The
above result suggests that, in order to find supersymmetric solutions, the effect of Fermi
fields must be switched on, which means that we must consider the effect of non-vanishing
fermion condensates.
A particularly simple and appealing solution can be found assuming that the only nonvanishing√fermion bilinear is the mixed gravitino-dilatino condensate, h λ̄ψM i. Let us define
vM = −( 2/8)hλ̄ψM i. It is a composite vector field and in general in a cosmological setting
it depends on time (note that while in global supersymmetry the fermion condensates are
space-time independent [85], this is not the case with local supersymmetry). Furthermore,
we look for solutions with HM N P = 0. In this case the equations hδλi = 0, hδψ M i = 0
give
ΓM (∂M φ − 8vM )η = 0 ,
(4.7)
1
N
DM η − 8vM + ΓM vN η = 0 .
2
(4.8)
74
4 Supersymmetric vacuums configuration
The integrability condition of eq. (4.8) is
A B
B A
RM N P Q ΓP Q − 2vA vB (g AB ΓM N + δN
Γ M − δM
Γ N)
−32fM N + 2(ΓM A DN vA − ΓN A DM vA ) η = 0 ,
(4.9)
where fM N = ∂M vN − ∂N vM and DM vA = ∂M vA − ΓB
M A vB . For the metric we make an
isotropic ansätz
ds2 = −dt2 + a2 (t)dxi dxi ,
i = 1, . . . , 9 ,
(4.10)
and we define as usual the Hubble parameter H(t) = ȧ/a. The strategy here is to find a
field configuration φ(t), H(t), vM (t) such that eqs. (4.7) and (4.9) are identically satisfied,
without requiring any condition on η. This is because eq. (4.9) is only the integrability
condition for eq. (4.8), and as such it is a necessary but not sufficient condition for unbroken
supersymmetry. If it is satisfied for any η we still have the freedom to choose η so that
also eq. (4.8) is satisfied.
Examining eqs. (4.7) and (4.9) we see that this is possible only if v i (t) = 0, i = 1, ..9.
Denoting vM =0 (t) ≡ σ(t), eq. (4.7) becomes simply φ̇ = 8σ. Eq. (4.9), for M = 0, N = i,
becomes
Ḣ − σ̇ + H(H − σ) = 0 ,
(4.11)
(H − σ)2 = 0 .
(4.12)
while for M = i, N = j we get
All these equations are identically satisfied by H(t) = σ(t). We now ask whether
H(t) = σ(t) ,
(4.13a)
φ̇(t) = 8σ(t)
(4.13b)
is a solution of the equations of motions, as we expect for a supersymmetric configuration.
As usual the equations of motion obtained with a variation with respect to bosonic fields
are automatically satisfied when the expectation value over the vacuum is taken, and only
the variation with respect to fermionic fields has to be checked.
We introduce the shifted dilaton φ̄ = φ − dβ, where β = log a and d = 9 is the number
of spatial dimensions, and we also retain the lapse function N in the metric, so that
ds2 = −N 2 dt2 + e2β dxi dxi . Restricting to homogeneous fields, the relevant part of the
action can be written as
"
!
√
Z
2
1
2
−φ̄ 1
2
˙
dte
−dβ̇ + φ̄ + 2 −
λ̄ψ0 φ̄˙
S=−
2
N
8
!
!2 
√
√
(4.14)
2
2
+2d −
λ̄ψ0 β̇ − 8
λ̄ψ0  .
8
8
The last term in the action (4.14) comes from the term ( λ̄ΓABC λ)(ψ̄ M ΓABC ψM ) in
eq. (4.3), making use of the Fierz identity ( λ̄ΓABC λ)(ψ̄ M ΓABC ψM ) = 96(λ̄ψ M )(λ̄ψM )
75
4.3 Unbroken supersymmetry by fermion condensate
1.0
V(σ)
0.5
0.0
-0.5
-2.0
-1.0
0.0
σ
1.0
2.0
Figure 4.1: A symmetry breaking potential for the composite field σ(t).
(see e.g. the appendix of ref. [83]). Instead, the terms ( λ̄ΓABC λ)(ψ̄ M ΓM AB ψC ) and
(λ̄ΓABC λ)(ψ̄A ΓB ψC ) in the action (4.3) are independent from ( λ̄ψ M )(λ̄ψM ) and their
condensates can be consistently set to zero.
Variating now the action with respect to N, φ̄, β and then taking the expectation value
of the terms λ̄ψ0 , (λ̄ψ0 )2 over the vacuum, we get the equations
d −φ̄
e (H − σ) = 0
(4.15)
dt
√
2
λ̄ψ0 )2 i = 0
(4.16)
φ̄˙ 2 − 9H 2 + 2σ φ̄˙ + 18σH − 8h(
8
√
˙ φ̄˙ + σ) − 9H 2 + φ̄˙ 2 + 2σ φ̄˙ + 18σH − 8h( 2 λ̄ψ )2 i = 0 .
(4.17)
2(φ̄¨ + σ̇) − 2φ̄(
0
8
For the configuration H = σ, φ̇ = 8σ (and therefore φ̄˙ = φ̇ − 9H = −σ) the equations
of motion are identically satisfied if h( λ̄ψ0 )2 i = hλ̄ψ0 i2 . Consistency therefore requires
that supersymmetry enforces this relation between the condensates. In general, it is well
known that relations of this kind are indeed enforced by supersymmetry; for instance, the
relation |hχ̄χi|2 = h|χ̄χ|2 i holds for the gaugino condensate in the case of super-Yang-Mills
theories [85] and in supergravity coupled to super-Yang-Mills [86].
It remains to discuss the dynamics of the condensate σ(t). This is a composite field
whose dynamics will be governed by an effective action which in principle follows from the
fundamental action (4.3). To assume that a condensate forms is the same as assuming
that the field σ(t) has an effective action with a potential V (σ) with the absolute minimum
at σ = σ̄ 6= 0, see fig. 4.1. Choosing as initial condition σ → 0 + as t → −∞, σ(t) will
evolve from σ = 0 toward the positive minimum of the potential, and it will make damped
76
4 Supersymmetric vacuums configuration
1.5
1.0
σ
0.5
0.0
75.0
t
125.0
Figure 4.2: The evolution of the field σ(t).
oscillations around σ = σ̄, the damping mechanism being provided by the expansion of the
Universe and possibly by the creation of particles coupled to the σ field. The qualitative
behaviour of σ(t) will be therefore of the form plotted in fig. 4.2. For illustrative purposes,
we have shown in fig. 2 the evolution of σ obtained assuming an effective action, in the
string frame, of the form
Z
−φ̄ 1 2
σ̇ − V (σ) ,
(4.18)
S ∼ dt e
2
where V (σ) = −σ 4 + (2/3)σ 6 is the potential shown in fig. 4.1. (We use units such that
the minimum is at σ̄ = 1.) This gives the equation of motion σ̈ − φ̄˙ σ̇ + V 0 = 0, with
−φ̄˙ = σ providing the friction term. However, the qualitative behaviour is independent
from these specific choices.
Since H(t) = σ(t) and φ̇ = 8σ(t), this solution corresponds to a cosmological model
that starts at t → −∞ from Minkowski space with constant dilaton and vanishing fermion
condensates, i.e. from the string perturbative vacuum, and evolves toward a de Sitter
metric H = const., with linearly growing dilaton. This is similar to the scenario found
in ref. [66], discussed in the previous chapter. In the present case the scale at which the
curvature is regularized is given by the fermion condensate σ̄ while in [66] it was given by
the α0 corrections. However, in the case studied in ref. [66], higher order α 0 corrections were
not under control, so that a definite statement about the effectiveness of the regularization
mechanism could not be made. In the present case, instead, the fact that σ, and therefore
H and φ̇, stops growing follows from the general requirement that the potential V (σ) be
bounded from below and has a minimum, as we expect for the effective potential derived
from any well-defined fundamental action as the action (4.3).
4.4 Conclusions
77
The deSitter solution should finally be matched to a standard radiation-dominated era.
For the matching, O(eφ ) corrections to the string effective action are probably important,
since φ̇ is positive and therefore at some stage e φ becomes large. When the gauge coupling
∼ eφ becomes strong, gaugino condensation is also expected to occur, suggesting that the
gaugino condensate might play a role in matching the de Sitter phase to a radiation
dominated era.1
It is also interesting to observe that, if at small σ the potential V (σ) behaves as V (σ) '
−σ 4 /(2c2 ), with c a positive constant, then at large negative values of time the solution of
the equation of motion for σ is σ(t) ' c/(−t) and therefore H(t) ' c/(−t), corresponding
to a super-inflationary stage of expansion, as it also happens for the solutions found in in
the bosonic sector of the model.
4.4
Conclusions
We conclude this chapter reminding that the solution that we have presented illustrates
a possible role of fermion condensates in a supersymmetric cosmology and provides a
novel mechanism for the regularization of the singularity of the pre-big bang cosmological
solutions.
Anyway we cannot deny that the solution we have shown has more an illustrative rather
than a realistic value, as the dynamics of the fermion condensate needs clarification. From
the point of view of reaching a realistic model, we still have to match the pre- and post-big
bang and moreover we have discussed an isotropic ten-dimensional solution not touching
the issue of the compactification of the extra dimensions. 2
Anyway we now have an additional mechanism beside the α 0 correction terms, to ensure
a non singular cosmology: there we had the problem of not mastering the full α 0 perturbative expansion, here we miss the physics concerning the dynamics of the condensate,
even if this regularization mechanism relies only on general properties of the condensate
potential.
1
One might also ask whether, including the gauge sector, the condensate σ has an effect on gauginos. However, in minimal supergravity coupled to super Yang-Mills theory in D = 10, the gauginogravitino-dilatino coupling is proportional to terms of the type χ̄ΓABC χψ̄M ΓABC ΓM λ, and the condensate
hψ̄M ΓABC ΓM λi is independent from hψ̄M λi.
2
Anisotropic cosmological model, in which three spatial dimensions expand and six get compactified,
can be obtained by switching on the effect of the three-form HABC [70, 71] or including the effect of a
dilatino condensate λ̄ΓABC λ, or a gravitino condensate such as ψ̄A ΓB ψC , or a gaugino condensate χ̄ΓABC χ.
They can separate 3 spatial dimensions from the remaining 6 if either the 3 indices A, B, C belong to the
3-dimensional space, or if they are holomorphic indices of a 6-dimensional internal complex manifold.
5 Loop corrections and graceful
exit
In the previous chapter we showed an example of a mechanism which can regularize the
pre-big bang singularity. At this stage it is however necessary to go beyond the lowest
order effective action for understanding how string theory cures the big-bang singularity
and the matching of pre-big bang cosmology with standard Friedmann-Robertson-Walker
(FRW) post-big bang cosmology. For instance in our example the growth of the dilaton is
still unbounded, thus pointing towards a region where quantum corrections are important.
Moreover it has been shown that a graceful exit is indeed possible, at least with ad hoc
invented corrections [87], thus it is natural to ask if motivated corrections, derived from
some actual string compactification, can equally well do the job.
One can imagine two possible scenarios. The first is that pertubative corrections
succeed in turning the regime of pre-big bang accelerated expansion into a decelerated
expansion, and at the same time the dilaton is stabilized. This should take place before
entering into a full strong coupling regime, so that perturbative results can still be trusted.
In the second scenario the evolution proceeds toward the full strongly coupled regime.
In this case one must take into account that at strong coupling and large curvature new
light states appear and then the approach based on the effective supergravity action plus
string corrections breaks down.
In this chapter we examine the effect of perturbative string loops on the cosmological
pre-big bang evolution on more “realistic” four dimensional actions. We study loop corrections derived from heterotic string theory compactified on a Z N orbifold and we consider
the effect of the all-order loop corrections to the Kähler potential and of the corrections to
gravitational couplings. It is important to go beyond the one-loop approximation to have
some control of the theory even when the coupling is of order one, case which is relevant
for the graceful exit problem. After discussing how far one can go within a perturbative
approach, we discuss the question of whether perturbative string theory is an adequate
tool for discussing string cosmology close to the big-bang singularity, or whether instead
non-perturbative string physics plays a crucial role.
5.1
Supersymmetric action in four dimensions
The general Lagrangian coupling D = 4 N = 1 supergravity to gauge and chiral multiplets
depends on three arbitrary functions of the chiral multiplet [88, 89]:
• The Kähler potential K(z, z̄) which is a real function determining the kinetic terms
78
5.2 Necessary condition for a graceful exit
79
of the chiral fields z according to
Lkin = Kz z̄ ∂µ z∂ µ z̄ ,
(5.1)
with Kz z̄ ≡ ∂ 2 K/∂z∂ z̄. K is called the Kähler potential because the manifold of the
scalar field z is Kähler , with metric K z z̄ .
• The superpotential W (z) which is a holomorphic function of the chiral multiplet.
The potential entering the lagrangian is given by
)
(
2
2
|W
|
V (z, z̄) = eK/MP l Dz W Kz−1
,
(5.2)
z̄ Dz W − 3
MP2 l
whith Dz W ≡ Wz + W Kz .
• The holomorphic function fab , which determines the gauge kinetic terms
a
a
Lgauge = Refab Fµν
F bµν + Imfab Fµν
F̃ bµν .
(5.3)
It contributes to the gauge part of the scalar potential for gauge charged particles
VD = Ref −1 ab (Kz , T a z)(Kz̄ , T b z̄) .
(5.4)
The complete potential V will be V = V F + VD .
In the next sections we shall consider the explicit form of these functions for a string
derived low energy effective action and look for cosmological solutions of the resulting
equations of motion.
5.2
Necessary condition for a graceful exit
String loop corrections have been much studied in the literature, especially for Z N orbifold
compactifications of the heterotic string [90, 91, 92, 93] and we can therefore ask whether,
at least in some compactification scheme, they fulfil the non-trivial properties needed for a
graceful exit. In particular, corrections to the Kähler potential are known at all loops; this
will be very important for our analysis, since in order to follow the cosmological evolution
into the strong coupling regime, a knowledge of the first few terms of the perturbative
expansion is not really sufficient, and one must have at least some glimpse into the structure
at all loops.
The motivation for our work [94] comes in part from the works [87, 95], where an
explicit example of graceful exit transition between the + and the − branch is showed by
the use of an ad hoc invented loop corrections to the low energy effective action.
Some simple criteria loop corrections must satisfy in order to trigger a graceful exit can
be understood by rewriting eq. (3.15) in the form (from now on we shall deal with 4
dimensions, then dropping from the dilaton the dimensional index)
p
φ̇ = 3H ± 3H 2 + eφ ρc
(5.5)
where δ 2 = 1/3 now (βi = 0) and we parametrized the corrections with an effective
quantum source term eφ ρc . To allow a continous transition between the two branches we
80
5 Loop corrections and graceful exit
must first have a negative effective energy source ρ c , then the derivative of the shifted
dilaton φ̄˙ ≡ φ̇ − 3H must go through a zero and finally the effective source must turn off.
Moreover on the − branch we have that the Einstein frame Hubble parameter
(5.6)
HE ≡ e−φ/2 H − φ̇/2
is positive whereas on the + branch it is negative, thus a complete transition requires a
change of sign of HE , i.e. a bounce of the Einstein cosmic scale factor.
In [87] a complete exit is achieved by the use of loop corrections of the appropriate sign
and functional form, thus showing that they can in principle trigger a complete graceful
exit, but this leaves open the question of whether this actually happens for the corrections
derived from at least some specific compactification of string theory. In particular the sign
of the corrections is important and this already is an interesting point to check against real
string derived corrections, and furthermore if they should also have a rather non-trivial
functional form that suppresses them at strong coupling.
5.3
The effective action with loop corrections
We consider the effective action of heterotic string theory compactified to four dimensions
on a ZN orbifold, so that one supersymmetry is left in four dimensions, and we restrict
to the graviton-dilaton-moduli sector. In a generic orbifold compactification there are
the untwisted moduli fields denoted by chiral multiplets U i and the diagonal untwisted
moduli fields Ti (non-diagonal moduli are included in the matter fields). The moduli
fields Ui determines the complex structure, i.e. the ‘shape’ of the compact space. We
shall neglect the fields Ui and we restrict to a common diagonal modulus, T i = T that
determines the overall volume of compact space.
At the fundamental level string theory compactified on orbifolds is invariant under
T -duality, which includes SL(2, Z) transformations of the common modulus T
T →
aT − ib
,
icT + d
(5.7)
with a, b, c, d ∈ Z and ad − bc = 1. These modular transformations are good quantum
symmetries, and therefore they must be exact symmetries also at the level of the loopcorrected low-energy effective action.
While at tree level the dilaton is inert under modular transformations, at one-loop
the cancellation of a mixed Kähler -Lorentz anomaly via a Green-Schwarz counterterm
requires that the dilaton transforms as [90]
S → S + 2κ log(icT + d) ,
(5.8)
where S denotes the chiral multiplet whose bosonic part is S = e −φ + ia, where φ as usual
is the dilaton field a the 4-dimensional axion dual to the antisymmetric tensor B µν ; κ is a
positive constant of order one which depends on the coefficient of the anomaly, see below.
Then S + S̄ + 2κ log(T + T̄ ) is modular invariant. We can therefore introduce a one-loop
corrected modular invariant coupling g 02 from [90]
1
1
= (S + S̄) + κ log(T + T̄ ) = e−φ + κσ ,
2
g02
(5.9)
5.3 The effective action with loop corrections
81
where the field σ is defined from ReT = (1/2)e σ (the factor 1/2 is not conventional but we
found it convenient). Loop corrections to the effective action can be computed directly as
an expansion in terms of the modular invariant coupling g 02 [90]. As we see from eq. (5.9),
g02 =
eφ
,
1 + κσeφ
(5.10)
and therefore an expansion in g02 provides a resummation and a reorganization of the
expansion in eφ . Note in particular that even when e φ is large, the expansion in g02 is still
under control if κσ 1, i.e. if κ log v 1, where v is the volume of compact space in
string units.
We include in the action terms with two derivatives and terms with four derivatives,
i.e. O(α0 ) corrections to the leading term. For both the two- and four-derivatives terms
we include modular invariant loop corrections. We discuss separately the two- and fourderivatives terms in subsections (5.3.1) and (5.3.2).
The superpotential W is independent of S, which is the string loop counting parameter,
thus it is not renormalized in string perturbation theory [96]. Moreover in the absence of
matter field the superpotential vanishes [97], implying that there are no self-interaction
between the moduli and therefore they represent flat directions in field space.
5.3.1
Terms with two derivatives
In the Einstein frame, where the gravitational term has the canonical Einstein-Hilbert
form, the action for the metric-dilaton-modulus system compactified to four dimensions is
Z
MP2 l
1
j
i µ
4 √
E
(5.11)
d x g R − Ki ∂µ z ∂ z̄j .
S0 =
8π
2
Here z i = (S, T ), Kij = d2 K/dz i dz̄j and K is the Kähler potential. The superscript
E reminds that this action is written in the Einstein frame. The Einstein-Hilbert term
√
gR is not renormalized at one-loop [93] and this non-renormalization theorem persists
to all orders in perturbation theory around any heterotic string ground state with at least
N = 1 space-time supersymmetry [98]. At tree level the Kähler potential is K tree =
− log(S + S̄) − 3 log(T + T̄ ). It does renormalize, and at one loop, for heterotic string
compactified on a ZN orbifold, becomes [90]
K1−loop = − log S + S̄ + 2κ log(T + T̄ ) − 3 log(T + T̄ ) ,
(5.12)
where κ = 3δ GS /(8π 2 ). For instance for Z3 orbifolds δ GS = C(E8 )/2 = 15, where C(E8 )
is the quadratic Casimir of E8 , and therefore κ ' 0.57. Eq. (5.12) holds at one-loop, i.e.
at first order in an expansion in 1/(S + S̄). In terms of the modular invariant coupling g 02
defined in eq. (5.9) we can write eq. (5.12) as
2
g0
K1−loop = log
− 3 log(T + T̄ ) .
(5.13)
2
Eq. (5.13) is the leading term of an expansion in g 02 of the all-order Kähler potential.
Indeed, the Kähler potential has been computed at all perturbative orders in ref. [90],
82
5 Loop corrections and graceful exit
under the assumption that no dilaton dependent corrections other than the anomaly term
are generated in perturbation theory. One defines implicitly the all-loop corrected coupling
g 2 from
1
1
2κ
1
= 2+
log( 2 ) + const .
2
g
3
g
g0
The all-loop corrected Kähler potential then reads [90]
2
g
κ 2 −3
K = log
1+ g
− 3 log(T + T̄ ) ,
2
3
(5.14)
(5.15)
and at g02 1 it reduces to eq. (5.13) plus terms O(g 02 log g02 ).
2 after
The coupling g 2 is just the effective gauge coupling that multiplies the term F µν
taking into account loop corrections [90] and, as we shall see in the next subsection, it also
multiplies the four derivative term. So the dilaton enters the action only through g 2 . We
therefore define a new field ϕ from
g 2 = eϕ
(5.16)
and we shall treat it as our fundamental dilaton field. Therefore our loop-corrected twoderivative action is given by eq. (5.11) where now z i = (S 0 , T ), Re S 0 = e−ϕ = g −2 and K
is given by eq. (5.15).
In the following, it will be convenient to work in the string frame. We define the string
E from g E = G e−ϕ .
frame metric Gµν in terms of the metric in the Einstein frame g µν
µν
µν
Note that we use ϕ rather than φ to transform between the two frames. At lowest order
in eφ of course this reduces to the standard definition, but beyond one-loop the definition
in terms of ϕ is more convenient.
Writing explicitly the kinetic terms of the dilaton and modulus field, the loop-corrected
two-derivative action in the string frame reads
Z
√
3
1
(5.17)
d4 x G e−ϕ R + (1 + eϕ G(ϕ)) ∂µ ϕ∂ µ ϕ − ∂µ σ∂ µ σ ,
S0 = 2
2
2κ4
where we have defined
G(ϕ) =
5.3.2
3κ
2
6 + κeϕ
.
(3 + κeϕ )2
(5.18)
Terms with four derivatives
The four-derivatives term at tree level
For the four derivative term we find convenient to work directly in the string frame. At
tree-level it can be written as [99, 100]
0
Z
√
1
α S + S̄ 4
d x G
Rµνρσ Rµνρσ + bRµν Rµν + cR2 . (5.19)
(S1 )tree = 2
8
2
2κ4
The ambiguity in the choice of order α 0 terms has already been discussed in sec. 3.3 and we
just remind that the study of the cosmological evolution using any specific action truncated
83
5.3 The effective action with loop corrections
at order α0 should then be considered as only indicative of the possible cosmological
behaviours. Here we shall take the point of view that, independently of these ambiguities,
a solution that, thanks to suitably chosen α 0 corrections, approaches asymptotically a
de Sitter phase with linear dilaton is a simple way to model a regularizing effect, which
may have a different and deeper physical motivation as we saw in the previous section.
Our choice for the form of the tree-level-four derivative term is the same used in
ref. [101]:
0Z
√
α
1
(S1 )tree = 2
d4 x −g e−φ R2GB − (∂φ)4 .
(5.20)
2κ4 8
Actually the order α0 part of the action may also involve a number of four-derivative terms
which depends also on ∂σ. We shall neglect these terms because, on the one hand they
make the action more complicated, and on the other hand they are basically irrelevant
to the dynamics, as it will be clear from the results of sec. 5.4. Actually, because of the
ambiguities intrinsic in a truncation at finite order in α 0 , it is not very meaningful to insist
on any specific form of the action, and it is more important to look for properties shared
at least by a large class of actions compatible with string theory.
The loop-corrected four-derivatives term
Let us first recall what happens in the slightly simpler case of a gauge coupling g a , where
the index a refers to the gauge group under consideration. The coupling 1/g a2 is idena F aµν . At tree level, this term only appears when we
tified as the coefficient of (1/4)Fµν
expand in components the superfield expression (−1/2)f ab (S)W a W b , where W a is the
a ; f (S)
chiral superfield containing Fµν
tree = Sδab is independent of the gauge group, and
ab
2
1/ga = ReS. At one-loop fab (S) gets a moduli-dependent renormalization [102]
fab (S)1−loop = S + δab ∆a (T, T̄ ) ,
(5.21)
For orbifolds with no N = 2 subsector, such as Z 3 and Z7 , ∆a (T, T̄ ) = δa is a moduliindependent constant, and there is no moduli-dependent one-loop correction. Beyond
one-loop, fab (S) is protected by a non-renormalization theorem [103]: by Peccei-Quinn
symmetry it cannot depend on the imaginary part of S, the axion, and as it is an holomorphic function of S then it cannot depend either on the real part S which involves the
dilaton.
Furthermore, at one loop a contribution to F µν F µν comes from the anomaly: in fact,
since the fermions in the supergravity-matter action are chiral, the tree level effective action
