Non-Perturbative Evaluation of Some QED Contributions to the Muonic Hydrogen N = 2 Lamb Shift and Hyperfine Structure Paul Indelicato

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Non-perturbative evaluation of some QED contributions to the muonic hydrogen n = 2 Lamb shift and hyperfine structure Paul Indelicato

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Paul Indelicato. Non-perturbative evaluation of some QED contributions to the muonic hydrogen n = 2 Lamb shift and hyperfine structure. 2012. hal-00743906v1

HAL Id: hal-00743906 https://hal.archives-ouvertes.fr/hal-00743906v1 Preprint submitted on 21 Oct 2012 (v1), last revised 1 Nov 2012 (v2)

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Non-perturbative evaluation of some QED contributions to the muonic hydrogen n = 2 Lamb shift and hyperﬁne structure

P. Indelicato∗ Laboratoire Kastler Brossel, École Normale Supérieure; CNRS; Université Pierre et Marie Curie - Paris 6; 4, place Jussieu, 75252 Paris CEDEX 05, France (Dated: October 21th, 2012) The largest contributions to the n = 2 Lamb-shift, fine structure interval and 2s hyperfine structure of muonic hydrogen are calculated by exact numerical evaluations of the Dirac equation, rather than by a perturbation expansion in powers of 1/c, in the framework of non-relativistic quantum electro- dynamics. Previous calculations and the validity of the perturbation expansion for light elements are confirmed. The dependence of the various effects on the nuclear size and model are studied.

PACS numbers: 31.30.jf,36.10.Ee,31.30.Gs

Despite many years of study, the proton charge radius tic QED. The wavefunction and operators are expanded has remained relatively poorly known. It has been de- in powers of the fine-structure constant, and the con- rived from measurements in electron-proton collisions tributions are obtained by perturbation theory. Hylton [1, 2] or from high-precision spectroscopy of hydrogen [42] showed that the perturbation calculation of the fi- [3–10] as described in the CODATA report in [11]. Tests nite size correction to the vacuum polarization in heavy of fundamental physics based on the progress in accu- elements gives incorrect results. Since a bound muon racy of spectroscopy of hydrogen and deuterium have is closer to the nucleus than a bound electron by a fac- been limited by the lack of an accurate value for the pro- tor mµ/m 207, its Bohr radius is slightly smaller than e ≈ ton radius. Moreover, the values for the proton radius the Compton wavelength of the electron oC = ~/mec by a obtained by different methods or different analyses of ex- factor m /α mµ 137/207 (α 1/137.036 is the fine struc- e ≈ ≈ isting experiments are spread over a range larger than the ture constant, me and mµ the electron and muon mass uncertainty quoted for the individual results. Two recent respectively). The Compton wavelength is the scale of measurements have resulted in a puzzle. The accurate QED corrections, and for the 2S level, the muon wave- determination of the 2S Lamb shift by laser spectroscopy function mean radius is only 2.6 times larger than the in muonic hydrogen provides a proton size with a ten electron Compton wavelength. times smaller uncertainty than any previous value and It is thus worthwhile to reconsider the largest correc- it differs by five standards deviations from the 2006 CO- tions that contribute to the 2S Lamb-shift in muonic hy- DATA value[12]. At the same time, a new, improved de- drogen using non-perturbative methods. In the present termination of the charge radius by electron scattering, work, we use the latest version of the MCDF code of performed at Mainz with the MAMI microtron, provides Desclaux and Indelicato [43], which is designed to cal- a value in good agreement with the value from hydro- culate properties of exotic atoms [44], to evaluate the gen and deuterium spectroscopy [13, 14]. Taking into exact contribution of the electron Uehling potential with account improved theory in hydrogen and deuterium Dirac wavefunctions including the finite nuclear size. In and the MAMI measurement lead the recently released the same way, we calculate the Källén and Sabry contri- 2010 adjustment [15] to differ by 6.9 standard deviation bution. between from the proton radius obtained from muonic Throughout this paper we will use QED units, ~ = 1, hydrogen. c = 1. The electric charge is given by e2 = 4πα. Many papers have been published in the last year, trying to solve this puzzle. A few are dealing with the calculation of the n = 2 level energies in muonic hydrogen. Several others are concerned with the effect of the I. NUMERICAL EVALUATION OF THE DIRAC EQUATION WITH REALISTIC NUCLEAR CHARGE internal structure of the proton on these energies [16–27]. DISTRIBUTION MODELS Others look at exotic phenomenon beyond the standard model [28–34]. Many contributions to the Lamb shift, fine, and hyper- A. Evaluation by the numerical solution of the Dirac fine structure of muonic hydrogen have been evaluated equation over the years; the results are summarized in [35–40] and in a recent book by Eides et al. [41]. Most of these We calculate higher-order finite size correction, start- calculations are done in the framework of nonrelativis- ing from the Dirac equation with reduced mass, as techniques for the accurate numerical solution of the Dirac equation in a Coulomb potential have been developed over a period of many years within theframework of the ∗ [email protected] Multiconfiguration Dirac-Fock (MCDF) method for the 2 atomic many-body problem [45–48]. using 8 and 14 points integration formulas due to The Dirac equation is written as Roothan. The two integration formulas provide the same result within 9 decimal places. α p + βµ + V (r) Φn (r) = n Φn (r), (1) · r N κµ E κµ κµ where α and β are the Dirac 4 4 matrices, VN(r) is the × B. Charge distribution models Coulomb potential of the nucleus, nκµ is the atom total energy, and Φ is a one-electron DiracE four-component spinor: For the proton charge distribution, two models are ex- " # tensively used. The first corresponds to a proton dipole 1 Pnκ(r) χκµ(θ, φ) (charge) form factor, the second is a gaussian model. Φnκµ(r) = (2) r i Qnκ(r) χ κµ(θ, φ) Here we also use uniform and Fermi charge distribu- − tiond and fits to experimental data [50, 51]. The analytic in which χκµ(θ, φ) is the two-component Pauli spherical distributions are parametrized so they provide the same spinor [46], n is the principal quantum number, κ is the mean square radius R. Moments of the charge distribu- Dirac quantum number, and µ is the eigenvalue of Jz. tion are defined by This reduces, for a spherically symmetric potential, to Z the differential equation: ∞ < rn >= 4π r2+nρ(r)dr, (7)  d κ  " # " # 0  VN(r) r + r  Pnκ(r) Pnκ(r)  − d  = ED , (3)  d κ  nκµ where the nuclear charge distribution ρ(r) = ρN(r)/(Ze) + V (r) 2µ Qnκ(r) Qnκ(r) dr r N − r is normalized by where Pnκ(r) and Qnκ(r) are the large and small radial Z Z ∞ components of the wavefunction, respectively, κ the (~r)d~r 4 (r)r2dr 1 (8) D ρ = π ρ = , Dirac quantum number, Enκµ is the binding energy, µr 0 is the muon reduced mass, µ = m M /(m + M )(m r µ p µ p µ for a spherically symmetric charge distribution. The and Mp are the muon and proton masses). √ 2 To solve this equation numerically, we use a 5 point mean square radius is R = < r >. predictor-corrector method (order h7) [48, 49] on a linear The potential can be deduced from the charge density mesh defined as using the well known expression: Z r Z rn 4πe 2 ∞ tn = ln + arn, (4) VN(r) = du u ρN(u) 4πe duuρN(u). (9) r0 − r 0 − r with tn = t0 + nh, and r0 > 0 is the first point of the mesh, The exponential charge distribution and correspond- corresponding to n = 0. This immediately gives t0 = ar0. ing potential energy are written Equation (4) can be inverted to yield r ξ tn e− W ar0e ρ (r) = Ze , r = , N 3 n a 8πξ tn r r ! dr W ar0e ξ c n = (5) 2 1 e− e− t ), VN(r) = Ze − , dtn a 1 + W ar0e n − r − 2ξ where W is the Lambert (or product logarithm) function. (n + 2)! ξn The wavefunction and differential equation between 0 < rn > = , (10) 2 and r0 are represented by a 10 term series expansion. For R a point nucleus, the first point is usually given by r0 = which gives ξ = √ . The gaussian charge distribution 2 2 3 10− /Z and h = 0.025. Here we use values down to r0 = and potential are given by 7 10− /Z and h = 0.002 to obtain the best possible accuracy. ( r )2 For a finite charge distribution, the nuclear boundary is e− ξ r N ρN(r) = Ze , fixed at the value N, where is large enough to obtain π3/2ξ3 sufficient accuracy. The mean value of an operator , O r that gives the first-order contributions to the energy, is erf ξ V (r) = Ze2 , calculated as N − r Z n+3 n ∞ h 2 2i 2Γ ξ ∆E = dr P(r) + Q(r) (r) n 2 O 0 O < r > = , (11) Z r √π 0 h i = dr P(r)2 + Q(r)2 (r) q 2 0 O where erf is the error function, and ξ = R. Other Z 3 ∞ dr h i similar expressions for the two above models can be + dt P(r)2 + Q(r)2 (r) (6) found in [52]. r0 dt O 3

results for RZ as a function of the charge and magnetic ieγµ − moment radii R and RM. Using (13) or (14) for the exponential or gaussian model, and Eq. (17), we get respectively

gµν 4 3 2 2 3 4 i q2 3R + 9R RM + 11R R + 9RR + 3R − RExp. = M M M (18) Z 3 2 √3(R + RM)

