Slides - Agenda INFN

CHIRAL PERTURBATION THEORY
OF HYPERFINE SPLITTING
IN MUONIC HYDROGEN
FRANZISKA HAGELSTEIN
INSTITUT FÜR KERNPHYSIK, UNIVERSITÄT MAINZ, GERMANY
1
Proton Size Puzzle
[1]
[2]
[2]
[3,4]
[1] J. C. Bernauer et al., Phys. Rev. Lett. 105, 242001 (2010).
[2] P. J. Mohr, et al., Rev. Mod. Phys. 84, 1527 (2012).
[3] R. Pohl, A. Antognini et al., Nature 466, 213 (2010).
[4] A. Antognini et al., Science 339, 417 (2013).
seven standard-deviation discrepancy (7σ) !!!
µH
[RE
= 0.84087(39) fm]
CD 2015, 29.06.2015
CODATA 2010
[RE
= 0.8775(51) fm]
2
Franziska Hagelstein
Although extracted from the same data, the frequency value of the triplet resonance, nt, is slightly
more accurate than in (6) owing to several improvements in the data analysis. The fitted line widths
are 20.0(3.6) and 15.9(2.4) GHz, respectively, compatible with the expected 19.0 GHz resulting from
the laser bandwidth (1.75 GHz at full width at half
maximum) and the Doppler broadening (1 GHz)
of the 18.6-GHz natural line width.
The systematic uncertainty of each measurement is 300 MHz, given by the frequency calth
ibration
uncertainty arising from pulse-to-pulse
LS
fluctuations in the laser and from broadening
effects occurring in the Raman process. Other
thsystematic corrections we have considered are
HFS
the Zeeman shift in the 5-T field (<60 MHz),
AC and DC Stark shifts (<1 MHz), Doppler
shift (<1 MHz), pressure shift (<2 MHz), and
black-body radiation shift (<<1 MHz). All these
typically important atomic spectroscopy systematics are small because of the small size of mp.
The Lamb shift and the hyperfine splitting.
From these two transition measurements, we
can independently deduce both the Lamb shift
(DEL = DE2P1/2−2S1/2) and the 2S-HFS splitting
(DEHFS) by the linear combinations (13)
sequence of Eqs. 4. The other terms arise from
the proton structure. The leading finite size effect
−5.2275(10)rE2 meV is approximately given by
Eq. 1 with corrections given in (13, 17, 18).
Two-photon exchange (TPE) effects, including the
proton polarizability, are covered by the term
DETPE = 0.0332(20) meV (19, 24–26). Issues
related with TPE are discussed in (12, 13).
The comparison of DELth (Eq. 7) with DELexp
(Eq. 5) yields
between the muon and the proton magnetic moments, corrected for radiative and recoil contributions, and includes a small dependence of
−0.0022rE2 meV = −0.0016 meV on the charge
radius (13).
The leading proton structure term depends
on rZ, defined as
μH Spectroscopy
E
E
= 206.0336(15)
= 22.9763(15)
rE ¼ 0:84087(26)2exp (29)th fm TPE
LS ð8Þ
¼E
0:84087(39) fm
5.2275(10) (R /fm) +
0.1621(10) (RZ /fm) +
radiative, relativistic and recoil effects
E
(pol)
EHFS ,
rZ ¼ ∫d3 r∫d3 r′ r′ rE (r)rM (r − r′ )
,
ð10Þ
with rM being the TPE
normalized proton magLS
netic moment distribution. The HFS polariz-
with
with
E
= 0.0332(20),
(pol)
EHFS = 0.0080(26),
finite size effects:
proton structure
1
3
hns þ hnt ¼ DEL þ 8:8123ð2ÞmeV
4
4
− 3:2480ð2ÞmeV ð4Þ
hn − hn ¼ DE
two-photon
exchange (TPE) effects, Finite size effects are included in DE and
including
theinclude
proton
DE . The numerical terms
the cal- polarizability
s
t
HFS
L
HFS
culated values of the 2P fine structure, the 2P3/2
hyperfine splitting, and the mixing of the 2P
states (14–18). The
finite proton size effects on
exp
the 2P fine and hyperfine structure are smaller
LSbecause of the small overlap
than 1 × 10−4 meV
exp
between the 2P
wave functions and the nucleus. Thus, their
uncertainties arising from
HFS
the proton structure are negligible. By using
the measured transition frequencies ns and nt
in Eqs. 4, we obtain (1 meV corresponds to
E
= 202.3706(23) meV,
E
= 22.8089(51) meV
CD 2015, 29.06.2015
A. Antognini, et al., Annals Phys. 331 (2013) 127–145.
3
Franziska Hagelstein
Finite Size Effects
Fermi - Energy:
HFS:
8 Z↵ 1 +  1
3 a3 mM n3
with Bohr radius
a = 1/(Z↵mr )
EF (nS) =
EnS (LO + NLO) = EF [1
2 Z↵mr RZ ]
wi
NLO becomes appreciable in μH
Lamb shift:
Enl (LO) =
l0
2⇡Z↵ 1
2
R
3 ⇡(an)3 E
EnS (NLO) =
J. L. Friar, Annals Phys. 122 (1979) 151.
wave function
at the origin
2
RE
3
RE(2)
CD 2015, 29.06.2015
Z↵ 3
RE(2)
3
4
3n a
4
d
2
= 6 lim
G
(Q
)
E
2
2
Q !0 dQ
⇢
Z 1
48
dQ
2
2
=
G
(Q
)
E
⇡ 0 Q4
1 2
1 + hr iE Q2
3
RAPID COMMUNICATIONS
PHYSICAL REVIEW A 91, 040502(R) (2015)
Breakdown of the expansion of finite-size corrections to the hydrogen Lamb shift
in moments of charge distribution
Franziska Hagelstein and Vladimir Pascalutsa
Institut für Kernphysik, Cluster of Excellence PRISMA, Johannes Gutenberg-Universität Mainz, D-55128 Mainz, Germany
(Received 13 February 2015; published 20 April 2015)
We quantify a limitation in the usual accounting of the finite-size effects, where the leading [(Zα)4 ] and
subleading [(Zα)5 ] contributions to the Lamb shift are given by the mean-square radius and the third Zemach
moment of the charge distribution. In the presence of any nonsmooth behavior of the nuclear form factor at
scales comparable to the inverse Bohr radius, the expansion of the Lamb shift in the moments breaks down. This
is relevant for some of the explanations of the “proton size puzzle.” We find, for instance, that the de Rújula
toy model of the proton form factor does not resolve the puzzle as claimed, despite the large value of the third
Zemach moment. Without relying on the radii expansion, we show how tiny, milli-percent (pcm) changes in the
proton electric form factor at a MeV scale would be able to explain the puzzle. It shows that one needs to know
all the soft contributions to the proton electric form factor to pcm accuracy for a precision extraction of the proton
2
charge radius from atomic Lamb shifts.
the finite-size effects are not always expandable in the moments of
charge distribution
• a tiny non-smoothness of the electric form factor
DOI: 10.1103/PhysRevA.91.040502
GE (Q ) at scales
PACS number(s): 31.30.jr, 14.20.Dh, 13.40.Gp, 11.55.Fv
comparable to the inverse Bohr radius can break down this expansion
I. INTRODUCTION
E
The proton structure is long known to affect the hydrogen
1
spectrum, predominantly
by an upward shift of the S levels
FF(1)expressed in terms of the root-mean-square2(rms) radius,
E"
2P 2S
!
0
RE = ⟨r 2 ⟩E , ⟨r N ⟩E ≡ d r⃗ r NρE (⃗r ),
(1)
=
Z
dQ w(Q) G (Q )
with
A Lorentz-invariant definition of the above moments is
given in terms of the electric form factor (FF), GE (Q2 ), as
#
#
d
2
2
2
#
(Z↵mr )(6a)
,
⟨r ⟩E = −6 42 GE (Q )5
4
2
#
2 =0
w(Q) = dQ (Z↵) Qm
rQ
$
% )2 +
"⇡
∞
[(Z↵m
dQ
48
1
r
G2E (Q2 ) − 1 + Q2 ⟨r 2 ⟩E . (6b)
⟨r 3 ⟩E(2) =
4
π 0 Q
3
Q2
Q2 ]
4
At the current level of precision, the eH Lamb shift sees only
the LO term, while in µH the NLO term becomes appreciable.
An immediate resolution of the eH vs µH discrepancy (also
known as the proton size puzzle) was suggested by de
Rújula [9], whose toy model for proton charge distribution
2(Zα)4 m3r 2
#EnS (LO) =
RE ,
(2)
yielded a large Friar radius, capable of providing the observed
3n3
µH Lamb shift using the RE value from eH. Shortly after, this
where Z = 1 for the proton, mr is the reduced mass. The
model was shown to be incompatible with the empirical FF
proton charge radius has thus been extracted from the hydrogen
GE extracted from ep scattering [10,11]. In this work we find
(eH) and muonic-hydrogen (µH) Lamb shifts, with rather
that the µH Lamb shift in de Rújula’s model is not described
CD 2015, 29.06.2015
5
Franziska Hagelstein
contradictory results:
correctly by the standard formulas of Eqs. (2) and (4). The
of the proton charge distribution ρE . At leading order (LO)
in the fine-structure constant α, the nth S level is shifted by
(cf. [1])
one needs to know all the “soft” (below several MeV) contributions
to proton electric FF to pcm accuracy
Structure Effects through TPE
proton structure effects at subleading orders arise through multi-photon
processes
“elastic” contribution:
finite-size recoil, 3rd Zemach moment (Lamb shift),
Zemach radius (HFS)
“polarizability” contribution:
“inelastic” contribution, “subtraction” term (Lamb shift)
“blob” corresponds to doubly-virtual Compton scattering (VVCS):
T µ⌫ (q, p) =
✓
g µ⌫ +
+
µ⌫
or: TA (q, p) =
CD 2015, 29.06.2015
1
M
µ ⌫
q q
q2
µ⌫↵
◆
T1 (⌫, Q2 ) +
q↵ S1 (⌫, Q2 ) +
1
M2
1
M2
✓
pµ
◆✓
p·q µ
⌫
q
p
q2
µ⌫ 2
q + qµ
i µ⌫↵
i
✏
q↵ s S1 (⌫, Q2 ) + 3 ✏µ⌫↵ q↵ (p · q s
M
M
6
⌫↵
q↵
q⌫
µ↵
◆
p·q ⌫
2
q
T
(⌫,
Q
),
2
2
q
q↵ S2 (⌫, Q2 )
s · q p ) S2 (⌫, Q2 )
Franziska Hagelstein
Eur. Phys. J. C (2014) 74:2852
DOI 10.1140/epjc/s10052-014-2852-0
Regular Article - Theoretical Physics
Chiral perturbation theory of muonic-hydrogen Lamb shift:
polarizability contribution
Jose Manuel Alarcón1,a , Vadim Lensky2,3 , Vladimir Pascalutsa1
1
Cluster of Excellence PRISMA Institut für Kernphysik, Johannes Gutenberg-Universität, Mainz 55099, Germany
Theoretical Physics Group, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK
3 Institute for Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya 25, 117218 Moscow, Russia
2
Z1
Z
2
2
2
2
2
2
Q
2⌫
T
(⌫,
Q
)
(Q
+
⌫
)T
(⌫,
Q
)
↵m
1
2
Received: 6 December 2013
/
Accepted:
9
April
2014
/
Published
online:
29
April
2014
2
EnS
= 2014.