leads, through triangle graphs, to one-loop anomalies. The type of anomaly depends on
the connections attached to the vertices. In particular, because of Kähler symmetry, under
which the Kähler potential transform as K → K + F + F̄ and which acts on the fermions
ψ I in the form of a chiral rotation
1 I
(F −F̄ )
ψ I → ψ I e− 4 ξ
,
(5.22)
chiral fermions are coupled to a UK (1) Kähler connection, which is a non-propagating
composite field. Modular transformations act on the effective action as a subset of Kähler
84
5 Loop corrections and graceful exit
transformations and therefore, if the theory is Kähler invariant, it is also invariant under modular transformations. Considering a triangle graph with one Kähler connection
and two gauge bosons attached at the vertices, we get a mixed Kähler-gauge anomaly,
a F̃ aµν . The anomaly can be represented in the effective theory with
proportional to Fµν
a non-local term whose (local) variation reproduces the anomaly. Because of supersymmetry, this effective non-local term, when expanded in component fields, together with
a F̃ aµν also contains a term proportional to F a F aµν which, restricting to a common
Fµν
µν
modulus, reads [90]
"
#
κ
2
1 a aµν
µ
−1 ∂ K̂
−
∂ T ∂µ T̄
K̂(T, T̄ ) − 2
F F
.
(5.23)
3
4 µν
∂T ∂ T̄
with K̂(T, T̄ ) = −3 log(T + T̄ ). We note that the effect of the triangle anomaly involving
the Kähler connection is different from the effect of anomalous U (1) gauge groups, whose
anomaly is cancelled by a Fayet-Iliopoulos kind of term [104].
In the limit of constant T this term becomes local, and gives an additional modulidependent one-loop contribution to the gauge coupling g a2 . Therefore, specializing for the
moment to a Z3 or Z7 orbifold,
S + S̄
1
=
+ κ log(T + T̄ ) + δa .
2
ga
2
(5.24)
The variation of the term κ log(T + T̄ ) under modular transformations is just the
anomaly, and the requirement of anomaly cancellation imposes the transformation law,
eq. (5.8), on the dilaton field. Note that in the case of Z 3 and Z7 orbifolds the modulidependent part of ∆a vanishes, and therefore it cannot contribute to the cancellation of
the anomaly; the cancellation comes entirely from the variation of S; so, in this case the
anomaly must be independent of the gauge group, as indeed checked in ref. [90].
At all loops, ga2 in eq. (5.24) is replaced by definition with the all-loop corrected effective
coupling, which is the quantity that appears in the all-loop corrected Kähler potential,
eq. (5.15) [90].
Let us now discuss four-derivative gravitational couplings, which are the ones relevant
for our analysis. In this case we have 3 couplings, multiplying R 2µνρσ , R2µν , R2 .
The situation is similar to the case of gauge couplings, and the contributions again
come from the threshold corrections and from the anomaly (in this case, a mixed Kähler
-gravitational anomaly, i.e. a triangle graph with one Kähler connection and two spin
connections attached to the vertices). There are however some complications: first of all,
only one combination of these operators, corresponding to the the square of the Weyl
tensor, i.e. to R2µνρσ − 2Rµν Rµν + (1/3)R2 , is obtained from a holomorfic function [105],
and the other two independent combinations are not protected by a non-renormalization
theorem. Since, in terms of superfields, the other two combinations that can be formed
depend only on R2µν and R2 , this means that R2µνρσ or R2GB are protected, while naked
R2µν , R2 terms are not. Furthermore, the ambiguity due to truncation at order α 0 that
we have discussed at tree level, persists of course at one and higher loops. It is then
clear that the most general action is very complicated. We have chosen to focus on the
loop corrections to the same operator that we have considered at tree level, i.e. to the
combination R2GB − (∂ϕ)4 , neglecting naked R2µν , R2 , (∂ϕ)4 terms. It is of course possible
85
5.4 The cosmological evolution
to extend our analysis including other operators, but we believe that our choice is sufficient
to illustrate the general role of loop corrections, while at the same time the action retains a
sufficiently simple form, and in particular the equations of motion remain of second order.
Our four-derivative action is therefore S 1 + Snl , where
1
S1 = 2
2κ4
α0
8
Z
√
d4 x G e−ϕ + ∆(σ) R2GB − (∂ϕ)4 ,
(5.25)
and Snl is the non-local contribution from the anomaly,
1
Snl = 2
2κ4
α0
8
2κ
3
Z
√
d4 x G R2GB −1
∂ 2 K̂ µ
∂ T ∂µ T̄
∂T ∂ T̄
!
.
(5.26)
In the following we shall neglect the non-local term. However, an effect of the anomaly
is still present, because it has also produced the local contribution necessary to turn e −φ
into the modular-invariant combination e −ϕ . Threshold corrections produce the function
∆,
∆(T, T̄ ) = −
b̂gr
4
log
(T
+
T̄
)|η(iT
)|
+ δgr .
4π 2
(5.27)
The constant δgr depends on the orbifold considered, and typical values are estimated in
ref. [106]. The constant b̂gr is related to the number of chiral, vector, and spin- 32 massless
super-multiplets, NS , NV , N3/2 respectively, by [107]
1
11
b̂gr = (−3NV + NS ) − (−3 + N3/2 )
6
3
(5.28)
and vanishes for orbifolds with no N = 2 subsector as Z 3 and Z7 ; η(iT ) is the Dedeknid
eta function (D.6). The anomaly produces also a term ∼ R µνρσ R̃µνρσ . However, below we
shall specialize to a metric of the FRW form, and in this background R µνρσ R̃µνρσ vanishes
identically (which allows us to look for solutions of the equations of motion with Im S = 0,
Im T = 0 [107]).
5.4
The cosmological evolution
We now restrict to an isotropic FRW metric with scale factor a(t) = e α(t) , and Hubble
parameter H = ȧ/a = α̇. We use H to denote the Hubble parameter in the string frame.
Another useful quantity is the Hubble parameter in the Einstein frame, H E , related to H
and ϕ by eq. (5.6) with φ substituted by ϕ, as we use ϕ to move from the Einstein to the
string frame. The shifted dilaton ϕ̄ is defined by ϕ̄˙ ≡ ϕ̇ − 3H. In the numerical analysis
we shall use units α0 = 2.
In this section we study the equations of motion, taking initial conditions of the prebig-bang type; we shall add various sources of corrections one at the time, in order to have
some understanding of the role of the various terms, and we shall compare with the above
picture.
86
5.4.1
5 Loop corrections and graceful exit
The evolution without loop corrections
First of all, we examine the behaviour of the system including α 0 corrections, but without
the inclusion of loop corrections. In this case φ = ϕ and our action reads
Z
√ −ϕ
1
3
α0
µ
µ
4
2
4
S= 2
R + ∂µ ϕ∂ ϕ − ∂µ σ∂ σ +
d x Ge
. (5.29)
RGB − (∂ϕ)
2
8
2κ4
If we neglect the modulus field σ, this action reduces to that considered in sec. 3.3
and then we know that, starting from initial conditions of the pre-big-bang type, the
solution has at first the usual pre-big-bang superinflationary evolution and then, when the
curvature becomes of order one (in units α 0 = 2), it feels the effect of the α0 corrections and
is attracted to a fixed point. Writing also the equation of motion for σ, one immediately
sees that there is an algebraic solution of the equations of motion with σ̇ = 0, and H, ϕ̇
constant and the same as in [66], i.e. H = 0.616 . . . , ϕ̇ = 1.40 . . ..
The numerical integration, see figs.5.1, shows that this solution is still an attractor of the
pre-big-bang solution.
For the discussion of the graceful exit, it is very convenient to display the solutions also
in the (H, ϕ̇) plane, following ref. [101]. In this graph, shown in fig. 5.2, four lines are of
special interest. In order of increasing slope, the first line is the (+) branch of the lowest
order solution (more precisely, this line corresponds to the lowest order solution only in
the limit σ̇ = 0, and the deviation of the initial evolution from it that we see in fig. 5.2 is
due to a non-vanishing initial value of σ̇). The second line corresponds to branch change,
i.e. ϕ̇ − 3H = 0. The third is the line where H E = 0, and as found in [87], it is necessary
that the evolution crosses also this line to complete the exit. Finally, we have the line
representing a (−) branch solution. We see from fig. 5.2 that the lowest order solution
ends up at a fixed point, after crossing the branch change line, but it is still in the region
HE < 0.
The solution shown in this subsection can be considered as the starting point of our
analysis; in the following subsections we shall see how the various loop corrections modify
this basic picture.
5.4.2
The effect of the loop-corrected Kähler potential
To begin our analysis we restrict to a Z 3 orbifold, so that threshold corrections vanish and
we also neglect the non-local term. The action that we use in this section is therefore
Z
√ −ϕ
3
1
4
R + (1 + eϕ G(ϕ)) ∂µ ϕ∂ µ ϕ − ∂µ σ∂ µ σ
S = 2 d x Ge
2
2κ4
(5.30)
α0
2
4
+
,
RGB − (∂ϕ)
8
with
G(ϕ) =
3κ
2
6 + κeϕ
.
(3 + κeϕ )2
(5.31)
We again restrict to isotropic FRW metric and homogeneous fields and write the equations
of motion for the fields ϕ(t), σ(t), α(t). Taking the variation with respect to σ, we get the
87
5.4 The cosmological evolution
1.5
.
ϕ
1.0
H
0.5
0.0
0.0
20.0
t
40.0
20.0
40.0
0.3
.
0.2
σ
0.1
0.0
0.0
t
Figure 5.1: H, ϕ̇ σ̇ vs. t for the classical action (5.29)
88
5 Loop corrections and graceful exit
1.0
0.8
H
0.6
0.4
0.2
0.0
0.0
0.5
.
ϕ
1.0
1.5
Figure 5.2: The evolution in the (H, ϕ̇) plane for action (5.29). The four lines, in
order of increasing slope, are the (+) branch, the branch change line (ϕ̄˙ = 0), the
bounce line (HE = 0), and the (−) branch, see the text.
equation of motion
d 3α−ϕ e
σ̇ = 0 .
dt
(5.32)
Therefore, if we take as initial condition σ̇ = 0, σ will stay constant. In this case the
non-local term in (5.26), vanishes at all times, and therefore, for this initial condition, no
approximation is made omitting it.
Before starting with the full numerical integration it is useful to make contact with
the general analisys of ref. [87] mentioned in sec. 5.2. We restrict to constant σ and we
write the Hamiltonian costraint in the form
6H 2 + ϕ̇2 − 6H ϕ̇ = eϕ (ρα0 + ρq ) ,
(5.33)
where
ρ α0
3 4
α0 −ϕ
3
6H ϕ̇ − ϕ̇
= e
2
4
(5.34)
is the contribution of the α0 corrections. The contribution of loop corrections is in the
function ρq which, from our action, turns out to be
2
ρq = −ϕ̇ G(ϕ) = −ϕ̇
2
3κ
2
6 + κeϕ
.
(3 + κeϕ )2
(5.35)
5.4 The cosmological evolution
89
In ref. [101] it was found that a graceful exit could be obtained with a loop correction
that gives ρq = −3f (ϕ)ϕ̇4 , with f (ϕ) a smoothed theta function going to zero, for large
ϕ, as e−16ϕ . This form of the correction was just postulated in ref. [87], but comparing
it with the string result, eq. (5.35), we find that, first of all, the sign comes out right,
which is of course non-trivial. The dependence is ∼ ϕ̇ 2 rather than ϕ̇4 since it comes
from a correction to the kinetic term and, most importantly, its behaviour at large ϕ is
different. In fact G(ϕ) resembles a smoothed theta function, which is also a non-trivial
and encouraging result, but it goes to zero only as e −ϕ , which just compensate the factor
eϕ in eq. (5.33). We shall see from the numerical analysis that this produces important
differences compared to ref. [87].
We now turn to the full numerical analysis, we restore σ as a dynamical field and we
set α0 = 2. The equations of motion obtained with a variation with respect to ϕ and α
are, respectively,
3
3
−6Ḣ(1 + H 2 ) + ϕ̈(2 + 2eϕ G + 3ϕ̇2 ) − 12H 2 − σ̇ 2 − ϕ̇4 − 6H 4 + 3H ϕ̇3 +
2
4
ϕ
2
ϕ 0
+6(1 + e G)H ϕ̇ − ϕ̇ (1 − e G ) = 0 ,
(5.36)
1
4Ḣ(1 − H ϕ̇) − 2ϕ̈(1 + H 2 ) + 6H 2 − 4H ϕ̇ + (1 − eϕ G)ϕ̇2 − ϕ̇4 − 4H 3 ϕ̇ +
4
3 2
2 2
+2ϕ̇ H + σ̇ = 0 ,
(5.37)
2
and together with eq. (5.32) they determine the evolution of the system. The variation
with respect to the g00 component of the metric produces a constraint of the initial data,
3
6H 2 + ϕ̇2 − 6H ϕ̇ − σ̇ 2 = eϕ (ρα0 + ρq ) ,
2
(5.38)
with ρα0 and ρq given in eqs. (5.34,5.35). The constraint is conserved by the dynamical
equations of motion. We used this conservation as a check of the accuracy of the integration
routine. Typically, the constraint is zero with an accuracy of 10 −5 .
The result of the numerical integration is shown in figs. 5.3. We see that at first loop
corrections are small and ϕ̇, H are the same as in fig. 5.2. At some stage loop corrections
become important and the solution settles to a new fixed point, again with ϕ̇ and H
constant. Instead, at least on the scale used, the evolution of σ̇ is indistinguishable from
the case without loop corrections, compare figs. 5.3b and 5.1b, because σ̇ is practically
zero when loop corrections become effective. The change of regime takes place when the
coupling g 2 is of order one, as can be seen from fig. 5.4, where we expand the region in
time where loop corrections become important and we plot H, ϕ̇ and the coupling g 2 .
From these plots, it might seem that after all the situation is not so different from
the tree level evolution, because in both cases the solution in the string frame eventually approaches a de Sitter phase with linearly growing dilaton. An important difference
however is found plotting the solution in the (H, ϕ̇) plane, see fig. 5.5. We see in fact
that the solution has crossed the line H E = 0 (and actually even the (−) branch line)
and therefore entered the region of parameter space where a graceful exit is in principle
possible. Plotting the evolution of H E shows again that the loop corrections due to the
Kähler potential produce a bounce in H E , see fig. 5.6.
Thus, loop corrections to the Kähler potential succeed in doing part of what loop
corrections are expected to do, i.e. they produce a bounce in H E and move the solution
90
5 Loop corrections and graceful exit
1.5
0.3
.
ϕ
.
1.0
0.2
σ
0.5
H
0.0
0.0
0.1
20.0
(a)
t
0.0
0.0
40.0
20.0
t
(b)
40.0
Figure 5.3: (a) The evolution of ϕ̇, H including the all-order loop corrections to the
Kähler potential, eq.(5.30); (b) the evolution of σ̇. Initial conditions are the same as
in the tree-level case and κ = 0.57.
1.5
.
ϕ
1.0
0.5
0.0
20.0
g
2
H
25.0
t
30.0
Figure 5.4: H, ϕ̇ and g 2 = eϕ as a function of time with the all-order loop corrections
to the Kähler potential. Compared to fig. 5.3a we have expanded the range of t where
loop corrections become important.
91
5.4 The cosmological evolution
1.0
0.8
H
0.6
0.4
0.2
0.0
0.0
0.5
.
ϕ
1.0
1.5
Figure 5.5: The evolution in the (H, ϕ̇) plane. The straight lines are as in fig. 5.2.
0.10
0.05
HE
0.00
−0.05
15.0
20.0
t
25.0
30.0
Figure 5.6: HE as a function of the string frame cosmic time with the all-order loop
corrections to the Kähler potetntial.
92
5 Loop corrections and graceful exit
into the region HE > 0. However, we also want to obtain a solution with H, ϕ̇ eventually
decreasing and we want to connect this solution to the (−) branch. We therefore turn to
threshold correction to see if they can produce this effect.
5.4.3
The effect of threshold corrections
We now turn on the moduli-dependent threshold corrections, so that the action becomes
Z
√
3
1
ϕ
µ
−ϕ
µ
4
R + (1 + e G(ϕ)) ∂µ ϕ∂ ϕ − ∂µ σ∂ σ +
S =
d x G e
2
2κ24
0
α −ϕ
+
.
(5.39)
e + ∆(σ) R2GB − (∂ϕ)4
8
The equations of motion are now (setting again α 0 = 2)
3
3
−6Ḣ(1 + H 2 ) + ϕ̈[2 + 2eϕ G + 3(1 + ∆)ϕ̇2 ] − 12H 2 − σ̇ 2 − ϕ̇4 − 6H 4 +
2
4
˙ ϕ̇3 + 6(1 + eϕ G)H ϕ̇ − ϕ̇2 (1 − eϕ G0 ) = 0 ,
+[3H(1 + ∆) + ∆]
(5.40)
˙ − 2ϕ̈(1 + H 2 ) + 6H 2 − 4H ϕ̇ + (1 − eϕ G)ϕ̇2 − 1 + ∆ ϕ̇4 +
4Ḣ[1 − H(ϕ̇ − ∆)]
4
3
˙ + 2(ϕ̇2 + ∆)H
¨ 2 + σ̇ 2 = 0 ,
−4H 3 (ϕ̇ − ∆)
(5.41)
2
and the constraint on the initial data is
3
6H 2 + ϕ̇2 − 6H ϕ̇ − σ̇ 2 = eϕ (ρα0 + ρq + ρqα0 ) ,
2
where ρα0 , ρq are given in eqs. (5.34,5.35) and
3
α0
3
4
˙
−6∆H
− ∆ϕ̇
.
ρqα0 =
2
4
(5.42)
(5.43)
As initial conditions for σ we take a value close to the self-dual point, σ(0) ' σ sd = log 2
(∆0 (σsd ) = 0), that is Re T ' 1, and we take σ̇ small (consistently with the fact that the
pre-big-bang evolution starts from the flat perturbative vacuum). We shall discuss later
the dependence on the initial conditions. With these choices, for a generic orbifold ∆(σ)
turns out to be practically constant during the course of the evolution (and for a Z 3 or Z7
orbifold ∆(σ) = δgr is exactly constant) and its value is determined by b̂gr and δgr ; taking
for instance δgr = 0, we have found nonsingular solutions in the range b̂gr ∈ [−20, 0), which
corresponds to ∆(σsd ) ∈ [−0.18, 0).
The evolution of the system under these conditions is shown in figs. 5.7. The behaviour
of H, ϕ̇ is quite remarkable: threshold corrections turn the de Sitter phase with linearly
growing dilaton into a phase with H, ϕ̇ decreasing! At the same time the modulus σ, and
therefore the volume of internal space, shows a rather elaborate dynamics, see fig. 5.7b.
These figures refer to a Z6 orbifold, for which κ ' 0.19. The same qualitative behaviour
is obtained for a Z3 orbifold, in which case ∆(σ) = δgr is exactly constant, and the same
results are also obtained for different, generic, values of κ.
The evolution in the (H, ϕ̇) plane is shown in fig. 5.8, and we see that the solution
approaches the (−) branch. From this figure we also see that the solution approaches
93
5.4 The cosmological evolution
1.5
.
ϕ
1.0
0.5
0.0
0.0
H
50.0
t
100.0
50.0
100.0
(a)
0.01
.
σ
0.00
−0.01
−0.02
−0.03
0.0
t
(b)
Figure 5.7: (a) The evolution of H and ϕ̇ with loop corrections to the Kähler potential
and threshold corrections. The initial conditions are H(0) = 0.015, ϕ(0) = −30, σ(0) =
0.69, σ̇(0) = 0.001, and ϕ̇(0) = 0.07067 . . . is then fixed by the Hamiltonian constraint;
the values of the parameters are κ = 0.19, b̂gr ' −4, δgr = 0. (b) The evolution of σ̇.
94
5 Loop corrections and graceful exit
1.0
0.8
H
0.6
0.4
0.2
0.0
0.0
0.5
.
ϕ
1.0
1.5
Figure 5.8: The evolution in the (H, ϕ̇) plane with loop corrections. The straight
lines are as in fig. 5.2.
at first the tree-level fixed point discussed in sec. 3.3, then corrections to the Kähler
potential and the threshold corrections become important about at the same time, so that
after leaving this fixed point the solution deviates immediately from the behaviour that it
has in the absence of thresholds corrections, shown in figs. 5.3, and it does not get close to
the fixed point marked by a cross in fig. 5.5. Instead, if we do not include the corrections
to the Kähler potential and we only switch on the threshold corrections, we found that
the solution never crosses the bounce line H E = 0, and therefore the corrections to the
Kähler potential are really an essential ingredient of our solution.
Fig. 5.9 shows instead the evolution of the coupling g 2 , and we see that the curvature
and the derivative of the dilaton start decreasing when g 2 ∼ 1, so that when the solution
is close to the (−) branch we are already at large g, and at this stage non-perturbative
effects are expected to become important. We shall discuss this point further in sec. 5.5.
Although it is appropriate to recall at this point that these results are obtained with
some specific choices of action and of initial conditions, it is certainly interesting to have
at least an example of such a behaviour, with choices well motivated by string theory.
To get some understanding of the dependence on the initial conditions we have run the
integration routine for many different values of σ(0) and σ̇(0). The shaded area in fig. 5.10
˙ where the behaviour is qualitatively the same as that
is the region of the plane (σ(0), σ(0))
shown above, while for initial conditions outside the shaded region the evolution in general
runs into a singularity. Considering that σ is at the exponent in Re T , the limitation on
σ(0) is not particularly strong, while the required values of σ̇(0) are of the same order as
the initial value of H. These initial conditions do not imply therefore any fine tuning.
To have a better understanding of these solutions, it is also useful to display the
95
5.4 The cosmological evolution
1.5
.
ϕ
1.0
0.5
g
2
H
0.0
45.0
50.0
55.0
t
60.0
Figure 5.9: H, ϕ̇ and g 2 against cosmic (string frame) time, with loop corrections.
0.025
0.015
.
σ
0.005
−0.005
−0.015
−0.025
0.7
1.0
1.3
σ
1.6
1.9
2.2
Figure 5.10: The shaded area indicates the region of initial conditions for which
the system has a nonsingular evolution, and the dot corresponds to the value actually
chosen in the solution displayed in figs. 5.7. We have displayed only the part of the
plane with σ > σsd = log 2, since modular invariance ensures that the figure is invariant
under the transformation σ → 2σsd − σ.
96
5 Loop corrections and graceful exit
8.0
.
ϕE
6.0
4.0
2.0
0.0
0.0
50.0
t
100.0
50.0
100.0
(a)
2.5
1.5
HE
0.5
−0.5
0.0
t
(b)
Figure 5.11: (a) ϕ̇E and (b) HE against string time, with loop corrections.
97
5.5 Transition to a D-brane dominated regime
corresponding Einstein-frame quantities. (We still plot them against string frame time t,
but the same qualitative behaviour is obtained against Einstein frame time t E ; the two
are related by dt = dtE exp(ϕ/2)). In figs. 5.11 we plot ϕ̇E = dϕ/dtE and HE . The latter
is particularly interesting and shows that in the Einstein frame our solution approaches
asymptotically a de Sitter inflation. This is of course very different from the result of [87]
or of sec.3.3, where de Sitter inflation takes place in the string frame.
5.5
Transition to a D-brane dominated regime
We now discuss the limitations on the validity of our solutions. As it is clear from fig. 5.9,
at large values of time we are deep into the strong coupling regime, g 2 1. Can we still
believe our solutions? In our action we have included the corrections to the Kähler potential at all perturbative orders, while other operators, like R and R 2µνρσ are protected by
non renormalization theorems. Therefore, despite the ambiguities that we have discussed
for the four-derivative terms, due in particular to naked R 2µν and R2 terms, one might be
tempted to argue that the solution is at least representative of the behaviour at strong
coupling. However, this point of view is untenable, and at some point the perturbative
approach itself breaks down.
To understand this point, it is useful to work in the Einstein frame. The two-derivative
part of our action then reads
S2E
M2
= Pl
16π
Z
√
1
3
µ
µ
d x g R − Zϕ ∂µ ϕ∂ ϕ − ∂µ σ∂ σ ,
2
2
4
(5.44)
with
ϕ
Zϕ = 1 − 2e G(ϕ) = 1 − 2e
ϕ
3κ
2
6 + κeϕ
.
(3 + κeϕ )2
(5.45)
At weak coupling eϕ G(ϕ) 1 and the kinetic term of the dilaton has the ‘correct’ sign.
However, as eϕ → ∞, eϕ G(ϕ) → 3/2 and Zϕ < 0; Zϕ vanishes at a critical value of g 2 = eϕ
given by
2
gcr
0.67
3 √
=
6−2 '
.
2κ
κ
(5.46)
2 it appears that the dilaton becomes ghost-like. We can rescale the dilaton
At eϕ > gcr
so that it has a canonically normalized kinetic term (−1/2)∂ µ ϕ∂µ ϕ, and in terms of the
rescaled dilaton the four-derivative interactions, and in general all interactions involving
the dilaton, become strong as we approach g cr , and formally diverge at the critical point.
This signals that the effective action approach that we have used breaks down and we
must move to a new description, where the light degrees of freedom are different. In string
theory the light degrees of freedom at strong coupling are given by D-branes. This suggest
that, if the cosmological evolution enters the regime e ϕ > gc2 , the effective action approach
that we have used breaks down, and we enter a new regime, which cannot be described
in terms of a classical evolution of massless modes of a closed string, and we must instead
resort to a description in terms of D-branes.
98
5 Loop corrections and graceful exit
More precisely, the condition Zϕ = 0 identifies the critical point only if ϕ̇, H can be
neglected. In fact, the equation of motion for ϕ in the Einstein frame reads (we insert for
future use also a potential V (ϕ))
Mϕ ϕ̈E = −3AHE ϕ̇E − V 0 ,
(5.47)
Mϕ = 1 − 2eϕ G(ϕ) − 3∆(σ)ϕ̇2E + . . . ,
(5.48)
A = 1 − 2eϕ G(ϕ) − ∆(σ)ϕ̇2E + . . . .
(5.49)
where
and
The dots denote tree-level α0 corrections (which are negligible at the later stage of the
evolution). We recall that we found regular solutions for ∆ < 0. So we see that, if we
include the effect of the term |∆|ϕ̇2E , the critical line is given by the condition M ϕ = 0 or
Zϕ + 3|∆|ϕ̇2E = 0 ,
(5.50)
rather than Zϕ = 0. Of course when ϕ̇E > 1 we should at least include all higher
powers in the α0 expansion. More importantly, in the regime where H 1 or ϕ̇ 1
(or when HE , ϕ̇E 1) the 10-dimensional heterotic theory has to be embedded into 11dimensional M-theory compactified on S 1 /Z2 , with the gauge group E8 × E8 splitted over
the two boundaries. The connection between the 11 and the 10-dimensional theory can be
inferred by admitting that the 10-dimensional metric G is derived from the 11-dimensional
one g(11) according to
g(11) =
GM N 0
2
0 R11
where R11 is the radius of S 1 . The dimensional reduction leads to