r ieF Q2 γν 2 q − RGauss = 2 R2 + R2 . (19) Z 3π M An other useful quantity, which appears in the estima- FIG. 1. Electromagnetic interaction between a lepton (narrow tion of the finite size correction to vacuum polarization line) and a nucleon (bold line). is the third Zemach’s moment Z D 3E 3 r = r ρ(2)(r)dr, (20) (2) The electric form factor is related to the charge distribution by where the convolved charge distribution is Z Z 2 iq r GE(Q~ ) = dre− ρ(r), (12) ρ (r) = ρ(r u)ρ(u)du. (21) · (2) − For the exponential model this leads to This can be rewritten in the more convenient form [17, 35], in the limit of large proton masses, 2 4 2 1 R 2 R 4 GE(Q~ ) = 1 q + q + (13) Z 2 ! R2q2 2 ≈ − 6 48 ··· D E 48 1 q 1 + r3 = dq G2 Q~ 2 1 + R2 . (22) 12 (2) π q4 E − 3 while for the Gaussian model one has It can be easily seen from Eqs. (13) or (14) that the ex- 2 4 pression is finite for q 0. 1 2q2 R R ~ 2 6 R 2 4 → GE(Q ) = e− 1 q + q + (14) We now turn to more realistic models, based on ex- ≈ − 6 72 ··· periment. A recent analysis of the world’s data on elas- The two models have an identical slope R2/6 as functions tic electron-proton scattering and calculations of two- of q2 for q 0 as expected (see, e.g, [53]). photon exchange effects provides an analytic expression In 1956,→ Zemach introduced an electromagnetic form for the electric form factors [51], given as factor, useful for evaluating the hyperfine structure en- P2 i ergy correction 2 1 + i=0 aiτ GE Q~ = , (23) P4 j Z 1 + j=0 bjτ ρem(r) = ρ(r u)µ(u)du, (15) − 2 where τ = q /(4Mp). The ai coefficients can be found in where µ(u) is the magnetic moment density. Both µ(u) Table I of Ref. [51]. A second work [50] uses a combi- nation of several spectral functions tacking into account and ρem are normalized to unity as in Eq. (8). The ¯ Zemach radius is given by several resonances and continua like the 2π, KKand ρπ continua. Here we use the fit resulting from the super- Z convergence approach from this work. This corresponds RZ = rZ = rρem(r)dr. (16) to a sum of 12 dipole-like functions, which are able to h i represent the experimental data with a reduced χ2 of The Zemach’s radius can be written in momentum space 1.8. A comparison of the electric form factor from both as [54, 55] works is presented on Fig. 3. It is clear that both experimental form factors and the dipole approximation   Z ~ 2 with an identical radius are very close. We can obtain 4 1  GM Q  R = − dq G Q~ 2 1 , (17) the charge radius and the next correction by performing Z 2  E  π q  1 + κp −  an expansion in q of the experimental form factors. We get for Ref. [51] where κp is the proton anomalous magnetic moment, and GM is normalized so that GM (0) = 1 + κp. The expo- 0.85032 0.85034 G Q~ 2 1 q2 + q4 + , (24) nential and gaussian models enables to obtain analytic E ≈ − 6 43.3909 ··· 4 and for Ref. [50] GE 1.0 Belushkin et al. 0.849952 0.849954 G Q~ 2 1 q2 + q4 + . (25) E ≈ − 6 38.793 ··· 0.8 Arrington et al.

These expansions are very close to the one for a dipole 0.6 form factors from Eq. (13). Dipole In order to compare different charge density models, 0.4 we have performed an analytic evaluation of the charge densities corresponding to [51], replacing (23) in (12) and 0.2 performing the inverse Fourier transforms, to obtain the corresponding charge distribution, depending on the set q of a coefficients. The corresponding densities are plot- 2 4 6 8 10 ted and compared to Fermi, Gaussian and exponential models. Distances are converted from GeV to fm using FIG. 3. Comparison of the electric form factor from Refs. [50, ~c = 0.1973269631 GeVfm in the density obtained from 51], with a dipole model with the same R = 0.850 fm as deduced the experiment. The charge distributions are compared from the experimental function. on Fig. 2. We plotted both ρ(r) and ρ(r)r2 to reveal the differences at long and medium distances. The experimental charge density is rather different from all three analytic distributions, while, once multiplied by r2, it is closer to the exponential distribution.

30 Experiment

25 Gaussian 20 Exponential 15 Fermi 10 FIG. 4. Feynman diagrams corresponding to the full vacuum 5 polarization contribution, and expansion in Zα. Diagram (a)

0 corresponds to the Uehling potential [Eqs. (26) and (28)]. Di- 0.0 0.5 1.0 1.5 2.0 agram (b) corresponds to the Wichmann and Kroll correction. The double line represents a bound lepton wavefunction, the wavy line a retarded photon propagator. The single line cor- 1.2 respond to a free electron-positron (or muon-antimuon) pair. Experiment The grey circles correspond to the interaction with the nucleus. 1.0 Gaussian 0.8 Exponential II. EVALUATION OF MAIN VACUUM POLARIZATION 0.6 AND FINITE SIZE CORRECTION Fermi 0.4 The Feynman diagram corresponding to the Uehling 0.2 approximation to the vacuum polarization correction is presented in Fig. 4 (a). The evaluation of the vacuum po- 0.0 0.0 0.5 1.0 1.5 2.0 larization can be performed using standard techniques of (perturbative) non-relativistic QED (NRQED) as described in [35, 37]. Here we use the analytic results of FIG. 2. Top: charge densities ρ(r), bottom: charge densities Klarsfeld [56] as described in [57] and numerical solu- r2 r ρ( ) for the experimental fits in Ref. [51], compared to Gaus- tion of the Dirac equation from Sec. I. In order to ob- sian, Fermi and exponential models distributions. All models are calculated to have the same R = 0.850 fm RMS radius as tain higher order effects, we solve the Dirac equation in deduced from the experimental function. a combined potential resulting from the finite nuclear charge distribution and of the Uehling potential. The logarithmic singularity of the Uehling potential at the origin for a point charge cannot be easily incorporated 5 in a numerical Dirac solver. In the case of a finite charge distributions, the singularity is milder, but great care must be exercised to obtain results accurate enough for our purpose. For a point charge, the Uehling potential, which represents the leading contribution to the vacuum polarization, is expressed as [56, 58, 59]

Z 2m rz α(Zα) ∞ 2 1 e e Vpn(r) = dz √z2 1 + − 11 − 3π − z2 z4 r 1 (26) FIG. 5. Feynman diagrams included in the Källen and Sabry 2 2α(Zα ) 1 2 V21(r) potential (Eq. (32)). See Fig. 4 for explanation of sym- = χ1 r − 3π r λe bols. where me is the electron mass, λe is the electron Compton wavelength and the function χ1 belongs to a family of expression for this potential has also been derived on functions defined by Ref. [64–67]. In the previous version of the mdfgme Z code, the Källèn and Sabry potential used was the one ∞ 1 1 1 x dze xz √z2 provided by Ref. [60], which is only accurate to 3 digits. χn( ) = − n + 3 1. (27) 1 z z 2z − The expression of this potential is for a point charge The Uehling potential for a spherically symmetric charge α2(Zα) 2 distribution is expressed as [56] V r L r 21( ) = 2 1( ), (32) Z π r λe 2α(Zα) 1 ∞ V11(r) = dr0 r0ρ(r0) where − 3 r 0 2 2 Z ∞ 2 8 χ2 r r0 χ2 r + r0 (28). L r dte rt f t × λe | − | − λe | | 1( ) = − 5 ( ) 1 3t − 3t The expression of the potential at the origin is given by 2 4 + + √t2 1 ln 8t t2 1 Z 3t4 3t2 − − 8α(Zα) ∞ 2 (33) V11(0) = dr0r0ρ(r0)χ1 r0 , (29) 2 7 13 − 3 0 λe + √t2 1 + + − 9t6 108t4 54t2 while it behaves at large distances as [60] ! 2 5 2 44 + + + ln √t2 1 + t , 2α(Zα) 1 2 2 2 2 9t7 4t5 3t3 − 9t − V11(r) = χ1 r + < r > χ 1 r − 3π r λe 3 − λe 2 4 2 and + < r > χ 3( r) + ... (30) 15 − λe   Z 2 √ 2 2 ∞  3x 1 ln x 1 + x ln 8x x 1  using the moments of the charge distribution (7). The f t dx  − − −  ( ) =  2  . energy shift associated with the potential (26) or (28) in t  x (x 1) − √ 2  − x 1 first order perturbation is calculated as − (34) 11,pn ∆E = n, l, κ, µr V11 n, l, κ, µr (31) nlκ | | The function f (t) can be calculated analytically in term of the ln and dilogarithm functions. Blomqvist [68] has where n, l, κ, µ is a wavefunction solution of Eq. (1), r shown that L (r) can be expressed as which depends on the reduced mass. It was shown 1 recently [61] that this method provides the correct in- 2 L (r) = g (r) ln (r) + g (r)log(r) + g (r), (35) clusion of the vacuum polarization recoil correction at 1 2 1 0 the Barker and Glover level [62]. Since we directly use and provided a series expansion of this function for small relativistic functions, the other corrections described in r. Fullerton and Rinker [60] provided polynomial ap- [61] are automatically included. proximations to the functions gi(r). Here we have numerically evaluated the function L1(r) to a very good accuracy, using Mathematica. We then fitted the coeffi- III. HIGHER ORDER QED CORRECTIONS cients of polynomials for the function gi(r). The results are presented in Appendix A. For x > 3, we have used A. Reevaluation of the Källèn and Sabry potential the functional form

r 3/2 2 5/2 The Källén and Sabry potential [63], is a fourth order e− a + b √r + cr + dr + er + f r L (r) = . (36) potential, corresponding to the diagrams in Fig. 5. The 1 r7/2 6