d⌫ is published
dq with open access at4Springerlink.com
n article
© The
Author(s)
i⇡ 3 This
Q (Q4 4m2 ⌫ 2 )
0
Eur. Phys. J. C (2014) 74:2852
Page 3 of 10 2852
1 The two-photon
Abstract TheFig.
proton
polarizability effect in the muoniccf. Table 1. The ‘elastic’ and ‘inelastic’ 2γ contributions are
exchange diagrams of elastic
scattering out as a prediction of baryon
hydrogen Lamblepton–nucleon
shift comes
well constrained by the available empirical information on,
calculated in this work in the
with corrections
zero-energy (threshold)
chiral perturbation
theory at leading order and our calcurespectively, the proton form factors and unpolarized struckinematics. Diagrams obtained
(a)
(b)
(c)
+3
from(pol)
these by crossing and
due to “elastic”
(2P
−
2S)
=
8
µeV.
This
result
is
lation yields !E
ture
functions.
The
‘subtraction’ contribution
must be mod-
time-reversal symmetry are
−1
drawn
consistent with included
mostbutofnotevaluations
based on dispersive sum
eled, and in principle one can make up
a modelFFs
where
the
proton
rules, but it is about a factor of 2 smaller than the recent result
effect is large enough to resolve the puzzle [8].
subtracted,
(f) observe that chiral perturbation theory
obtained in heavy-baryon chiral perturbation(d)theory. We also (e) In this work we
i.e., “polarizability”
find that the effect of !(1232)-resonance excitation on the
(χ PT) contains definitive predictions
for all of the above
5 ) proton structure effects, hence no modelLamb shift is suppressed, as is the entire contribution of the
mentioned O(αem
contribution alone
of course that χ PT is an adequate themagnetic polarizability; the electric polarizability
dominates. (h)ing is needed, assuming
(g)
(j)
ory of the low-energy nucleon structure. Some of the effects
Our results reaffirm the point of view that the proton structure
two scalar amplitudes:
Focusing on the O( p 3 ) corrections (i.e., the VVCS amplitude
were already assessed in the heavy-baryon variant of the theeffects, beyond ofthe
charge radius, are too small to resolve
the
corresponding to the graphs in Fig. 1) we have explicitly verµ
ν
CD 2015,
29.06.2015
Franziska Hagelstein
P P
ified that the resulting
NB amplitudes
the dispersive
µν
µν
2
2
2
2
ory (HBχ
PT),satisfy
namely:
Nevado and Pineda [11] computed
the
‘proton
radius puzzle’.
=
TPE in μH Lamb Shift: Polarizability Contribution
dddddddddd
dddddddddddddd
dddddddddddddddd
ddddddddddddddddddddddddddd
dddddddddddddddχdd
dddddddddddddddddddd
dddddddddddddddd
dddddddddddddddddddddd
ddχddddd
ddddddddddddddddddd
ddχdddddd
dddddddddddddddddd
dχddddd
dddddddddddddddddddd
ddd
ddd
ddd
ddd
ddd
ddddd
∆ddd
ddd
dd
dd
ddµddd
B𝜒Pt result is in good agreement with calculations based on dispersive sum rules
CD 2015, 29.06.2015
8
𝜒Pt Calculations for
TPE in μH Lamb Shift
Nevado & Pineda
HB PT
Alarcón et al.
B PT
(subt)
E2S
(inel)
E2S
(pol)
E2S
(el)
E2S
E2S
[2] A. Pineda, Physical Review C71 (2005) 065205.
[3] C. Peset, A. Pineda (2014).
[4] J. M. Alarcon, V. Lensky, V. Pascalutsa, Eur. Phys. J. C74 (2014) 2852.
Alarcón et al.
HB PT
3.0
5.2
8.2(+1.2
2.5 )
18.5(9.3)
10.1(5.1)
28.6
a prediction
[1] D. Nevado, A. Pineda, Phys. Rev. C77 (2008) 035202.
1.3
19.1
17.85
Peset & Pineda
HB PTa
26.2(10.0)
8.3(4.3)
34.4(12.5)
at LO and NLO (including pions and deltas)
Table 1: Summary of available PT calculations for the TPE correction to the
Lamb shift in µH. Energy shifts are given in µeV, M 1 is given as ⇥10 4 fm3 .
“polarizability” contribution:
2⌫ 2
T 1 (⌫, Q ) = T 1 (0, Q ) +
⇡
2
2
T 2 (⌫, Q ) =
⇡
2
CD 2015, 29.06.2015
2
1
⌫0
1
⌫0
2
⌫0 2Q
d⌫ 0 2
⌫ + Q2
0
d⌫
T (⌫
⌫0 2
0
T (⌫
0
0
, Q2 )
,
⌫2
subtraction function must be modelled!