#
"
Z
(11)
2
R
F
p
− δ(x10 = ∂(S 1 /Z2 )) 6 =
16πS11 =
d11 x g(11)
L9P l
LP l
Z
2
2
√
R11
(∂R11 )
F
10
(10)
d x G
R
+
− 6 ,
2
L9P l
R11
LP l
(5.51)
(5.52)
involving the Ricci scalar R and the gauge field strength F . The 11-dimensional Planck
length LP l and the radius R11 can be rewritten in terms of the heterotic string length λ H
s
and coupling g ≡ ehϕi/2 according to
LP l = g 1/3 λH
s ,
R11 = gλH
s .
(5.53)
In fig. 5.12 (adapted from ref. [108]) on the vertical axis we show H, in the string
frame. This is an indicator of the curvature and therefore of the typical energy scale of
the solution. One might as well use ϕ̇, but of course precise numerical values here are
not very important. In this graph we prefer to use the string frame quantity H because
99
5.5 Transition to a D-brane dominated regime
1.5
D=10
D=11
M−theory
α’
1.0
corrections
H
11−D L
Pl
0.5
11−D SuG
ra
string
loops
S−duality
0.0
0
1
2 φ/2
e
3
4
Figure 5.12: The “phase diagram” of M-theory compactified on S 1 . See the text
for explanations of the various lines. The cosmological solution found in sect. 5.4.3 is
marked by the arrows.
in this case the α0 corrections become important when H ∼ 1, while in terms of H E this
condition becomes e−ϕ HE2 ∼ 1.
The solid line H ∼ 1/g separates the region where an effective 10-dimensional description is possible, from the truly 11-dimensional regime. The region just above the
line labelled 11D-SuGra is described by 11-dimensional supergravity, while above the line
labelled 11-D LP l we are in the full M-theory regime. Of course, again, the position of
the line separating the full M-theory regime from the 11-D supergravity regime is only
indicative, and we have arbitrarily chosen its position so that it meets the curve H = 1/g
exactly at H = g = 1.
On the 10-dimensional side we have also drawn the line given by eq. (5.50), which is
another critical line where a change of regime occurs. When ϕ̇ E is not small, the form of
this curve is only indicative. The label ‘S-duality’ means that, crossing this line, we enter
a regime where the light degrees of freedom are related to the original ones by weak-strong
coupling duality. On the same figure we display the solution of fig. 5.7, labelled by the
arrows. The solution for H will eventually decrease, but this only happens at very large
values of g (see fig. 5.12), and we see that the solution enters the 11-dimensional domain
before it starts decreasing.
Finally, we found that it is not possible to stabilize the dilaton in our solution at
a minimum of a potential. In fact at the later stage of the evolution the tree level α 0
corrections are neglegible, as we see in fig. 5.7a, and M ϕ ' 1 − 2eϕ G(ϕ) + 3|∆|ϕ̇2E . If we
would stabilize ϕ around the minimum of the potential, it should first oscillate around
the minimum and at the inversion points ϕ̇ E = 0, so that here the coefficient of ϕ̈E in
100
5 Loop corrections and graceful exit
1.0
HE
0.0
Zϕ
-1.0
52.0
53.0
t
54.0
Figure 5.13: The evolution of Zϕ ≡ 1−2eϕ G(ϕ) close to the point where HE becomes
positive, against string frame time.
eq. (5.47) becomes ' Zϕ . As shown in fig. 5.13, this quantity is negative after we cross
the HE > 0 line. As we discussed, this is not a problem for the consistency of the solution
as long as ϕ̇ is not small (in fact, fig. 5.12 shows that the limitation on the validity of
the solution is rather given by the crossing into the 11-dimensional region), but it is clear
that no consistent solution with ϕ̇ E = 0 can be obtained trying to stabilize the dilaton
with a potential. In fact, if we try to force ϕ̇ E to a small value, the coefficient of ϕ̈ E in
eq. (5.47) becomes approximately equal to Z ϕ , which at this stage is negative. Therefore
the evolution runs away from the minimum of the potential. Numerically, we have found
that, including a potential in the numerical integration of the equations of motion, when
the solution approaches the minimum of the potential the numerical precision, monitored
by the constraint equation, degrades immediately and the solution explodes.
Therefore, in our scenario, the problem of the dilaton stabilization can only be solved
after the solution enters in the non-perturbative regime.
5.6
Conclusions
In this chapter we have tried to penetrate into the strong coupling regime of the cosmological evolution derived from string theory. This regime is crucial for an understanding
of the big-bang singularity in string theory, but since loop corrections do not tame the
growth of the coupling while remaining within the weak coupling domain, it is clear that a
knowledge limited to, say, one-loop corrections is of little use, and we really need to have
at least a glimpse into the structure of the corrections at all perturbative orders. Luckily,
for the effective action of orbifold compactifications of heterotic string theory, supersym-
5.6 Conclusions
101
metry and modular invariance impose strong constraints on the form of the corrections
at all orders. In particular, the kinetic terms of the dilaton is known exactly, while other
operators, like R and R2µνρσ , are protected by non-renormalization theorems. Therefore,
in spite of some ambiguities in the choice of the four-derivative terms, present both at tree
level and for their loop corrections, one can try to investigate string cosmology beyond the
weak coupling domain, and to obtain at least some indications of what a well motivated
stringy scenario looks like.
As a first step, we have therefore tried to push this perturbative approach as far as
possible, following the evolution even in the strong coupling domain g 1. We have found
solutions with interesting properties, that in the string frame start with a pre-big bang
superinflationary phase, go through a phase with H, ϕ̇ approximately constant and of order
one in string units, (a phase that replaces the big-bang singularity) and then match to a
regime with H, ϕ̇ decreasing. Probably the main element that is missing from this part of
the analysis is the inclusion of non-local terms. These might model the backreaction due
to quantum particles production, which might play an important role in the graceful exit
transition [51]. Unfortunately, these are quite difficult to include in a numerical analysis.
Despite some nice properties, the cosmological model that we have presented still have
some unsatisfactory features, and in particular the dilaton could not be stabilized with a
potential, and so this model cannot be the end of the story.
On the other hand we have found that, if we look at our solution from the broader
perspective of 11-dimensional theory, it ceases to be valid as soon as we enter into the
strong coupling region, even if one includes perturbative corrections at all orders. Thus,
we think that our analysis reveals quite clearly the direction that should be taken to
make further progress. When we move toward large curvatures we meet critical lines in
the (H, g) plane, beyond which D-branes becomes the relevant degrees of freedom. Here
we have found another critical line at strong coupling; beyond this line the light modes
relevant for an effective action approach are interpreted as D-branes. The combination of
these critical lines, shown in fig. 5.12, and the behaviour of our solutions, also displayed
on the same graph, suggest that the evolution enters unavoidably the regime where new
descriptions set in. The understanding and the smoothing of the big-bang singularity
therefore requires the use of truly non-perturbative string physics.
6 The generalized second law in
string cosmology
The analysis exposed so far has made a massive use of the tools provided by string theory,
within the framework of the low energy effective action, for a better understanding of the
big bang singularity. We have found a consistent picture even if not free of difficulties so we
now turn [109] to general thermodynamics considerations, which were first applied to the
study of cosmological singularities by Bekenstein [110] in the context of Einstein’s general
relativity. We propose that accounting for geometric and quantum entropy, accompanied
by a generalized second law (GSL) of thermodynamics, i.e. demanding that entropy never
decreases, should be added to supplement string theory and show that under certain
conditions GSL forbids cosmological singularities.
Geometric entropy is related to the existence of cosmological horizon, whereas the
quantum one is related to the existence of field fluctuactions. During the PBB phase field
fluctuactions give a negative contribution to the derivative of the entropy, leading to a
violation of the GSL for solutions which eventually end up into singularities. We interpret
this by observing that the GSL of thermodynamics forbids cosmological solutions to run
into singularities.
Our discussion is close to that of [111], roughly summarized below, and inspired from
[112], where the idea that geometric and quantum entropy should be added, and be accompanied by GSL was introduced.
6.1
Entropy bounds and geometric entropy
The Bekenstein bound [113] states that for any physical system of maximal radius R and
energy E, its entropy cannot exceed
SBB = 2πER ,
(6.1)
and the bound is saturated by a black hole of mass E and size equal to its Schwarzschild
radius R = 2GN E, being the black hole entropy Sbh given by [113]
Sbh =
A
,
4GN
(6.2)
where A is the area of the horizon of the black hole. The fact that a black hole’s entropy is
proportional to its area has lead to the formulation of the holographic principle [114, 115],
102
6.1 Entropy bounds and geometric entropy
103
which roughly speaking states that the degrees of freedom of a black hole are stored on
its event horizon, thus they are on a surface.
Applying Bekenstein’s bound (6.1) to our visible Universe extrapolated back at the
Planck epoch tP l , assuming for semplicity that it has always been radiation dominated
and that it evolved adiabatically, we obtain
SBB ∼ E(tP l )R(tP l ) ∼ ρ0 H0−4 ∼ 10120 ,
(6.3)
whereas the entropy in the present observable Universe is approximately given by
S0 ∼ (ργ0 )3/4 H0−3 ∼ 1090 ,
(6.4)
neglecting the small mismatch between ρ γ0 and ρ0 ∼ 104 ργ0 . In [111] to recompose this
huge discrepancy between the bound and the actual value of the entropy it is suggested
that the right bound to impose is a modified version of the Bekenstein one, the Hubble
Bekenstein bound (HEB), according to which the maximum entropy that a region of space
can achieve corresponds to the sum of the entropies of the black holes that can fit that
region. In a cosmological context a black hole with radius bigger than the Hubble length
cannot exist, as H −1 correspond to the scale of causal connection, then in a region of size
R in a Universe with Hubble parameter H the right bound to impose is
R 3 H −2
,
(6.5)
SHEB (R, H) =
H −1
GN
being H −2 /GN the entropy of a H −1 -sized black hole and (RH)3 their number in a
region of volume R3 (from now on we drop numerical factors). Considering our observable
Universe at the Planck time, its size was H 0−1 (tP l ) ≡ H0−1 (tP l /t0 )1/2 ∼ 10−30 H0−1 thus
leading to a HEB
3 −2
−1
H (tP l )
H0 (tP l )
∼ 1090 ,
(6.6)
SHEB (H0 (tP l ), H(tP l )) ∼
H −1 (tP l )
GN
which is saturated by S0 given in (6.4) at the Planck epoch. The HEB eventually grows
during the FRW to equal the BB (6.3) today 1 , thus it fails to be saturated now because it
was saturated at the Planck epoch, giving a natural explanation of the mismatch in scale
between (6.3) and (6.4). This kind of reasoning may also explain the arrow of time, as
time increases in the direction that allows the HEB to grow, as once saturated it cannot
decrease without violating the second law of thermodynamics 2 .
It should be noted that by applying the HEB to any epoch successive to the Planck
era we would have failed in saturating it, as considering an epoch characterized by a
temperature T , when our Universe size was H 0−1 (T ) ≡ H0−1 T0 /T , would have lead to
−1
3 −2
H0 (T )
T
H (T )
SHEB (H0 (T ), H(T )) =
= SHEB
< SHEB = S0 , (6.7)
H −1 (T )
GN
H(T )
1
In a radiation dominated FRW Universe S(H0 (t), H(t)) ∝ t1/2 .
We already mentioned to the entropy considerations made by Penrose, see for instance [116], where it
is argued that primordial Universe was in a state of such unnaturally low entropy because it fulfilled by
some reason the Weyl curvature hypothesis, i.e. it was in a state with vanishing Weyl curvature tensor. The
arrow of time then emerged as the Universe evolved in the direction of reaching a configuration with more
entropy, like the one characterized by matter collapsed into black holes. Our approach can be considered an
extension of this argument to PBB, where the Weyl curvature hypothesis is replaced by the PBB postulate.
2
104
6 Generalized second law of thermodynamics
where here SHEB with no arguments denote the quantity (6.6). We shall apply a similar
way of reasoning to the PBB scenario where the previous analysis applies from the big
bang on, but before turning to that we introduce the notion of geometric entropy.
To introduce the notion of geometric entropy we refer to a simple example made in
[117]. Let us consider a quantum massless scalar field in otherwise empty space, in its
ground state. Tracing over the field degrees of freedom located inside an imaginary sphere
of radius R the density matrix ρout depending only over the outer degrees of freedom is
obtained. Analogously ρin can be obtained by tracing over outer degrees of freedom and
both ρin and ρout turn out to have the same eigenvalues 3 , a part from a possible unbalance
in the zero ones, thus the entropy S = −Trρ ln ρ can only depend on the common feature
between the inner and the outer space: the shared boundary. This example seems to
suggest that an actual geometric entropy rather than a bound on forms of entropies can
be associated with boundary surfaces.
6.2
Geometric and quantum entropy
Geometric entropy has been calculated for special systems, but we assume that it is a
general property of a system with a cosmological horizon, resulting from the existence
of causal boundaries in space-time: entropy is tied to the lack of information due to the
fact that we have no acess to what is going on beyond the cosmological horizon. The
concept of geometric entropy is closely related to the holographic principle. For a system
with a cosmological horizon, geometric entropy within a Hubble volume is given roughly,
ignoring numerical factors, by the area of the horizon, as it is shown in the famous paper
by Gibbons and Hawking [118] for spacetimes with a negative cosmological constant (antide Sitter). The geometric entropy S g has origin in the existence of a cosmological horizon
[118, 117, 112].
The second source of entropy we will focus on is quantum entropy S q , associated with
quantum fluctuations. Changes in Sq take into account “quantum leakage” of entropy,
resulting from the phenomenon of freezing and defreezing of quantum fluctuations as their
characteristic length stretches out (freezes) or becomes shorter than (defreezes) the Hubble
length, see sec. A.3. For example, quantum modes whose wavelength is stretched by an
accelerated cosmic expansion to the point that it is larger than the Hubble length, become
frozen (“exit the horizon”), they are lost as dynamical modes and they do not contribute to
entropy; conversely quantum modes whose wavelength becomes smaller than the Hubble
length during a period of decelerated expansion, thaw (“reenter the horizon”) and become
dynamical again [119, 120, 121].
Consistently to our previuos discussion, for a given scale factor a(t) and a Hubble
parameter H(t) = ȧ/a, the number of Hubble volumes within a given comoving volume
V = a3 (t) is given by the total volume divided by one single Hubble volume n H =
a3 (t)/|H(t)|−3 . If the entropy within a given horizon is S H , then the total entropy is given
by S = nH S H . We remind that in the string frame G N ∝ eφ , φ being the 4-dimensional
dilaton. We shall discuss only flat, homogeneous, and isotropic string cosmologies in the
3
This can be checked
the ground state |0i as |0i =
`
´ by realising that` decomposing
´
obtain (ρin )ij = ψψ † ij and (ρout )ab = ψ T ψ ∗ ab .
P
ia
ψia |iiin |aiout we
105
6.2 Geometric and quantum entropy
string frame, whose lowest order effective action (3.1) we re-write for the sake of clarity
Z
√
1
(6.8)
d4 x Ge−φ R + (∂φ)2 .
Slo = 2
2κ4
In ordinary cosmology, geometric entropy within a Hubble volume is given by its area
(in Planck units) SgH = H −2 G−1
N , and therefore specific geometric entropy is given by
sg = nH S H /a3 = |H|G−1
[112].
A possible expression for specific geometric entropy in
N
string cosmology is obtained by substituting G N = eφ , leading to
sg = |H|e−φ .
(6.9)
Reassurance that sg is indeed given by (6.9) is provided by the following observation. The
action Slo can be expressed in a (3 + 1) covariant form, using the 3-metric g ij , the extrinsic
curvature Kij , considering only vanishing 3−Ricci scalar and homogeneous dilaton, by
Z
√
Slo = d3 xdt gij e−φ −3Kij K ij − 2g ij ∂t Kij + K 2 − (∂t φ)2 ,
(6.10)
which is invariant under the symmetry transformation
gij → e2λ gij ,
φ → φ + 3λ ,
for an arbitrary time dependent λ. From the variation of the action
Z
p
δS = d3 xdt Gij e−φ 4K λ̇
(6.11)
(6.12)
the current and conserved charge Q can be read off
Q = 4a3 e−φ K .
(6.13)
The symmetry is exact in the flat homogeneous case, and it seems plausible that it is a
good symmetry even when α0 corrections are present [66, 53]. With definition (6.9), the
total geometric entropy
Sg = a3 |H|e−φ , ,
(6.14)
is proportional to the corresponding conserved charge. Adiabatic evolution, determined
by ∂t Sg = 0, leads to (3.15), which can be rewritten as
Ḣ
− φ̇ + 3H = 0 ,
H
(6.15)
satisfied by the (±) vacuum branches of string cosmology.
Quantum entropy for a single field in string cosmology is, as in [120, 112], given by
sq =
Z
kmax
kmin
d3 kf (k) ,
(6.16)
106
6 Generalized second law of thermodynamics
where for large occupation numbers f (k) ' ln n k . The ultraviolet cutoff kmax is assumed
to remain constant at the string scale. The infrared cutoff k min is determined by the
perturbation equation
p
00 !
s(η)
ψk00c + kc2 − p
ψ kc = 0 ,
(6.17)
s(η)
where as usual η is conformal time (defined by dη = dt/a), 0 = ∂η , and kc is the comoving
momentum
related to physical momentum k(η) as k c = a(η)k(η). Modes for which kc2 ≤
√ 00
√s are “frozen”, and are lost as dynamical modes. The “pump field” s(η) = a 2m e`φ ,
s
depends on the background evolution and on the spin and dilaton coupling of various
fields for some m, l [51, 122]. We
are interested in solutions for which a 0 /a ∼ φ0 ∼ 1/η,
√ 00
and therefore, for all particles √ss ∼ 1/η 2 . It follows that kmin ∼ H. In other phases of
cosmological evolution our assumption does not necessarily hold, but in standard radiation
domination (RD) with frozen dilaton all modes reenter the Hubble length. Using the
reasonable approximation f (k) ∼ constant, we obtain as in [112],
∆Sq '= −µ∆nH .
(6.18)
The parameter µ is positive, and in many cases proportional to the number of species
of particles, taking into account all degrees of freedom of the system, perturbative and
non-perturbative. The main contribution to µ comes from light degrees of freedom and
therefore if some non-perturbative objects, such as D branes become light they will make
a substantial contribution to µ.
6.3
The generalized second law
We now turn to the generalized second law of thermodynamics, taking into account geometric and quantum entropy. Enforcing dS ≥ 0, and in particular,
∂t S = ∂ t Sg + ∂ t Sq ≥ 0 ,
leads to an important inequality,
H −2 e−φ − µ ∂t nH + nH ∂t H −2 e−φ ≥ 0.
(6.19)
(6.20)
When quantum entropy is negligible compared to geometric entropy, GSL (6.20) leads to
φ̇ ≤
Ḣ
+ 3H ,
H
(6.21)
yielding a bound on φ̇, and therefore on dilaton kinetic energy, for a given H, Ḣ. Bound
(6.21) was first obtained in [111], and interpreted as following from a saturated HEB, as
explained in sec. 6.1.
When quantum entropy becomes relevant we obtain another bound. We are interested
in a situation in which the universe expands, H > 0, and φ and H are non-decreasing,
107
6.4 Application to the pre-big bang scenario
and therefore ∂t H −2 e−φ ≤ 0 and ∂t nH > 0. A necessary condition for GSL to hold is
that
e−φ
H2 ≤
,
(6.22)
µ
bounding total geometric entropy
3
He
−φ
e− 2 φ
≤ √ .
µ
(6.23)
We stress that to be useful in analysis of cosmological singularities (6.22) has to be considered for perturbations that exit the horizon. We note that if the condition (6.22) is
satisfied, then the cosmological evolution never reaches the nonperturbative region, allowing a self-consistent analysis using the low energy effective action approach.
It is not apriori clear that the form of GSL and entropy sources remains unchanged
when curvature becomes large, in fact, we may expect higher order corrections to appear.
For example, the conserved charge of the scaling symmetry of the action will depend in
general on higher order curvature corrections. Nevertheless, in the following we will assume
that specific geometric entropy is given by eq. (6.9), without higher order corrections,
and try to verify that, for some reason yet to be understood, there are no higher order
corrections to eq. (6.9). Our results are consistent with this assumption.
6.4
Application to the pre-big bang scenario
We turn now to apply our general analysis to the PBB string cosmology scenario, in
which the Universe starts from the perturbative vacuum of heterotic string theory and
then undergoes a phase of dilaton-driven inflation (DDI) described in sec. (3.3), joining
smoothly at later times standard RD cosmology. As already pointed out a key issue
confronting this scenario is whether can the graceful exit transition from DDI to RD
be completed. In particular, it has been showed that curvature can be bounded by an
algebraic fixed point behaviour when both H and φ̇ are constants and the universe is in a
linear-dilaton de Sitter space but it is clear that another general theoretical ingredient is
missing, and we now suggest that the generalized second law of thermodynamics (6.19) is
that missing ingredient.
We shall study numerically examples of PBB string cosmologies to verify that the
overall picture we suggest is valid in cases that can be analyzed explicitly. We first consider
α0 corrections to the lowest order string effective action,
Z
1
1
2
−φ
4 √
R + (∂φ) + Lα0 ,
d x −ge
(6.24)
S= 2
2
2κ4
where
L α0
1 µν
α0 1 2
4
2
µν
R + A (∂φ) + Dφ (∂φ) + C R − g R ∂µ φ∂ν φ ,
=
2 2 GB
2
(6.25)
with C = −(2A + 2D + 1), is the most general form of four derivative corrections that
leads to equations of motion with at most second (time) derivatives. Action (6.24) leads
108
6 Generalized second law of thermodynamics
to equations of motion
−3H 2 + φ̄˙ 2 − ρ̄ = 0 ,
σ̄ − 2Ḣ + 2H φ̄˙ = 0 ,
λ̄ − 3H − φ̄˙ 2 + 2φ̄¨ = 0 ,
2
(6.26a)
(6.26b)
(6.26c)
where ρ̄, λ̄, σ̄ are effective sources parameterizing the contribution of α 0 corrections [123].
Parameters A and D should have been determined by string theory, however, at the
moment, it is not possible to calculate them in general. If A, D were determined we could
just use the results and check whether their generic cosmological solutions are non-singular,
but since A, D are unavailable at the moment, we turn to GSL to restrict them.
First, we look at the initial stages of the evolution when the string coupling and H are
very small. We find that not all the values of the parameters A, D are allowed by GSL.
The condition σ̄ ≥ 0, which is equivalent to GSL on generic solutions at the very early
stage of the evolution, if the only relevant form of entropy is geometric entropy, leads to
the following condition on A, D
40.05A + 28.86D ≤ 7.253 .
(6.27)
The values of A, D which satisfy this inequality are labeled “allowed”, and the rest are
“forbidden”. In [123] a condition that α 0 corrections are such that solutions start to turn
towards a fixed point at the very early stages of their evolution was found by imposing in
eq. (5.5) that ρc > 0 leading to
61.1768A + 40.8475D ≤ 16.083 ,
(6.28)
and such solutions were labeled “turning the right way”. Both conditions are displayed
in fig. 6.1. They select almost the same region of (A, D) space, a gratifying result, GSL
“forbids” actions whose generic solutions are singular and do not reach a fixed point. We
further observe that generic solutions which “turn the wrong way” at the early stages of
their evolution continue their course in a way similar to the solution presented in fig. 6.2.
We find numerically that at a certain moment in time H starts to decrease, at that point
