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Physics Letters B
www.elsevier.com/locate/physletb
Improved estimates of the nuclear structure corrections in ßD
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O.J. Hernandez3'0, C. Jia *. S. Baccaa'b, N. Nevo Dinurc, N. Barnea
a TR1UMF, 4004 Wesbrook Mall, Vancouver, BC V6T2A3, Canada
b Department of Physics and Astronomy, University of Manitoba, Winnipeg, MB R3T 2N2, Canada c Racah Institute of Physics, The Hebrew University, Jerusalem 9190401, Israel
A R T I C L E I N F 0
A B S T R A C T
Article history:
Received 20 June 2014
Received in revised form 21 July 2014
Accepted 21 July 2014
Available online 25 July 2014
Editor: W. Haxton
Keywords: Muonic atoms Nuclear polarizability Chiral potential
We calculate the nuclear structure corrections to the Lamb shift in muonic deuterium by using state-of-the-art nucleon-nucleon potentials derived from chiral effective field theory. Our calculations complement previous theoretical work obtained from phenomenological potentials and the zero range approximation. The study of the chiral convergence order-by-order and the dependence on cutoff variations allows us to improve the estimates on the nuclear structure corrections and the theoretical uncertainty coming from nuclear potentials. This will enter the determination of the nuclear radius from ongoing muonic deuterium experiments at PSI.
© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.
1. Introduction
The root-mean-square charge radius of the proton was recently determined by spectroscopic measurements of the 2S-2P atomic shift, i.e., the Lamb shift (LS) [1], in muonic hydrogen [2,3], where the proton is orbited by a muon instead of an electron as in ordinary hydrogen. With respect to the C0DATA-2010 compilation [4], which is based on the combined electron proton scattering data and the spectroscopic measurements in the ordinary hydrogen atom, the accuracy was improved ten-fold and a proton radius value smaller by 7 standard deviations was observed. This large deviation between the muonic and the electronic measurements constitutes the so-called "proton radius puzzle". It has attracted a lot of attention since 2010, from both theoretical and experimental viewpoints. Several beyond-the-standard-model theories, including lepton universality violations, have attempted to solve this puzzle (see e.g. [5] for a review). For example, the authors of Refs. [6-8] investigated the possibility of the existence of new interaction mediators that can explain not only the proton radius puzzle, but also the (g — 2) muon anomaly. As yet, none of these theories have been either verified or ruled out by experiments. Alternative explanations are being sought after either through novel aspects of hadronic structure [9-11], or from renewed analyses of the electron scattering data, e.g., Refs. [12,13]. To date, no com-
* Corresponding author.
E-mail addresses: javierh@triumf.ca (O.J. Hernandez), jichen@triumf.ca (C. Ji), bacca@triumf.ca (S. Bacca), nir.nevo@mail.huji.ac.il (N. Nevo Dinur), nir@phys.huji.ac.il (N. Barnea).
monly accepted explanation of the puzzle has been found. Various new dedicated experiments have been planned to measure electron [14,15] and muon [16] scattering on the proton. In addition, experimental reexamination of ordinary hydrogen spectroscopy is under way, e.g., Ref. [17]. A complementary experimental program based on high-precision spectroscopic measurements on various muonic atoms aims to study the systematics of the discrepancy with ordinary atoms as a function of the atomic mass A and charge number Z. In particular, the CREMA collaboration [18] plans to measure the Lamb shift and isotope shifts in several light muonic atoms. The deuteron is the lightest compound nucleus, made up of one proton and one neutron, and it plays an important role in few-body nuclear physics. The Lamb shift of its muonic atom, ¡D, is currently being measured at PSI. This measurement will provide a solid and independent test of the systematic uncertainties in the ¡H experiment. Furthermore, assuming a new interaction mediator that violates lepton universality, the ¡D experiment may help to constrain the possible couplings of this new interaction to the proton and the neutron. Therefore, it is important to compare the deuteron charge radius (r^y2 extracted from the ¡D Lamb shift with the values determined from previous and ongoing experiments on eD scattering [19,20], as well as from the precision measurements on the H/D isotope shift [21,22].