, Q2 ) + L (⌫ 0 , Q2 )
⌫0 2 ⌫2
9
TPE in μH HFS
EHFS = [1 +
QED +
p
weak
Fermi - Energy:
+
with
TPE in HFS:
TPE
E
EFp
=
m
1
2(1 + {)⇡ 4 i
Z
1
0
FSE ] EF
FSE
d⌫
Z
dq
=
(Q4
EF (nS) =
Z
+
1
4m2 ⌫ 2 )
p
recoil
(
2Q
8 Z↵ 1 +  1
3 a3 mM n3
+
2
Q2
⌫
pol
2
S1 (⌫, Q2 ) + 3
⌫
S2 (⌫, Q2 )
M
)
• TPE effect on the HFS is
completely constrained by
empirical information
• a B𝜒Pt calculation of the
HFS in μH will put the
reliability of both 𝜒Pt and
dispersive calculations to the
test
CD 2015, 29.06.2015
⇢
Z 1
2
2
2
M
Q
F
(Q
)G
(Q
)
d⌫ 0
1
M
2
0
2
S1 (⌫, Q ) = 8⇡↵
+
g
(⌫
,
Q
)
1
4
2
2
0
2
2
Q
4M ⌫
⌫
⌫0 ⌫
⇢
Z 1
2
2
M
F
(Q
)G
(Q
)
d⌫ 0 g2 (⌫ 0 , Q2 )
2
M
2
S2 (⌫, Q ) = 8⇡↵ M ⌫
+
02
Q4 4M 2 ⌫ 2
⌫2
⌫0 2
⌫0 ⌫
10
Franziska Hagelstein
TPE in μH HFS
TPE
E
EFp
Z
1
Z
(
2
2
2Q
⌫
m
1
1
⌫
2
=
d⌫ dq 4
S1 (⌫, Q ) + 3 S2 (⌫, Q2 )
4
2
2
2
2(1 + {)⇡ i 0
Q
4m ⌫
Q
M
Z 1
Z 2⇡
Z ⇡ Z ⇡
m
2
=
dQ
d
d✓
d
sin
sin ✓ ⇥
4(1 + {)⇡ 4 0
0
0
0
⇢
Q
3iQ cos
2
2
2
⇥ 2
(2
+
cos
)S
(⌫,
Q
)
+
S
(⌫,
Q
)
1
2
Q + 4m2 cos2
M
)
“polarizability” / “inelastic” contribution:
⇢
1
M Q2 F1 (Q2 )GM (Q2 )
d⌫ 0
0
2
S1 (⌫, Q ) = 8⇡↵
+
g
(⌫
,
Q
)
1
02
2
Q4 4M 2 ⌫ 2
⌫
⌫
⌫0
1
2⇡↵
d⌫ 0
Born
2
2
2
= S1 (⌫, Q ) +
F2 (Q ) + 8⇡↵
g1 (⌫ 0 , Q2 )
0
2
2
M
⌫
⌫0 ⌫
⇢
1
2
2
M
F
(Q
)G
(Q
)
d⌫ 0 g2 (⌫ 0 , Q2 )
2
M
2
S2 (⌫, Q ) = 8⇡↵ M ⌫
+
02
Q4 4M 2 ⌫ 2
⌫2
⌫0 2
⌫0 ⌫
1
d⌫ 0
Born
2
0
2
= S2 (⌫, Q ) + 8⇡↵M ⌫
g
(⌫
,
Q
)
2
02
2
⌫
⌫
⌫0
2
2
CD 2015, 29.06.2015
11
1
Pol
Franziska Hagelstein
Page 3 of 10 2852
Eur. Phys. J. C (2014) 74:2852
Pion-Nucleon-Loop
=
Contribution
Fig. 1 The two-photon
exchange diagrams of elastic
lepton–nucleon scattering
calculated in this work in the
zero-energy (threshold)
kinematics. Diagrams obtained
from these by crossing and
time-reversal symmetry are
included but not drawn
cutoff dependence:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(j)
of two scalar amplitudes:
EHFS ê EFermi @ppmD
T µν (P, q) = −g µν T1 (ν 2 , Q 2 ) +
Pµ Pν
T2 (ν 2 , Q 2 ),
M 2p
(5)
with P the proton 4-momentum, ν = P · q/M p , Q 2 = −q 2 ,
P 2 = M 2p . Note that the scalar amplitudes T1,2 are even
functions of both the photon energy ν and the virtuality Q.
Terms proportional to q µ or q ν are omitted because they
vanish upon contraction with the lepton tensor.
Going back to the energy shift one obtains [12]:
30
"E nS
20
αem φn2 1
=
4π 3 m ℓ i
!
!∞
3
d q dν
0
In this work we calculate the functions T1 and T2 by
extending the Bχ PT calculation of real Compton scattering [26] to the case of virtual photons. We then split the
amplitudes into the Born (B) and non-Born (NB) pieces:
(B)
Ti = Ti
20
40
60
D1
.
QêM
(7)
The Born part is defined in terms of the elastic nucleon form
p
factors as in, e.g. [13,27]:
#
"
4π αem Q 4 (FD (Q 2 )+ FP (Q 2 ))2
(B)
2
2
T1 =
− FD (Q ) , (8a)
Mp
Q 4 −4M 2p ν 2
#
"
16π αem M p Q 2
Q2 2 2
(B)
2
2
T2 =
F (Q ) . (8b)
FD (Q )+
Q 4 − 4M 2p ν 2
4M 2p P
80
-10
DPol
(NB)
+ Ti
D2
100
(NB)
T1
=
(ν 2 , Q 2 )
(NB)
T1 (0,
2ν 2
Q )+
π
2
!∞
σT (ν ′ , Q 2 )
dν ′ ′2
,
ν − ν2
Fermi - Energy:
(9a)
ν0
(ν 2 , Q 2 )
1 Z↵ 1 + 
!∞ = ′ 2 2
′
2
EF (2S)
ν
Q σT (ν ′ , Q 2 )=
+ σ22.8054
2
L (ν , Q )
=
dν ′ ′23 a32 mM ′2
,
π
ν +Q
ν − ν2
(NB)
T2
meV
(9b)
ν0
(Q 2 − 2ν 2 ) T1 (ν 2 , Q 2 ) − (Q 2 + ν 2 ) T2 (ν 2 , Q 2 )
×
. (6)
Q 4 [(Q 4 /4m 2ℓ ) − ν 2 ]
10
Focusing on the O( p 3 ) corrections (i.e., the VVCS amplitude
corresponding to the graphs in Fig. 1) we have explicitly verified that the resulting NB amplitudes satisfy the dispersive
sum rules [28]:
In our calculation the Born part was separated by subtracting the on-shell γ N N pion loop vertex in the one-particlereducible VVCS graphs; see diagrams (b) and (c) in Fig. 1.