Ḣ = 0 and particle production effects are still extremely weak, and therefore (6.21) is the
relevant bound, but (6.21) is certainly violated.
We have scanned the (A, D) plane to check whether a generic solution that reaches
a fixed point respects GSL throughout the whole evolution, and conversely, whether a
generic solution obeying GSL evolves towards a fixed point. The results are shown in
fig. 6.1. Clearly, the “forbidden” region does not contain actions whose generic solutions
go to fixed points. Nevertheless, there are some (A, D) values located in the small wedges
near the bounding lines, for which the corresponding solutions always satisfy (6.21), but
do not reach a fixed point, and are singular. This happens because they meet a cusp
singularity.
Cusp singularities can be characterized as follows. As we have a system of first order
differential equation in H and φ̇, it can be represented as
Ḣ
b1 (H, φ̇)
,
(6.29)
=
A(H, φ̇)
b2 (H φ̇)
φ̈
109
6.4 Application to the pre-big bang scenario
10
D
6
2
−5
−3
−1
−2
1
3
A
5
−6
−10
Figure 6.1: Two lines, separating actions whose generic solutions “turn the right
way” at the early stages of evolution (red-dashed), and actions whose generic solutions
satisfy classical GSL while close to the (+) branch vacuum (blue-solid). The dots
represent (A, D) values whose generic solutions reach a fixed point, and are all in the
”allowed” region.
0.20
H
0.15
0.10
.
φ
0.05
0.2
0.4
0.6
0.8
1.0
Figure 6.2: Typical solution that “turns the wrong way”. The dashed line is the (+)
branch vacuum.
110
6 Generalized second law of thermodynamics
H
0.6
0.5
0.4
0.3
0.2
.
φ
0.1
−0.7 −0.5 −0.3 −0.1
0.1
0.3
0.5
Figure 6.3: Graceful exit enforced by GSL on generic solutions. The horizontal line
is bound (6.22) and the curve on the right is bound (6.21), shaded regions indicate
GSL violation.
where A is a field dependent 2 × 2 matrix. Cusp singularities are due to the vanishing of
the determinant of A. Consistency requires adding higher order α 0 corrections when cusp
singularities are approached, which we shall not attempt here.
If particle production effects are strong, the quantum part of GSL adds bound (6.22),
˙ plane, the region above a straight
which adds another “forbidden” region in the (H, φ̄)
line parallel to the φ̄˙ axis. The quantum part of GSL has therefore a significant impact on
corrections to the effective action. On a fixed point φ is still increasing, and therefore the
bounding line described by (6.22) is moving downwards, and when the critical line moves
below the fixed point, GSL is violated. This means that when a certain critical value of
the coupling eφ is reached, the solution can no longer stay on the fixed point, and it must
move away towards an exit, because of the effect of quantum corrections.
The full GSL therefore forces actions to have generic solutions that are non-singular,
classical GSL bounds dilaton kinetic energy and quantum GSL bounds H and therefore,
at a certain moment of the evolution Ḣ must vanish (at least asymptotically), and the
curvature is bounded. If cusp singularities are removed by adding higher order corrections,
as might be expected, we can apply GSL with similar conclusions also in this case. A
schematic graceful exit enforced by GSL is shown in fig. 6.3.
We conclude that the use of thermodynamics and entropy in string cosmology provides
model independent tools to analyze cosmological solutions which are not yet under full
theoretical control. Our result indicate that if we impose GSL, in addition to equations
of motion, non-singular string cosmology is quite generic.
7 Higgs-graviscalar mixing in
type I string theory
In this chapter we change subject leaving cosmological issues aside. We now investigate
a possible phenomenological consequence of interest for the standard model of particle
physics derived by admitting that the string theory extra dimensions are presently not
string scale-sized but bigger, as in the Antoniadis, Arkani-Hamed, Dimopulous and Dvali,
large extra dimensions scenario [124, 125]. This kind of scenario, exploits dramatically
the feature, possessed by perturbative type I string theory, that the string scale can be
separated from the Planck scale by even many order of magnitudes, as suggested by
(2.129).
In this framework we will study [126] the mixing between brane fluctuations, or branons
for short, and closed string modes, such as the graviton, graviphotons and the dilaton or
other graviscalars. Since branons are generically gauge non singlets, such a mixing can
arise from trilinear couplings of the form σ 2 h, involving two open and one closed string
modes that we denote σ and h, respectively. Upon identifying σ with the Standard Model
Higgs scalar, the above coupling induces a Higgs-graviscalar mixing proportional to the
Higgs vacuum expectation value (VEV). It has been suggested in a paper by Giudice,
Rattazzi and Wells [127] that this mixing leads to an invisible width of the Higgs that may
be observable experimentally. Indeed, since the Higgs is much heavier than the spacing of
the bulk Kaluza-Klein (KK) modes, it would feel a coupling to a quasi-continuous tower
of states, leading to a disappearance amplitude rather than to oscillations.
In the context of the effective field theory, the required trilinear coupling σ 2 h was
postulated to emerge from an Rσ 2 term, where R is the curvature scalar formed by the
pull-back metric on the D-brane world volume. Its coefficient ξ cannot be fixed by the
effective field theory and should be of order unity in order to obtain a visible effect.
However, in the conformal case, one obtains a small value, ξ = 1/6, dictated by the
conformally coupled scalar in four dimensions.
Here we study the branon-bulk mixing in type I string theory and we compute in
particular the trilinear coupling involving two open and one closed string states. Our
results are obtained in supersymmetric theories but remain valid in non supersymmetric
D-brane models, where supersymmetry is mainly broken only on the world-volume of some
D-branes, located for instance on top of anti-orientifold planes [34, 128]. More precisely,
there are three possibilities for the Higgs field that we analyse separately.
In the first case, the Higgs scalar is identified with an excitation of an open string
having both ends on the same collection of parallel D-branes (Dirichlet-Dirichlet or DD
open strings in the transverse directions). To lowest order, the effective action can then
111
112
7 Higgs-graviscalar mixing
be obtained by an appropriate truncation of an N = 4 supersymmetric theory. In the
Abelian case, it is given by the Born-Infeld action, depending on the pull-back of bulk
fields on the D-brane world volume. Expanding in normal coordinates one finds that
although no Rσ 2 term is strictly speaking generated, there is a quadratic coupling of
branons to the internal components of the Riemann tensor, ∂ m ∂n hµµ σ n σ m which induces
a Higgs-graviscalar mixing. The effect in the invisible width can be summarized in terms
of an effective parameter ξ which is of order unity in the case of δ = 2 large transverse
extra dimensions of (sub)millimeter size. The compatibility of this coupling with the
conformal symmetry of D3-branes can be explained by analyzing the explicit form of the
corresponding conformal transformations.
The second possibility is when the Higgs corresponds to an open string with one end
on the Standard Model branes and the other end on another D-brane extended in the bulk
(Neumann-Dirichlet or ND strings). In this case, the branon interactions do not emerge
from a Born-Infeld action but can be extracted directly by evaluating the corresponding
string amplitudes involving twist fields. An explicit computation of the 3-point function
shows that the branon coupling to the Riemann tensor now vanishes but it remains the
mixing with the graviphoton. As a result, the invisible width is much smaller than in the
previous case.
In the third case, the Higgs lives on a brane intersection, corresponding to an open
string stretched between two orthogonal D-branes transverse to the large dimensions (ND
string in non bulk directions). The Higgs-graviscalar mixing in this case vanishes.
In sec. 7.1 we briefly introduce the AADD large extra-dimension scenario, in sec. 7.2
we consider the first case where the Higgs is a DD state living on the brane and we derive
the coupling between branons and closed string modes by expanding the Born-Infeld and
Chern-Simons action [129], which will be extended to the non-Abelian case in sec. 7.3. In
sec. 7.4 we comment on the compatibility of the result obtained in the previous sections
with the conformal symmetry of the D3-brane effective action in the α 0 → 0 limit. In
sec. 7.5 the disappearance amplitude for the Higgs is computed, which is also analyzed
through a one-loop computation in sec. 7.6. In sec. 7.7 the analysis is extended to the
cases where the Higgs emerges as an excitation of a ND open string, stretched between
two orthogonal branes and in sec. 7.8 we conclude and comment about the detectability
of this effect.
7.1
The large extra dimension scenario
An interesting possibility to address the gauge hierarchy problem is when the string scale
lies well below the Planck scale, possibly in the TeV region [125, 130]. Starting from
relation (2.129), or from the Einstein-Hilbert action we see that an internal volume V sub
string-sized can give an effective lower dimensional Planck mass bigger than the string
scale. By T-duality this is equivalent to a super string-sized extra volume in the presence
of a brane orthogonal to the extra dimensions, as it can be realized at once by looking at
the dimensional reduction (truncation) of the Einstein-Hilbert action
Z
MPδ+2
l(4+δ)
16π
d
4+δ
MP2 l(4) Z
√
√
(4+δ)
x gR
→
d4 x g V R(4) .
16π
(7.1)
113
7.1 The large extra dimension scenario
By localizing the gauge charged matter to a lower dimensional sub-space an effective lower
dimensional Planck mass different from the higher dimensional one can be obtained. This
assumptions allow to trade the hierarchy problem, namely “Why is the Planck scale so
big compared to the electroweak scale?” for the problem “Why is the extra dimension
size stabilized to such a huge value?”, leaving the possibility of obtaining completely new
solutions to it. The scenario of large extra dimensions we consider is within the framework
of perturbative type I string theory with the Standard Model localized on a collection of
D-branes, in the bulk of δ extra large compact dimensions of size R (some other extra
dimensions might be string-sized). Standard Model degrees of freedom are described by
open strings ending on the D-branes, while gravity corresponds to closed strings that
propagate also in the bulk.
The relation between higher and lower dimensional Planck masses is given roughly by
MP2 l(4) ∼ (Rδ )δ MPδ+2
l(4+δ) ,
(7.2)
which leads, once the known value for the 4-dimensional Planck mass M P l(4) has been
inserted, to
δ+2
MP l(4+δ) δ
31/δ−17
.
(7.3)
Rδ ∼ 10
cm ×
1TeV
The existence of large extra dimensions is constrained firstly by the fact that the gravitational force follows the 1/r 2 law down to at least the millimiter range. So the more
interesting and still realistic case is the one with δ = 2, which allows a higher dimensional
scale in the TeV region and R ∼ mm.
Other model independent constraints comes from the cosistency on the neutrino flux
from supernova SN1987A with the cooling rate predicted by stellar collapse models,
putting an upper bound to the energy loss through emission of gravitons quantified by
[131]
R/(1mm)
< 10−5 ,
MP l(4+δ) /(1TeV)
δ = 2,
(7.4)
whereas for δ = 6 the stringest (model-independent) bound comes from the process
e+ e− → γ+ missing energy which gives the constraint, see for instance [132],
R/(1mm)
< 10−9 ,
MP l(4+δ) /(1TeV)
δ = 6.
(7.5)
For δ = 4 both physical effect give an analogous constraint R/M < 10 −8 mm/TeV.
Another bound is derived from the requirement that the energy lost by leakage of
matter into the extra dimensions is negligible with rescpect to the energy lost in the
standard cooling because of the cosmological expansion, condition which is fulfilled by the
Universe at temperature T roughly given by
!1/(δ+1)
MPδ+2
6δ−9
MP l(4+δ) (δ+2)/(δ+1)
l(4+δ)
δ+1
MeV ×
(7.6)
∼ 10
T <
MP l(4)
1TeV
which can be checked by comparing the energy lost by cooling dρ/dt ∝ T 6 /MP l with the
loss by leakage into extra dimensions dρ/dt ∝ T 7+δ /MP2+δ
.
l(4+δ)
114
7 Higgs-graviscalar mixing
7.2
Branons’ effective action
We consider the effective field theory on a single Dp-brane, i.e. with U (1) gauge group,
which is given by the sum of Born-Infeld (2.169) and Chern-Simons (2.170) actions which
we rewrite:
Z
r
SBI = −Tp d x e
− det G̃µν + B̃µν + 2πα0 Fµν ,
Z
X
0
SCS = µp dp+1 x eB̃+2πα F ∧
C̃ (p+1) ,
p+1
−Φ̃
(7.7)
(7.8)
p
where Fµν is the field strength of the Abelian world-volume gauge field, and T p , µp are
the tension and Ramond-Ramond (R-R) charge of the Dp-brane. The closed string fields
are the pull-back of the bulk fields to the D-brane world volume:
G̃µν = Gµν + Gmµ ∂ν σ m + Gmν ∂µ σ m + Gmn ∂µ σ m ∂ν σ n ,
m
m
m
(7.9a)
n
B̃µν = Bµν + Bmµ ∂ν σ − Bmν ∂µ σ + Bmn ∂µ σ ∂ν σ ,
(7.9b)
Φ̃ = Φ ,
C̃µ(p+1)
0 ...µp
=
Cµ(p+1)
0 ...µp
+ ∂ µ0 σ
m
(p+1)
Cmµ
1 ...µp
m
+ ∂ µ0 σ ∂ µ1 σ
n
(p+1)
Cmnµ
2 ...µp
(7.9c)
,
(7.9d)
where G, B, Φ and C are the metric, two index antisymmetric tensor, dilaton and the
R-R (p + 1)-form potential, respectively. Here, we define the transverse coordinates of
the brane as our σ fields and an implicit antisymmetrization over indices µ 0 , µ1 , . . . , µp in
(7.9d) is understood.
Recasting the Born-Infeld action (7.7) into the Einstein frame the following action is
obtained
(E)
SBI
= −Tp
Z
d
p+1
xe
p−3
Φ̃
4
r
− det g̃µν + e−Φ̃/2 B̃µν + 2πα0 e−Φ̃/2 Fµν .
(7.10)
No rescaling is needed for the Chern-Simons action as it is metric independent.
(E)
Expanding SBI + SCS around a flat Minkowski space,
gM N = η M N + h M N ,
(7.11a)
BM N = b M N ,
(7.11b)
Φ=φ,
(7.11c)
115
7.2 Branons’ effective action
one obtains [129]
(p − 3)
µ0 ...µp
1 µ
L1 = −Tp
φ + hµ ± µp Cµ(p+1)
,
(7.12)
...µ
p
0
4
2
(p + 1)!
µ
p−3 m
0
µν
m µ
m ∂m hµ
+
σ ∂m φ + πα bµν F
L2 = −Tp ∂µ σ h m + σ
2
4
(7.13)
µ0 ...µp
0
(p−1)
m
(p+1)
m
(p+1)
,
+ πα F ∧ C
±µp σ ∂m Cµ0 ...µp + (p + 1)σ ∂µ0 Cmµ1 ...µp
(p + 1)!
1 h
(∂ρ Aµ ∂ ρ Aν + ∂µ Aρ ∂ν Aρ − 2∂µ Aρ ∂ρ Aν ) hµν −
L(3,N S 2 ) = 2
2gY M
µ
ρ σ
σ
ρ hµ
ρ σ
σ
ρ p−7
(∂ρ Aσ ∂ A − ∂ρ A ∂σ A )
− (∂ρ Aσ ∂ A − ∂ρ A ∂σ A )
φ
2
4
Tp
1
∂µ σm ∂ν σ m − ηµν ∂ρ σm ∂ ρ σ m hµν − hmn ∂ρ σ m ∂ ρ σ n −
+
(7.14)
2
2
hµµ
p−3
p−3
m n
ρ m
φ − σ σ ∂m ∂n
−
φ+
∂ σ ∂ρ σm
4
4
2
i
2 (∂µ σ n ) σ m ∂m hµn − 2πα0 (2bµm F µν ∂ν σ m + σ m Fµν ∂m bµν ) ,
1 m n
(p+1)
L(3,R2 ) = ±µp
+
+ (p + 1)σ m ∂µ0 σ n ∂m Cnµ
σ σ ∂m ∂n Cµ(p+1)
1 ...µp
0 ...µp
2
(p + 1)p
(p + 1)p
(p+1)
∂µ0 σ m ∂µ1 σ n Cmnµ
+
(2πα0 )Fµ0 µ1 σ m ∂m Cµ(p−1)
+
2 ...µp
2 ...µp
2
2
(7.15)
(p + 1)p(p − 1)
(p−1)
+
(2πα0 )Fµ0 µ1 ∂µ2 σ m Cmµ
3 ...µp
2
µ0 ...µp
(p + 1)p(p − 1)(p − 2)
(p−3)
0 2
(2πα ) Fµ0 µ1 Fµ2 µ3 Cµ4 ...µp
,
8
(p + 1)!
where the ± signs correspond to the two choices of the D-branes R-R charge (branes or
anti-branes) and µ0 ...µp is the usual antisimmetric tensor density.
The non kinetic terms in the above expressions (with no spacetime derivative on σ)
are obtained by retaining the terms up to quadratic level of the Taylor expansion
∞
X
(σ m ∂ym )k k=1
k!
e
p−3
Φ
4
√
g ∓ C (p+1) (y n )
ym =0
.
(7.16)
This shows that the branons experience a non derivative interaction in a nontrivial background, which can be interpreted as a potential V br for the position of the brane
p−3 √
(7.17)
Vbr ≡ Tp e 4 Φ g ∓ C (p+1) .
We expect that in a supersymmetric background the Neveu-Schwarz Neveu-Schwarz (NSNS) and the Ramond Ramond (R-R) fields give mutually compensating contribution to
the potential term: we shall check this fact in sec. 7.4, using the supergravity description
of branes.
Let us consider now the trilinear Lagrangian (7.14). It corresponds to the closed string
116
7 Higgs-graviscalar mixing
linearization of the following non linear Lagrangian, quadratic in NS open string modes:
LN S = −e
√
g
Tp
Fµν Fρσ g µρ g νσ +
(∇µ σ m ∇ν σ n g µν gmn −
2
2
4gY M
i
σ m σ n Rµmµn + 2πα0 (2bµm Fµν ∂ ν σ m + Fµν σ m ∂m bµν )) ,
p−3
Φ
4
1
(7.18)
where we introduced the (gravitational) covariant derivative over σ fields
n
∇µ σ m = ∂ µ σ m + Γ m
nµ σ .
(7.19)
The gravitational connection is given by
Γm
nµ =
1 mM
g
(gnM,µ + gM µ,n − gnµ,M ) ,
2
(7.20)
where column denotes differentiation as usual.
We thus found, besides the expected Yang-Mills kinetic terms, a potential of interaction
between branons and the bulk closed string states. Note that the potential term in (7.18)
vanishes in a trivial background; it generates interactions of σ m with higher KK modes
of the bulk fields. The above results can be also obtained by a direct computation of
corresponding on shell string amplitudes.
7.3
Non-Abelian generalization
In the non-Abelian case, we cannot rely on the Born-Infeld action to obtain the effective
field theory. Instead, one can compute the relevant string amplitudes. The 0 and 1-closed
string amplitudes have been treated in detail in sec. 2.6 and 2.10.1.
7.3.1
One open-one closed string on the disk
The N S vertex operator in the (0)-picture is 1
(0)
VN S (v, k) = go ta 2α0
−1/2
a
iẊ M + 2α0 k · ψψ M eik·X .
vM
(7.21)
Contracting it with the closed string vertex (2.181), using correlators (2.183) and the first
of (2.55), we obtain the amplitude
AN S,N S 2
gc
=
go
2
α0
(k2N ηRM − DN R k2M ) v
1/2
aR M N
0
2
Tr(ta )2−α mKK ×
1
ar M N
+ k1r DM N σ ,
2
(7.22)
1
To be precise the vertex operator involve for ∂t X for a NN coordinate and ∂n X for a DD one, where
∂t (∂n ) denotes derivative with respect to the tangent (normal) direction to world-sheet boundary. For the
sake of a just computation it is sufficient to derive XL,R with respect to its argument and to keep account
of the correlators (2.26).
117
7.3 Non-Abelian generalization
where k1(2) is the closed(open) string momentum and m KK is the Kaluza-Klein mass of
the closed string particle
mKK = k1m k1m = −k1µ k1µ ,
(7.23)
and the matrix DM N is defined as in (2.184).The on-shell conditions are
k1µ + k2µ = 0 ,
k12 = k22 = 0 ,
(7.24)
which for massless σ implies vanishing m KK for the closed string particle. Restricting
(7.22) to irreducible representations of the brane Lorentz group we obtain
gc 2 1/2 −α0 m2
a mν
KK Tr(ta ) 2ik
2
+ σ m ik1m hµµ ,
2ν σm h
0
go α
gc 1 (p − 3) −α0 m2
KK Tr(ta ) [σ m ik
A(φ, σ) = −i
2
1m φ] ,
go α0 1/2
2
2gc 2 1/2
0 2
A(b, A) = i
Tr(ta )2−α mKK [ik2µ Aaν bµν ] .
0
go α
A(h, σ) = −i
(7.25a)
(7.25b)
(7.25c)
The amplitudes (7.25) vanish on shell, as in this case k 2µ = −k1µ and k1m = 0 and then
σ m k1ν hνm = σ m kM hM
m = 0 because of the on shell conditions (2.182).
Let us consider now the interesting case of two branes (which in the oriented theory
correspond to a U (2) gauge group) with the presence of a Wilson line in the direction X µ̄ ,
say, proportional to t3 . The Wilson line can be parametrized by
W (lµ̄ ) = eilµ̄
H
µ̄
σ X ⊗t3
dσ ∂2πα
0
(7.26)
and charged states under a Wilson line have momenta k µ̄ which are shifted according to
k µ̄ → k µ̄ +
ql
,
2πα0
being q the charge of the state. For instance in the case of a U (2) gauge group, the states
corresponding to the Chan-Paton factors τ ± ≡ t1 ± it2 have charges q = ±1 according to
[t3 , τ ± ] = ±τ ± .
(7.27)
If the direction µ̄ is compact with radius R 0 the Kaluza-Klein index along µ̄ is n +
qlR0 /(2πα0 ), thus the state charged under t3 acquire mass, whereas the state corresponding
to the Chan-Paton factor 11 (the identity) and t 3 stay massless. Performing a T-duality
transformation along X µ̄ the Wilson line is traded with an interbrane separation and it is
now parametrized by the T-dual of (7.26)
WT (lm̄ ) = expilm̄
H
m̄
τ X ⊗t3
dσ ∂2πα
0
.
(7.28)
Charged state under WT have shifted windings w → w + ql/(2πR) (with R = α 0 /R0 the
dual radius) as the geometrical picture of two branes separated by a distance l suggests.
118
7 Higgs-graviscalar mixing
In the presence of a brane separation background the disk amplitude must be weighted by
WT according to [133]
H
∂τ X
3
h. . .iWT = h. . . eil dσ 2πα0 ⊗t i .
If we are dealing with states that have a Kaluza-Klein momentum index n the integral in
(7.28) is
I
ilm̄ nm̄
∂τ X m̄
=
.
(7.29)
ilm̄ dσ
2πα0
R
The amplitude (7.22) is modified being now weighted by
lm̄ nm̄
lm̄ nm̄
Tr (ta ) cos
+ iTr ta t3 sin
R
R
(7.30)
and the σ’s with Chan-Paton factor t 1,2 , or equivalently τ ± acquire mass
m2σ =
lm̄ lm̄
.
4π 2 α0 2
(7.31)
We see that the corrections vanishes for branons with Chan-Paton factors t 1,2 and in
the case the ones with Chan-Paton factor 11 and t 3 stay massless the correction (7.30) is
trivial2 .
We now turn to the RR case. Contracting an open string vertex operator in the
(−1)-picture
(−1)
a −ϕ M ik·X
VN S (v, k) = vM
e ψ e
(7.32)
with the R-R vertex operator (2.190) in the (−1/2, −1/2)-picture, the following NS,R-R
amplitude is obtained
AN S,R2 =
igc
Tr(ta )he−φ e−φ/2 e−φ̃/2 ihψ N S CΓM0 ...Mp+1 S̃i ×
0
α go
a
heik2 X eik1 XL eik1 XR ivN
FM0 ...Mp+1 =
a
gc
i√
FM0 ...Mp+1 ,
Tr(ta )Tr CΓM0 ...Mp+1 (C −1 ΓN M T )T vN
2α0 go
(7.33)
where M is defined as in (2.186) and the correlators
hψ M (z1 )Sα (z2 )S̃β̇ (z̄2 )i = (2z12 z12̄ )−1/2 (z22̄ )−3/4 Cγ M
hψ M (z1 )Sα (z2 )S̃β (z̄2 )i = (2z12 z12̄ )−1/2 (z22̄ )−3/4 Cγ M
αγ
Mβ̇γ
p even (7.34a)
αγ
M βγ
p odd , (7.34b)
and the first of (2.55) have been used. Using the gamma matrix identities
ΓM ΓM1 ...Mn = ΓM M1 ...Mn + ng M [M1 ΓM2 ...Mn ] ,
ΓM1 ...Mn ΓM = ΓM1 ...Mn M + (−1)n+1 ng M [M1 ΓM2 ...Mn ] ,
2
(7.35)
To give a mass to the σ with t3 Chan-Paton factor an additional Wilson line, not commuting with t3 ,
should be turned on, but this will break supersymmetry invalidating our string construction.
119
7.3 Non-Abelian generalization
and specializing to different polarization vectors we obtain
igc
m
(p+1)
m (p+1)
√ σ ik1m Cµ0 ...µp + (p + 1)ik2µ0 σ Cmµ1 ...µp ,
A(C, σ) = ±
α0 go 2
igc
√ p(p + 1)ik2µ0 Aµ1 Cµ(p−1)
,
A(C, A) = ±
2 ...µp
α0 go 2
(7.36a)
(7.36b)
where antisymmetrization over the greek indices µ 0 . . . µp is understood and the correpondence between ±’s in the amplitude and in the defnition of M is again given by (2.193).
After the rescalings (2.194), (2.195) and
√
gY M = go / 2α0 ,
√ σ → σ/ πgo 2α0 ,
√
Aµ → Aµ 2α0 /go = Aµ /gY M ,
bM N → −bM N /(2κ) ,
(7.37)
(7.38a)
(7.38b)
(7.38c)
we obtain that these amplitudes are in agreement in the α 0 → 0 limit with (7.13) in the
Abelian case.
7.3.2
Two open-one closed string amplitude
The 3-point amplitude with two open strings and one closed string involve one kinematical
invariant variable t, which is given in terms of the open string momenta k 2 and k3 and of
the momentum k1 of the closed string by
(k2 + k3 )2 = 2k2 · k3 = −t ,
k1µ · k1µ = −k1m · k1m = −t ,
2
(7.39)
2
(k2 + k1 ) = (k3 + k1 ) = t ,
where the last product is understood to be over the full 10-dimensional space,
such that
√
k2,3 have non-vanishing components along the brane directions only, and −t is the KK
mass of the closed string particle. The string amplitude calculation for two open string
excitations and one closed NS-NS particle gives the following amplitude
AN S,N S−N S = −igc Tr(ta tb ) ηM N DRS /(2α0 ) − ηM N DRS k1 Dk1 /4+
0
0 Γ(1/2)Γ(−α t/2 − 1/2)
Dk1R Dk1S ηM N /4] 2−α t
+
Γ(−α0 t/2)
[2DN S k2R Dk1M − 2DN S k3R Dk1N − DN S k1M Dk1R − 2ηM R k2S k1N +
(7.40)
2ηN R k2S k1M − ηM R Dk1N Dk1S + ηM R DN S (k1 Dk1 ) + 2DRS k1M Dk1N +
0
−α0 t Γ(1/2)Γ(−α t/2 + 1/2)
2k2R k3S ηM N ] 2
+ (1, M ) ↔ (2, N ) v aM v bN RS .
Γ(−α0 t/2 + 1)
120
7 Higgs-graviscalar mixing
The specific amplitudes in the α0 → 0 limit are
b hµν − ik σ m ik µ σ hν −
A(h, σ, σ) = igc πTr(ta tb ) 2ik2µ σ am ik3ν σm
2µ
3 m ν
(7.41a)
2ik2µ σ am ik3µ σ an hmn − 4ik2µ σ an σ am ik1m hµn − σ am σ an ik1m ik1n hµµ ] + (2 ↔ 3)
i 3−p
h
b
b
A(φ, σ, σ) = igc πTr(ta tb ) ik2 σ am ik3 σm
+ σ am σm
ik1m ik1n × √ φ + (2 ↔ 3) (7.41b)
2 2
A(h, A, A) = 2igc πTr(ta tb ) ik2µ Aaα ik3ν Abα + ik2α Aaµ ik3α Abν −
i
(7.41c)
2ik2µ Aaα ik3α Abν − 21 ηµν (ik2α Aaβ ik3α Abβ − ik2α Aaβ ik3β Abα ) hµν + (2 ↔ 3)
i 7−p
h
A(φ, A, A) = igc πTr(ta tb ) ik2µ Aaν ik3µ Abν − ik2µ Aaν ik3ν Abµ × √ φ + (2 ↔ 3)(7.41d)
2 2
A(b, σ, A) = 2igc πTr(ta tb ) 2 ik2µ Aaν − ik2ν Aaµ ik3ν σ bm bµm +
(7.41e)
σ am ik2µ Abν − ik2ν Abµ ik1m bµν ,
√
where we used Γ(−1/2) = −2Γ(1/2) = −2 π and the limit for α0 t → 0
2
Γ(−t/2) → − .
t
After the rescaling (2.194), (2.195) and (7.38) the previous amplitudes are reproduced by
the expansions (7.14).
Using, in addition to the ones already shown, the correlator [134]
(i)
J RM
hψ ψ (z1 )ψ (z2 )Sα (z3 )S̃β (z̄3 )i =
hψ N (z2 )Sα (z3 )S̃β (z̄3 )i =
z1 − z i
i=2,3,3̄