The extraction of the nuclear charge radius from LS measurements relies heavily on theoretical input. For the deuteron, the 2S-2P energy transition is related to {r2)XJ2 by [23]
m3a , 2
AE = Sqed + ápol + ázem +--rr— (r.
chid '
http://dx.doi.Org/10.1016/j.physletb.2014.07.039
0370-2693/© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.Org/licenses/by/3.0/). Funded by SCOAP3.
where a is the fine-structure constant and mr is the reduced muon mass. Quantum electrodynamic (QED) corrections Sqe0, as well as nuclear structure corrections Spol and SZem, are obtained from theoretical calculations. Sqe0, which originates from vacuum polarization, lepton self-energy, and relativistic recoil effects, is known with very good accuracy [24]. The uncertainty of the extracted radius is by far limited by the uncertainty of the nuclear structure corrections. These corrections arise from the two-photon exchange (TPE) (see e.g. Fig. 1 in Ref. [25]), in which the virtual photons transfer energy and momentum to the nucleus. The TPE contribution is traditionally separated into the sum of the elastic zemach term SZem, proportional to the third Zemach moment [23], and the inelastic polarization term Spol, i.e., STPE = SZem + Spol. The inelastic part is further separated into Spol = Sp^ + S^, the sum of the nuclear polarization Spo and the intrinsic nucleon polarizability S^.
For fD, these nuclear structure corrections have been most recently estimated by Carlson et al. [26] using forward dispersion relations to analyze experimental elastic deuteron form factors and inelastic electromagnetic deuteron scattering data. Due to lack of data in the most relevant low-energy quasielastic regime, this analysis suffers from a 35% uncertainty. Theoretical estimates of nuclear corrections were recently made by Friar [27], utilizing the zero-range model. The uncertainty was roughly estimated to be 1-2% due to missing higher-order corrections, mainly from two-body operators and S-D mixing in the deuteron ground state [28]. However, this uncertainty is obtained by dimensional analysis, and is thus only an approximation. A pioneering calculation of STPE in fiD was made by Pachucki [29], who used a high precision nucleon-nucleon (NN) force of phenomenological nature, namely the AV18 potential [30]. Pachucki quoted ~1% uncertainty on STPE, mostly due to higher-order atomic-physics terms in the a expansion, but did not include the uncertainty from nuclear potentials.
In order to obtain an improved estimate, one may rely on microscopic calculations based on nuclear Hamiltonians constructed with state-of-the-art NN potentials [30-32]. The NN force is an effective potential emerging from the low-energy quantum chro-modynamics (QCD). As such it depends on the adopted resolution scale, a set of fitting parameters, and truncation orders. Realistic NN potentials are typically fitted to reproduce NN scattering data (mostly np and pp) with high accuracy x2 ^ 1 up to pion production threshold. In order to provide a rigorous estimate of the theoretical uncertainty one should consider the effect of varying all the fitting parameters within a certain confidence interval. Such a task is at the moment out of hand. Alternatively, one could explore various realistic NN potentials as a sample of the possible model space.
An early attempt to estimate the theoretical error of the nuclear polarization in f D using several of the NN forces available at that time was made by Leidemann and Rosenfelder [33], who found a potential dependence less than 2%. However contributions from Coulomb distortion were missing in that calculation.
The purpose of this Letter is to provide new calculations and improved estimates of the theoretical uncertainty in the nuclear structure corrections using state-of-the-art nuclear potentials from chiral effective field theory (x EFT) [31,32]. The x EFT nuclear potentials result from a systematic expansion of the interaction in powers of a soft momentum scale over a hard momentum scale Q/Ax [31]. The soft scale Q is associated with the typical momenta of the system or dynamics under consideration, and the hard scale Ax determines at which scale the effective theory breaks down. The short range part of the x EFT potential stems from contact terms in the Lagrangian, which are controlled by free parameters in the x EFT Lagrangian, commonly known as low energy constants (LECs). Their values are set by fitting the interaction
to the NN scattering data at minimum x 2/datum for given resolution scales, determined by the regularization cutoff A ~ O(Ax). For the chiral forces better x2/datum and smaller A dependence in predictions is obtained with the increase of the expansion order. One can improve the description of the np scattering data, going from a x 2/datum of 36.2 at next-to-leading order (NLO) calculation up to 1.1 at next-to-next-to-next-to-leading order (N3LO) in the energy range of 0-290 MeV [34]. By performing a systematic study in x EFT order-by-order and by varying the resolution scale A, we can better assess the theoretical error for Stpe stemming from nuclear forces.