The coefficients are also given in Appendix A. of the Dirac energy splitting (the 2p1/2 and 2s level are To obtain the finite nuclear size correction, we use the exactly degenerate for a point nucleus). known expression for a spherically-symmetric charge I evaluated the different quantities on a grid of proton distribution [60] sizes ranging from 0.3 to 1.2 fm, with steps of 0.025 fm (80 points). I also evaluated the contribution for the muonic 2 Z α (Zα) ∞ 2 V r dr r r L r r hydrogen proton size and the CODATA 2010 proton size. 21( ) = 2 0 0ρ( 0) 0( 0 ) π r 0 λe | − | The first few terms of the dependence of the relativistic 2 energy on the moments of the charge distribution where L0( r + r0 ) , (37) given by Friar [52]. For a Gaussian charge distribution − λe | | he finds for a s state where Coul. 2 3 2 2 ∆E = a r + b r (2) + c r Z x h i h i h i (41) L (x) = duL (u). (38) + d r2 log r + e r2 2 log r . 0 − 1 h ih i h i h i I use this as a guide to fit my numerical results. Using our approximation to L1(x) in Eqs (35) and (36), we As a first example a 3-parameter fits provides obtain the following approximate expressions for L0(x). For x 3, the expression is very similar to the one for ∆ECoul. (R) = 5.19972 R2 + 0.0351289 R3 ≤ 2p1/2 2s1/2 L1(x). One obtains − − 0.0000534235 R4 meV (42) − L (r) = rh (r) ln2(r) + rh (r)log(r) + h (r), (39) 0 2 1 0 A better fit is provided by

The expression for the functions hi are given in Appendix Coul. 2 3 ∆E2p 2s (R) = 5.19990R + 0.0355905R B. For the asymptotic function, given for x > 3, we inte- 1/2− 1/2 − grate directly Eq. (36), which yield (ﬁxing the integration 0.000488059R4 + 0.000172334R5 (43) constant so that L0(r) is 0 at inﬁnity) − 0.0000245051R6 meV − L (r) = 41.1352787432251923 0 Using Friar functional form with only one log R term, I

obtain 5.1094977559522696 8.05074798111erf √r − Coul. 2 3 ∆E2p 2s (R) = 5.199365R + 0.03466100R 1.028091975364Ei( r) 1/2− 1/2 − − − 4 r + 0.00007366037R e− (40) 2 02809197536r3/2 4 54214815071r2 5 + 5/2 . + . 0.00001720960R (44) r − − 6 6 0.494718704003r + 0.98439728916 √r + 1.198332 10− R − × ! + 0.0002677236R2 log R meV. 0.344009752879 − The function with two logarithmic terms and close values of the BIC and χ2 criteria is given by where Ei(r) is the exponential integral. Coul. 2 ∆E2p 2s (R) = 5.199337R 1/2− 1/2 − 3 4 IV. NUMERICAL RESULTS + 0.03458139R + 0.0001092856R (45) + 0.0002788380R2 log R A. Finite size correction to the Coulomb contribution 0.00004957598R4 log R − Obtaining the accuracy required from the calculation Criteria for the quality of the fit are plotted in Fig. 6. I on ED , which has a value of 632.1 eV, while the Lamb 2 2κµ ≈ use both the reduced χ and a Bayesian information cri- shift is 0.22 eV with an aim at better than 0.001 meV, terion (BIC) to evaluate the improvement in the value is a very≈ demanding task. For a point nucleus, we get when increasing the number of parameters [69]. We ob- 2 2 exact degeneracy for the 2s and 2p1/2 Dirac energies. The tain a coefficient for R which is 5.1999 meV/fm and a best numerical accuracy was obtained generating the coefficient for R3 equal to 0.03559− meV/fm3. Using our 3 2 wavefunction on a grid with r0 = 2 10− and h = 2 numerical solutions we also find 5.19972 meV/fm and 3 × × 3 − 10− . This corresponds to 8700 tabulation points for 0.032908 meV/fm for the Gaussian model. Borie [39] the wavefunction, with around≈ 2800 points inside the finds 5.1975 meV/fm2 and 0.0347 meV/fm3 for an ex- − 3 proton. I checked that variations in r0 and h do not ponential model, and 0.0317 meV/fm for a Gaussian change the final value. The main finite nuclear size effect model, in reasonable agreement with the result pre- on the 2p1/2–2s energy separation comes from the sum sented here. 7

0 shift in Ref. [71]. In the Gaussian model, I find æ -500 3 æ Polynomial D E 32R r3 = 1.960R3, (48) -1000 (2) ≈ BIC 3 √3π à Polynomial+Log -1500 æ æ leading to R = 0.920 fm and an energy shift of æ ì Polynomial+2 Logs -2000 æ à à æ 0.0256 meV, still in agreement. One can perform a more æ − ì à advanced calculation, using the experimental charge 2 3 4 5 6 7 8 degree distribution from Ref. [51], as given in (23). I find D E r3 = 2.45 fm3, significantly lower than Friar and Sick’s 0.01 æ value. This lead to R = 0.864 fm for the exponential

æ Polynomial 10-5 model and R = 0.889 fm for the Gaussian model, provid- 2

Χ ing shifts of 0.0226 meV and 0.0232 meV respectively, -8 10 à Polynomial+Log in closer agreement− to Borie’s work.− In a recent paper, De æ

æ 10-11 Rùjula [16] claims that the discrepancy found between æ ì Polynomial+2 Logs æ à à æ the charge radii obtained from hydrogen and muonic -14 æ 10 ì à 2 3 4 5 6 7 8 hydrogen could be due to the fact that theoretical calcu- Degree lations use too simple a dipole model to represent the nucleus. He builds a “toy model” composed of the sum FIG. 6. BIC criterium (top) and reduced χ2 (bottom) as func- of two dipole function corresponding to two resonances tion of the degree of the polynomial and of the logarithmic with different masses. In his model the third moment of dependence used in the fit of the energy. the charge distribution is much higher than what is derived from a dipole model, enabling to mostly resolve the discrepancy between charge radii obtained from muonic and normal hydrogen. He gets In the same way, I evaluated the finite size correction D E to the fine structure. r3 = 36.6 7.3 43R3, (49) (2) ± ≈ Coul. ∆E2p 2p (R) = 8.41563570 3/2− 1/2 using R from muonic hydrogen. I use the fit to the exper- 0.00005192R2 imental form factor from Ref. [51] as given in Eq.(23) to − (46) 7 3 check the result from Ref. [16] against an experimental + 1.1818650 10− R × determination. I obtain 9 4 1.19528126 10− R meV. D E − × r3 2.4485 3.98 0.8503, (50) (2) ≈ ≈ × The constant term is in perfect agreement with Borie’s value 8.41564 meV [70] (Table 7). very close to the dipole model value of Eq. (47). Using It is interesting to explore at this stage the influence the recent MAMI experiment, combined with data from of the charge distribution shape on the Coulomb and [51], Distler et al. [22] obtain vacuum polarization contribution. Friar and Sick [71] D E have evaluated the third Zemach moment from Eq. (20), r3 2.85(8) 4.18 0.8803, (51) (2) ≈ ≈ × using the proton-electron scattering data available in 2005. Using a model-independent analysis, they find The conclusions from Ref. [16], which depend on an D E r3 = 2.71 0.13 fm3, leading to an energy shift of overly large third moment of the charge distribution are (2) ± thus not supported by experiment. 0.0247 0.0012 meV. Using the Fourier transform of the− exponential± (13) distribution in Eq. (22), I find B. Finite size correction to the Uehling contribution D E 35 √3R3 r3 = 3.789R3, (47) (2) 16 ≈ For the vacuum polarization we obtain, for a point D E nucleus, showing that r3 is proportional to R3 and justifying (2) 11 pn 11 pn the fit in R2 and R3 performed to derive the coefficients ∆E , ∆E , = 205.028201 meV, (52) 2p1/2 2s1/2 above. Equation (47) is in exact agreement with the result − that can be obtained from Eq. (15) in [16], but Eq. (16) to be compared with 205.0282 meV in Ref. [39]. in the same work is not correct (the denominator should Pachucki [37] obtained 205.0243 meV as the sum of the be 256, not 64). The value obtained by Friar and Sick non-relativistic 205.0074 meV and first order relativistic corresponds to R = 0.894 fm. In that case our energy 0.0169 meV corrections. If I calculate the difference be- shift is 0.0250 meV in good agreement with the energy tween (52) and Pachuki non-relativistic value, I obtain a − 8 difference of 0.0208076 meV. This is in excellent agree- the proton size, I can also evaluate the finite size correc- ment with the value provided in Ref. [72], Eq. (6), tion to the Källén and Sabry contribution. A direct fit to 0.020843 meV. the numerical data gives a result of the form To achieve this result we used the mesh parameters described in the previous section, and checked by varying ∆E21,fs ∆E21,fs = 1.508097 0.00021341293R2 2p1/2 − 2s1/2 − them so that the results were stable within the decimal 6 3 + 7.3404895 10− R (57) places provided here. For finite nuclei, I use the same × 7 4 parameters as in the previous section. Again changes in 5.0291143 10− R meV. − × r0 and h do not change the final value. We get and ∆E11,fs ∆E11,fs = 205.0282076 0.02810970909R2 2p1/2 2s1/2 21,fs 21,fs 10 2 − − ∆E ∆E = 0.0000414300 9.25489 10− R + 0.0007111893365R3 (53) 2p3/2 − 2p1/2 − × 11 3 + 2.33622 10− R 0.00003572368803R4 × 12 4 − 1.80063 10− R meV. − × The constant term is in excellent agreement with the one (58) in Eq. (52). This result must be combined to Eq. (45) to obtain values that can be compared with the literature. I for the fine structure, to be compared with 0.00004 meV obtain: in Ref. [70].

