Two-photon exchange correction to the Lamb shift and hyperfine splitting of S levels

We evaluate the two-photon exchange corrections to the Lamb shift and hyperfine splitting of S states in electronic hydrogen relying on modern experimental data and present the two-photon exchange on a neutron inside the electronic and muonic atom. These results are relevant for the precise extraction of the isotope shift as well as in the analysis of the ground state hyperfine splitting in usual and muonic hydrogen.

We evaluate the two-photon exchange corrections to the Lamb shift and hyperfine splitting of S states in electronic hydrogen relying on modern experimental data and present the two-photon exchange on a neutron inside the electronic and muonic atom. These results are relevant for the precise extraction of the isotope shift as well as in the analysis of the ground state hyperfine splitting in usual and muonic hydrogen.
We evaluate the correction to the Lamb shift of S energy levels E LS following Refs. [8,13]. It can be expressed as a sum of three terms: the Born contribution E Born , the subtraction term E subt and the inelastic correction E inel . To evaluate the dimensionless forward unpolarized amplitude, we always normalize the TPE contributions to the energy E 0 : where M is the proton mass, m is the lepton mass, |ψ nS (0)| 2 = α 3 m 3 r /(πn 3 ) is the non-relativistic squared wave function of the hydrogen atom at origin with the reduced mass of the lepton and proton bound state m r = M m/(M +m), α is the fine-structure constant, and n is the principal quantum number. The inelastic contribution E inel can be expressed as an integral over the unpolarized proton structure functions F 1 and F 2 [13]: with the photon energy ν γ = (p · q) /M , the virtuality Q 2 = −q 2 and kinematical notations: The photon-energy integration starts from the pionnucleon inelastic threshold ν inel thr : where m π denotes the pion mass. The weighting func-arXiv:1808.09204v2 [hep-ph] 19 Sep 2018 tionsγ 1 andγ 2 are given bỹ The contribution E subt from the forward Compton scattering subtraction function T subt 1 0, Q 2 is defined according to Refs. [8,13,15,20] as with and determined mainly by the value of the magnetic polarizability β M entering the low-energy expansion of T 1 as The left part of the TPE effect from proton form factors is called the Born correction E Born [8,13]: with the Dirac (F D ), Sachs electric (G E ) and magnetic (G M ) form factors. The kinematical factor γ 2 is given by In this term, we expand the electric form factor in terms of charge radius at low momentum transfer following evaluation of the Zemach correction in Ref. [39], and the third Zemach moment contribution in Refs. [54,55], and connect regions of large and small momentum transfer. We take an average of Refs. [2,4] for a central value of the charge radius and estimate its uncertainty as a half of the difference between results in Refs. [2,4]. We evaluate the Born contribution exploiting form factors of Refs. [3,4] and take the unpolarized proton structure functions from the fit of Refs. [56][57][58]. We present results for TPE corrections to the Lamb shift (LS) of the ground state in electronic and muonic hydrogen in Tab. I. The Born TPE is around 1.3 times larger than the leading third Zemach moment effect [8]. The contribution from the subtraction function in electronic hydrogen is roughly two times smaller than the Born correction and larger than the estimate of Ref. [59], where the smaller value of the proton magnetic polarizability β M = (1.9 ± 0.5) × 10 −4 fm 3 , compared to the current p.d.g. quotation β M = (2.5 ± 0.5) × 10 −4 fm 3 [67], was used and the Q 2 -dependence of the subtraction function was assumed but not justified by data or theory. The inelastic correction to the Lamb shift is almost twice larger than the Born contribution and 1.3 times larger than the result in the logarithmic approximation [8]. Our estimate is 1.1 times smaller than the calculation of Ref. [9] and agrees with an update of Ref. [59] within uncertainties. In Ref. [9], the inelastic contribution was described by Regge model. The model of structure functions as a sum of resonances with nonresonant background was used in Ref. [59], while the result in Tab. I is based mainly on the fit of precise JLAB experimental data in the resonance region of Refs. [56,57]. Note that the inelastic two-photon effect in electronic hydrogen is in agreement within errors with the dispersive calculation of Ref. [60] which is based mainly on the photoabsorption cross section data modified by empirical elastic form factors. The Born correction in muonic hydrogen is accidentally in a reasonable agreement with Ref. [8], where the dipole parametrization of proton form factors was used, and slightly smaller than the previous estimate of Ref. [13], where we have combined proton state contributions in Ref. [13]   Studying the isotope shift in light atoms, it is instructive to know also the two-photon effect due to the scattering