2. Calculation details
To calculate the deuteron polarization we essentially follow the derivation provided in our recent paper [25]. Within the multipole expansion formalism we take into account the leading electric dipole term sO0 , the Zemach related contribution SZ3, and other multipole corrections including a monopole sR2), a quadrupole sQ)
and an interference between two rank-1 operators S
small parameter in the expansion is n = Vmr!md where mr is the reduced muon mass and md is the deuteron mass. According to our power counting, the contribution of a term with superscript (1) (or (2)) to Spo is suppressed by n (or n2) relative to that of the
leading S® term. We also include the Coulomb distortion corrections S®, relativistic longitudinal and transverse corrections S(L0) and S®, magnetic dipole corrections S M, and finite-nucleon-size corrections including S^, S(R1¡np and S^l. Therefore, we calculate
SAol as
SpAol = + sC0) + oL0 + of) + SM0)] + S
, rS(2) , S(2) , S(2) 1 I rS(1) IS(1) +S(2)1 (2)
+ LSR2 + 0Q + 0D1D3j + LSz 1 + SR1np + SNS J • (2)
Detailed formulas relating (most of) these corrections are found in [25] and are not repeated here. Except for sZ13), S^ and sR^, which are ground-state observables, each of the above contributions can be written as a sum of terms of the form
Sa = dm la (M)ga (m),
where ga(o)) is an energy-dependent weight function (different for each of the corrections), «th is the threshold excitation energy, and Sa (a) is the response function. Sa (a) is given by
la (M) =Y \{ f | O a\0}\2S(Ef - E0 - M)
where O a is the transition operator (that can be different in the above corrections), |0) and | f} are the ground and final eigenstates of the Hamiltonian H with eigenvalues E0 and Ef respectively. The integration in Eq. (4) carries over the excited scattering states of the deuteron.
The Coulomb corrections S^0) used here contain only the term of order a6 lna. Therefore we have
SC0) = -2n m3ra6
' ^ft 2m a2 dm — ln —-—Sd1(m), m M
where S01(a) is the electric-dipole response function with operator 01 = RpYi(Rp) [25]. The terms of higher orders in a, which
were included in SC ) of Ref. [25], only correct ¿Ao by <0.25%, and are thus neglected in this analysis.
Because the deuteron is a nucleus with the total angular momentum J 0 = 1 in the ground state, the magnetic term S(M), which is negligible in the ¡4He+, has to be included here. It relates to the magnetic response function SM1(a) by
*(°) 1 m3 s/ gp — gn
sm = 3m3a l-mr
■Smi(oJ),
where gp = 5.586, gn = —3.826, and mp is the proton mass. The magnetic dipole (M1) operator is defined by sp —sn [29].
The sum of the terms S^ and S^ cancels exactly the elastic Zemach term SZem:
ázem = — [ 43 + 4l]
= —mça5
+ — À <r>(2)
1/2 _ ch>P = 2
.24 ^ (2) + ( J2
where p = 4.120 fm-1 and X = 0.01935 fm2 are used Refs. [25,27] to reproduce the proton charge radius (r2 0.8409 fm [3] and the neutron charge radius squared (rcu m -0.1161 fm2 [35]. The n-th moment (rn)(2) is defined by
(rn)(2) = jjdRdR'pp(R)|R - R fpp (R'),
with pp denoting the point-proton density in deuteron. The cancellation of 5Zem was explicitly applied in Refs. [27,29], whose results for nuclear structure effects are provided as the combination of 5pol and 5Zem. In this work, we will also provide our final result as a combination of the two.