∆E11C,fs ∆E11C,fs = 205.0282076 5.227446248R2 2p1/2 − 2s1/2 − 3 V. HIGHER-ORDER VACUUM POLARIZATION + 0.03529257801R CORRECTIONS + 0.00007356191826R4 (54) + 0.0002788380236R2 log(R) A. Higher-order vacuum polarization 0.00004957597920R4 log(R) − The term named “VP iteration”, which correspond to Fig. 7, is given by Eq. (215) of Ref. [38] The R2 coefficient can be compared to the one in Ref. [72] 2 Table III, which has the value 5.2254 meVfm− , which 4 α 2 − ∆E (2s) = 0.01244 (Zα)2 µ c2 (59) contains additional recoil corrections. VPVP 9 π r For the Uelhing correction to the fine structure, I obtain in the same way: where µr =94.96446 MeV for muonic hydrogen (using [11]). This adds 0.15086 meV to the Lamb-shift for 11,fs 11,fs muonic hydrogen. The Uehling potential under the form ∆E2p ∆E2p = 0.0050157881 3/2 − 1/2 used in Sec.II can be introduced in the potential of the 7 2 1.1662334 10− R − × (55) Dirac equation (3) when solving it numerically. This 9 3 + 2.2741334 10− R amounts to get the exact solution with any number of × 10 4 vacuum polarization insertions as shown in Fig. 7. The 1.5308196 10− R meV, − × numerical methods that we used are described in Ref. [56, 57]. Because of the Logarithmic dependence of the where the constant term is again in perfect agreement point nucleus Uehling potential at the origin, we do not with Borie’s value 0.0050 meV [70] (Table 7). calculate the iterated vacuum polarization directly for point nucleus. We instead calculate for different mean square radii and charge distribution models, and fit the C. Finite size correction to the Källén and Sabry curves with f (R) = a + bR2 + cR3 + dR4. All 4 models contribution provides very similar values. The final value is

We can then evaluate the Källén and Sabry contribu- ∆E11,loop,fs ∆E11,loop,fs = 0.15102161 21 21 2p1/2 − 2s1/2 tion ∆E2p ∆E2s using V21 calculated following Sec. 1/2 − 1/2 0.000098409804R2 III A, with good accuracy, using our numerical wave- − (60) 6 3 functions. For a point nucleus I obtain + 7.9038238 10− R × 7 4 7.2004764 10− R meV. ∆E21,pn ∆E21,pn = 1.508097 meV, (56) − × 2p1/2 − 2s1/2 The value calculated in Ref. [37] is 0.1509 meV and the in agreement with the result of Ref. [37], 1.5079 meV, one in Ref. [39] is 0.1510 meV in very good agreement and in excellent agreement with the one from Ref. [39], with the present work. The method employed here pro- 1.5081 meV. Using the wavefunctions calculated with vides in addition the proton size dependence for this 9

a b

+ +...

a b

FIG. 7. Feynman diagrams obtained when the Uehling potential is added to the nuclear potential in the Dirac equation. A double line represents a bound electron wavefunction or propagator and a wavy line a retarded photon propagator. The grey c d circles correspond to the interaction with the nucleus. Diagram (b) correspond to Fig. 5 and third term in Eq. (25) in Ref. [73] FIG. 8. Lower order Feynman diagrams included in the Käl- lén and Sabry V21(r) potential, when the Uehling potential is included in the diﬀerential equation. See Figs. 4 and 7 for ex- correction, which was not calculated before. For the ﬁne planation of symbols. Diagrams (a) and (b) exactly correspond structure, I obtain in the same way to diagrams (a) and (b) in Fig. 5 and Eq. (25) in Ref. [73]

11,loop,fs 11,loop,fs 6 ∆E ∆E = 2.33197 10− 2p3/2 − 2p1/2 × ﬁne structure this correction is very small: 9 2 1.77101 10− R − × (61) 21 11,fs 21 11,fs 8 12 2 10 3 ∆E2p× ∆E2p× = 3.75754 10− 3.49318 10− R + 9.36429 10− R 3/2 − 1/2 × − × × 13 3 10 4 + 2.58244 10− R 1.79951 10− R meV. × − × 14 4 2.74201 10− R meV. − × (63)

B. Other higher-order Uehling correction C. Wichmann and Kroll correction Since we include the vacuum polarization in the Dirac equation potential, all energies calculated by perturba- We use the approximate potentials as presented in tion using the numerical wavefunction contains the con- Refs. [68, 76] to evaluate the Wichmann and Kroll [77] tribution of higher-order diagrams where the external V13 correction to the Uehling potential. The correspond- legs, which represent the wavefunction, can be replaced ing diagram is shown on Fig. 4 (b). This contribution by a wavefunction and a bound propagator with one, is given together with the light-by-light scattering dia- or several vacuum polarization insertion. For example grams of Fig. 9 in Refs. [73, 75, 78]. We ﬁnd for a point the Källèn and Sabry correction calculated in this way, nucleus, the exact value, and a size correction, given by contains correction of the type presented in Fig. 8. This correction is given by 13,fs 13,fs 8 2 ∆E ∆E = 0.0010170628 + 5.5414179 10− R 2p1/2 − 2s1/2 − × 21 11,fs 21 11,fs 10 3 ∆E × ∆E × = 0.0021552 5.1356872 10− R 2p1/2 2s1/2 − × − 11 4 6 2 1 9364450 10− R meV 1.32976 10− R . , − × (62) − × 8 3 (64) + 9.4577 10− R × 9 4 8.5185 10− R meV, to be compared to the value given in Ref. [75] (Table III) − × of 0.001018(4) meV. In the lowest order approximation, − 7 the diagram on Fig. 9(a) provides an energy shift of with a 10− meV accuracy. This correction is part of the 2 three-loop corrections form Ref. [73, 74]. The diagrams ∆E13/Z [75, 78]. For the ﬁne structure, this correction is in Fig. 8 correspond to diagrams (a) and (b) of Fig. 5 comparable to the contribution from Eq. (63): in Ref. [73] and (e) (upper left) and (f) in Fig. 2 of 13,fs 13,fs 8 ∆E ∆E = 4.21088 10− Ref. [75]. The sum of contributions of the diagram (a) 2p3/2 − 2p1/2 − × 13 2 and (b) is 0.00223, in good agreement with our all-order + 4.41081 10− R × (65) fully relativistic result. The three loop diagram Fig. 5 15 3 8.49036 10− R (c) Ref. [73] and Fig. 2 (g) Ref. [75] is included in the − × 15 4 all-order contribution obtained by solving numerically + 1.81474 10− R meV. the Dirac equation with the Uëlhing potential. For the × 10

and 4 !3 ! α (Zα) µr 4 1 mµ ∆EµSE,2p3/2 = mµ ln k0(2P) + π 8 mµ −3 12 µr a b ! ! 29 mµ 2 2G + ln 2 α + α 2p3/2 (α) . (69) 90 α µr

The Bethe logarithms are given by ln k0(2S) = 2.811769893 and ln k (2P) = 0.030016709 [80]. The 0 − the remainders are given by G s(α) = 31.185150(90), 2 − G2p1/2 (α) = 0.97350(20) and G2p3/2 (α) = 0.48650(20) [79, FIG. 9. Feynman diagrams for the light-by-light scattering. See 81]. One then− gets the exact muon self-energy− for each Figs. 4 and 7 for explanation of symbols. state. For the 2s state, this gives 0.675150 meV instead of 0.675389 meV. For the 2p I get 0.00916882 meV and 1/2 − for 2p3/2, 0.008393377 meV in place of 0.01424054 meV and 0.00332838 meV respectively, if one would use only − the low order A40 term. The ﬁnite size correction is given by perturbation theory [11] Eq. (54) FIG. 10. Feynman diagrams for the muon self-energy. See Figs. 23 4 and 7 for explanation of symbols. ESE NS(R, Zα) = 4 ln 2 α(Zα) NS(R, Zα), (70) − − 4 E where ([11] Eq. (51)) D. Muon radiative corrections !3 2 2 2 µr (Zα) ZαR R Z m 1. Muon self-energy NS( , α) = 3 µ , (71) E 3 mµ n oC

Highly accurate self-energy values for electronic is the lowest-order finite nuclear size correction to the atoms and point nucleus are known from Ref.[79]. The Coulomb energy. Here oC = 1.867594282 fm is the muon self-energy correction, represented in Fig. 10 is conve- Compton wavelength. Equation (71) provides NS(R) = 2 E niently expressed by the slowly varying function F(Zα) 5.19745R for muonic hydrogen in agreement with Refs. defined by [37, 39]. The self-energy correction to the Lamb shift with α (Zα)4 finite-size correction is then E mc2F Z ∆ SE = 3 ( α), (66) π n SE,fs SE,fs 2 ∆E2p ∆E2s (R) = 0.68431882+0.000824R meV, (72) where m is the particle mass. 1/2 − 1/2 − The recoil corrections to F(Zα) are described in detail and to the fine structure: in [11]. The dependence in the reduced mass has to be in- ∆ESE,fs ∆ESE,fs = 0.017562197 meV. (73) cluded leading to the following expressions, specialized 2p3/2 − 2p1/2 for the n = 2 shells: It should be noted that in Ref. [39], the R2-dependent 4 !3 α (Zα) µr 2 4 10 part of the 2s self-energy is much larger than what is ∆EµSE,2S = mµc ln k0(2S) + π 8 mµ −3 9 given in Eq. (72). This value was checked indepen- ! dently by using an all-order calculation with finite size, 4 mµ 139 following the work of Mohr and Soff [82]. The results of + ln 2 + 2 ln 2 πα 3 α µr 32 − this calculation agree reasonably well with Eq. (72) and ! 67 16 ln 2 mµ 2 is given by [83] + + ln 2 α 30 3 α µr H 2 ESE NS(Z = 1, R) = 0.68431882 0.001677053R !!2 − − − m 3 µ 2 +0.0009996784R ln 2 α − α µr 0.0004785372R4 2 − 5 +α G2s(α) , (67) +0.00008785574R meV. (74)