with ν0 = m π + (m 2π + Q 2 )/(2M p ) the pion-production
threshold, m π the pion mass, and σT (L) the tree-level cross
section of pion production off the proton induced by transverse (longitudinal) virtual photons, cf. Appendix B. We
hence establish that one is to calculate the ‘elastic’ contribution from the Born part of the VVCS amplitudes and
the ‘polarizability’ contribution from the non-Born part,
in accordance with the procedure advocated by Birse and
McGovern [13].
Substituting the O( p 3 ) NB amplitudes into Eq. (6) we
obtain the following value for the polarizability correction:
(pol)
"E 2S
= −8.16 µeV.
(10)
This is quite different from the corresponding HBχ PT result
for this effect obtained by Nevado and Pineda [11]:
(pol)
"E 2S (LO-HBχ PT) = −18.45 µeV.
(11)
We postpone a detailed discussion of this difference till
Sect. 4.
123
CD 2015, 29.06.2015
12
Franziska Hagelstein
𝜒Pt Results for
TPE in μH HFS
Fermi - Energy:
EF (2S) =
l
π0
N
⇡
EHFS
= 0.44 ± 0.04 µeV
Page 3 of 10 2852
Eur. Phys. J. C (2014) 74:2852
⇡N loops
EHFS
⇡ & ⇡N
EHFS
loops
Fig. 1 The two-photon
exchange diagrams of elastic
lepton–nucleon scattering
calculated in this work in the
zero-energy (threshold)
kinematics. Diagrams obtained
from these by crossing and
time-reversal symmetry are
included but not drawn
= 0.85 ± 0.42 µeV
=
= 1.29 ± 0.42 µeV
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(j)
of two scalar amplitudes:
T µν (P, q) = −g µν T1 (ν 2 , Q 2 ) +
dχddddd
ddddddddddddddddddddddd
dd
dd
dd
dd
dd
∆ddddddddµddd
CD 2015, 29.06.2015
Pµ Pν
T2 (ν 2 , Q 2 ),
M 2p
(5)
Focusing on the O( p 3 ) corrections (i.e., the VVCS amplitude
corresponding to the graphs in Fig. 1) we have explicitly verified that the resulting NB amplitudes satisfy the dispersive
sum rules [28]:
T1 (ν 2 , Q 2 )
with P the proton 4-momentum, ν = P · q/M p , Q 2 = −q 2 ,
!∞
′
2
P 2 = M 2p . Note that the scalar amplitudes T1,2 are even
2ν 2
(NB)
2
′ σT (ν , Q )
=
T
(0,
Q
)
+
dν
,
(9a)
1
functions of both the photon
ν and
the virtuality
′2 − ν 2
ν042509.
[1]energy
C. E.
Carlson,
et Q.
al., Phys. Rev. A83 π(2011)
µ
ν
ν
0
Terms proportional to q or q are omitted because they
(NB)
vanish upon contraction with
lepton
T2 (ν 2Rev.
, Q 2 ) A78 (2008) 022517.
[2]the C.
E.tensor.
Carlson, et al., Phys.
Going back to the energy shift one obtains [12]:
!∞
2 σ (ν ′ , Q 2 ) + σ (ν ′ , Q 2 )
ν ′ 2 QSoc.
2
T Opt. Eng.
L 6165 (2006) 0M.
[3] R. Faustov, et al., Proc.
SPIE
=
dν ′ Int.
,
(9b)
∞
′2 + Q 2
′2 − ν 2
!
!
π
ν
ν
αem φn2 1
3 [4] A. Martynenko et al., Nucl.
ν0 Phys. A703 (2002) 365–377.
"E nS =
d q dν
4π 3 m ℓ i
2 )/(2M ) the pion-production
0
ν0 =A71
mπ +
(m 2π + Q022506.
p
[5] A. Martynenko, Phys.with
Rev.
(2005)
(Q 2 − 2ν 2 ) T1 (ν 2 , Q 2 ) − (Q 2 + ν 2 ) T2 (ν 2 , Q 2 )
threshold, m π the pion mass, and σT (L) the tree-level cross
×
. (6)
section of pion production off the proton induced by transQ 4 [(Q 4 /4m 2ℓ ) − ν 2 ]
verse (longitudinal) virtual photons, cf. Appendix B. We
In this work we calculate the functions T1 and T2 by
hence establish that one is to calculate the ‘elastic’ conextending the Bχ PT calculation of real Compton scattertribution from the Born part of the VVCS amplitudes and
ing [26] to the case of virtual photons. We then split the
the ‘polarizability’ contribution from the non-Born part,
amplitudes into the Born (B) and non-Born (NB) pieces:
in accordance with the procedure advocated by Birse and
McGovern [13].
(B)
(NB)
.
(7)
Ti = Ti + Ti
into Eq.
(6) we
Substituting the O( p 3 ) NB amplitudes
13
Franziska
Hagelstein
(NB)
dddddddddd
ddddddddddddddddddddddd
dddddddddddddddddddd
dddddddddddddddddddd
ddd
1 Z↵ 1 + 
= 22.8054 meV
3 a3 mM
ddd
ddd
Dispersive Calculations
for TPE in μH HFS
Reference
Carlson et al.a
FF
AMT
AS
Kelly
MAMI
combinedb
RZ
[fm]
1.080
1.091
1.069
1.045
Z
[ppm]
7703
7782
7622
p
recoil
[ppm]
931
931
931
Faustov et al.c
Martynenko et al.d
Experiment
Dipole
1.022
7180
[1] C. E. Carlson, et al., Phys. Rev. A83 (2011) 042509.