Unlike the ¡4He+ case of Ref. [25], a pp charge correlation is not present in ¡D, as it contains only one proton. On the other hand, sRVr^, which denotes a contribution from the np charge correlation, has to be included, and is calculated as
sRlp = Xm4a5{°\\R p — Rn ||°>,
where the operator |Rp — Rn| denotes the relative distance between the proton and the neutron.
To calculate |0) and | f) we expand them on the harmonic oscillator (HO) basis [36] and then diagonalize the Hamiltonian matrix, whose dimension is set by Nmax = 2n +1, where n is the HO quantum number and I is the relative angular momentum between the proton and the neutron. We use the HO basis because it is very flexible and allows the use of both local potentials in coordinate space, like the AV18, as well as non-local forces, like those derived in x EFT, within the same framework. From the diagonal-ization of the Hamiltonian matrix we obtain a set of eigenstates |l) and eigenvalues E. Once the size of the basis is large enough, the lowest eigenvalue state coincides with the ground-state of the deuteron. All the other states |i) can be regarded as a discretization of the two-body continuum. In terms of this discrete basis, Eq. (3) becomes a sum over the transition probabilities from the ground state to the discretized excited states with weighted functions ga as
Sa =J2 MOa10)|2ga(Mi), (10)
where a)/x = E/x — E0 is the excitation energy, and the finite sum runs over all the calculated states. As proven in Ref. [37], even if one is approximating continuum states with discrete states,
Table 1
Deuteron electromagnetic observables, as calculated by different potential
models.
E° r2 1/2 Vstr >d Qd aE Pm
[MeV] [fm] [fm2] [fm3] [fm3]
NLO(4°°, 7°°) 2.1647 1.975 °.27°7 °.652 °.°7°6
NLO(55°, 7°°) 2.1794 1.974 °.2745 °.647 °.°696
N2LO(45°, 7°°) 2.2°22 1.97° °.2711 °.64° °.°695
N2LO(55°,6°°) 2.189° 1.971 °.2749 °.644 °.°694
N2LO(6°°,7°°) 2.1999 1.97° °.2747 °.64° °.°69°
N3LO(45°,7°°) 2.2189 1.986 °.2659 °.637 °.°695
N3LO(55°,6°°) 2.2196 1.979 °.2673 °.635 °.°693
N3LO(6°°,7°°) 2.2235 1.975 °.2692 °.633 °.°689
N3 LO-EM 2.2246 1.974 °.275° °.633 °.°684
AV18 2.2246 1.967 °.2697 °.633 °.°679
Eq. (10) becomes exact when the size of the basis becomes large enough (with increasing Nmax). Furthermore, for the two-body problem the basis size remains tractable so that a direct diago-nalization is always possible and one does not need to introduce the Lanczos algorithm as in Ref. [37].
We calculate the deuteron ground-state properties and polariz-ability using the AV18 potential [30] and NN forces from xEFT. The AV18 results serve as checks with Ref. [29] and the xEFT results are genuine new. We use chiral potentials developed by Epelbaum et al. [38] at different order in xEFT, starting from NLO to N3LO. For these, a certain range of cutoff variation will be explored, going from A = 400 to 600 MeV for the cutoff to regularize the Lippmann-Schwinger equation [39] and from A = 600 to 700 MeV for the spectral function cutoff (see [38] for details). Hereafter, we shall refer to these potentials either individually as NkLO(A, A) or collectively as NkLO-EGM. We will also use the chiral potential at N3LO developed by Entem and Machleidt at a fixed cutoff A = 500 MeV [34] (N3LO-EM). By doing so, we will provide a systematic study of the convergence and control the theoretical uncertainty.
3. Results
To estimate the possible spread in STPE due to the variation in nuclear Hamiltonian models, we first present few deuteron ground-state observables and follow their evolution with the nuclear forces. Our numerical results for the energy E0, the root-mean-square structure radius {rjtr )]/2, the quadrupole moment Qd, the electric polarizability aE, and the magnetic susceptibility jiM are presented in Table 1. The first three are also presented graphically in Fig. 1. With the AV18 and N3 LO-EM we reproduce the experimental binding energy, 2.224573(2) MeV [40], up to 5 decimal digit as in Refs. [30] and [34]. For the chiral NkLO(A, A) potentials we use the cutoff values provided by Epelbaum [41]. We note that the range of the explored cutoffs does not coincide exactly with the cutoff range spanned in Ref. [38]. Only for N2LO there is an exact one-to-one correspondence of cutoff sets used; there we agree with Epelbaum's values within the third decimal digit for energy, radius and quadrupole moment.