4 !3 ! α (Zα) µr 4 1 mµ 2. Muon loop vacuum polarization ∆EµSE,2p1/2 = mµ ln k0(2P) π 8 mµ −3 − 6 µr ! ! 103 mµ The vacuum polarization due to the creation of virtual 2 2G + ln 2 α + α 2p1/2 (α) , (68) 180 α µr muon pairs is represented by the same diagram 4 (a) 11 and same equations (28) as vacuum polarization due to VI. EVALUATION OF THE RECOIL CORRECTIONS electron-positron pairs, replacing the electron Compton wavelength by the muon one. For S states, it is given by The relativistic treatment of recoil corrections is de- [11] Eq. (27),[35] Eq. (32) scribed in, e.g., [11], Eq. (10). The analytic solution of the Dirac equation for a point nucleus and a particle of 4 !3 α(αZ) 4 5 µr E ns m c2 mass m is given by µVP( ) = 3 + πα µ , (75) − πn −15 48 mµ 2 ED = mc f (n, j) (80) in which higher order terms in Zα have been neglected. For the 2s Lamb shift in muonic hydrogen it gives with 0.01669 meV and is included in Refs. [37, 39] and [84] 1 f (n, j) = s . (81) Eq. (2.29) for the first α correction. As it is a sizable con- (Zα)2 1 + q !2 tribution, and the muon Compton wavelength, which 2 n j+ 1 + (j+ 1 ) (Zα)2 represent the scale of QED corrections for muons is of − 2 2 − the order of the finite nuclear size (1.9 fm), one could ex- The recoil can then be included by evaluating [62, 85] pect a non-negligible finite size contribution. Using the 2 2 numerical procedure described in Sec. II, replacing the EM = Mc + µ c f (n, j) 1 r − electron Compton wavelength by the muon one in Eq. 2 f (n, j) 1 µ2c2 (28), I obtain + − r (82) 2M µ11,fs µ11,fs 4 3 2 ∆E ∆E = 0.01671487464 1 δl,0 (Zα) µr c 2p1/2 − 2s1/2 + − , (83) κ(2l + 1) 2n3M2 0.00005279721702R2 p − where M = m + M . If one expands the previous equa- + 0.00001269866912R3 µ p tion in power of (Zα), one would find that the terms of or- 6 4 5.360546098 10− R der up to (Zα)4 are identical to what is given in Ref. [62]. − × + 0.00001717649157R2 log(R) We also compared the numerical results from our numerical approach for point nucleus, as described in Sec. 6 4 + 2.047113814 10− R log(R) meV, × I A to what can be obtained by using directly (80) and (76) find excellent agreement. Below, we will make exclusive use of the direct numerical evaluation of the Dirac equa- where the constant term is in excellent agreement with tion. The relativistic corrections to Eq. (83) associated (75) and the R dependence explicit. From Ref. [11], Eq. with motion of the nucleus are called relativistic-recoil (55), one obtains correction. The correction to order (Zα)5 and to all orders in mµ/M is given by [11, 41, 85, 86] fs 3 2 p E VP(ns) = α(Zα) NS(R, α) = 0.00020758R meV (77) µ 3 2 5 ( 4 E µ c (Zα) δl 1 8 E5 n l r ,0 k l n RR( , ) = 3 ln 2 ln 0( , ) for the 2s level. This term is about 4 times larger than mµMp πn 3 (Zα) − 3 the numerical coefficient for R2 in Eq. (76). δl,0 7 2δl,0 Using the wavefunction evaluated with the Uehling an,l − 9 − 3 − M2 m2 potential in the Dirac equation, I also obtain the value of p − µ the sum of diagrams with one muon vacuum polariza- " ! !# ) 2 mµ 2 Mp tion loop and any number of electron loops on each side, Mp ln mµ ln (84) as in Fig. 7, with one loop being a muon loop: × µr − µr

µ11 11,fs µ11 11,fs where × ×     ∆E2p ∆E2s = 0.00005346 n 1/2 − 1/2   2 X 1 1  7 2 an l = 2 ln  +  + 1  δl 7.6035 10− R , −  n i  − 2n ,0 − × (78) i=1 7 3 + 3.2548 10− R × 1 δl,0 8 4 + − . (85) 5.4392 10− R meV. l(l + 1)(2l + 1) − × This muonic vacuum polarization is a small contribution This correction corresponds to the diagrams in Fig. 11. to the ﬁne structure The next order of the relativistic recoil corrections is given for s states by µ11 11,fs µ11 11,fs 7 ∆E × ∆E × = 1.67794 10− 2p3/2 2p1/2 6 ( − × mµ (Zα) 7 10 2 E6 ns m c2 2.10861 10− R RR( ) = 3 µ 4 ln 2 − × (79) Mp n − 2 11 3 + 8.51427 10− R ) × 11 1 11 4 ln , (86) 1.38884 10− R meV. 2 − × −60π (Zα) 12

bound particle coordinates. The vector spherical har- Y (0) r J 2 monic 1q ( ˆ) is an eigenfunction of and Jz, deﬁned as [88, 90–93]

(0) X Y (rˆ) = Y11q (rˆ) = C 1, 1, 1; q σ, σ, q Y1,q σ (rˆ) ξσ 1q − σ − FIG. 11. Feynman diagrams corresponding to the Relativistic (94) recoil correction (84). The heavy double line represents the proton wave function or propagator. The other symbols are where C j1, j2, j; m1, m2, m is a Clebsh-Gordan coeffi- explained in Fig. 4. cient, Y1,q are scalar spherical harmonic and ξσ are eigen- 2 vectors of s and sz, the spin 1 matrices [88, 90–93]. The and for l 1 states by reduction to radial and angular integrals is presented ≥ in various works [88–90]. In heavy atoms, the hyper- 6 (" # fine structure correction due to the magnetic moment 6 mµ (Zα) 2 l(l + 1) E (nl) = mµc 3 RR M n3 − n2 contribution is usually calculated for a finite charge dis- p tribution, but a point magnetic dipole moment (see, e.g., ) 2 [88, 89]). When matrix elements non-diagonal in J are . (87) ×(4l2 1) (2l + 3) needed, one can use [94] for a one-particle atom − Z Using Eqs. (82) to (87), I obtain gα ∞ P (r)Q (r) + P (r)Q (r) EHFS A dr 1 2 2 1 ∆ M1 = 2 , (95) 2Mp r ∆ERec. ∆ERec. (R) = 0 + 0.0574706 0.0449705 0 2p1/2 − 2s1/2 − (88) = 0.0125001meV, where g = µp/2 = 2.792847356 for the proton, is the anomalous magnetic moment, A is an angular coefficient and to the fine structure: ( ) ∆ERec. ∆ERec. = 0.0000051 0.0862059 + 0 I j1 F 2p3/2 2p1/2 − − (89) j2 I k I+j +F = 0.0862008 meV. A = ( 1) 1 ! − − I 1 I This is in excellent agreement with the results from Ref. I 0 I [70]. − ! J 1 p j1 1 j2 ( 1) 1− 2 (2J + 1)(2J + 1) π (l , k, l ) , × − 1 2 1 0 1 1 2 2 − 2 VII. EVALUATION OF SOME ALL-ORDER HYPERFINE (96) STRUCTURE CORRECTIONS where π (l1, k, l2) = 0 if l1 + l2 + 1 is odd and 1 otherwise. The expression of the hyperfine magnetic dipole op- The ji are the total angular momentum of the i state for erator can be written as the bound particle, li are orbital angular momentum, I is the nuclear spin, k the multipole order (k = 1 for the Hh f s = ecα A(r) = ecα A(r), (90) magnetic dipole contribution described in Eq. (95)) and F − · − · the total angular momentum of the atom. The difference with between ∆EHFS values calculated with a finite or point µ0 µ r nuclear charge contribution is called the Breit-Rosenthal A(r) = × , (91) 4π r3 correction [95]. To consider a finite magnetic moment distribution, one µ where is the nuclear magnetic moment and we have uses the Bohr-Weisskopf correction [96]. The correction assumed a magnetic moment distribution of a point par- can be written [97] ticle for the nucleus. It is convenient to express Hh f s using vector spherical harmonics. On obtains [87–90] Z BW gα ∞ 2 ∆E = A drnrnµ(rn) 1 1 − 2Mp 0 Hh f s = M T , (92) Z r (97) · n P (r)Q (r) + P (r)Q (r) dr 1 2 2 1 where 2 , × 0 r r α Y (0) (rˆ) 8π 1q where the magnetic moment density µ(r ) is normalized T 1 (r) = ie · , (93) n − 3 r2 as

1 Z and M representing the magnetic moment operator ∞ 2 T 1 drnrnµ(rn) = 1. (98) from the nucleus. The operator acts only on the 0 13