[2] C. E. Carlson, et al., Phys. Rev. A78 (2008) 022517.
[3] R. Faustov, et al., Proc. SPIE Int. Soc. Opt. Eng. 6165 (2006) 0M.
[4] A. Martynenko et al., Nucl. Phys. A703 (2002) 365–377.
[5] A. Martynenko, Phys. Rev. A71 (2005) 022506.
pol
1
2
[ppm]
[ppm]
[ppm]
351(114) 370(112) 19(19)
353
353
470(104) 518
48
460(80)
58
514
E2S HFS
[ppm]
[meV]
6421(140) 22.8123
6498
22.8105
6338
22.8141
22.8187
22.8146(49)
FSE
22.8138(78)e
1.082(37)
22.8089(51)
a QED,
higher-order and other small corrections included in E2S HFS are taken from Marrad ).
tynenko. The Zemach term includes radiative corrections: Z = 2↵mr RZ (1 + Z
b slightly moved average of the selected form factors
c The calculation is based on experimental data for the nucleon polarized structure functions
obtained at SLAC, DESY and CERN.
d The calculation is based on experimental data for the nucleon polarized structure functions
obtained at SLAC, DESY and CERN.
e Adjusted value; the original value, 22.8148(78) meV, is corrected by adding
1µeV, because the conventions of “elastic” and “inelastic” contributions, applied in Martynenko, are
inconsistent.
Fermi - Energy:
EF (2S) =
1 Z↵ 1 + 
= 22.8054 meV
3 a3 mM
Table 1: Summary of available dispersive calculations for the TPE correction
to the HFS in µH.
CD 2015, 29.06.2015
14
Franziska Hagelstein
Δ-Exchange Contribution
TPE effect on the Lamb shift is dominated by the electric dipole
polarizability ↵E1, while the contribution from the magnetic dipole
polarizability M 1 is suppressed
Z 1
⇥
⇤
↵ 2
dQ
(pol)
2
2
2
3 3
EnS =
w(⌧
)
T
(0,
Q
)
T
(0,
Q
)
=
1/(⇡n
a )
1
2
with
n
n
2
⇡
Q
0
low-energy theorem:
2
2
lim
T
(⌫,
Q
)
=
4⇡
⌫
(↵E1 +
1
2
⌫,Q !0
2
2
lim
T
(⌫,
Q
)
=
4⇡
Q
(↵E1 +
2
2
⌫,Q !0
M 1)
+ 4⇡ Q2
M 1)
+ O(q 4 )
M1
+ O(q 4 )
Eur. Phys. J. C (2014) 74:2852
Does the Δ-excitation contribute
significantly to the TPE effect in HFS?
⇢
3e
L=
N̄ T3 igM (@µ
2M (M + M )
CD 2015, 29.06.2015
⌫ )F̃
µ⌫
gE
5 (@µ
15
Fig. 3 ThegC
!(1232)-excitation mechanism. Double line represe
µ⌫
)F
i of the5 !↵ (@↵ ⌫ @⌫ ↵ )@µ F µ⌫ + H.c.,
⌫propagator
M
!∞
Franziska Hagelstein
gM = 2.88
Polarizabilties:
Δ-Exchange Contribution
gE =
1.04
gC =
2.6
scalar dipole polarizabilties:
4
fm3
���
��
����
�
��
6.7 ⇥ 10
��
��
3
�����
��
���
������������������
=
fm
����
��
M1
2
e2 g M
⇡
2
2⇡ M+
4
��
��
↵E1
2
e2 g E
=
⇡ 0.1 ⇥ 10
3
2⇡M+
���
�
��
��
��
��
��
��
��
�
��
��
��
��
��
���
���
���
���
������������������
spin polarizabilties:
2
0
=
e
2
4⇡M+
2
LT
e
=
3
4⇡M+
✓
✓
2
gE
2
M+
+
2
gM
2
M
2 M
gE
+ gE gM
M+ M
M
⇡ ( 0.138 + 0.023) ⇥ 10
CD 2015, 29.06.2015
4gM gE
M+
◆
4
4
fm =
dddddddd
⇡
4
2.7 ⇥ 10
1
+ gE gC
M
0.114 ⇥ 10
fm4
◆
16
4
dχdddddd
ddddddddddddddddddd
dχdddεd
dddddddddddddddddddd
fm
4
dddd
dddd
dddd
δdddddddddddddddd
Franziska Hagelstein
Conclusions
Why disagreement in HFS ???
effect of the Δ-excitation
might not be negligible
empirical information on
polarized (spin) structure
functions is limited • little data on g
2
problem in B𝜒PT?