The electromagnetic observables in Table 1 (all but E 0) have been calculated assuming that nucleons are point-like and using only one-body operators with no relativistic corrections (RC) or meson-exchange currents (MEC). The nonrelativistic one-body operators correspond to the dominant contributions to the electromagnetic observables in the x EFT expansion.
The charge radius is related to the structure radius by finite nucleon size corrections as
r<"h)d
= irlr)d + tfhl
> -2.17
s -2.19
4— 1.980
<N 1.975
<N F 0.278
o- 0.270
AV18 N3LO EM
N LO EM AV18
N3 LO EM AV18
Table 3
Nuclear polarization contributions to the 2S-2P Lamb shift AE [meV] in ^D with different potentials.
Ref. [29] AV18 N3 LO-EM N3LO-EGM
S(0) c -1.910 -1.907 -1.912 (-1.911,-1.926)
i?> 0.035 0.029 0.029 (0.029, 0.030)
i<° - -0.012 -0.012 -0.013
sC) 0.261 0.262 0.262 (0.262, 0.264)
iM 0.016 0.008 0.007 0.007
sm iZ 3 - 0.357 0.359 (0.359, 0.363)
i® i(2) iR2 0.045 0.042 0.041 0.041
i(2) 0.066 0.061 0.061 0.061
i(2) iD1D3 -0.151 -0.139 -0.139 (-0.139, -0.140)
iNS i(d iZ 1 - 0.064 0.064 (0.064, 0.065)
i(d inp - 0.017 0.017 0.017
i(2) iNS - -0.015 -0.015 -0.015
a pol izem <tl + «Z em -1.638 -1.235 -0.421 -1.656 -1.237 -0.423 -1.661 (-1.236, -1.246) (-0.424, -0.428) (-1.660, -1.674)
Fig. 1. Deuteron energy (a), LO structure radius (b) and quadrupole moment (c) at different chiral orders for the various choices of the cutoffs [A, a}. Compared are the results with the AV18 and N3LO-EM potentials and the experimental values (gray band).
Table 2
Deuteron ground state properties and electric polarizability without and with RC + MEC compared to experimental data.
<r2 )1/2 vstr Id [fm] Qd [fm2] aE [fm3]
N3 LO-EM This work 1.974 0.2750 0.633
+ RC + MEC 1.978 0.2851
AV18 This work 1.967 0.2697 0.633
+ RC + MEC - 0.2752
Experiment 1.97507(78)3 0.285783(30)4 0.70(5)5 0.61(4)6
1 Ref. [34]. 2 Ref. [30]. 3 Ref. [22]. 4 Ref. [46]. 5 Ref. [47]. 6 Ref. [48].
where the structure radius is separated into the leading component from the nonrelativistic one-body current and the subleading part from RC + MEC. In the x EFT language, RC and MEC (also called two-body currents) enter at higher order in the Q/Ax expansion with respect to the leading component, specifically at N2LO and N3LO, respectively, see e.g. [42]. Similar corrections enter in other electromagnetic observables as well. Relativistic and two-body corrections should be evaluated consistently within a theory as to satisfy gauge invariance. They have already been derived in x EFT, see e.g. [43-45], where pion-exchange and contact two-nucleon currents appear, but are not applied in this work. Instead, following Refs. [30,34], we show in Table 2 that, adding phenomenologically
RC and MEC corrections, the calculated values for <rjtr )d and Qd are consistent with the most recent experimental results [22,46].'