Borie and Rinker [98], write the total diagonal hyper- where φC(0) is the unperturbed Coulomb wavefunction fine energy correction for a muonic atom as at the origin for a point nucleus. Replacing into Eq. (102) and keeping only first order terms, we get (Eq. 2.8 of 4πκ F(F + 1) I(I + 1) j(j + 1) gα ∆Ei,j = − − Ref. [99] corrected for a misprint): 2 1 2M κ 4 p Z − Z r ∞ P (r)Q (r) dr 1 2 dr r2µ (r ). (99) 2 n n BR n D E × 0 r 0 Z 2 2 ∆E = Sp Sµ φC(0) HFS −3 · where the normalization is different: Z ! Z Z ∞ 3 ∞ 2 1 2αmµ ρ(u) u r µ(r)dudr , d rnµBR(rn) = 4π drnrnµBR(rn) = 1. (100) × − | − | 0 0 Z ! This means that µ (r) = µ(r)/(4π). Evaluation of the = EF 1 2αmµ ρ(u) u r µ(r)dudr , BR − | − | Wigner 3J and 6J symbols in (96) give the same angular factor than in Eq. (95). (104) The equivalence of the two formalism can be easily checked: starting from (99) and droping the angular factors, we get, doing an integration by part where EF is the well known HFS Fermi energy. Trans- forming Eq. (104) using r r + u, etc. Zemach obtains Z Z r ∞ P (r)Q (r) → dr 1 2 dr r2µ(r ) r2 n n n 0 0 "Z r Z r # Z P (t)Q (t) ∞ ∆E = EF 1 2αmµ rZ , (105) dr r2 r dt 1 2 HFS − h i = n nµ( n) 2 0 0 t 0 Z Z rn ∞ 2 P1(r)Q2(r) dr r µ(r ) dr with rZ given in Eq. (16). The 2s state Fermi energy is n n n 2 h i − 0 0 r given by Z ∞ P (r)Q (r) dr 1 2 = 2 0 r 4 3 Z Z rn (Zα) µr ∞ P1(r)Q2(r) 2s dr r2 r dr EF = gp . (106) n nµ( n) 2 , (101) 3 mpmµ − 0 0 r where we have used (98). We thus find that the for- mula in Borie and Rinker represents the full hyperfine structure correction, including the Bohr-Weisskopf part. In 1956, Zemach [99] calculated the fine structure energy of hydrogen, including recoil effects. He showed A. Hyperfine structure of the 2s level that in first order in the finite size, the HFS depends on the charge and magnetic distribution moments only through the Zemacs’s form factor defined in Eq. (15). In order to check the dependence of the hyperfine The proton is assumed to be at the origin of coordinates. structure on the Zemach radius and on the proton fi- Its charge and magnetic moment distribution are given nite size, I have performed a series of calculations for in terms of charge distribution ρ(r) and magnetic mo- a dipolar distribution for both the charge and magnetic ment distributions µ(r). Zemach calculate the correction moment distribution. We can then study the dependence in first order to the hyperfine energy of s-states of hy- of the HFS beyond the first order corresponding to the drogen due to the electric charge distribution. The HFS Zemach correction. I calculated the hyperfine energy BW energy is written as splitting ∆EHFS (RZ, R) = EHFS(R) + EHFS (R, RM) numer- Z ically. I also evaluate with and without self-consistent 2 D E Z 2 inclusion of the Uëhling potential in the calculation, to ∆EHFS = Sp Sµ φ(r) µ(r)dr (102) −3 · | | obtain all-order Uëhling contribution to the HFS energy.

φ the non-relativistic electron wavefunction and Sx are We calculated the correction ∆EHFS (RZ, R) for several the spin operators of the electron and proton. If the value of RZ between 0.8 fm and 1.15 fm, and proton sizes magnetic moment distribution is taken to be the one of a ranging from 0.3 fm to 1.2 fm, by steps of 0.05 fm, which point charge, µ(r) = δ(r), the integral reduces to φ(0) 2. represents 285 values. The results show that the cor- The first order correction to the wavefunction due| to the| rection to the HFS energy due to charge and magnetic nucleus finite charge distribution is given by moment distribution is not quite independent of R as Z ! one would expect from Eq. (105), in which the finite size contribution depends only on RZ. We fitted the hyper- φ(r) = φC(0) 1 αmµ ρ(u) u r du , (103) 2s − | − | fine structure splitting of the 2s level, EHFS (RZ, R) by a 14

1.4 function of R and RZ, which gives: 0.001 2s EHFS (RZ, R) = 22.807995 1.2 -0.001 0.0022324349R2 + 0.00072910794R3 − 4 0.000065912957R 0.16034434RZ 1.0 − − L RR 0.00057179529 Z fm − H 0.00069518048R2R 0.8 − Z 0.0005 3 0.00018463878R RZ radius − (107) 2 0.6 -0.0005 + 0.0010566454RZ 2 + 0.00096830453RRZ Zeamch 2 2 0.4 + 0.00037883473R RZ 0.00048210961R3 − Z 0.2 3 0.00041573690RRZ − 0 4 -0.0015 + 0.00018238754RZ meV. 0.0 0.0 0.2 0.4 0.6 0.8 1.0 The constant term should be close to the sum of the Fermi charge radius HfmL energy 22.80541 meV and of the Breit term [100]. the HFS correction calculated with a point-nucleus Dirac wave- FIG. 12. difference between Borie’s value and Eq. (107) result function for which I find 22.807995 meV. When setting as a function of the charge and Zemach radii (meV). the speed of light to infinity in the program I recover exactly the Fermi energy. The Breit contribution is thus 0.002595 meV, to be compared to 0.0026 meV in Ref. [40] (Table II, line 3) and 0.00258 meV in Ref. [70]. Mar- scribed as a function of RZ and R as tynenko [40] evaluates this correction, which he names “Proton structure corrections of order α5 and α6”, to be 2s,VP 2 0.1535 meV, following [35]. He finds the coefficient EHFS (RZ, R) = 0.074369030 + 0.000074236132R − 1 for the Zemach’s radius to be 0.16018 meVf− , in very 3 6 4 − + 0.00013277334R 8.0987285 10− R good agreement with the present all-order calculation − × 1 1 0.0017880269R 0.00017204505RR 0.16034 meVf− . Borie’s value [70] 0.16037 meVf− is − Z − Z − − 2 even closer. The difference between Borie’s value and 0.00037499458R RZ Eq. (107) is represented in Fig. 12 as a function of the − 0.000070355379R3R charge and Zemach radii. The maximum difference is − Z around 1 µeV. 0.00022093411R2 + 0.00035038656RR2 − Z Z In Ref. [35], the charge and magnetic moment dis- + 0.00020554316R2R2 + 0.00025100642R3 tributions are written down in the dipole form, which Z Z 0.00017200435RR3 corresponds to (13), − Z 4 0.000061266973RZ meV. 2 − GM q Λ4 (109) G q2 = = , (108) E 2 1 + κp Λ2 + q2 with Λ = 848.5MeV. This leads to R = 0.806 fm as in It corresponds to the diagrams presented in Fig. 13. Ref. [2] and RZ = 1.017 fm using this definition for The size-independent term 0.07437 meV corresponds the form factor in Eq. (17). Moreover there are re- to the sum of the two contributions represented by coil corrections included. Pachucki [35] finds that the the two top diagrams in Fig. 13 and is given as HFS pure Zemach contribution (in the limit mp ) is ∆E = 0.0746 meV in Ref. [40]. The term → ∞ 1loop-after-loop VP 0.183 meV. In Ref. [101], the Zemach corrections is ∆EHFS = 0.0481 meV corresponds to a vacuum polar- − 4 1γ,VP given as δ(Zemach) EF = 71.80 10− EF, for RZ = × − × 1 ization loop in the HFS potential [35, 40, 98], which is not 1.022 fm. This leads to a coefficient 0.1602 meVfm− , in − 1 evaluated here. Corrections present in Ref. [70] not in- excellent agreement with our value 0.16036 meVfm− . − cluded in Eqs. (107) and (109) gives an extra contribution The effect of the vacuum polarization on the 2s hyper- of fine structure energy shift as a function of the Zemach and charge radius have been calculated for the same set 2s,HO of values as the main contribution. The data can be de- EHFS = 0.10287 meV. (110) 15

Combining Eqs. (107), (109) and (110), I get E2s (R , R) = 22.985234 0.0021581988R2 HFS Z − + 0.00086188128R3 0.000074011685R4 − 0.16213237R − Z 0.00074384033RR − Z 0.0010701751R2R − Z 3 0.00025499415R RZ (111) − 2 + 0.00083571133RZ 2 + 0.0013186911RRZ 2 2 + 0.00058437789R RZ 0.00023110319R3 − Z 0.00058774125RR3 − Z 4 + 0.00012112057RZ meV.