on a single neutron [49,61]. We repeat the Lamb shift calculation without the subtraction of pure Coulomb part and leading charge radius (∼ G E (0)) contribution in Eq. (10) in case of the neutron. Note that a special care has to be taken applying these results to nuclei, since we nor-malize to the energy E 0 of Eq. (2) which changes going to the nucleus. We exploit form factors from Refs. [62][63][64][65][66], use the fit of Christy and Bosted [56] for the unpolarized structure functions, and estimate the subtraction function following Ref. [20] with the neutron magnetic polarizability β M = (3.7 ± 1.2) × 10 −4 fm 3 from p.d.g. [67] and the Reggeon residue according to Refs. [58,68,69]. We present results in Tab  en and µn systems has a different sign to electronic and muonic hydrogen cases. For a neutron with zero charge, the elastic Friar term is relatively small compared to the positively charged proton, and the main contribution comes from the neutron magnetic form factor resulting in a positive sign and relatively small uncertainty. We obtain the central value averaging over the form factor parametrizations and estimate the uncertainty as a difference between the largest and smallest results. As in Ref. [49], the inelastic corrections for proton and neutron coincide within errors. We double the uncertainty for the inelastic contribution in case of the neutron compared to the proton. Note that the resulting two-photon exchange effect in µH is roughly four times larger than in µn system: E µH LS ≈ 4E µn LS , as it has been estimated in Refs. [49,70]. The main uncertainty in the two-photon correction is due to the pure knowledge of the forward Compton scattering subtraction function. However, it can be improved exploiting the chiral perturbation theory predictions [15,17,19], constraints at high energy [22,71] as well as the phenomenological studies of the difference between the subtraction function for protons and neutrons [69,72], and by improved extraction of the neutron magnetic polarizability [73][74][75][76].
For the hyperfine splitting correction E HFS , we use definitions of Refs. [25,[38][39][40]. The result is given by a sum of Zemach E Z , recoil E R and polarizabillity E pol terms: where the relative to the leading Fermi splitting E F : with the proton magnetic moment µ P , contributions are given by with ρ(τ ) = τ − τ (1 + τ ), F P is the Pauli form factor, g 1 ν γ , Q 2 and g 2 ν γ , Q 2 are the spin-dependent inelastic proton structure functions. The Zemach correction E Z is obtained by scaling with the reduced mass from the averaged over electric and magnetic radii result of Ref. [39]. The recoil E R and polarizability E pol contributions are evaluated following the same steps as in Ref. [39]. The proton spin structure functions parametrization is based on Refs. [77][78][79][80][81]. The experimental value of the hyperfine splitting in muonic hydrogen is taken from Ref. [2] and in electronic hydrogen from Refs. [26][27][28][29][30][31][32][33][34][35][36].
In Tab. III, we provide the hyperfine-splitting TPE contributions as well as extractions from the experimental data exploiting radiative corrections of Refs. [25,37,[82][83][84][85][86][87][88][89]. All corrections to the hyperfine splitting in electronic hydrogen are three orders of magnitude above the Lamb shift contributions. As well as in muonic hydrogen [39], they slightly differ to the previous estimates of Ref. [23] due to the inclusion of the recent form factor measurements [3,4]. Theoretical estimates of the hyperfine-splitting correction are within errors of the phenomenological extraction from measurements.
Additionally, we provide an update of Ref. [40] for the absolute value of the hyperfine-splitting energy E HFS,µH nS in muonic hydrogen removing axial-vector mesons [90] from the analysis and accounting for the vacuum polarization graphs with elastic and inelastic proton structure  In experimental extractions, the first uncertainty is the error of radiative corrections and measurement, and the second one contains a possible αEHFS error from higher orders.
An improved calculation of two-photon diagrams with QED corrections on fermion lines, graphs with three exchanged photons as well as evaluation of the two-photon contributions in non-forward kinematics can reduce the uncertainty further.
We presented the current knowledge of the TPE correction to S energy levels. The Lamb shift results can be useful in future extractions of the isotope shift, while the contributions to the hyperfine splitting can help to tune and analyze forthcoming 1S HFS measurements in µH [41][42][43][44].
We acknowledge Krzysztof Pachucki for advice given during the manuscript preparation and Marcin Kalinowski for useful discussion. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through Collaborative Research Center "The Low-Energy Frontier of the Standard Model" (SFB 1044). The author would like to acknowledge the Mainz Institute for Theoretical Physics (MITP) for its hospitality and support.