0
0 0
RM β δ α0
RM α δ β
RN η M − η M N η R
N
Γ
Γ
0
η
α
δ
β
β
α
P
P α0 β

 ×
δα δβ + P 
+
(7.42)
z1 − z 2
2
z1 − z 3
z1 − z̄3
R
M
X
N
hψ P (z2 )Sα0 (z3 )S̃β 0 (z̄3 )i ,
(i)
where J N R is the representation over the field at z i of the Lorentz generator ψ R ψ M ,
we can compute the amplitude between two NS open string excitations and a R-R closed
string one, which is
Γ(1/2)Γ(−α0 t/2 + 1/2)
vN FM0 ...Mn ×
Γ(−α0 t/2 + 1)
2α0
2ik1m Tr[Cγ M0 ...Mn M γ N ]v m + i(k3ρ − k2ρ )Tr[Cγ M0 ...Mn M γ ρM N ]vM ,
A2N S,R2 =
igc
0
Tr(ta tb )2−α t
1/2
(7.43)
121
7.4 Conformal invariance
which reduces to in the limit α0 → 0
h
8igc π
A(C, σ, σ) = ± 1/2 Tr(ta tb ) σ m σ n ik1m ik1n Cµ(p+1)
+
0 ...µp
α0
2(p + 1)σ
am
ik3µ0 σ
bn
(p+1)
ik1m Cnµ
1 ...µp
+ p(p + 1)ik2µ0 σ
am
ik3µ1 σ
an
(p+1)
Cmnµ
2 ...µp
8igc π
A(C, A, A) = ± 0 Tr(ta tb )×
α
h
i
(p + 1)p(p − 1)(p − 2)ik2µ0 Aaµ1 ik3µ2 Abµ3 Cµ(p−3)
+ (2 ↔ 3) ,
4 ...µp
h
16igc π
A(C, A, σ) = ± 1/2 Tr(ta tb )
p(p + 1)σ am ik2µ0 Abµ1 ik1m Cµ(p−1)
2 ...µp
α0
i
(p−1)
(p + 1)p(p − 1)ik2µ0 Aaµ1 ik3µ2 σ m Cmµ
,
3 ...µp
i
(7.44a)
+ (2 ↔ 3) ,
(7.44b)
(7.44c)
where again the matching between the sign choice in (7.44)’s and (2.186) is set by (2.193)
and antisymmetrization over the µ’s indexes is understood. After the usual redefinitions
these amplitudes can be derived by (7.15), generalized to non trivial Chan-Paton factors.
We remember moreover that in a presence of a Wilson line W T (li ) the amplitude should
be weighted by a factor
Tr ta tb cos (lm nm /R) + iTr(ta tb t3 ) sin (lm nm /R) .
(7.45)
Finally we note the range of validity of our computation: we took the limit α 0 t → 0 to
neglect higher derivative interaction and as we considered tree level amplitudes we can
trust our result only if eΦ 1.
7.4
Conformal invariance
It is known that the gauge field theory on a D3-brane is N = 4 supersymmetric, which is
conformal invariant. It is also known that conformally coupled scalar fields in 4 dimensions
exhibit a ξRσ 2 interaction with ξ = 1/6 and R the 4-dimensional Ricci scalar. One may
be worried why the calculations exposed so far do not show the expected ξRσ 2 coupling
for p = 3, which is also a source of Higgs-graviscalar mixing. In fact we shall argue below
that the conformal symmetry is realized in a rather different way. Moreover in sec. 7.5
we shall show that the potential interaction we obtained in Eq. (7.18) gives rise still to a
disappearance amplitude for the Higgs that can be parametrized in terms of an effective
ξ that we shall compute.
Looking back to (7.18), we can check that conformal invariance of the gauge fields
for p = 3 (and no other values of p) is obtained in the usual way, using a conformal
transformation on the brane
gµν → ĝµν = Ω2 gµν ,
(7.46)
with the dilaton Φ and the gauge field A µ inert. For the scalar branons the situation is
different as they also couple to graviphotons and graviscalars. Here we show that conformal
invariance on a flat 3-brane is achieved if (7.46) is supplemented by
gmn → ĝmn = Ω−2 gmn ,
(7.47)
122
7 Higgs-graviscalar mixing
with the branons unaltered. Indeed, it is easy to see that (7.46) and (7.47) applied together
correspond to a conformal symmetry of the action derived from (7.18) for p = 3 in a trivial
background. Moreover in the presence of a R-R field, applying the transformations
Cµ0 ...µp → Ĉµ0 ...µp = Ω4 Cµ0 ...µp ,
2
Cmµ1 ...µp → Ĉmµ1 ...µp = Ω Cmµ1 ...µp ,
Cmnµ2 ...µp → Ĉmnµ2 ...µp = Cmnµ2 ...µp ,
(7.48a)
(7.48b)
(7.48c)
with gµm inert, one can show that the conformal symmetry is exact provided the back√
ground is chosen so that the potential (7.17) vanishes and that ghµm0 = Cmµ1 ...µp µ0 ...µp /p!.
Here, we dropped for simplicity the (p + 1) superscript on the R-R form C (p+1) .
The effect of the ξRσ 2 term is thus replaced by other interaction which at the quadratic
level becomes σ m σ n Rµmµn . In relation to these effects, one may wonder about the argument in [135], where the N = 4 super Yang-Mills theory was considered on S 4 rather then
on R4 and it was claimed that the flat direction of σ is lifted by the curvature coupling
Rσ 2 , thus making the path integral to converge, at least if the metric is close enough
√
to the one of a four sphere, which has R > 0. In our case the (σ m ∂m )k ( g + C) or its
quadratic expansion may equally well do the job for the case of a four sphere embedded
in a higher dimensional spacetime. It would be interesting to check this explicitly.
Amusingly enough, there is at least one case in which the potential (7.17) vanishes
in a non-trivial way, thus preserving the conformal symmetry. It corresponds to the
supergravity background induced by some parallel Dp-branes. The background is given,
in the string frame, for p < 7, by [136]
1
1
ds2 = H − 2 (dxµ dxµ ) + H 2 (dy m dym ) ,
eΦ = H
3−p
4
,
(7.49)
Cµ0 ...µp = µ0 ...µp H −1 ,
Q
1
H =1+
,
7 − p r 7−p
√
where r ≡ y m ym . The solution (7.49) depends on the parameter Q, with dimensions of
[length]7−p , defined by
Z
Z
(p+1)
d ∗ dC
=
∗dC = S8−p Q ,
(7.50)
⊥
∂⊥
where S8−p is the volume of the (8 − p)-dimensional sphere of unit radius. The classical
parameter Q is related to the microscopic string parameter µ p by3
Q = N µp 2κ2 /S8−p ,
3
This is derived using the terms in the supergravity action that involve the R-R form
Z
Z
1
d10 x(dC)2 − µp
C.
S=− 2
4κ
branes
Thus, the equation of motion is
Z
which, compared to (7.50), gives (7.51).
⊥
d ∗ dC = 2κ2
X
branes
µp ,
(7.51)
123
7.5 Higgs-graviscalar mixing
where the integer N counts the number of D-branes and the classical limit is recovered at
N → ∞. Using that µp ∝ go−2 , κ ∝ gc and that go2 ∝ gc ∝ ehΦi ≡ gs we have also
Q ∼ N gs (2πα0 )
7−p
2
.
(7.52)
In this supersymmetric background the potential felt by a stuck of N 0 test Dp-branes
with tension and charge Tptest (assuming N N 0 so that the test branes alter negligibly
the background they are plunged into) is
Vbr = Tp H −1 − µp H −1 = 0
(7.53)
where we used the BPS relation Tp = µp . We also note that in this case the conformal transformations (7.46), (7.47) and (7.48) can be described at once by defining the
conformal transformation of the function H
H(r) → Ĥ(r) = Ω−4 H(r) .
(7.54)
If the background is non supersymmetric and cancellation (7.53) does not hold (as in the
case of a test antibrane) conformal invariance is broken.
Finally, we check the limit of validity of the supergravity solution (7.49). We should
have α0 R 1 for the curvature scale R, and the weak coupling condition e Φ 1. In fact,
on the background (7.49) we have
(
3−p
r→0
H 02
−→ Q−1/2 r 2
R ∼ 5/2 r→∞
2
H
−→ Q/r 8−p

3−p (7−p)(p−3)
 r→0
4
Q
3−p
4
−→ 7−p
r
eΦ = H 4
r→∞
 −→
1 + 3−p Q
4(7−p) r 7−p
Thus, in the r → ∞ limit the curvature vanishes and the coupling is bounded for every p,
whereas in the r → 0 limit curvature (coupling) blows up for p > 3 (p < 3). However, in
the p ≤ 3 case, the curvature is bounded by R max
2
Rmax ∝ Q− 7−p
(7.55)
and thus, for p ≤ 3, α0 corrections can be taken under control for any value of r by sending
Q → ∞ in a way that α0 Q−2/(7−p) → 0 (for p = 3, the AdS5 × S5 geometry is obtained
in this way). If we plonge a brane into a nontrivial general background, relation (7.53)
generally won’t hold and a potential for the position of the brane will be generated. 4
Hence, everything appears to be consistent even in the absence of a ξRσ 2 term.
7.5
Higgs-graviscalar mixing
We shall now show how in our scenario a mixing may take place between branons and a
graviscalar. The mixing is triggered by the trilinear coupling σ 2 h in (7.14) if σ acquires an
4
We also note that the condition that the curvature induced by a brane is small is also relevant for the
consistency of the string computations previously exposed, which require a flat background to be reliable.
124
7 Higgs-graviscalar mixing
expectation value [127]. Before we analyse the mixing, we discuss first the Abelian case of
a single brane, where the graviphoton absorbs the branon and acquires a (localized) mass.
For this purpose we need the expansion of the Born-Infeld action (7.10) at the quadratic
level of the NS-NS closed string modes:
#
"
1 p−3 2 2 p−3 µ 1
1 µ 2 1
µν
µν
φhµ + bµν b
h
φ +
, (7.56)
LN S 2 = −Tp
− hµν h +
8 µ
4
2
4
8
4
which can also be checked by computing the relative string scattering amplitudes [137].
It might have been expected the appearance of a mass-term for the graviphoton, as
the presence of the brane breaks translational invariance. The graviphoton indeed becomes massive and eats the U (1) part of the branons, but this is not manifest with the
parametrization of the metric that we used, g M N = ηM N + hM N . Actually, with this
parametrization, the field hµm is not the graviphoton unless one is restricted to the lowest
order approximation. The graviphoton V µ m is defined by parametrizing the metric in the
following way
(10)
ds2 = gM N dxM dxN = gµν dxµ dxν + gmn dxm + Vµ m dxµ (dxn + Vν n dxν ) ,
or equivalently
gM N =
gµν + gmn Vµm Vνn Vµn
Vmν
gmn
.
(7.57)
Then Vµ m can be identified with the graviphoton since the ten dimensional coordinate
transformation
xm → xm0 = xm + ξ m
becomes equivalent to the gauge transformation
Vµ m → V µ m 0 = V µ m + ∂ µ ξ m .
The resulting bulk kinetic terms for the dimensionally reduced theory, omitting the terms
involving graviscalars and dilaton, is
µ νn
1 p
1
ν µn
m
m
(p+1)
(7.58)
Lbulk = 2 |g| R
− gmn ∂µ Vν − ∂ν Vµ (∂ V − ∂ V ) .
2κ
4
Expanding the Born-Infeld action (7.10) over the metric (7.57) we obtain, up to quadratic order in the fields
" 2 1
√
1
p−3
p−3 2
1
0
L2N S 2 = −Tp g
hµµ +
hµµ +
φ +
Vµm + ∂µ σ m +
φ −
2
2
2
8
2
(7.59)
2
1 i
p−3
1
1
µν
µ
0
hµν h + σ ∂i hµ +
φ +
bµν + 2πα Fµν
,
4
2
2
4
where we arranged the terms involving the branons and the graviphotons in a perfect
square, showing that for each m the U (1) branon is eaten by the corresponding graviphoton
125
7.5 Higgs-graviscalar mixing
which becomes massive [138]. Its mass m gp is given by5
m2gp =
16πTp
,
(MP l )p−1
(7.60)
where MP l is the lower dimensional Planck mass on the p-brane. This eating mechanism
is T-dual of the mechanism which makes the antisymmetric tensor b µν massive by eating
the U (1) world-volume gauge field Aµ [139], triggered by the last term in (7.59). In
fact, a massless two-index antisymmetric tensor in (p + 1) dimensions has 21 (p − 1)(p − 2)
components and it absorbs the (p − 1) components of a gauge field through the last term
of (7.59) to make a massive antisymmetric tensor field with 12 p(p − 1) components.
The terms quadratic in hµµ and hµν are due to the cosmological constant term associated
to the brane tension. The additional interaction σ i ∂i (hµµ +(p−3)φ/2) can be interpreted as
a mixing between the longitudinal mode of the graviphoton (involving only the branon with
the identity 11 Chan-Paton factor) and the Kaluza-Klein excitations of the zero helicity
part of the graviton and dilaton. Using canonically normalized σ (the normalization of h
is fixed by (2.172a)), this mixing is given by
p−1
p−3
1
i
µ
0
2
φ
(7.61)
mgp MP l σ ∂i hµ +
Lmix = − √
2
2 16π
times possibly the factor (7.30) in the case a brane separation is present. We note that
there is also a similar mixing with excitations of the R-R sector, whose amplitude is equal
in magnitude and opposite in sign to the previous one. The equality of the magnitude
of the NS-NS and R-R mixing contributions is not surprising, as unitarity relates these
mixing amplitudes to the imaginary part of the one loop two branon point-function, which
must vanish in a supersymmetric background, as it will be explicitly checked in sec. 7.6.
In the remaining part of this section we shall focus on the NS-NS sector.
The mixing (7.61) vanishes on-shell unless the σ m is massive for some direction m̄ and
involves only the 11 Chan-Paton σ field. We then assume that σ m̄ 3 acquires a mass mσ
and we turn on a T-dual Wilson line (an interbrane separation) along the m-th direction
proportional to t3 , who has the effect of multiplying the mixing term (7.61) by (7.30).
Using then the propagators (2.172) to obtain
p−3 2
64π
1
hµµ hνν +
.
(7.62)
φφ = p−1 2
2
k
+
m2KK
MP l
In this case with mσ 6= 0 and with a Wilson line turned on also the term σ m ∂µ hµm will give
a contribution to the mixing. Using the additional propagators, derived from (2.172a),
µ ν
hm̄
hn̄ =
g µν gm̄n̄
,
k 2 + m2KK
MPp−1
l
16π
ν = 0,
hµµ hm̄
(7.63)
(7.64)
and assuming that mσ 1/R so that the branons is not able to resolve the discreteness
of the Kaluza-Klein spectrum, we have
Z
2 + m2
1 2
dδ k km̄
σ
2
Σ(p ) = mgp Vlar
(7.65)
2
(2π)δ p2 + k 2 + i
5
In our analysis we implicitly assumed that the Kaluza-Klein scale 1/R mgp otherwise bulk terms
may induce mixing and mass terms of comparable strength to (7.60).
126
7 Higgs-graviscalar mixing
where δ is the number of large extra dimension and V lar their volume and we have averaged
to 1/2 the squared sinus function. Σ contains the contribution from the insertion of KK
modes in the branon propagator, which reads:
Gσ (p2 ) = −
p2
+
m2σ
1
.
+ Σ(p2 ) + i
(7.66)
The imaginary part of the self-energy Σ above is related to the decay amplitude Γ of the
branon. Using
1
= πδ(x) ,
lim Im
→0
x + i
the type I relation for the theory on a p + 1 brane in 10 dimensions [140] (which will be
justified in sec. 7.6)
MPp−1
l =
2
M 5−p V̄ (2π)p−3 ,
α2Y M s
(7.67)
with V̄ the reduced volume defined by
V̄ ≡ Ms9−p Π9i=p+1 Ri ,
and Ms the string scale, and letting the branon a mass m σ we finally obtain
Γ=
πb(δ) 2 δ−1
1
Im Σ(p2 = m2σ ) =
m m Sδ−1 Vlar /(2π)δ =
mσ
8 gp σ
4πα2Y M πb(δ) Tp
mσ δ−1
Sδ−1 ,
(2π)p−3 4 Ms6−p Ms
(7.68)
where Sδ−1 is the volume of the (δ − 1)-dimensional sphere (we note that α Y M has
dimensions (mass)3−p ) and b(δ) ≡ 1 + 1/δ. This decay amplitude depends crucially on the
brane tension, which we can assume to be at the TeV scale so that m gp ∼ 10−4 eV .
Actually to be able to make a more definite prediction, instead of considering the
Lagrangian (7.61) with a Wilson line term (7.30), we can start from the trilinear coupling
we found in secs. 7.2-3
p−3
1 m n
µ
φ − (∂µ σ n ) σ m ∂m hµn
(7.69)
L = − σ σ ∂m ∂n hµ +
4
2
and assuming that one of the branons gets a mass m σ and a non-vanishing VEV v, we
substitute
σ m̄ = v + ρm̄
(7.70)
and obtain the mixing term
Lmix
1 m̄ 2
p−3
µ
φ + vρm̄ ∂µ ∂m̄ hµm̄ .
= − vρ ∂m̄ hµ +
2
2
(7.71)
In principle this term is not the full story but just the first term in an expansion in power
of σ · ∂ giving terms roughly like
X
(σ · ∂)n h
Lσh = MPp+1
(7.72)
a
n
l
n(p+1)/2
MP l
n
127
7.5 Higgs-graviscalar mixing
for some numerical coefficient an . Substituting (7.70) in the previous expression the total
mixing term would be obtained. The modification induced by the Wilson line to the
two-point amplitude keeps account of this resummation, but we can safely guess that for
v/Ms < 1 the first term of the series, which is the above (7.71), does not fail in giving the
right order of magnitude of the effect.
Using then contractions (7.62) and (7.63) and again assuming that m σ 1/R (large
extra-dimensions), we obtain the following correction to the branon self-energy from (7.71)
Σ(p2 ) = v 2
16πV
MPp−1
l
Z
4 + k2 k2
dδ k km̄
m̄
,
δ
2
(2π) p + k 2 + i
(7.73)
where δ is the number of large extra dimensions and V their volume. Using then (7.67)
we have
4πα2Y M πa(δ) mσ v 2
1
Γ=
Im Σ(p2 = m2σ ) =
mσ
(2π)p−3 2 Ms5−p
mσ
Ms
δ
Sδ−1 ,
(7.74)
where Sδ−1 is the volume of the (δ − 1)-dimensional sphere of unit radius and we defined
a(δ) ≡
1
δ+5
3
+ =
.
δ(δ + 2) δ
δ(δ + 2)
(7.75)
This result is the same as the previous (7.68), a part from a numerical coefficient of order
unity, provided that Tp is substituted with v 2 m2σ .
Let us compare this effect with the one obtained by substituting ξσ 2 R/2 in the (7.69),
R being the induced (3 + 1)-dimensional Ricci term and ξ = 1/6 corresponding to the
conformal case for a massless scalar. Such term, as suggested in [127], can be rewritten as
c
Lξ = √
H (Tξ )µµ
3 8πMP l
(7.76)
where H is a canonically normalized scalar field defined in terms of the metric by [141]
1
H≡
c
hm
m
km kn
hmn
+
ki2
,
c ≡ (3(δ − 1)/(δ + 2))1/2 is a constant and (Tξ )µµ = 6ξ(σ 2 ) is the trace of the part of the
energy momentum tensor of the scalar σ which depends on ξ. In this framework defining
a “fundamental” scale
MD ≡
MP2 l(4+δ)
8πVlar
!1/(δ+2)
(7.77)
the disappearance amplitude is (p = 3) [127]
Γ = πc2 ξ 2 v 2
m1+δ
σ
S
.
2+δ δ−1
MD
(7.78)
128
7 Higgs-graviscalar mixing
Let us note that the parametrization of H in terms of h mn becomes singular for δ = 1, as in
that case H is eaten by the massive graviton who thus acquire a zero helicity state 6 . Then
for δ = 1 strictly speaking there is no graviscalar H, but the Higgs can still mix, as shown
in (7.5), with the zero helicity components of the massive graviton and graviphotons.
Comparing (7.74) with (7.78) we see they are equivalent if the identification
3p+δ−7
MD
= (2π)p−3 Msp+δ−1 /(4πα2Y M )
(7.79)
p
is made, thus providing the “string prediction” ξ = a(δ)/2/c. The two terms in the
expression (7.75) of a(δ) correspond to the contributions from the mixing with the graviscalars (first term) and with the graviphotons (second term).
Thus, we see that despite the absence of a ξσ 2 R term in the effective action, a mixing
can nevertheless take place with an effective ξ given by
s
δ+5
.
(7.80)
ξ=
6δ(δ − 1)
This mixing becomes maximal for the case of δ = 2 large extra dimensions, where ξ =
p
7/12 ' 0.76, leading to a possible observable invisible width for the Higgs [127]. For
δ > 2, the effective ξ decreases and varies between ξ ' 0.47 for δ = 3 and ξ ' 1/4 for
δ = 6.
7.6
Two open string cylinder amplitude
In this section we check the presence of the linear coupling between branons and graviscalars by cutting the non planar one loop (cylinder, see fig. 7.1) amplitude with one open
string attached to each end of the cylinder. The resulting amplitude admit a double representation: a tree level exchange of a closed string excitation or a loop of open string
particles.
Figure 7.1: One loop open string diagram with two external states. Non-planar
diagram.
The one loop amplitude for two open string excitations is given by
(0)
(0)
A1loop = hBVN S (0)VN S (π + iwt)icyl ,
6
(7.81)
Indeed classifying the (p + 1 + δ)-dimensional massless graviton in terms of representations of the
Lorentz group SO(1, p) a higher dimensional graviton with vanishing Kaluza-Klein mass corresponds to
(p + 1)(p − 2)/2 helicity-2 states belonging to the lower dimensional massless graviton, δ(p − 1) helicity-1
states belonging to graviphotons and δ(δ + 1)/2 graviscalars. For mKK 6= 0 eating mechanisms are active
and the resulting degrees of freedom rearrange to give (p + 2)(p − 1)/2 helicity states for the massive spin-2
graviton, (δ − 1)p helicity states for the massive graviphotons and δ(δ − 1)/2 graviscalars.
129
7.6 Two open string cylinder amplitude
being t the modulus of the cylinder and the two vertex operators have been attached to
different ends of the cylinder, the position of the first has been fixed and integration over
the second vertex position is understood. The ghost insertion B is given by
Z
Z
1
1
1
1
2
i
dσ dσ bww (σ )∂t gw̄w̄ =
dσ 1 dσ 2 bww (w) = π .
B=
(b, ∂t g) =
4π
π
2πt
where we used that gww = 1/2, ∂t gww = 1/(2t) and we chose the parametrization of the
cylinder so that 0 < σ 1 < π, 0 < σ 2 < 2πt, its border being σ 1 = 0, π. The insertion of
the c gost is trivial (i.e. the Jacobian for fixing the coordinate of one of the two vertex
operators is 1) and as only the odd spin structures contribute to the amplitude there is
no superghost insertion.
We note that strictly speaking amplitude (7.81) we are looking for had better to vanish
as otherwise it would give a renormalization of open string excitations wave functions,
which is forbidden as our model is a supersymmetric truncation of a N = 1 D = 10 model
leading to N = 4 in D = 4. A non vanishing result is obtained by considering separately
the contribution of intermediate closed string excitations belonging to the NS-NS and R-R
sector, which cancel each other in a supersymmetric model. We shall need the cylinder
Green function
 w 2
(w − w̄)2