In Ref. [40], the equivalent expression is FIG. 13. Feynman diagrams corresponding to the evaluation E2sMart. (R ) = 22.9857 0.16018R meV, (112) of the hyperfine structure using wavefunctions obtained with HFS Z − Z the Uehling potential in the Dirac equation. The grey squares while it is correspond to the hyperfine interaction. E2sBorie (R ) = 22.9627 0.16037R meV, (113) HFS Z − Z in Ref. [70]. Using a Zemach’s radius of 0.9477 fm in Eq. (114) (112), needed to reproduce entry 11 in Table II of Ref. [40], one obtains 22.8148 meV as expected. In Eq. (113), In the same way, Eqs. (46), (55), (58), (61), (63), (65), (73), it gives 22.8107 meV. Using the same Zemach radius (79), and (89) lead to the fine structure interval (which and Eq. (111) I obtain 22.8104 meV with the muonic include the recoil corrections (89), included in Table II hydrogen proton radius value and 22.8103 meV with for the 2s Lamb shift) the CODATA one, in excellent agreement with Borie’s ETot,fs ETot,fs(R) = 8.3520516 value. All three values are in agreement with the result 2p3/2 − 2p1/2 22.8146(49) meV in Ref. [19]. In a recent work, how- 0.00005206070R2 ever, the use of Form factors in the Breit equations leads − 7 3 + 1.70858 10− R to smaller finite size corrections, leading to 22.8560 meV × (115) 8 4 [102]. A comparison between some of these results is 4.30422 10− R − × presented in Table I. 8 5 + 1.54724 10− R × 9 6 2.13593 10− R meV. VIII. EVALUATION OF MUONIC HYDROGEN n = 2 − × TRANSITIONS Martynenko [105] finds ETot,fs ETot,fs = 8.352082 meV 2p3/2 − 2p1/2 for the fine structure. A. Lamb shift and fine structure A number of terms not included in Eqs. (114) are presented in Table II together with the relevant references. The results presented in this work for the Lamb shift Combining Eq. (114) with the sum of the contributions (Eqs. (45), (54), (57), (60), (62), (64), (72), (76), (78)) can contained in Table II, I obtain the final 2s 2p1/2 energy: be summarized in the following proton-size dependent − equation: ETot,fs ETot,fs(R) = 206.046613695 5.226988678R2 2p1/2 2s1/2 Tot,fs Tot,fs − − E E (R) = 206.0209137 5.226988678R2 3 2p1/2 − 2s1/2 − + 0.03532068001R + 0.03532068001R3 + 0.00006692700063R4 + 0.00006692700063R4 + 0.0002962967640R2 log(R) + 0.0002962967640R2 log(R) 0.00004751147090R4 log(R) meV. − 0.00004751147090R4 log(R) meV. (116) − 16

TABLE I. Comparison of contributions to the 2s hyperﬁne structure from Refs. [40, 70] and the present work (meV) for 5 RZ = 1.0668 fm [2] as used in Ref. [40]. Note that in this reference, the proton structure correction of order α (item # 6) may combine the Zemach correction and the recoil correction (# 24). VP: Vacuum Polarization.

# Ref. [40] Ref. [70] This work Fermi energy 1 22.8054 22.8054 Dirac Energy (includes Breit corr.) 2 22.807995 Vacuum polarization corrections of orders α5, α6 in 2nd-order 3 0.0746 0.07443 perturbation theory VP1 All-order VP contribution to HFS, with finite magnetisation distribution 4 0.07244 finite extent of magnetisation density correction to the above 5 0.00114 Proton structure corr. of order α5 6 0.1518 −0.17108 0.17173 Proton structure corrections of order α6 7 −0.0017 − − Electron vacuum polarization contribution+ proton structure corrections of order α6 8 −0.0026 contribution of 1γ interaction of order α6 9− 0.0003 0.00037 0.00037 VP2EF (neglected in Ref. [40]) 10 0.00056 0.00056 muon loop VP (part corresponding to VP2 neglected in Ref. [40]) 11 0.00091 0.00091 Hadronic Vac. Pol. 12 0.0005 0.0006 0.0006 Vertex (order α5) 13 0.00311 0.00311 Vertex (order α6) (only part with powers of ln(α) - see Ref. [103] ) 14 −0.00017 −0.00017 Breit 15 0.0026− 0.00258 − Muon anomalous magnetic moment correction of order α5, α6 16 0.0266 0.02659 0.02659 Relativistic and radiative recoil corrections with 17 0.0018 proton anomalous magnetic moment of order α6 One-loop electron vacuum polarization contribution of 1γ interaction 18 0.0482 0.04818 0.04818 5 6 of orders α , α (VP2) finite extent of magnetisation density correction to the above 19 0.00114 0.00114 One-loop muon vacuum polarization contribution of 1γ interaction of order α6 20 0.0004− 0.00037− 0.00037 Muon self energy+proton structure correction of order α6 21 0.001 0.001 Vertex corrections+proton structure corrections of order α6 22 0.0018 0.0018 “Jellyfish” diagram correction+ proton structure corrections of order α6 23− 0.0005− 0.0005 Recoil correction Ref. [104] 24 0.02123 0.02123 Proton polarizability contribution of order α5 25 0.0105 Proton polarizability Ref. [104] 26 0.00801 0.00801 Weak interaction contribution 27 0.0003 0.00027 0.00027 Total 22.8148 22.8129 22.8111

This can be compared with the result from from E. Borie term. This lead to the ﬁnal result [70] ETot,fs ETot,fs(R) = 8.351988025 2p3/2 − 2p1/2 EBorie,fs EBorie,fs(R) = 206.0579(60) 5.22713R2 0.00005206070023R2 2p1/2 − 2s1/2 − − (117) 7 3 + 0.0365(18)R3 meV, + 1.708581114 10− R × (120) 8 4 4.304222172 10− R − × and Carroll et al. [106] 8 5 + 1.547238809 10− R × 9 6 Car.,fs Car.,fs 2 2.135927044 10− R meV. E E (R) = 206.0604 5.2794R − × 2p1/2 − 2s1/2 − (118) + 0.0546R3 meV.

B. Transitions between hyperﬁne sublevels An extra recoil contribution is given in Ref. [72] for the ﬁne structure, corresponding to entry #9 in Table II The energies of the two transitions observed experi- for the Lamb shift, mentally in muonic hydrogen are given by

VPRec. VPRec. F=2 F=1 E2p E2p = 0.00006359 meV. (119) E E = E2p E2s + E2p E2p 3/2 − 1/2 − 2p3/2 − 2s1/2 1/2 − 1/2 3/2 − 1/2 3 p 1 (121) + E2 3/2 E2s , This term correspond to corrections beyond the full Dirac 8 HFS − 4 HFS 17

TABLE II. Contributions to the Lamb shift not included in Eq. (114) (meV). The uncertainty on the proton polarization value used in Ref. [12] has been increased by a factor of 10, according to the discussion in Ref. [26].

# Contribution Reference Value Unc. 1 NR three-loop electron VP (Eq. (11), (15), (18) and (23)) [73] 0.00529 2 Virtual Delbrück scattering (2:2) [75, 78] 0.00115 0.00001 3 Light by light electron loop contribution (3:1) [75, 78] -0.00102 0.00001 4 Mixed self-energy vacuum polarization [35, 84, 107] -0.00254 5 Hadronic vacuum polarization [108–110] 0.01121 0.00044 6 Recoil contribution Eqs. (82) and (83) [11, 36, 62, 85] 0.05747063 7 Relativistic recoil of order (Zα)5 Eq. (84) [11, 37–39, 41] -0.04497053 8 Relativistic Recoil of order (Zα)6 Eq. (86) [11, 37] 0.0002475 9 Recoil correction to VP of order m/M and (m/M)2 in Eq. (4) [72] -0.001987 10 Proton Self-energy [35, 37, 41, 111] -0.0108 0.0010 11 Proton polarization [18, 37, 109, 112, 113] 0.0129 0.0040 12 Electron loop in the radiative photon [98, 114–116] -0.00171 of order α2(Zα)4 13 Mixed electron and muon loops [117] 0.00007 14 Rad. Recoil corr. α(Zα)5 [61] 0.000136 5 15 Hadronic polarization α(Zα) mr [109, 110] 0.000047 16 Hadronic polarization in the radiative [109, 110] -0.000015 2 4 photon α (Zα) mr 17 Polarization operator induced correction [110] 0.00019 5 to nuclear polarizability α(Zα) mr 18 Radiative photon induced correction [110] -0.00001 5 to nuclear polarizability α(Zα) mr Total 0.0256 0.0041

and Using the results presented above I get

F=2 F=1 2 E E (RZ, R) = 209.92451 5.2265012R 2p3/2 − 2s1/2 − + 0.035105381R3 + 0.000085386880R4 8 5 + 1.5472388 10− R F=1 F=0 × E2p E2s = E2p1/2 E2s1/2 + E2p3/2 E2p1/2 9 6 3/2 − 1/2 − − 2.1359270 10− R 5 p 3 (122) − × E2 3/2 + E2s + δEF=1 . + 0.040533092Rz − 8 HFS 4 HFS HFS + 0.00018596008RRz + 0.00026754376R2Rz + 0.000063748539R3Rz (124) 0.00020892783Rz2 − 0.00032967277RRz2 − Here we use the results from [105] for the 2p states: 0.00014609447R2Rz2 − + 0.000057775798Rz3 + 0.00014693531RRz3 0.000030280142Rz4 − + 0.00029629676R2 log(R) 4 2p1/2 0.000047511471R log(R) E = 7.964364 meV − HFS meV. 2p3/2 (123) EHFS = 3.392588 meV F=1 δEHFS = 0.14456 meV. This can be compared with the result from 18

U. Jentschura [84] perfine splitting on the proton charge distribution and Zemach radius has been evaluated as well. ∆EJents.,F=2 ∆EJents.,F=1 = 209.9974(48) The discrepancy between the proton size deduced 2p3/2 2s1/2 − (125) from muonic hydrogen and the one coming from CO- 5.2262R2 meV, − DATA is slightly enlarged when tacking into account all the newly calculated effects. It is changed from 6.9 σ to s using the 2 hyperfine structure of Ref. [40]. 7.1 σ. For the other transition I obtain