B𝜒Pt result is smaller than calculations based on
dispersive sum rules
dχddddd
ddddddddddddddddddddddd
HFS
dddddddddd
ddddddddddddddddddddddd
dddddddddddddddddddd
dddddddddddddddddddd
dd
dd
dd
dd
dd
ddd
ddd
ddd
∆ddddddddµddd
B𝜒Pt result is in good agreement with calculations based on
dispersive sum rules
dddddddddd
dddddddddddddd
dddddddddddddddd
ddddddddddddddddddddddddddd
Lamb Shift
dddddddddddddddχdd
dddddddddddddddddddd
dddddddddddddddd
dddddddddddddddddddddd
ddχddddd
ddddddddddddddddddd
ddχdddddd
dddddddddddddddddd
dχddddd
dddddddddddddddddddd
ddd
ddd
ddd
ddd
ddd
ddd
dd
dd
∆ddddddddddµddd
CD 2015, 29.06.2015
17
Franziska Hagelstein
Backup Slides
CD 2015, 29.06.2015
18
Franziska Hagelstein
PROTON FORM FACTOR IN
HYDROGEN LAMB SHIFT
FF(1)
E2P 2S
wE HQL GEp HQ2 L ¥ 104
2
=
Z
1
0
1 ê amH
1 ê aeH
0
Yukawa-type potential:
dQ w(Q) GE (Q2 )
Z↵ 1
VY (r) =
r ⇡
ep data
-2
-4
mH
eH
-6
10
-5
0.001
10
convergence
radius of the
expansion is limited by t0
0.1
2
4
Q2
5 4 2 (Z↵mr )
(Z↵) mr Q
4
⇡
[(Z↵mr )2 + Q2 ]
•
FH, V. Pascalutsa, Phys. Rev. A91 (2015) 040502
•
“soft” contributions to the proton or
lepton electric form factor could be able
to explain the proton size puzzle ?!?
FRANZISKA HAGELSTEIN
t0
dt
e
t
p
r t
Im GE (t)
h2P1/2 |VY | 2P1/2 i h2S1/2 |VY | 2S1/2 i
Z
(Z↵)4 m3r 1
Im GE (t)
=
dt p
2⇡
( t + Z↵mr )4
t0
1
(Z↵)4 m3r X ( Z↵mr )k k+2
=
hr
iE
12
k!
k=0
=
Q@GeVD
with w(Q) =
1
FF(1)
2S =
EE2P
-7
Z
(Z↵)4 m3r ⇥ 2
hr iE
12
⇤
Z↵mr hr3 iE + O(↵6 )
the finite-size effects are not always
expandable in the moments of charge
distribution
a small variation in the form factor around
the inverse Bohr radius scale may lead to
significant effects !!!
INSTITUT FÜR KERNPHYSIK, UNIVERSITÄT MAINZ, GERMANY
Theory of μH Lamb Shift
th
ELS
= 206.0668(25)
5.2275(10) (RE /fm)
theory uncertainty:
2
2.5 µeV
numerical values reviewed in: A. Antognini et al., Annals Phys. 331, 127-145 (2013).
dddddddddd
subleading effects of
proton structure
proposed to resolve
the puzzle
ddddddddddd
ddddddddddddddddddddddddd
µdddddddµdd
310 µeV
ddddddddddd
ddddddddddddddddddddddddddddd
dddddd
ddd
dddd
ddddd
ddddddddddd
ddddddddd
V (2
dddddddddd
ddddddddddddddddddddddddd
dddddd
CD 2015, 29.06.2015
)
(2 )
(2 )
= Velastic + Vpolariz.
A. De Rujula, Phys. Lett. B693 (2010)
ddddd
dddd
dd
ddddd
ddd
20
dddd
ddddd
G. A. Miller, Phys. Lett. B718 (2013)
Franziska Hagelstein
TPE in μH Lamb Shift
Lamb shift:
EnS
↵m
= 3
i⇡
2
n
Z1
d⌫
0
Z
Q2
dq
2
n
with the hydrogen wavefunction at the origin
unitarity relations:
2
4⇡ ↵
f1 (⌫, Q2 ) = ⌫ T (⌫, Q2 )
M
4⇡ 2 ↵
Q2 ⌫ ⇥
2
2
Im T2 (⌫, Q ) =
f2 (⌫, Q ) = 2
T +
⌫
⌫ + Q2
Im T1 (⌫, Q2 ) =
L
⇤
2
(⌫, Q )
2⌫ 2 T1 (⌫, Q2 ) (Q2 + ⌫ 2 )T2 (⌫, Q2 )
Q4 (Q4 4m2 ⌫ 2 )
= 1/(⇡n3 a3 )
1 2
GM (Q2 ) (1 x)
2
⇤
1 ⇥ 2 2
f2el (x, Q2 ) =
GE (Q ) + ⌧ G2M (Q2 ) (1
1+⌧
f1el (x, Q2 ) =
x)
“elastic” contribution:
2⌫ 2
T 1 (⌫, Q ) = T 1 (0, Q ) +
⇡
2
T 2 (⌫, Q ) =
⇡
2
2
1
⌫0
1
⌫0
2
⌫0 2Q
d⌫ 0 2
⌫ + Q2
0
d⌫
T (⌫
⌫0 2
0
T (⌫
0
0
T1Born (⌫, Q2 ) =
, Q2 )
,
⌫2
, Q2 ) + L (⌫ 0 , Q2 )
⌫0 2 ⌫2
⇥
2
2
⇤2
4⇡↵
M
)
“elastic” and “inelastic”
contributions are wellconstrained by empirical
information
subtraction function must be modelled!
CD 2015, 29.06.2015
4
Q F1 (Q ) + F2 (Q )
2
2
F
(Q
)
1
Q4 4M 2 ⌫ 2
⇢
2
16⇡↵M
Q
Q2 2 2
Born
2
2
2
T2
(⌫, Q ) = 4
F1 (Q ) +
F (Q )
Q
4M 2 ⌫ 2
4M 2 2
“polarizability” contribution:
2
(
21
Franziska Hagelstein
Dispersive Calculations
for TPE in μH Lamb Shift
Pachucki
M1
(subt)
E2S
(inel)
E2S
(pol)
E2S
1.56(57)
1.9
13.9
12(2)
E2S
(el)
23.2(1.0)
E2S
35.2(2.2)
a Adjusted
Martynenko
1.9(5)
2.3
16.1
13.8(2.9)
Carlson &
Vanderhaeghen
3.4(1.2)
5.3(1.9)
12.7(5)
8 7.4(2.0)
>
< 27.8
29.5(1.3)
>
:
30.8
36.9(2.4)
(subt)
[1] K. Pachucki, Phys. Rev. A60 (1999) 3593–3598.