, it  − α0
Gcyl (w) = −α ln θ1
2π
8πt
0
(7.82)
and the fermionic two point correlator on the cylinder
hψ M (w)ψ N (0)i|a = η M N
θ 0 1 (0, it)θa (w/(2π), it)
,
θa (0, it)θ1 (w/(2π), it)
(7.83)
for the generic spin structure a. The R-R contribution to the amplitude in the closed
channel representation is (we omit the Chan-Paton factor which is just Tr(t 1 )Tr(t2 ) and
the polarization tensor 1M , 2N )
Z 2π
Z
X πl α0 n2 θ4 (w/(2π), il) 2α0 k1 k2
πgo2 V̄k 3−2p 2(4−p) ∞
dw
e 2 R2
2
π
dl
AR−R =
2α0 V̄⊥
lη 3 (il)
0
0
n∈Zδ
("
 #
 2 

 2 θ24 (0, il)
w
w 
2

02 N M
0 2

×
α ∂w ln θ4
, il  + α k1L k2L ∂w ln 
(7.84)
θ4 2π , il 
2π
2η 12 (il)
2
2
N M
M N θ3 (w/(2π), il) θ2 (0, il)
02
+4α k1 k2 − k1 k2 η
,
θ42 (w/(2π), il) 2η 6 (il)
where we defined the adimensional volume of the space parallel and perpendicular to the
brane
Vk
V⊥
(2πR)9−p
V̄k ≡ p+1 ,
V̄⊥ ≡ 9−p =
,
9−p
α0 2
α0 2
α0 2
We used (D.8) and for simplicity all the DD dimensions are assumed to be R-sized and all
the NN ones are infinite. The NS-NS contribution is right the opposite of this. Expanding
now the θ’s, retaining only the first terms in the massive modes and positing ν ≡ w/2π,
130
7 Higgs-graviscalar mixing
we have
Z 1
Z
X πl α0 n2 πl 0
vk 2(2−p) 2(5−p) ∞
dν
dl
e 2 R 2 e 2 α k1 k2
2
π
AR−R =
v⊥
0
0
n∈Rδ
i
h
nh
×
e2πiν e−πl − e−2πiν e−πl 1 + 2α0 k1 k2 ln 1 − e2πiν e−πl − e−2πiν e−πl
2 −2α0 k1N k2M e2πiν e−πl − e−2πiν e−πl
8 1 + 16e−2πl
(7.85)
o
,
−4α0 k1M k2N − η M N k1 k2 1 + 4e2πiν e−πl + 4e−2πiν e−πl + 24e−2πl
go2
which once recast in the form
AR−R
Z ∞
α0 η M N k1 k2 − k1N k2M 2
κ
=
dl
p+1
2π 2 α0 go2 (p)
0
X
cn el(α k1 k2 −n
0
2 α0 /R2 −4n
) (7.86)
n∈Zδ ,n∈Z
clearly displays a closed string mode expansion (the c n are numerical coefficient, c0 = 1)
using that
Z ∞
1
0 2
2
.
e−lα (k +m ) dl = 0 2
α (k + m2 )
0
The gravitational coupling κp+1 which appears in (7.86) is the correct one
r
2
κp+1 = κ10
V⊥
(7.87)
provided that (remember gc = κ10 /(2π), see (2.195))
2
go2 (p = 9)
= (2π)9/2 2α0 ,
gc
(7.88)
which is in agreement
with (13.3.31) of [29], remember (7.37), (2.195) and (2.197). The
√
additional factor 2 in (7.87) is due to the halving of the volume because
of the presence
√
of the orientifold. In the case of an unoriented theory κ p+1 = κ10 / V⊥ holds7 . From the
previous (7.88) and using (7.37) the relation (7.67) can be derived, where α Y M ≡ gY2 M /4π.
Considering instead inclusive quantitities, i.e. summing over the Kaluza-Klein tower
before integrating over l, which can be made by using (D.8), in the R α 0 1/2 limit it is
obtained
Z
∞ X l(k1 k2 −4n/α0 ) πR2 δ/2 1
κ2p+1
N M
MN
dl
e
k1 k2
, (7.89)
AR−R = 2 0 2 k1 k2 − η
2π α go
α0
lδ/2
0
n∈Z
which is divergent in the l → 0 region (for δ ≥ 2), thus seeming to imply that inclusive
quantities do not possess a smooth limit in the v ⊥ → ∞ limit.
Performing instead in (7.86) the l integral first the amplitude becomes
AR−R
X
κ2p+1
1
MN
N M
=
η
k
k
−
k
k
,
1
2
1
2
2πα0 go2
k
k
−
n2 /R2
1 2
δ
n∈R
7
We note that in both oriented and unoriented case in our conventions (2.195) holds.
(7.90)
7.6 Two open string cylinder amplitude
131
where we have neglected all massive modes retaining only the n = 0 term to make more
transparent the connection with (7.65) and (7.68). In fact (7.68) can be recovered from
(7.90) once a small imaginary part is given to the denominator, the sum over n converted
m (v ) 6= 0 and k k = m2 6= 0,
into an integral, the momenta extrapolated so to fulfil k 1,2
2,1 m
1 2
σ
2
0
rescaling (7.38a) is performed and finally remembering that T p = 1/(2π α go2 (p)).
To understand how string theory tackle the l → 0 singularity it should be noted that
in the l → 0 limit massive string modes cannot be neglected any more in the closed string
representation. If we want to rely on a field theory interpretation of the amplitude in
terms of few light (massless) degrees of freedom also in the l → 0 limit, we have to resort
to the open channel representation of the amplitude.
As already described in sec. 2.9 for the vacuum one loop amplitude, also the amplitude
(7.81) can be written in the open string channel by using the properties of the θ-functions
shown in app. D. The Ramond part of the amplitude in the open string representation is
!2α0 k1 k2
Z 2πt
Z ∞
w
X
θ1 2π
, it (w−w̄)2
πgo2
1
−2πtw2 R2 /α0
e
dw
e 16πt
dt
AR = − 0 Vk
2α
η 3 (it)
(8π 2 α0 t)(p+1)/2 w∈Zδ
0
0
!
("
 w 2
2


0
0 (w − w̄)
MN 2
, it  − α
×
η
∂w −α ln θ1
2π
8πt
2 # 4
(7.91)
 w 2
2
(w − w̄)
θ2 (0, it)


+k1N k2M ∂w −α0 ln θ1
, it  − α0
2π
8πt
2η 12 (it)
θ 2 (w/(2π), it) θ22 (0, it)
2
+4α0 k1N k2M − η M N k1 k2 12
.
θ2 (w/(2π), it) 2η 6 (it)
Expanding the theta functions and retaining only the massless modes we have
Z 1
Z ∞
1−p
0
dν e2πtν(1−ν)α k1 k2 η M N k1 k2 − k1M k2N (1 − ν)ν ,
(7.92)
dt t 2
AR ∝
0
0
where in the discrete
√ sum in (7.91) all but the zero winding term has been neglected as in
our model R α0 . The contribution of the NS sector is the opposite of the RR one in
a supersymmetric model. In a non supersymmetric model in which the R-R and NS-NS
part of the 1-loop amplitude add up instead of cancelling each other, the introduction of
tachyons in the open string channel is unavoidable.
This amplitude can be read as a field theory one considering the Feynman trick
Z 1
1
1
dx
=
ab
(a
+
(b
− a)x)2
0
and the useful formula (2.167) which enable to re-write the field-theory one loop amplitude
for loop-circulating states with masses m 1 and m2 as
Z
dp+1 k
1
1
p+1 (p + k)2 + m2 k 2 + m2 =
(2π)
2
Z
Z ∞ Z 1 1
p+1
d k
2
2
2
2
(7.93)
dx t e−t(k +(p +2kp)x+m1 x+m2 (1−x)) =
dt
p+1
(2π)
0
0
Z ∞ Z 1
1
2
2
2
dt
dx t(1−p)/2 e−t(p x(1−x)+m1 x+m2 (1−x)) .
(p+1)/2
(4π)
0
0
132
7 Higgs-graviscalar mixing
We can summarize by saying that non singularity of the string amplitude is achieved
thanks to the possibility of a dual description: the l integral should then be cut at some
point l0 and pasted with a t integral cut at t0 = 1/l0 , as each representation of the integral
is manifestly
R ∞ convergent
R ∞ respectively in the l → ∞ and t → ∞ limit. Given the amplitude
A1loop = 0 dlAcl = 0 dtAop , its manifestly convergent representation is
Z ∞
Z ∞
A1loop =
dl Acl +
dt Aop .
l0
7.7
1/l0
Higgs on branes intersection
In this section we study the case where the Higgs lives on a branes intersection, corresponding to an open string with mixed Neumann-Dirichlet (ND) boundary conditions in
four internal directions. We will distinguish two subcases, depending on whether one of
the two orthogonal branes extend (partly) in the bulk of large extra dimensions.
We thus consider the coupling between two ND open string modes and a closed string
NS-NS state. We shall consider first the oriented theory. As we cannot use now the
Born-Infeld action, a string calculation is the only way to compute this coupling.
The kinematics of the problem is the same with the one described in (7.39). The vertex
operator for a NS open string state χ is, with one end on D5-branes and the other end on
D9-branes, is, in the (−1)-picture,
(−1)
V59
= go taa0 χα e−ϕ ∆S α eikX ,
(7.94)
where the Chan-Paton factor index a(a 0 ) transforms in the (anti-)fundamental of the
D5(9)-branes gauge group. The operator ∆ is the product of twist fields associated to the
four internal coordinates with mixed ND boundary conditions, S α is the corresponding
spin field, and χα selects the internal spinor helicity. This vertex operator has the same
expression as the left (supersymmetric) part of the vertex for a massless heterotic twisted
state of a Z2 orbifold [142]. For a 95 state, one has the same operator with χ α replaced
by (χ̄)α ≡ (χα )† and S α replaced by Sα . The NS-NS closed string state vertex operator
(in the (0, 0)-picture) is given by
α0
α0
2gc
(0,0)
N
M
ikX
M
N
¯
i∂ X̃ + k · ψ̃ ψ̃
e
eikX̃ .(7.95)
VN S 2 (ζ, k) = − 0 ζM N i∂X + k · ψψ
α
2
2
The relevant correlators between the twist field ∆ and X is (for left-movers) [143]:
q