Comparing our aE calculated with the AV18 potential with the result of [49] we get an agreement of about 0.15%. Our N3LO-EGM and N3 LO-EM results agree within 1% among the AV18 result and the pion-less EFT results of [50]. For the magnetic susceptibility using the AV18 potential we obtain //M = 0.0679 fm3, which
1 In Ref. [22], (rs2tr)J2 is extracted from the H-D isotope shift measurement,
which does not explicitly depend on the measurement of <r2ch)p.
agrees within 0.1% with previous calculations using the same potential and within 1% with other calculations using different potentials [51]. The latter is comparable with the sensitivity to the NN potential that we observed in Table 1. In fact, our N3LO-EGM and N3 LO-EM results for //M scatter at ~1% and are within few percent of the results based on pion-less EFT from Ref. [50].
In Fig. 1, we first show the binding energy. One observes that as the chiral order gets higher, convergence is approached and the band due to the cutoff variation becomes order-by-order smaller. One also notes, that despite the fact that the deuteron binding energy is a prediction of NkLO-EGM potentials [38], it agrees well with the values from the AV18 and N3 LO-EM potentials, where the deuteron binding energy has been included in the fit [30,34]. Overall, they deviate from the experimental value by only 0.4%.
For the structure radius the situation is different. While quite a nice convergence in the chiral order is achieved, the band obtained by the cutoff variation becomes larger at N3LO. Interpreting the cutoff band as an estimate of the theoretical error, it means the error is larger for the highest chiral order, which also results in a larger deviation from the experimental value. This was already observed in Ref. [38] and attributed to the effect of two-body currents. For <rjtr)^/2 and Qd we find that the values obtained with the chiral potential N3 LO-EM and AV18 fall marginally out of the cutoff band spanned by the N3LO-EGM potentials. This could imply that a slightly different parameterization of the two-body currents is needed for each potential. If we introduced the two-body operators, they would come with new low-energy constants (LECs). The new LECs would have to be calibrated in a way to compensate the cutoff dependence, thus reducing the theoretical errors.
A full compilation of the nuclear structure corrections is displayed in Table 3. We first discuss the calculation with the AV18 potential, comparing our results with Pachucki [29]. As already pointed out in [25] we find very good agreement with Pachucki for the leading order dipole and Coulomb correction terms. Small but non-negligible differences appear in the relativistic corrections. These differences can be traced to the leading-order truncation in the w/mr expansion of the weight function ga made in [29], whereas we have used the full form given in [25]. Our results for 422\ S(2) and 8(2lD3 are smaller than those obtained by Pachucki in [29] by ~8%, although the same formulas are used. As an independent check, we use the deuteron electric-quadrupole response
-1.65 -1.66 -1.67 -1.68 -1.69
-1.90 -1.91 -1.92 -1.93 -1.94
0.0078
I 0.0074 Sis 0.0070 0.0066
N LO N LO
Ns LO EM
Ns LO EM
Fig. 2. Total polarization (a), leading order dipole contribution (b) and magnetic dipole contribution (c) at different chiral orders for the various choices of the cutoffs [a, a} in comparison to the results with the AV18 and N3LO-EM potentials.
S£ol = -1.239 ± 0.005(1ct) meV, and 5zem as
SZem = -0.424 ± 0.003(1ct) meV,
where the standard deviation represents an estimate of the uncertainty in the high precision nuclear Hamiltonians at a 1a level and can be used to estimate confidence intervals for the polarization and the Zemach moment.
From N2LO to N3LO in the x EFT expansion, the value of SZem + SpAol changes by 0.3%. This can be considered the systematic error due to the x EFT truncation and needs to be included in the total error budget.
The atomic physics error from further corrections of order (Za)6 was estimated by Pachucki to be 1% [29].
Combining all the uncertainties in a quadrature sum, the overall accuracy of SZem + SAol is expected to be about 1.16%.