[2] A. Martynenko, Phys. Atom. Nucl. 69 (2006) 1309–1316.
[3] C. E. Carlson, M. Vanderhaeghen, hep-ph/1101.5965 (2011).
[4] M. C. Birse, J. A. McGovern, Eur. Phys. J. A48 (2012) 120.
[5] M. Gorchtein, et al., Phys. Rev. A87 (2013) 052501.
Birse &
McGovern
3.1(5)
4.2(1.0)
12.7(5)b
8.5(1.1)
Gorchtein
et al.a
24.7(1.6)c
24.5(1.2)
33(2)
39.8(4.8)
2.3(4.6)
13.0(6)
15.3(4.6)
(el)
values; the original values, E2S
= 3.3 and E2S = 30.1, are based on a
di↵erent decomposition into the “elastic” and polarizability contributions.
b taken from Carlson & Vanderhaeghen
c Result taken from Carlson & Vanderhaeghen with reinstated “non-pole” Born piece.
Table 1: Summary of available dispersive calculations for the TPE correction to
the Lamb shift in µH. Energy shifts are given in µeV, M 1 is given as ⇥10 4 fm3 .
CD 2015, 29.06.2015
22
T 1 (0, Q2 )
lim
= 4⇡
Q2
Q2 !0
M1
⇤8
M 1 (Q ) = M 1
(⇤2 + Q2 )4
2
Franziska Hagelstein
TPE in μH HFS
HFS:
EHFS = [1 +
QED
p
weak
+
Fermi - Energy:
EF (nS) =
+
FSE ] EF
with
TPE in HFS:
TPE
E
EFp
=
m
1
2(1 + {)⇡ 4 i
Z
1
d⌫
0
Z
dq
8 Z↵ 1 +  1
3 a3 mM n3
FSE
(Q4
=
1
4m2 ⌫ 2 )
Z
(
p
recoil
+
2Q
2
⌫
Q2
2
+
pol
S1 (⌫, Q2 ) + 3
⌫
S2 (⌫, Q2 )
M
)
⇢
Z 1
2
2
2
M
Q
F
(Q
)G
(Q
)
d⌫ 0
1
M
2
0
2
S1 (⌫, Q ) = 8⇡↵
+
g
(⌫
,
Q
)
1
02
2
Q4 4M 2 ⌫ 2
⌫
⌫
⌫0
⇢
Z 1
2
2
M F2 (Q )GM (Q )
d⌫ 0 g2 (⌫ 0 , Q2 )
2
S2 (⌫, Q ) = 8⇡↵ M ⌫
+
02
Q4 4M 2 ⌫ 2
⌫2
⌫0 2
⌫0 ⌫
unitarity relations:
4⇡ 2 ↵
M ⌫2 h Q
2
Im S1 (⌫, Q ) =
g1 (⌫, Q ) = 2
LT +
⌫
⌫ + Q2 ⌫
4⇡ 2 ↵M
M 2⌫ h ⌫
2
2
g2 (⌫, Q ) = 2
Im S2 (⌫, Q ) =
LT
⌫2
⌫ + Q2 Q
2
TT
i
2
(⌫, Q ),
TT
i
(⌫, Q2 )
g1el (x, Q2 ) = 1/2 F1 (Q2 )GM (Q2 ) (1
g2el (x, Q2 ) =
1/2 ⌧ F
2 (Q
2
x)
)GM (Q2 ) (1
x)
TPE effect on the HFS is completely constrained by empirical information
CD 2015, 29.06.2015
23
Franziska Hagelstein
HFS Formalism
Z
8↵mr
=
⇡
Z
1
0
2
dQ
Q2

GE (Q2 )GM (Q2 )
1+{
p
recoil
FSE
=
Z
+
p
recoil
pol
E
=
EFp
pol
1
2
CD 2015, 29.06.2015
Z
2↵mr RZ
1
Z
1
G
(Q
)
↵m
1
dQ2
M
2
2
2
dQ F2 (Q )
(⌧
)F
(Q
)
1
l
2
2
1
+
{
2(1
+
{)M
⇡
Q
0
0
⇢ Z 1
↵mM
1
dQ2
2
2
+
3
[ 2 (⌧p )
2 (⌧l )] F2 (Q )GM (Q )
2
2
2
2(1 + {)(M
m )⇡
Q
0

Z 1
p
p
4 ⌧p
(⌧
)
4
⌧l
dQ2
1 (⌧p )
1 l
2
2
F
(Q
)G
(Q
)
+
1
M
2
Q
⌧p
⌧l
0
2↵mr 1
=
M2 ⇡
TPE
+
1 ⌘
2
8↵mr
⇡
Z
1
0
dQ2
Q2
↵m
=
[ 1 + 2] = 1 + 2
2(1 + {)⇡M
⇢
Z 1
Z 1
dQ2
d⌫ Q4 1 (⌧ ) 4m2 ⌫ 2 1 (⌧l )
2
2
2
=
(⌧
)F
(Q
)
+
4M
g
(⌫,
Q
)
1
l
1
2
2
2
4
2
2
Q
Q
4m ⌫
0
⌫0 ⌫
Z 1
2 Z 1
dQ
d⌫ Q4 [ 2 (⌧ )
2 (⌧l )]
2
2
= 12M
g
(⌫,
Q
)
2
2
2
4
2
2
Q
Q
4m ⌫
0
⌫0 ⌫
24
Franziska Hagelstein