z13 z24
1
−
M
N
0
z14 z23
h∆(z1 )∆(z2 )XL (z3 )XL (z4 )i
α MN 
 ,
q
(7.96)
=− η
ln
h∆(z1 )∆(z2 )i
2
1 + z13 z24
z14 z23
where zi denote the corresponding world-sheet positions. For right-movers, the correlator
is the same provided one substitutes L, z i with R, z̄i . The correlator between two ∆’s is
h∆(z1 )∆(z2 )i =
1
(z1 − z2 )1/2
.
(7.97)
133
7.7 Higgs on branes intersection
Note that the normalization coefficient g o (it is understood go for p = 5) in front of
the vertex operator (7.94) is the same with the normalization of untwisted open string
states. This can be checked by comparing the χ 2 A2µ amplitude and the exchange interaction χAχχAχ which leads to internal propagation of a ND state. The χ α field carries an
index which labels the spinor representation of the internal SO(4) and the GSO projection
forces it to be a Weyl spinor. Hence, it has two helicity states forming the fundamental
representation of SU (2) (usually called SU (2) R ), rather then the full SO(4). This representation is pseudoreal, two-dimensional in the complex sense and four-dimensional when
viewed over the real numbers. In the oriented theory the two χ’s correspond to the two
independent excitations described by 59 and 95 states which together make up the bosonic
content of an N = 1 hypermultiplet in six dimensions.
In the unoriented theory, we expect just one complex boson (the bosonic content of
half of a hypermultiplet) as 59 and 95 modes are correlated. This is obtained [144] via
a projection which involves SU (2)R as well as the 5-brane gauge index, being the gauge
group Sp(k). The projection is a reality condition which can be applied to the spinor
χ as the representation (2k, 2) of Sp(k) × SU (2) R is real (the 2k of Sp(k) being also
pseudoreal).
The relevant correlators involving the spinor fields S α in 4 internal dimensions are
[134]:
hSα (z1 )S β (z2 )ψ M ψ N (z3 )i = − 12 ΓM N
β
α
1/2
z12
,
z31 z32
−1/2 −1
z34 ×
hS (z )S β (z2 )ψ M (z3 )ψ N (z4 )i = 21 (z32 z42 z31 z41 )−1/2 z12
i
h α 1
β
δ M N δαβ (z32 z41 + z31 z42 ) − ΓM N α z12 z34 ,
(7.98a)
(7.98b)
δαβ η M R η N S − η M S η N R
hSα (z1
+
2
3
4 )i = −
2
z12
z34
(7.98c)
β
−1
−1/2
M
R
N
S
N
S
M
R
M
S
N
R
N
R
M
S
η Γ +η Γ
−η Γ
−η Γ
(2z34 ) (z32 z42 z31 z41 )
,
α
)S β (z
)ψ M ψ N (z
)ψ R ψ S (z
where −iΓM N /2 = −i[ΓM , ΓN ]/4 is the Lorentz generator in the spinor representation.
This correlators can be used to compute the 3-point amplitude, which in the α 0 t → 0 limit
becomes:
2 µ ν
ν µ
A2N D,N S 2 = 2igc π −k2 · k3
(k k − k3 k2 ) δαβ ζµν
+
+
+
πt 3 2
(7.99)
† 3
k1r rm β
µ
ν
rn β
2
+
(Γ )α (k2 − k3 ) ζmν + (Γ )α (k2 − k3 ) ζµn
χα χβ ,
4
η µν
k2µ k3ν
k3µ k2ν
where χ2,3 is the χ-field with momentum k2,3 . The amplitude8 above displays a pole
term in t due to the χχAAb exchange interaction that has to be subtracted in order to
extract the contact terms. Using (2.195) and the usual rescaling (2.194a,b) one obtains
8
Note that in contrast to the SO(3, 1) case, for Euclidean SO(4) spinors, the quantity χ† ξ is scalar
provided χ and ξ have the same chirality.
134
7 Higgs-graviscalar mixing
the trilinear Lagrangian:
L2N D,N S 2
1
=−
2
hµµ p − 3
+
φ +
−∂µ χ̄∂ν
+ ∂ χ̄∂χ
2
4
i
1
∂n hµm (∂ µ χ̄Γmn χ − χ̄Γmn ∂ µ χ) .
4
χhµν
(7.100)
No contact interaction with bµν is found, neither a potential coupling to the internal
components of the Riemann tensor, as in the untwisted DD case we studied in secs. 7.2,7.3.
However, besides the standard kinetic terms we find still a coupling of the ND open string
modes to the KK excitations of the graviphoton, arising through the spin connection in
the gravitational covariant derivative
1 mn
Γmn χ .
∇gr
µ χ = ∂ µ χ + ωµ
4
(7.101)
Here, ωµmn is the standard spin connection (with one index parallel and two orthogonal
to the D5-brane) which is given in terms of the vielbein e aµ by
ωµmn =
1
1 ρm σn
1 νm
m
e
∂µ enν − ∂ν enµ − eνn ∂µ em
e (∂ρ eσi − ∂σ eρi ) eiµ ,
ν − ∂ ν eµ − e
2
2
2
whose first order expansion around flat space, g M N = ηM N + hM N , is
ωµmn = hµ[m,n] .
(7.102)
The connection part of the covariant derivative is completed by gauge terms to make the
full covariant derivative
1
(7.103)
∇µ χ = ∂µ χ + ωµmn Γmn χ + ig5 Aµ − ig9 A0µ χ ,
4
where Aµ (A0µ ) is the D5 (D9) world-volume gauge field with gauge coupling g 5 (g9 ). Finally,
open string excitations σ and χ have also non-derivative (D-terms) interactions [29]
gY2 M
i mn 2
m
n
[σ , σ ] − χ̄Γ χ
LD = −
(7.104)
4
2
in the normalization of (7.100).
We consider now the possibility of mixing between χ and closed string modes. In the
case where the Higgs, identified with χ, lives on an intersection of two orthogonal branes,
both transverse to the submillimeter bulk (e.g. D3 and D7, or D5 and D5’), no mixing
is generated between χ and closed string states. On the other hand, in the case where
one of the two orthogonal branes extends in the bulk, a mixing is induced, as can be
seen from the effective Lagrangian (7.100), between χ and the longitudinal component
of the corresponding graviphoton in the bulk. As in the DD case, in order to obtain a
quadratic coupling between closed and open string states, one of the χ’s must acquire a
non-vanishing vacuum expectation value.
Note that a VEV of χ along a supersymmetric flat direction, i.e. when the D-term
(7.104) vanishes, gives rise to the well-known Higgs branch which provides a string realization of a non-Abelian soliton [144] that we are not interested in here. We consider
135
7.7 Higgs on branes intersection
instead a real vacuum expectation value for χ, with non-vanishing D-term that breaks
supersymmetry, and we study the effective field theory obtained by expanding around the
VEV v, χ1 = v + χ01 , where χ1 is one of the two complex bosons. Dropping the prime
from χ01 and assuming the ND conditions to be along the directions 6̂ . . . 9̂ (orthogonal to
the 5-brane), we have up to quadratic order in χ and σ:
L0D = −
gY2 M h 4
v + 4v 3 Re(χ1 ) + v 2 4(Reχ1 )2 + 2χ†1 χ1 + 3χ†2 χ2 +
4
i
,
+v 2 [σ 6̂ , σ 9̂ ] + [σ 7̂ , σ 8̂ ]
(7.105)
where fields are canonically normalized. Note the appearance of a cosmological constant
and of a tadpole for χ1 . This is anyway only an effective approach and other potential
terms may be generated when supersymmetry is broken.
On the other hand, χ can also obtain a mass in a supersymmetric way, avoiding the
cosmological constant and tadpole-like terms, as in (7.105). This is achieved by turning
on a Wilson line for the gauge fields with polarization parallel to the 5-branes, or if we
T-dualize, by separating lower and higher dimensional branes by giving an expectation
value to one (or some) of the branons orthogonal to both branes. This corresponds to
moving in the so-called Coulomb branch of the theory.
The χ1 expectation value determines the mixing terms between the χ field and the
corresponding graviphoton
1 Lmix = − v ∂[6̂ h9̂]µ + ∂[7̂ h8̂]µ ∂ µ Imχ1 .
4
(7.106)
Using (7.63), one finds for the bosonic field Imχ 1 ,
v 2 8πV
Σχ (p ) =
8 MPp−1
l
2
Z
2
dδ k k 2 km̄
(2π)δ p2 + k 2
(7.107)
and consequently, using (7.67), one finds the following invisible width
4πα2Y M π v 2
1
Γχ =
Im Σ(p2 = m2χ ) =
mχ
mχ
(2π)p−3 32δ Ms5−p
mχ
Ms
δ
Sδ−1 .
(7.108)
Hence, the resulting effective parameter ξ in this case reads
1
ξ=
4
s
δ+2
,
6δ(δ − 1)
(7.109)
which is significantly smaller than in the DD case, studied in sec. 7.5. Indeed, the highest
value obtained for δ = 2 is ξ ' 1/7. Due to the fact that the graviphoton and its KK tower
form a quasi-continuum set of states, this result is not altered if we consider unoriented
type I models in which an orbifold projection takes the zero mode of the graviphoton out
of the spectrum.
136
7.8
7 Higgs-graviscalar mixing
Conclusions
In this chapter we investigated the possibility of mixing between the Higgs, identified as
an open string excitation, and closed string states from the bulk (graviscalars), when the
fundamental string scale is in the TeV region. We found that such a mixing can occur,
leading to a possible observable invisible decay width of the Higgs, only when the Higgs
lives on the Standard Model world-brane and corresponds to a DD open string with both
ends on parallel D-branes.
The experimental sensititvity is restricted to the case in which the disappearance
amplitude is a consistent fraction of the total decay width of the Higgs boson Γ tot
σ which
tot
in the standard model is Γσ SM ∼ 17 MeV, 32 MeV, 400 MeV, 1 GeV, 4 GeV, 10 GeV,
for a Higgs mass mσ ∼ 150, 155, 170, 190, 245, 300 GeV [145]. For m σ > 2mW the decay
width raises sharply because the decay channel W + W − opens up. Thus the experimental
sensitivity to this effect is limited to the case the string scale is small enough, which
quantitatively means (for p = 3)
Ms <
4π 2 α2Y M c2 ξ 2 v 2 mδ+1
σ Sδ−1
tot
Γσ
1
δ+2
.
(7.110)
In the optimistic case, ξ = 1 (which excludes the ND case), δ = 2 and α Y M = 0.1 this
effect is interesting for a string mass M s <∼ few TeV (this limit is almost insensible
to the Higgs mass for δ = 2, provided m σ < 150 GeV) which is on the limit of the
phenomenological constraints discussed in sec. 7.1 (indeed ruled out if we believe the
constraint 7.4).9 Moreover it should be remembered that an invisibly decaying Higgs
boson is not necesessarily an evidence of large extra dimensions, but other not standard
model interactions may provide additional decay states.
From the theoretical point of view, although our analysis was done in the context of
supersymmetric type I theory, our results remain valid in non supersymmetric D-brane
models where supersymmetry is broken (mainly) in the open string sector, using appropriate combinations of branes with (anti)-orientifolds that preserve different amount of
supersymmetries. The reason is that in these cases, the effective action can be obtained
by a corresponding truncation of a supersymmetric action.
We thus showed that if the Higgs can be modelled as a DD open string excitation this
effect, though small, should be present as the interaction terms which produce it are due
to arguments based on general covariance.
9
We note that the experimental bounds costrain the higher dimensional Planck mass MP l(4+δ) , which
is related to the string mass Ms through (7.77) and (7.79).
8 Conclusions
As string theory provides a unified theoretical framework for all the known interactions in
nature (and embarassingly many more) we consider of fundamental importance to test it
in situations of phenomenological interest. In particular we have studied its relevance for
cosmology within the context of the pre-big bang scenario and an application to particle
physics through the computation of an invisible decay width of the Higgs in the context
of the large extra-dimensions scenario.
The pre-big bang model has several attractive features from the phenomenological
point of view, and also from the theoretical side it allows to separate the issue of the
initial conditions from that of the cosmological singularity: the Universe emerges from a
cold and decoupled state, far simpler than the hot plasma of the big bang. Nevertheless
the graceful exit problem is still unsolved, i.e. the pre-big bang inflating phase cannot
be smoothly connected to a decelerated expansion, despite this two kinds of solutions to
the lowest level effective action always appear in pair. There are indications that the
singularity can be avoided through the inclusion of α 0 corrections which modify Einstein
gravity or by the effect of a fermion condensate which can preserve supersymmetry of
the classical cosmological solutions, even if in both cases there are some problems. In
the α0 correction case the full perturbative series is in principle important and not just
the first few terms, moreover not all the terms of the effective action can be fixed by
comparing the string amplitude with the field theory one, as some terms give vanishing
contributions to n-point scattering amplitudes for low enough n. To solve this issue one
should consider n-point scattering amplitudes with higher n, with the consequent almost
unsolvable technical problems in extracting from those amplitudes the relative effective
action. In the case the solutions are regularized because a fermion condensate swithces
on, it is the dynamics itself of the condensate which is not known.
We then turned our attention to a systematic study of the loop corrections to the low
energy effective action derived by some realistic string compactifications, also motivated
by previous works showing that ad hoc invented loop corrections can realize a graceful
exit. We obtained the non trivial result that the loop corrections point towards the right
direction, making the curvature decrease on the solutions, but they failed in keeping
the dilaton in the “moderate” coupling regime. Indeed for the case of the dilaton Kähler
potential the all order loop form of the quantum corrections are known but even this turns
out to be not enough because the growth of the dilaton is not stopped and eventually the
solutions enter the strong coupling regime, which, at low curvature, is described by elevendimensional supergravity. In this regime even a knowledge of all-loop corrections to the
ten-dimensional theory is not enough, because the degrees of freedom are different from
the one we started with.
137
138
7 Higgs-graviscalar mixing
From this analysis a quite consistent picture has emerged, which nevertheless is still
plagued with some difficulties, so we turned our attention to general thermodynamics
arguments, which can be relevant in ruling out a singularity. To achieve this goal we have
postulated the existence of a geometric entropy, which, consistently with a cosmological
version of the holographic principle, is associated to the existence of a scale of causal
connection in cosmology, the Hubble scale: microphysics cannot act between objects whose
separation is bigger than the Hubble length. Our analysis shows that considering the low
energy action with α0 corrections, the solutions which are singular violate the generalized
second law of thermodynamics before reaching the singularity while well-behaved solutions
do not. Moreover once we add to the entropy the contribution of the field quantum
fluctuactions, which is decreasing in the pre-big bang because fluctuactions wavelengths
get bigger than the Hubble size and they “freeze”, the generalized second law is violated
even on non-singular solutions when the coupling becomes of order one unless the solutions
move towards a low curvature regime, thus enforcing a graceful exit. If string theory then
is compatible with our version of the second law of thermodynamics we expect that some
features of the general theory (non-perturbative α 0 corrections, for instance and loop
corrections) will realize this kind of graceful exit.
From the point of view of particle physics, we studied in a concrete type I string theory
setup the mixing between open and closed strings in the presence of D-branes and in the
context of the large extra-dimensions scenario. In this scenario matter and gauge fields
are confined to lower dimensional D-branes, whereas gravity is free to propagate in the
full bulk of the theory and it is consistent to assume that the size of the extra dimensions
orthogonal to the brane, which are not felt by the gauge particles but only by gravity,
can be as large as a millimiter. The Planck mass is a derived quantity, resulting from
the volume of the extra dimensions and from the string scale which can be as low as the
TeV scale. Open strings represent gauge charged matter and gauge fields while closed
string excitations describe particles endowed with gravitational (and Ramond-Ramond)
interactions only. The mixing between open and closed strings is of phenomenological
interest as we can identify among the open string scalar excitations the Standard Model
Higgs and consequently we computed the mixing amplitude between the Higgs on one side
and graviscalars and the zero-helicity part of the graviphoton on the other side. If the
extra-dimensions are very large the Higgs experiences a coupling to a continuum of KaluzaKlein states, which does not lead to an oscillation but to a disappearance amplitude for the
Higgs itself. We have proposed different ways of modelling the Higgs as a string excitation
in a direction with DD or ND boundary conditions, obtaining in each case different results.
We admit that the model we investigated is not realistic as it is supersymmetric and
our Higgs is not exactly endowed with all the interactions that a Standard Model Higgs
should be endowed with, but it can be said that the interactions we found are still present
when supersymmetry is broken, for instance, in some string constructions involving antiorientifold planes and possibly anti-D-branes, and that they are also required by general
covariance arguments, implying that they are general and solid.
To conclude we finally remark that a better understanding of which is the “right”
vacuum selected by string theory is fundamental to study phenomenology, both in a cosmological and in a particle physics context, but we believe that while a rationale for the
vacuum choice is still missing it is not useless to study phenomenology in some concrete
string framework.
Acknowledgments-Ringraziamenti
I cannot but start by thanking my advisor Michele Maggiore, who, beside teaching me
many things about string cosmology and string theory, has also constantly displayed a
highly contagious enthusiasm for physics. I am very grateful to Rami Brustein for the
discussions and the work we did together and his warm hospitality at Ben Gurion University in Beer Sheva. I wish that his country may find soon a fair and lasting peace. I
am also very grateful to Ignatios Antoniadis for introducing me in the charming world of
large extra-dimensions and the kind hospitality he offered me at the École Polytechnique
in Paris and at CERN in Geneva. Moreover I cannot forget the long-lasting collaboration
with Stefano Foffa.
I would like to thank Fawad Hassan and Riccardo Rattazzi for their patience in answering my often not-too-smart questions and for their warm encouragement. I am also
grateful to Graham Ross and Ian Kogan for their kind hospitality at the theoretical physics
department of Oxford and for helpful discussions.
Thanks are due to other physicists with whom I had useful discussions: Riccardo
Apreda, Alessandra Buonanno, Roberto Contino, Paolo Creminelli, Massimiliano Gubinelli, Simone Lelli, Biagio Lucini, Laura Mersini, Giuseppe Policastro, Anna Rissone,
Alessandro Tomasiello, Carlo Ungarelli, Carlo Angelantonj, Adi Armoni, Rowan Killip,
Herve Partouche, Alejandro Ibarra, Bayram Tekin and Martin Sloth.
Now I come to thank those whith whom I might not have shared many physics discussions but other not less important life experiences, like football for instance, who include
some of the above mentioned and Davide, Michele, Claudio, Emanuele, Alberto, Davide,
Iacopo, Valentina, Alessandro, Tommaso, Anna, Mario, Patrizia, Dario, Lorenzo, Oliver,
Antonella, Onofrio, Donatella, Maura, Marj, Alejandro, Andrea, Luigi, Davide, Ettore,
Marco, Lorenzo, Donato, Francesco, Jose, Nikos, Matteo, Tommaso, Fabio, Silvia, Valerio,
Francesca, Alessandra, Davide, Walter, Sergio and many more. A special thanks goes to
Davide, without whose practical help the end of this work would have been even more
delayed than it has been already.
I also wish to thank all the friends who contributed to make the periods of my life I
spent abroad unforgettable. Thanks then to Alberto, Anna, Boriana, Claudia, Domenico,
Gabriele, Hristo, Ivica, Lorena, Markus, Mauro, Michele, Milena and Raoul for the happy
Paris days, to Alejandro, Antonios, Bayram, Cinzia, Geza, Graziano, Ramon, Sasha,
Shinshuke, Stavros with whom I shared my Oxford life and to Anna, Antonello, Donatella,
Dario, Monica e Tommaso without whom the long dark nordic winter would not have been
so warm.
Un caldo grazie va anche a tutti miei amici di Fano che nonostante la lontananza non
mi hanno fatto mai sentire un estraneo nella mia città.
139
140
Infine ma non certo per ultimi vorrei ringraziare i miei genitori e mio fratello Bruno,
il cui appoggio non solo materiale non mi è mai mancato nonostante la lontananza fisica.
Helsinki, 22nd April 2002
Appendix A
Perturbations in inflationary
cosmology
A.1
The Bogolubov coefficients
Considering a bosonic quantum field operator ψ, it can be expanded over a set of orthonormal function {fi } according to
Z
ψ(x) = dµk ak fk (x) + a†k fk∗ (x) ,
(A.1)
where the orhotnormality conditions are
Z
↔
1
dΣ(d) fk ∂ fk∗0 = iδkk0 ,
(fk , fk0 ) ≡
2k ∂V
(fk , fk∗0 ) = 0 .
(A.2)
The fk ’s are solution of the corresponding field equation, the Klein-Gordon equation, for
instance, for a spin-0 field. In the case of a Minkowski space f k (x) = eiωk t eikx and the
explicit form of the measure is
µk =
d3 k
.
(2π)3 2ωk
(A.3)
According to the usual quantization procedure the Fock vacuum |0i is defined by
ak |0i = 0
∀k .
(A.4)
The canonical commutator relations that the creator and annihilation operators inherit
from ψ are1
[ak , a†k0 ] = 2ωk δ 3 (k − k 0 ) ,
[ak , a0k ] = [a†k , a†k0 ] = 0 .
(A.5)
In a curved space the concept of particle is generically not well-defined. The most
natural choice, which anyway is not always possible, is that the normal modes are the ones
1
Note that here we use a Lorentz invariant measure dµk , normal modes
fk and oscillator
√
√ operators ak .
The usual notation is obtained by rescaling dµk → 2ωk dµk , fk → fk / 2k and ak → ak / 2k.
141
142
Appendix A
who reduce to positive-negative frequency modes with respect to a globally Minkowskian
time coordinate, or at least the ones that reduce to them at high enough k, where the
spacetime curvature can be neglected.
The Fock vacuum |0i depends heavily on the choice of the {f k } set, making the notion
of particle highly non covariant. A different choice F k can be expanded over the complete
set {fk }, or vice versa, according to
X
Fk (x) =
(A.6a)
αkk0 fk0 (x) + βkk0 fk∗0 (x) ,
k0
fk (x) =
X
k0
α∗k0 k Fk0 (x) − βk0 k Fk∗0 (x) ,
(A.6b)
where the normalization of the basis functions imply
αα† − ββ † = 1 ,
αβ T − βαT = 0 .
(A.7)
The corresponding creation and annihilation operators are then related by a Bogolubov
transformation [146, 147]
a = α T A + β † A† ,
(A.8a)
†
†
(A.8b)
A = α a−β a ,
T
†
∗
∗ †
(A.8c)
†
(A.8d)
a = β A+α A ,
†
A = −βa + αa .
The Fock vacua relative to the two sets of operator are different, as for instance the number
operator built out of the A, A† has expectation value over the a-defined vacuum
X
|βkk0 |2 .
(A.9)
h0|A†k Ak |0i =
k0
This has the remarkable consequence that the vacuum for one oscillator set may be highly
populated from the other set point of view. In the case of fermions the normalization
condition |α|2 + |β|2 = 1 constrains the occupation number N to be always N < 1.
A.2
Density perturbations
In sec. 1.4 we claimed that the intial density perturbation in the radiation dominated
phase are related by (1.34) to the density perturbation in the inflationary phase. To show
that this is the case let us consider small perturbation over an expanding homogeneous
isotropic Universe. The time derivative of the spatial 3-curvature k mode is related to the
pressure perturbation δpk through
δpk
.
ρ+p
(A.10)
∇2 Φ = 4πGN δρ ,
(A.11)
(3)
Ṙk = −H
Defining the gravitational potential Φ by
143
A.2 Density perturbations
which once written in terms of Fourier modes becomes
2 kc 2
δρ ,
Φk = −
3 aH
(A.12)
allows (A.10) to be rewritten as
1 dR(3)
2 δpk
=
H d ln t
3 δρk
"
kc
aH
2
Φk
(3)
(1 + w)Rk
#
,
(A.13)
where as usual w ≡ p/ρ and kc is the constant comoving momentum related to the physical
one k, which red-shifts as the Universe expands, by k c /a = k. Considering a perturbation
which is super-Hubble sized, kc < aH, as |δpk /δρk | ∼ 1 or less, the 3-curvature changes
(3)
negligibly if Φk . (1 + w)Rk , which is indeed the case as we are going to show.
Perturbing the continuity equation (1.10)
δ̇Φk = −3(ρ + p)δHk − 3Hδρk
(A.14)
is obtained, where δHk is defined by the perturbation of (1.8a)
8πGN
2
2HδHk ≡
δρk −
3
3
kc
a
2
(3)
Rk ,
(A.15)
(3)
and expressing δHk in terms of Rk
2
5 + 3w
(3)
Φ̇k +
Φk = −(1 + w)Rk
3H
3
(A.16)
is obtained. This equation during any era when w is constant has the non decreasing
solution
Φk = −3
(3)
1 + w (3)
R ,
5 + 3w k
(A.17)
(3)
showing that indeed Φk ∼ (1 + w)Rk and then that Rk is roughly constant when k
is super-Hubble sized. But then up to numerical factors also Φ k /(1 + w) stays constant,
implying via (A.12) that δρ/(ρ + p) is constant at Hubble scale crossing.
The perturbation wavelength becomes longer than the Hubble length when an accelerated expansion or contraction (ä/ȧ > 0) takes place, thus during inflation wavelengths
become super-Hubble sized and during the radiation (or matter) dominated epoch they
become sub-Hubble sized. This happens because wavelengths λ behave as λ ∝ a whereas
the Hubble length H −1 ∝ a3(1+w)/2 , the deSitter phase corresponding to w = −1 and
constant H = HdS . Thus during an expanding phase the perturbation wavelengths grow
faster or slower then the Hubble length depending on the expansion being accelerated or
decelerated (w ≷ −1/3). The condition for Hubble scale crossing can be simply stated
to be kc η ∼ 1 if, as it is in general a(η) ∝ η α for some α. Only perturbations whose
wavelength is short enough k > k1 ∼ Hmax never become super-Hubble scale sized, being
2
Hmax
the maximum curvature reached in the accelerated phase.
144
Appendix A
We now compute the right-hand side of eq.(1.34). The density perturbations we are
interested in first are the ones induced by the quantum fluctuation of the inflaton field.
Let’s consider quantum fluctuation in a generic ψ field
h0|ψk ψk0 |0i =
fk fk∗0 3
δ (k − k 0 ) ≡ h|δψk |2 iδ 3 (k − k 0 ) .
2ωk
(A.18)
In a cosmological bakground, i.e. considering the homogeneous and isotropic 4-dimensional
metric
ds2 = dt2 − a2 (t)dx2 = a2 (η)(dη 2 − dx2 )
(A.19)
the Klein-Gordon equation is (a dot means derivative with respect to the cosmic time t,
a prime stands for derivative with respect to the conf ormal time η)
a00 ψk
1
00
2
2
= 0.
(A.20)
ψk = ψ̈k + 3Hψk + k ψk = 3 (aψk ) + kc −
a
a a2
The deSitter space is described by the scale factor
a(η) = −
1
HdS η
η < η1 < 0 ,
(A.21)
or equivalently a(t) ∝ eHdS t , with constant HdS , and in deSitter space the normal mode
solutions to (A.20) are
1
i
fk (η, x) = √
eikc η−ikc x ,
(A.22)
1+
k
η
2a(η)
c
they reduce to the Minkowski space solution, apart from the factor 1/a(η), in the case
kc 1/η. The mean square value of the field in deSitter space at Hubble scale crossing
(that is when kc /a = ȧ/a = a0 /a2 , which is the relevant epoch for the description made in
sec. 1.4) is
H2
h|δψk |2 ik=H = dS
,
2kc3
(A.23)
and then the spectral power Pψ (k) is defined by
Z
d(ln k)Pψ (k) ≡
Z
d3 k
h|δψk |2 i .
(2π)3
(A.24)
At Hubble scale crossing it is
Pψ (k) =
HdS
2π
2
.
(A.25)
Considering the inflaton during its slow roll evolution as a massless field in a deSitter spacetime, this is the magnitude of the square of its quantum fluctuation, which is related to
the density perturbation present in the Universe when it is radiation dominated.
145
A.2 Density perturbations
We have now gathered all the elements to derive (1.35), where we have also used
δρk = δχk V 0 + δ χ̇k χ̇k , δ χ̇k ∼ (δχk )2 , (1 + w)ρ ∼ χ̇2 and (δχk )2 ≡ Pχ (k) when the
perturbation wavelength crosses the Hubble length in the inflationary phase. The ratio
H 2 /χ̇ entering (1.35) depends on the specific solution to the inflaton equation of motion.
A well studied case correspond to the new inflationary model, where the potential to which
the inflaton χ is subject is the one-loop, zero-temperature Coleman-Weinberg potential
VCW (χ) = Bσ 4 /2 + Bχ4 ln(χ2 /σ 2 ) − 1/2 ,
(A.26)
where B ∼ α2GU T . It is flat near the origin so that H ∼ V 1/2 /MP l ∼ const. In the slow
roll regime (1.32) the function χ(t) can be inverted to give
Z χ
Z t(χ)
V (χ̃)
3H 2 1
1
Hdt =
dχ̃ '
N (χ) ≡
(A.27)
− 2 ,
0
2λ2 χ2i
χ
χi V (χ̃)
ti
where λ ' |4B ln(χ2 σ 2 )| ∼constant. With this explicit solution the perturbation eq. (1.35)
gives
δρk
H 2 3/2
' λ1/2 Nk ,
(A.28)
∼
ρ
χ̇ k=H
where Nk denotes N at the time when k = H, which is only logarithmically dependent on
k. In the case of chaotic inflation the potential is simply
V (χ) = λχ4
(A.29)
(and also V = m2χ χ2 would work). The slow roll conditions are matched provided that the
inflaton is initially displaced from the minimum by a trans-Planckian amount, χ i > MP l ,
which still gives a sub-Planckian energy density if χ i . MP l /λ1/4 . The solution at early
time can be written as
π
(A.30)
N (χ) ' 2 χ2i − χ2 ,
MP l
which gives again (A.28) up to numerical factors. Astrophysical scales k A ∼ (10 −
10−1 Mpc)−1 which become sub-Hubble sized in the radiation dominated epoch when
H = Hin ∼ H0 × (102 − 104 )
NkA = 1/2 ln(HdS /Heq ) + 2/3 ln(Heq /Hin ) ∼ 56 + 1/2 ln(HdS /1013 GeV) ,
(A.31)
thus compelling λ . 10−14 . HdS ∼ 1013 GeV corresponds to a temperature T ∼ (H dS MP l )1/2
∼ 1016 GeV. A different implementation of the new inflationary model allows to consider
a potential of the type [27]
V (χ) = ∆4 − m2χ χ2 1 + c ln χ2 /MP2 l ,
(A.32)
where ∆ is the vacuum energy sustaining inflation and m χ the inflaton (tachyonic) mass.
This potential has a maximum at the origin and the inflaton evolves away from it, thus
making low powers of χ to dominate in the early stages of inflation. Assuming that a
linear term in the potential is forbidden the χ-dependent terms in (A.32) are naturally
146
Appendix A
generated by supersymmetry breaking, with a typical value m χ = β∆2 /MP l , where a
numerical coefficient β of order 10−1 or smaller allows to fulfil the slow-roll conditions
(1.32). Logarithmic corrections have been added to the potential to keep account that
on general grounds radiative corrections make m χ to depend logarithmically on χ once
supersymmetry is broken. Inflation eventually ends when nonrenormalizable terms in the
potential become relevant, as it happens when χ grows enough. For this model of inflation
we have [27]
1
1 + c + 2c ln χi
(A.33)
N (χ) =
ln
4cm2χ
1 + c + 2c ln χ
and the resulting amount of density perturbations is
H 2 ∆2
δρk
∼
,
'
ρ
χ̇ k=H
βMP l χH
(A.34)
where χH is the inflaton value at the time of Hubble scale crossing. Thus the flatness of
the spectrum is preserved as χH depends only mildly on k (χH ∝ k β ) and in this model
the above mentioned fine-tunig problem of the inflaton potential parameter is solved as
the vacuum energy ∆ can assume any value, including those that allow to fulfill the COBE
bound (1.35), depending on the value χ e of the inflaton at the end of inflation.
The same line of reasoning can be applied to the pre-big bang scenario. During the
pre-big bang phase spin-2 gravitational perturbations, for instance, fulfil (A.20) with a(η)
replaced by aE (η), the Einstein frame scale factor, given by
1/(p−1)
η
,
aE = η1
η < η1 < 0 ,
(A.35)
which is independent of δ (δ is defined in (3.11)) and from now on we specialize to p = 3.
The normal mode solutions of (A.20) are proportional to Hankel function H (2,1) (also
called Bessel functions of the third kind)
r
1
πkc η (2)
fk =
H0 (kc η) .
(A.36)
a(η)
2
(2,1)
The Hankel functions Hν
(Yν ) kind by
are defined in terms of Bessel functions of first (J ν ) and second
Hν(2,1) = Jν ∓ iYν ,
(A.37)
and their asympotic expansion is [148]
lim H (2,1) (x)
x→∞ ν
=
r
2 ∓ix
e
,
πx
(A.38)
up to a ν-dependent phase. Thus at Hubble scale crossing, which is at k c η ∼ 1, we have
from (A.18)
h|δφk |2 i ' h|δhk |2 i ∼ η1 ,
(A.39)
147
A.2 Density perturbations
where the constant η1 is the time of transition from the PBB to the FRW radiation
dominated phase and we thus assume it to be of order of the string scale. According to
(A.24) we obtain
Ph , (k) ∼ (kc /kc1 )3 ,
(A.40)
where kc1 ∼ η1−1 is the maximum comoving momentum.
Let us consider now the general case of a field ψ whose normal modes solve (A.20)
with a ∼ (η/η1 )α . It will be
p
fψk ∼ (η/η1 )−α kc ηHν(2) (kc η)
(A.41)
with ν = (α(α − 1) + 1/4)1/2 = |α − 1/2|. The quadratic fluctuations at Hubble scale
crossing (kc η ∼ 1) will be given by (A.18)
−1
h|δψk |2 ik=H ∼ k1c
(kc /kc1 )2α−1 .
(A.42)
As explained in [149] this is not the end of the story as we have also to consider the
fluctuations in the momentum conjugate to ψ, namely π = a 2 ψ 0 . If ψ satisfy eq.(A.20),
π satisfy an analogous equation with with a replaced by a −1 , which gives the πk normal
modes
p
(A.43)
fπk ∼ (η/η1 )α−1 kc1 kηHµ(2) (kη) + . . . ,
with µ = |α + 1/2| where . . . stand for other terms which are not more important than
the displayed one at Hubble scale crossing, leading to
h|δπk |2 ik=H ∼ k1c (kc /kc1 )1−2α .
(A.44)
This result implies that Pπ (Pψ ) is the relevant quantity if 2α − 1 > 0 (1 − 2α > 0), as it
has more power at scales k < k1c . Then the relevant power spectrum turns out to be
Pψ,π ∝ k 3−2|1/2−α| .
(A.45)
Moreover we expect that zero point vacuum fluctuations satisfy
ha−2 |δπ|2 i = ha2 |(∇δψ)|2 i ,
(A.46)
which is consistent with our result.
This procedure is applicable also to the case that the exact analitic form of the normal
mode is not known, as its dependence on k can be inferred as in (A.42) and (A.44) and
the perturbations at Hubble scale crossing in different phases are related as in (1.34). If
the explicit solution is known a straightforward matching between the normal modes in
two different cosmological phases can be obtained as in [150]. The results for the spectral
power of different species of massless excitations of the heterotic string theory compactified
to D = 4 are summarized in tab. A.1.
A remark is needed to justify our semiclassical approximation, involving field quantization on curved spacetime, which seem to have poor theoretical basis. The approximation can be considered reliable as quantum effects are usually importatnt at atomic
scale rBohr ∼ 10−8 mm, whereas quantum gravity effects are dominant at the Planck scale
lP l ∼ 10−32 mm (or at least at 10−12 mm in the most optimistic model with low string
scale, as explained in sec. 7.1), thus allowing a few order of magnitude-wide window to
which our semiclassical approach can be applied.
148
A.3
A.3.1
Appendix A
Particle perturbations
Quantum description
We can analyze further the issue of fluctuations in an expanding Universe. This issue is of
cosmological interest as the background Universe evolution undergoes a change of phase,
thus implying that the solutions to the Klein-Gordon equation change. For instance if we
allow a sudden change of phase at η = η 1 from the deSitter phase described by (A.21) to
the radiation dominated one described by the scale factor
a(η) =
η − 2η1
,
HdS |η1 |
η > η1 ,
(A.47)
equivalent to a(t) ∝ t1/2 , and whose normal modes are
fk (η, x) =
1 ikη−ikx
e
,
a(η)
(A.48)
the Bogolubov coefficients to pass from modes (A.22) to the above ones are
|βk |2 =
1
a4 (η1 )H 4
.
=
4
4k 4
4k 4 η1
(A.49)
In the case of pre-big bang, see sec. 3.1, the equation of motion for spin-2 gravitational
perturbations δh, for 3 spatial dimensions is
1
ã00 δh
2
00
2
¨
˙
˙
δh + 3H δh − φ̇δh + k δh = 3 (δhã) + kc −
= 0,
(A.50)
ã
a ã2
where ã = ae−φ/2 and φ denotes the dilaton as usual. Also other kinds of fluctuations can
be studied in the PBB scenario, and the method is completely analogous as they satisfy
an equation identical to the above with a convenient ã which generically will be a function
of the scale factor, the dilaton and the internal moduli β i .
As already shown in the previous section, see (A.41), for gravitons during pre-big bang
(p = 3) the normal mode solutions are proportional to Hankel functions of index zero.
The Bogolubov coefficient for the transition to the radiation dominated phase can be
computed as in [49] but instead of showing the exact computation, we prefer to expose a
simple classical approximation which does not fail in getting the core of the result.
A.3.2
Classical description
We are interested to see how normal modes of different phases are projected over one
another. Let us consider the perturbation equation (A.50). Once k c2 ã00 /ã it admits the
approximate solutions (dropping the dependence on x)
δh1,2 (η) =
1 ±ikη
e
,
ã(η)
(A.51)
which are also the exact solutions in the radiation dominated phase, as in that case
ã00 (η) = 0 for all physical cases (see tab. A.1 for a list of ã’s, remember that the dilaton is
frozen in the radiation dominated phase).
149
A.3 Particle perturbations
In the opposite regime kc2 ã00 /ã the approximate solutions are [151]
Z η
dη 0
ψ1 (η) = cost ,
ψ2 (η) =
.
ã2 (η 0 )
(A.52)
As already explained in sec. A.2, during an inflationary expansion or an accelerated
contraction (i.e. when ä has the same sign of ȧ) physical lengths tend to become bigger
than the Hubble length as d(a/H −1 )/dt = ä, whereas during the decelerated radiation
dominated era H −1 ∼ t ∼ a2 , implying that the Hubble length grows faster than physical
lengths. Qualitatively this is also true for the modified Hubble length H̃ ≡ ã0 /ã2 .
Assuming a(η) ∝ η α , while the perturbation wavelength is shorter then the Hubble
length (kc > a0 /a) its amplitude is decreasing as 1/a, see solution (A.51), and when it
is bigger then the Hubble length the amplitude stays frozen, solution ψ 1 of (A.52), or
behaves like ψ2 ∝ a1/α−2 , whichever dominates.
Hence, denoting ãout (ãin ) the modified scale factor at the epoch at which the perturbation wavelength crosses the (modified) Hubble length to become bigger (smaller) than
it, the fact that they enter the regime (A.52) instead compared to the usual (A.51) means
that they are effectively amplified by a factor ã out /ãin if solution ψ1 is dominating or
ãin /ãout if ψ2 is dominating. This semiclassical analysis thus suggests that [149]
ãout ãin
,
.
(A.53)
βk ' Max
ãin ãout
Here considering the momentum conjugate to the fundamental perturbation field yields
no new result.
In particular for a transition from a phase to a phase characterized by ã ∼ η α to a
phase ã ∼ η β , remembering that at Hubble scale crossing k c η ∼ 1 we have
βk ' Max
(k/k1 )−β (k/k1 )−α
,
(k/k1 )−α (k/k1 )−β
=
k
k1
|α−β|
,
(A.54)
being k1 the maximum amplified momentum, corresponding roughly to the (square root
of the) maximum curvature scale achieved in the cosmological evolution.
A useful quantity to estimate the amount of energy density stored in perturbations is
[45]
Ωper (k) '
1 k3
1 dρper (k)
=
k|βk |2 × Npol ∝ k 4−2|α−β| ,
ρC d ln k
ρC 2π 2
(A.55)
where Npol is the number of polarization avalaible for the involved field (for instance
Npol = 2 for the graviton and Npol = 1 for the dilaton). Using (A.49) and remembering
the definition of ρC (1.13), in the standard inflationary case (A.55) for gravitational waves
reduces to
8
k14
HdS 2
1
2
−4 2
h Ωgw (k) '
∼ 10 deSitter ,
(A.56)
2 m2 1 + z
3π 2 HdS
MP l
eq
Pl
−1 factor as during the matter domiwhere we posit k1 = HdS and we accounted for a zeq
nated period the energy density in gravitational waves redshifts as a −4 whereas the critical
150
Appendix A
density as a−3 . A value of < 1 takes account of a smearing of the transition from deSitter
to radiation domination (we can assume as a typical value . 10 −1 ). Using (A.54) for
the pre-big bang scenario (α = 1/2, β = 1) [79] we have for gravitational waves
8
k14
Ωgw (k) '
3π Hs2 MP2 l
k
k1
3
1
∼ 10−4 2
1 + zeq
2 Hs
MP l
k
k1
3
PBB ,
(A.57)
where is defined as k1 = Hs and Hs denotes the Hubble scale at PBB-Radiation dominated transition, which must be of the order of the string scale and it is redshifted to the
present frequency
f1 '
Hs
2π
λs /
teq
1/2
aeq
' 4 × 107 kHz
a0
Hs
0.15MP l
1/2
.
(A.58)
Actually for modes that cross the Hubble length in the matter dominated phase the
spectrum is different as in that phase ,
a(η) ∝ η 2
(A.59)
and then for k < keq the spectrum gets corrected by a factor (k/k eq )−2 , being keq the
wavenumber of the perturbation crossing the Hubble scale at the epoch of matter-radiation
equality, thus giving a kink to the spectrum in the very low frequency part for
k < keq ' Heq ' 102 H0 ' 3 × 10−15 Hz .
(A.60)
In both cases the spectrum is cut off at k = k 1 as for higher frequencies the condition
kc a00 /a is never matched and then they are not amplified, whereas the lower cutoff
is given by the the wavenumber corresponding to the present Hubble scale, as bigger
wavelengths are frozen and do not contribute to energy density.
Comparing (A.45) with (A.55), we note that the spectral slope of Ω per and Pper are
equal if the perturbations re-enter the (modified) Hubble length when ã ∝ η (β = 1), as
it is for gravitons and axions provided that during PBB ã(η) ∝ η α with α ∈
/ (1/2, 1).
In heterotic string theory there are many other fields that can be amplified, as summarized in tab. A.1, taken from [51]. These fields are derived from the low energy effective
2 ω , where
action (2.125), with dB substituded by dB − κ 210 /g10
3
2
ω3 = Tr A ∧ dA + A ∧ A ∧ A ,
3
(A.61)
A is the gauge field potential and with the decomposition