The proton two-photon exchange contribution to the ¡D Lamb shift is estimated to be —0.043(3) meV [29]. However, Ref. [26] suggests that only the inelastic part of such contribution is related to SpNol. Including also the neutron effect, the overall SpNol is esti-
mated as in Ref. [26] to be -0.027(2) meV.3 Therefore, we provide a total nuclear/hadron two-photon exchange contributions to ¡D Lamb shift as
function calculated by Arenhovel [52,53] from the Paris potential and obtain Sq = 0.060 meV, which is only 1% smaller than our corresponding value.2 This 1% discrepancy is also consistent with the difference of deuteron structure radius between the two calculations and compatible with the sensitivity to different potentials. Our calculated magnetic dipole contribution Sis ~50% smaller than that in [29], despite the fact that the same formula, i.e., Eq. (6), is used. As noted above, our values for the related magnetic susceptibility stand in line with previous calculations. We have also integrated the deuteron M1 response function from Arenhovel [52,53] and obtained S(M) = 0.0067 meV.
The evolution with EFT orders of the total nuclear elastic/in-
Sipe = S^i + fem + S^ = -1.690 ± 0.020 meV
elastic contribution S^ + SZem, the dipole component S(°, and the magnetization term Sare presented in Fig. 2, using the same values of the cutoffs (A, A) as in Table 1. We find similar convergence pattern and cutoff dependence for SAol + SZem and S^. This is a reflection of the dipole dominance in SpAol. We also note that the cutoff bands increase with the chiral order, similarly to what we observe for <r|tr)^/2 and Qd. In fact, we found strong correlation between <rjtr)^/2 and S(° (see also [54]). For the magnetic correction SjM), we observe that the cutoff dependence is large starting from the NLO and it is of comparable size as at the N3LO. This is again due to the effect of the missing two-body currents, which appear already at NLO in the case of magnetic transition [42]. However, given the size of this term, such further refinements are negligible here.
Comparing the spread of each term in Table 3 due to the potentials, we find that the variation of SpAol with potential models
mostly comes from the variation in the dipole term S(°. In fact the contribution of all other terms to the spread in SpAol is about 1 |aeV. Averaging over all potentials we can estimate sAo| as
The elastic contribution SZem depends on the input of the proton radius. However, due to cancellation of SZem in the elastic and inelastic parts of STPE, the only proton-radius dependence in STPE comes from the S^l term in 8^. Using the CODATA value 0.8775 fm [4], instead of the proton radius 0.8409 fm from the ¡ H experiment [3], will change SZem by -0.007 meV, but STPE by only -0.0016 meV. This is only 0.1% of STPE, thus negligible in the quadrature sum of the error.
4. Conclusions
A solid understanding of the theoretical indetermination is crucial to constrain any explanation of the proton radius puzzle based on physics beyond standard model. In this work we have made an attempt to constrain the nuclear structure corrections to the ¡D Lamb shift utilizing state-of-the-art nuclear potentials derived from chiral EFT. Considering the contributions to the nuclear polarization up to order O (n3), where n = ^Jm^/md, we obtain an estimate for Stpe = —1.690 meV. Combining the spread in the results due to the nuclear potentials (0.5%) and the convergence with x EFT orders (0.3%), we evaluate the nuclear physics error to be about 0.6%. This should be added in quadrature to the ~1% error coming from atomic physics.
Our predicted STPE differs from that obtained by Pachucki [29] by only 0.6%. However, this excellent agreement can be regarded as partly accidental, since differences of several individual contributions are larger than 0.6%.
Our improved estimates will benefit the determination of
2 1 /2
<rch )d from the ongoing ¡D Lamb shift measurement.
The difference is calculated with Sq in more digits than is presented in Table 3.
3 Refs. [10,55] suggested a different separation of the inelastic part of the nucleon
two-photon exchange contributions to the ¡H Lamb shift. It shifts correspond-
ingly SpNol in ¡D by 0.009 meV, and thus change the central value of Eq. (14) to
-1.681 meV. The size of this correction is within the uncertainty given in Eq. (14).
Acknowledgements [22]
We thank Hartmuth Arenhovel for providing the original data [24 in Ref. [52], and Evgeny Epelbaum for offering the xEFT potential
routine. We also thank Franz Kottmann, Judith A. McGovern, and [26
Michael C. Birse for useful discussions. This work was supported [27
in part by the Natural Sciences and Engineering Research Council [28
(NSERC) (Grant number 386418), the National Research Council of [29
Canada, the Israel Science Foundation (Grant number 954/09), and [31
the Pazy Foundation. [32
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