F=1 F=0 2 0.91 E E (RZ, R) = 229.66172 5.2286594R 2p3/2 − 2s1/2 − 3 + 0.035967212R 0.89 + 0.000011416693R4 0.12159928R 0.00055788025RR 0.87 − Z − Z 2 0.00080263129R RZ electron scat. − CODATA 3 0.85 0.00019124562R R µH LS − Z 2 Lattice QCD + 0.00062678350RZ 0.83 2 radius RMS(fm) charge proton + 0.00098901832RRZ 2 2 0.81 + 0.00043828342R RZ 0.00017332740R3 − Z 3 0.79 1960 1970 1980 1990 2000 2010 2020 0.00044080593RRZ − Year 4 + 0.000090840426RZ 2 + 0.00029629676R log(R) FIG. 14. Plot of the proton size as a function of time and 0.000047511471R4 log(R) method. − meV. (126) ACKNOWLEDGMENTS

Using Eq. (124), a Zemach radius of 1.0668 fm from The author wishes to thank Randof Pohl, Eric-Olivier Ref. [40] and the transition energy from Ref. [12], I Le Bigot, François Nez, François Biraben and several obtain a charge radius for the proton of 0.84091(69) fm other members of the CREMA collaboration for nu- in place of 0.84184(69) fm in Ref. [12] and 0.8775(51) fm merous and enlightening discussions and Jean-Paul De- in the 2010 CODATA fundamental constant adjustment. sclaux for help in implementing some new corrections This is 7.1 σ (using the combined σ) from the 2010 CO- in the MCDF code. Special thanks go to Peter Mohr for DATA value. A summary of proton size determinations several invitation to NIST where part of this work was is presented in Table III and Fig. 14. performed, many discussions and for critical reading of part of the manuscript. I also thank Michael Distler for providing me with the charge density deduced from the IX. CONCLUSION MAMI experiment and Krzysztof Pachuki for his sug- gestion to introduce the logarithmic contribution to the In the present work, I have evaluated finite-size de- fits. I thank also Franz Kottmann and Aldo Antognini pendent contributions to the n = 2 Lamb shift in muonic for a critical and detailed reading of the manuscript. hydrogen, to the fine structure and to the 2s hyperfine The Feynman diagrams presented in the figures are re- splitting. The calculations were performed numerically, alized with JAXODRAW [128]. This research was partly to all order in the finite size correction, in the framework supported by the Helmholtz Alliance HA216/EMMI. of the Dirac equation. High-order size contributions to Laboratoire Kastler Brossel is “Unité Mixte de Recherche the Uelhing potential and to higher-order QED correc- n◦ 8552” of École Normale Supérieure, CNRS and Uni- tions been evaluated. The full dependance of the 2s hy- versité Pierre et Marie Curie.

[1] L. N. Hand, D. G. Miller, and R. Wilson, Rev. Mod. Phys. [2] G. G. Simon, C. Schmitt, F. Borkowski, and V.H. Walther, 35, 335 (1963). Nuclear Physics A 333, 381 (1980). 19

TABLE III. Proton size determinations (fm). e−p: electron-proton scattering, µH: muonic hydrogen, ChPt: Lattice QCD corrected with Chiral perturbation theory. The three last lines uses the transition frequency from Ref. [12]

Hand et al. [1] 0.805 0.011 e−p ± Simon et al. [2] 0.862 0.012 e−p ± Mergel et al. [118] 0.847 0.008 e−p ± Rosenfelder [119] 0.880 0.015 e−p ± Sick 2003 [120] 0.895 0.018 e−p ± Angeli [121] 0.8791 0.0088 e−p ± Kelly [122] 0.863 0.004 e−p ± Hammer et al. [123] 0.848 hydrogen, e−p CODATA 06 [11] 0.8768 0.0069 Hydrogen, e−p ± Arington et al. [51] 0.850 e−p 0.004 Belushkin et al. [50] SC approach 0.844 +−0.008 e−p 0.008 Belushkin et al. [50]pQCD app. 0.830 +−0.005 e−p Wang et al. [124] 0.828 ChPt Pohl et al. [12] 0.84184 0.00067 µH ± Bernauer et al. [14] 0.879 0.008 e−p ± CODATA 2010 [15] 0.8775 0.0051 Hydrogen, e−p ± Adamušˇcínet al. [125, 126] 0.84894 0.00690 e−p This work (using R = 1.045 fm)[127] 0.84081 ± 0.00069 µH Z ± This work (using RZ = 1.0668 fm)[40] 0.84091 0.00069 µH Using Jentschura [84] 0.84169 ± 0.00066 µH Using Borie 0.84232 ± 0.00069 µH ±

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L1 (1980). Appendix A: Coefficients for the numerical evaluation of [98] E. Borie and G. A. Rinker, Review of Modern Physics 54, the Källén and Sabry potential for a point nucleus 67 (1982). [99] A. C. Zemach, Phys. Rev. 104, 1771 (1956). [100] G. Breit, Phys. Rev. 35, 1447 (1930). The functions defined in Eq. (35) are given here. We [101] E. V.Cherednikova, R. N. Faustov, and A. P.Martynenko, find, for x 3, the functions valid for a point nucleus: Nuclear Physics A 703, 365 (2002). ≤ [102] F. Garcia Daza, N. G. Kelkar, and M. Nowakowski, Jour- 39 nal of Physics G: Nuclear and Particle Physics , 035103 8 (2012). g0(r) = 0.00013575124407339550307r [103] S. J. Brodsky and G. W. Erickson, Phys. Rev. 148, 26 (1966). 0.00012633396034194731891r7 [104] C. E. Carlson, V. Nazaryan, and K. Griffioen, Phys. Rev. − 6 A 78, 022517 (2008). + 0.0023754193119115541914r [105] A. P. Martynenko, Physics of Atomic Nuclei 71, 125 0.0052460271878852635132r5 (2008). − [106] J. D. Carroll, A. W. Thomas, J. Rafelski, and G. A. Miller, + 0.16925588925254111005r4 (A1) Phys. Rev. A 84, 012506 (2011). 0.25201860708873574898r3 [107] U. Jentschura and B. Wundt, Eur. Phys. J. D 65, 357 (2011). − [108] J. L. Friar, J. Martorell, and D. W. L. Sprung, Phys. Rev. + 0.95109984162919008905r2 A 59, 4061 (1999). 2.0864972181198001792r [109] A. Martynenko and R. Faustov, Physics of Atomic Nuclei − 63, 845 (2000). + 1.6459704071917522632, [110] A. Martynenko and R. Faustov, Physics of Atomic Nuclei 64, 1282 (2001). [111] R. J. Hill and G. Paz, Phys. Rev. Lett. 107, 160402 (2011). [112] R. Rosenfelder, Physics Letters B 463, 317 (1999). [113] A. Martynenko, Physics of Atomic Nuclei 69, 1309 (2006). 8 [114] H. Suura and E. H. Wichmann, Phys. Rev. 105, 1930 g1(r) = 0.000078684672329473358699r − (1957). 0.0012293141869424835524r6 [115] A. Petermann, Phys. Rev. 105, 1931 (1957). − [116] R. Barbieri, M. Caffo, and E. Remiddi, Lettere al Nuovo 0.097906849416525020713r4 (A2) Cimento 7 (1973). − 0.41666290189975666225r2 [117] E. Borie, Helvetica Physica Acta 48, 671 (1975). − [118] P. Mergell, U. G. Meissner, and D. Drechsel, Nuclear + 0.13769050748433509769 Physics A 596, 367 (1996). [119] R. Rosenfelder, Physics Letters B 479, 381 (2000). [120] I. Sick, Physics Letters B 576, 62 (2003). and [121] I. Angeli, Atomic Data and Nuclear Data Tables 87, 185 (2004). [122] J. J. Kelly, Phys. Rev. C 70, 068202 (2004). g (r) = 0.000012756169252850100497r8 [123] H. W. Hammer and U.-G. Meißner, Eur. Phys. J. A 20, 469 2 − (2004). + 0.017425498169562658160r4 (A3) [124] P. Wang, D. B. Leinweber, A. W. Thomas, and R. D. Young, Phys. Rev. D 79, 094001 (2009). + 0.44444444460943167625 . [125] C. Adamušˇcín,S. Dubniˇcka,and A. Z. Dubniˇcková,Nu- clear Physics B - Proceedings Supplements 219-220, 178 (2011). For x > 3, we fitted the coefficients in Eq. (40) to the [126] C. Adamuscin, S. Dubnicka, and A. Z. Dubnickova, numerical values. We obtain Progress in Particle and Nuclear Physics in press (2012). [127] A. V. Volotka, V. M. Shabaev, G. Plunien, and G. Soff, Eur. Phys. J. D 33, 23 (2005). a = 4.3942926509010 [128] D. Binosi, J. Collins, C. Kaufhold, and L. Theussl, Com- puter Physics Communications 180, 1709 (2009). b = 10.059551479890 − c = 5.5493632222582 d = 5.3327556570422 e = 9.0762837836987 − f = 5.1094977559523 .

Using these functions we reach an agreement to 9 decimal place with both the result of the numerical evaluation and the expansion from [68]. 22

Appendix B: Coeﬃcients for the numerical evaluation of the and Källén and Sabry potential for a ﬁnite nucleus.

The coeﬃcients for the functions deﬁned in Eq. (39) that we obtained are listed below:

h (r) = 0.00001608988362060r9 0 − + 0.000015791745043r8 0.00036443366062r7 − + 0.0008743378646r6 g (r) = 0.44444444460943167625 2 − 0.038046259798r5 (B1) 0.0034850996339125316319r4 (B3) − − 4 6 8 + 0.063004651772r 1.4173521392055667219 10− r . − × 0.36332915853r3 − + 1.04324860906r2 2.39716878893r + 2.005566300 −

g1(r) = 0.7511983817345282548 + 0.13888763396658555408r2 + 0.020975409736870016795r4 (B2) + 0.00017561631242035479319r6 6 8 + 9.057708511987165794 10− r ×