GM N = 

BM N = 
Gµν + Gmn Vµm Vνn Gmn Vµn
Gmn Vνm
Bµν
−Wνm +
Bmn Vνn
Gmn
Wµm −

,
Bmn Vµn
Bmn
(A.62a)

.
(A.62b)
151
A.3 Particle perturbations
Particle
ã
d ln ã
d ln η
d ln Ω
d ln k
d ln P
d ln k
Gravitons
ae−φ4 /2
1/2
Axions
aeφ4 /2
5δ−1
2(1−δ)
Het. gauge fields
e−φ4 /2
1−3δ
2(1−δ)
Vµa
e−φ4 /2+βa
2β−3δ+1
2(1−δ)
Wµa
e−φ4 /2−βa
1−3δ−2ζa
2(1−δ)
Bab
ae−φ4 /2+βa +βb
3
3−7δ 4 − 2 2(1−δ) 1−3δ 4 − 2 2(1−δ)
a −3δ+1 4 − 2 2ζ2(1−δ)
a
4 − 2 1−3δ−2ζ
2(1−δ) 3


 1−3δ 
3 − 2  1−δ 


 δ 
3 − 2  1−δ



 a −δ 
3 − 2  ζ1−δ



 a
3 − 2  δ+ζ
1−δ 
1
2
+
ζa +ζb
1−δ
a +ζb
3 − 2 ζ1−δ
+ζb
3 − 2 ζa1−δ
Table A.1: Spectral slopes of energy density Ω and power spectrum P of the amplified
perturbations in a Universe undergoing a transtion from a PBB phase to a radiation
dominated era. It is also displayed the field ã entering the perturbation equation (A.50)
expressed in terms of the string scale factor a, the 4-dimensional dilaton φ4 and the
internal moduli βi . ζi > 0 as internal dimensions are contracting, see (3.11). The fields
are the bosonic massless modes of heterotic string theory corresponding to N = 1
SUGRA coupled to N = 1 SYM in D = 10 compactified to D = 4. From [51].
The pseudoscalar axions a is defined as the 4-dimensional dual to the antisymmetric
NS-NS B-field according to
H µνρ ≡ µνρσ eφ4 ∂σ a ,
(A.63)
and Vµa and Wµa are Kaluza-Klein gauge deriving respectively from the metric and the
antisymmetric tensor. The axion σ owns is name to the fact that it couples linearly to the
gauge and Kaluza-Klein vector field strengths through a µνρσ Fµν Fρσ term, and also the
internal components of the antisymmetric 2-tensor B mn are often called axions because
they exhibit an analog coupling to the metric Kaluza-Klein gauge field strengths. However,
despite their name and the fact that they couple (gravitationally) to the gauge topological
charge, none of them have to be necessarily identified with the “invisible axion” which is
involved in the solution of the strong CP problem.
We have then seen as a variety of spectral slopes for perturbation is possible in the
PBB scenario, depending on the dynamics of the internal dimensions. Amplifications
of the vacuum fluctuations of other massless fields is also possible by taking into account
different string theories than the heterotic one, see for instance [152] for a type IIB analysis.
Appendix B
The moduli problem
The analysis of the previous section applies also to generic moduli, denoted by χ, i.e. scalar
fields with gravitational interaction, and gravitinos (even if for strictly massless gravitinos
there is no gravitational production as they are conformally coupled [153]) which will be
gravitationally produced by the accelerated cosmic expansion and which will be present at
the beginning of the radiation era. As no massless field apart from the photon is known at
present the moduli must acquire a mass, thus potentially arising the usual moduli problem.
Gravitationally interacting particle with mass m χ have a decay width
Γχ ∼
m3χ
.
MP2 l
(B.1)
This implies that depending on their mass, moduli will decay before nucleosynthesis,
mχ & 104 GeV, between nucleosynthesis and today, 100 MeV. m χ . 104 GeV or still
have to decay today, mχ . 100 MeV. The abundance of any particle species χ can be
conveniently parametrized by
Yχ ≡ nχ /sr ,
(B.2)
being nχ its density and sr ∼ (ρr )3/4 the entropy of the cosmological fluid, which is
sensible to relativistic degrees of freedom only. As n χ ∝ sr ∝ a−3 , Yχ is constant during
the expansion.
Once the moduli contribution to the cosmological energy density ρ χ ∝ a−3 , which we
assume for the moment to happen at a temperature T ∼ m χ , it will start to decrease less
than the radiation energy density eventually dominating the Universe when the expansion
rate reaches the value Hdom given by
Hdom = Y 2
m2χ
,
MP l
(B.3)
and at its decay, which happens roughly when H ∼ Γ χ ≡ Hrh , it will release an amount
of entropy sχ which can be conveniently parametrized as
sχ − s r
sχ
∆s
≡
'
'
sr
sr
sr
Hdom
Hrh
1/2
' Yχ
152
MP l
mχ
1/2
∼ 106 Yχ
mχ −1/2
, (B.4)
104 GeV
153
The moduli problem
being sr the previously present entropy of the Universe and in the numerical estimate we
took account of the fact that sr ∼ g(T )T 3 and g(Tdom )/g(Trh ) ∼ 10.
If the moduli mass is high enough that they decay before nucleosynthesis, the only phenomenological bound may come from baryosynthesis, i.e. it should be checked that the
reheating temperature after the moduli decay is not too low so to allow a new baryosynthesis
m
3/2
χ
.
(B.5)
Trh ' (Hrh MP l )1/2 = TeV
105 TeV
For light long-lived moduli, mχ < 100MeV, the bound that they not affect the expansion rate at the nucleosynthesys epoch is that
Yχ <
0.1MeV
mχ
for mχ < 100MeV .
(B.6)
which is at most Yχ < 10−3 . If the moduli are stable the bound apply to any mass, even
if it is unlikely that very heavy moduli can be stable.
The most relevant bound has to be imposed in the case of intermediate mass range,
for which moduli decay between nucleosynthesis and the present day. Indeed too high
moduli abundance will create too many photons from their decay which may dissociate
deuterium, lowering its primordial abundance below observations unless Y χ < 10−13 , say,
for a safe limit [60, 61].
Abundances from gravitational particle production violate this bound as they predict
3/4
Yχ ∼ Ωχ /g∗ ∼ 10−3 − 10−4 , where g∗ accounts for the number of relativistic degrees of
freedom [62]. The way out of this problem is to assume that at some point entropy is
released in the Universe so that Yχ decreases massively. This may happen for instance
as a massive species starts to dominate the Universe and eventually decays into photons
and other relativistic particles, as explained above, but the amount of entropy (B.4) is not
enough.
To improve the situation a different scenario can be proposed in which the contribution
ρχ of the χ species starts behaving like ρ χ ∝ a−3 not when the temperature drops below
its mass, but when the expansion rate becomes of the order of its mass (H ∼ m χ ). This
is a natural assumptions as it can be seen for instance by considering a Klein-Gordon-like
equation of motion for the moduli in an expanding Universe
χ̈ + 3H(t)χ̇ + m2χ χ = 0 ,
(B.7)
which has oscillatory solutions as soon as the expansion rate drops below m χ (we assumed
for the potential V = m2χ χ2 ). In this oscillatory regime ρχ and the pressure pχ of the
modulus are
ρχ =
1 2
χ̇ + m2χ ,
2
pχ =
1 2
χ̇ − m2χ ∼ 0 ,
2
(B.8)
implying that ρχ ∝ a−3 (t) like for non relativistic matter. As there is no reason to expect
that at early epoch the modulus will sit at the bottom of its potential, when for instance
the typical energy of the Universe E ∼ (HM P l )1/2 > mχ , we will assume that at the
beginning of the oscillations it is displaced from the minimum by an amount χ 0 . After the
154
Appendix B
oscillations start the modulus will not contribute to the entropy of the Universe, which
lays in relativistic particle only, and it will dominate over the background radiation density
when the cosmic scale factor reach the value a dom given by
adom = aχ
MP l
χ0
2
,
(B.9)
where aχ denotes the scale factor at the epoch of the turning on of the oscillations, i.e.
when H = mχ . At tdec ∼ 1/Γ the modulus will decay and it will reheat the Universe by
releasing a huge amount of entropy decaying into relativistic particles and the scale factor
at this reheating epoch is
arh = adom
χ0
MP l
8/3 mχ
MP l
−4/3
,
(B.10)
and thus the entropy release can be measured in this case by
∆s
srh
'
'
sr
sr
arh
adom
3/4
' 10
14
mχ −1 χ0 2
.
104 GeV
MP l
(B.11)
After reheating the primordial abundances of the moduli can be diluted to value as low
(rh)
as Yχ
given by
Yχ(rh) =
−1
m
Yχ
χ
,
∼ 10−14
∆s
106 GeV
(B.12)
thus solving the moduli problem.1
We note that for inflation the moduli problem is far less severe, as in that case the
inflation scale HdS is not necessarily tied to the Planck mass, so that primordial moduli
abundances, being proportional to (H dS /MP l )3/2 can be far smaller than in the PBB case,
where the relevant scale is the string one, which is linked within one order of magnitude
to the Planck scale, at least in the heterotic string framework.
1
This mechanism works equally well in diluting the potentially dangerous black holes that may form as
a consequence of Ωper approaching 1. The fact that black holes evaporate solve only partially the problem
of a black hole dominated Universe, as in the evaporation they may recreate dangerous moduli.
Appendix C
Superstring
The action (2.38) is the gauge fixed form of the action
SSP
1
=−
4πα0
Z
dτ dσe
α0 M α
ψ̄ ρ ∇α ψM + 2χ̄α ρβ ρα ψ M ∂β XM +
2
(C.1)
1
M
β α
(2)
αβγ
ψ̄M ψ χ̄α ρ ρ χβ + R + χ̄α ρ ∇β χγ ,
2
γ ab ∂a X M ∂b XM +
where the terms involving the two-dimensional gravitational degrees of freedom (the graviton and the gravitrino) have been introduced. We denoted by e ab the zweinbein, by e its
determinant and by χα the gravitino. The gauge choice eab = δba and χα = 0 makes the
former (C.1) to collapse into (2.38).
The previous (C.1) has a superconformal symmetry, the supersymmetric extension of
the conformal symmetry of bosonic string. The conservation of the supercurrent (2.40)
emerges as the gravitino equation of motion much like the Virasoro costraint is derived
from the two-dimensional graviton equation of motion.
The super-Virasoro algebra can be written in terms of the Fourier modes of the energymomentum tensor Lm and of those of the supercurrent Gm and it is given by
c
[Lm , Ln ] = (m − n)Lm+n + (m3 − m)δm,−n ,
12
c
{Gm , Gn } = 2Lm+n + (4m2 − 1)δm,−n ,
12
m − 2n
[Lm , Gn ] =
Gm+n .
2
(C.2a)
(C.2b)
(C.2c)
The L, G mode expansions for the X, ψ theory are
1 X
1X M
M
: αm−n αM n : +
(2r − m) : ψm−r
ψM r :
2
4
n∈Z
r∈Z(+1/2)
X
M
Gr =
αn ψM r−n .
Lm =
(C.3a)
(C.3b)
n∈Z
The ghost and superghost theory satisfy an algebra analogous to (C.2) and the relative
155
156
Appendix C
Lg , Gg mode expansions are
1
(m + 2r) : βm−r γr : ,
2
n∈Z
r∈Z(+1/2)
X 1
g
Gr = −
(2r + n)βr−n cn + 2bn γr−n .
2
Lgm =
X
(m + n) : bm−n cn : +
X
(C.4a)
(C.4b)
n∈Z
The superghost energy momentum tensor in the bosonized version is
Tsg = ∂ϕ∂ϕ − i∂ 2 ϕ .
(C.5)
Appendix D
Theta functions
The theta functions θ1..4 can be represented as summations
∞
X
θ3 (ν, τ ) =
θ4 (ν, τ ) =
θ2 (ν, τ ) =
qn
2 /2
zn ,
n=−∞
∞
X
2 /2
zn ,
(D.1b)
2 /2
z n−1/2 ,
(D.1c)
(−1)n q n
n=−∞
∞
X
n=−∞
∞
X
q (n−1/2)
θ1 (ν, τ ) = i
(D.1a)
(−1)n q (n−1/2)
2 /2
z n−1/2 ,
(D.1d)
n=−∞
where z ≡ e2πiν and q ≡ e2πiτ . They also have a product representation
θ3 (ν, τ ) =
θ4 (ν, τ ) =
∞
Y
m=1
∞
Y
m=1
(1 − q m )(1 + zq m−1/2 )(1 + z −1 q m−1/2 ) ,
(D.2a)
(1 − q m )(1 − zq m−1/2 )(1 − z −1 q m−1/2 ) ,
(D.2b)
∞
Y
θ2 (ν, τ ) = 2q 1/8 cos πν
θ1 (ν, τ ) = 2q 1/8 sin πν
(1 − q m )(1 + zq m )(1 + z −1 q m ) ,
(D.2c)
(1 − q m )(1 − zq m )(1 − z −1 q m ) .
(D.2d)
m=1
∞
Y
m=1
Their modular transformations are
θ3 (ν, τ + 1) = θ4 (ν, τ ) ,
(D.3a)
θ4 (ν, τ + 1) = θ3 (ν, τ ) ,
(D.3b)
θ2 (ν, τ + 1) = e
iπ/4
θ2 (ν, τ ) ,
(D.3c)
θ1 (ν, τ + 1) = e
iπ/4
θ1 (ν, τ )
(D.3d)
157
158
Appendix D
and
2
θ3 (ν, −1/τ ) = (−iτ )1/2 eiπν τ θ3 (ντ, τ ) ,
θ4 (ν, −1/τ ) = (−iτ )
1/2 iπν 2 τ
e
θ2 (ν, −1/τ ) = (−iτ )1/2 e
θ1 (ν, −1/τ ) =
(D.4a)
θ2 (ντ, τ ) ,
(D.4b)
iπν 2 τ
θ4 (ντ, τ ) ,
1/2 iπν 2 τ
−i(−iτ ) e
θ1 (ντ, τ ) .
(D.4c)
(D.4d)
They satisfy the identities
θ34 (0, τ ) − θ44 (0, τ ) − θ24 (0, τ ) = 0 ,
(D.5a)
θ1 (0, τ ) = 0 .
(D.5b)
The Dedekind eta function is
η(τ ) = q 1/24
∞
Y
m=1
(1 − q m ) =
∂ν θ1 (0, τ )
2π
1/3
,
(D.6)
whose modular transformations are
η(τ + 1) = eiπ/12 η(τ ) ,
η(−1/τ ) = (−iτ )
1/2
η(τ ) .
(D.7a)
(D.7b)
Another useful identity is
X
n∈Z
e
−πtα0 n2 /(2R2 )
=
r
2R2 X −2πw2 R2 /(tα0 )
e
.
α0 t
w∈Z
(D.8)
Appendix E
Supersymmetry transformation on
PBB solutions
The notation here is as follows: capitol latin letters run from 0 to 9, small latin indices
i, j, k are 1, 2, 3, small latin indices m, n, p run from 4 to 9. H = dB, which is correct a
the leading order in α0 , or on a background with topologically trivial gravity and gauge
field configuration.
Setting to zero all fermionic expectation values the boson supersymmetry variations
are identically satisfied whereas for fermion variations we have
1
1
1
δψM = DM η − ΓM 6 ∂φη + ΓM H − HM ,
16 √
96
8
√
2
2
δλ = −
6 ∂φη +
Hη .
8
48
(E.1a)
(E.1b)
The condition for hδλi = 0 is simply
6 6 ∂φη = Hη ,
(E.2)
which, substituted into (E.1a), leads to
1
δψM = DM η − HM .
8
(E.3)
We now restrict ourselves to the ansätz
ds2 = −dt2 + a2 (t)hij dxi dxj + gmn (t, y)dy m dy n ,
(E.4)
which is the most general compatible with the maximal symmetry of the 3-space except
for the dependence of the scale factor a from the internal coordinates, which we do not
allow (zero mode approximation). h ij is the metric of a maximally symmetric 3-space.
We recall that in general relativity the condition for a general tensor T µν... to be
invariant with respect to a transformation of coordinates
xµ → x0µ = xµ + ξµ
(E.5)
generated by the Killing vector ξµ can be cast, to first order in , into the following form
δµσ T ρ ν . . . + δνσ T µ . . .ρ = δµρ T σ ν . . . + δνρ T µ . . .σ
159
(E.6)
160
Appendix E
if ξµ is a Killing vector tangent to a maximally symmetric space. In particular for a tensor
completely antisymmetric in n indices which are tangent to a maximally symmetric space
of dimension N the previous (E.6) implies, see [154],
(N − n)Hijk . . . = 0 ,
| {z }
(E.7)
n indices
where the tensor can possibly have an uncostrained number of additional indices not tangent to the maximally symmetric subspace. This condition turns out to be very restrictive
for the possible tensors that can survive. We then specialize to the ansätz (E.4) for the
metric and for the antisymmetric tensor to
√
(E.8)
Hijk = c hijk ,
where h ≡ det hij and HiM N = 0 unless M, N ∈ 1, 2, 3 (with M 6= N ). Moreover the
Bianchi identity dH = 0 has to be imposed which forces the parameter c in (E.8) to
be effectively constant and we assume that none of the field depends on the internal
dimensions.
Taking into account derivatves of H M N P , the only components who survive after imposing (E.8) and the Bianchi identity are D 0 Hijk , Di H0jk ∝ ȧ/aHijk , Dm Hnpq ∝ Γ0mp Hnp0
(here Γ0mp is a Christoffel connection and not a gamma matrix).
In this context the integrability condition (4.9), that we rewrite here
2RM N P Q ΓP Q + (DN HM ) − (DM HN ) − HM R Q HN RS ΓQS η = 0 ,
(E.9)
becomes, for M = 0 and N = i,
ȧ
ä
−2 Γ0i + 2 Hijk Γjk = 0 .
a
a
(E.10)
Observing that (Γ0i )−1 ∝ Γ0i and that Γ0i × Hijk Γjk is a sum of antihermitian operator
which is then an antihermitian operator, it cannot have real eigenvalue except the null
one, implying that
ä = 0 .
Considering instead M = i, N = j we have
2
ȧ
k
c2
ȧ
2
+
Γ
+
Γij − 2 Hijk Γk0 = 0
ij
2
2
6
a
a
a
a
(E.11)
(E.12)
which combined with the previous (E.11) excludes any possibility of non trivial cosmological evolution but the Milne Universe, which is a reparametrization of the Minkowski
space (a(t) = t in a spatially open, k = −1, Universe defines the Milne Universe).
Thus to respect supersymmetry under our starting assumptions (essentially homogeneity and isotropy in 3 spatial dimensions) a Minkowskian (3+1)-dimensional space seems
unavoidable.
List of Figures
1.1
1.2
1.3
2.1
2.2
3.1
3.2
3.3
3.4
4.1
4.2
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
6.1
6.2
6.3
7.1
CMBR anisotropy spectrum from Boomerang data . . . . . . . . . . . . . .
Primordial plasma driven oscillation in the adiabatic and isocurvature case
Evolution of the cosmic scale factor in a pure radiation Universe . . . . . .
String scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
One loop string diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scale factor duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gravitational wave spectrum by PBB inflation . . . . . . . . . . . . . . . .
Non singular pre-big bang solution . . . . . . . . . . . . . . . . . . . . . . .
Constraints on the string cosmological model parameter by overproduction
of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A symmetry breaking potential for the condensate σ(t) . . . . . . . . . . . .
The evolution of the field σ(t). . . . . . . . . . . . . . . . . . . . . . . . . .
H, ϕ̇ σ̇ vs. t for the classical action . . . . . . . . . . . . . . . . . . . . . . .
H vs. ϕ̇ for the classical action . . . . . . . . . . . . . . . . . . . . . . . . .
Evolution of ϕ̇, H and σ with loop corrected Kähler potential . . . . . . . .
Evolution of ϕ̇, H and g 2 = eϕ with loop corrected Kähler potential . . . .
H vs. ϕ̇ with the loop-corrected Kähler potential . . . . . . . . . . . . . . .
HE vs. t with the loop-corrected Kähler potential . . . . . . . . . . . . . . .
Evolution of H, ϕ̇ and σ̇ vs. t (loop corrections) . . . . . . . . . . . . . . .
The evolution in the (H, ϕ̇) plane (loop corrections) . . . . . . . . . . . . .
H, ϕ̇ and g 2 against cosmic time (loop corrections) . . . . . . . . . . . . . .
Initial condition for a good solution . . . . . . . . . . . . . . . . . . . . . . .
ϕ̇E and HE against string time (loop corrections) . . . . . . . . . . . . . . .
Phase diagram of M-theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
Evolution of Zϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fixed point solutions in the A, D plane . . . . . . . . . . . . . . . . . . . .
Wrong-way solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Graceful exit enforced by GSL on generic solutions . . . . . . . . . . . . . .
One loop open string diagram with two external states . . . . . . . . . . . .
161
3
4
7
29
44
57
61
65
68
75
76
87
88
90
90
91
91
93
94
95
95
96
99
100
109
109
110
128
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