Muonic hydrogen and the proton size

We reexamine the structure of the $n=2$ levels of muonic hydrogen using a two-body potential that includes all relativistic, recoil and one loop corrections. The potential was originally derived from QED to describe the muonium atom and accounts for all contributions to order $\alpha^5$. Since one loop corrections are included, the anomalous magnetic moment contributions of the muon can be identified and replaced by the proton anomalous magnetic moment to describe muonic hydrogen with a point-like proton. This serves as a convenient starting point to include the dominant electron vacuum polarization corrections to the spectrum and extract the proton's mean squared radius $r_p=\sqrt{\langle r^2\rangle}$. Our results are consistent with other theoretical calculations that find that the muonic hydrogen value for $r_p$ is smaller than the result obtained from electron scattering.

(Dated: June 25, 2018) We reexamine the structure of the n = 2 levels of muonic hydrogen using a two-body potential that includes all relativistic, recoil and one loop corrections. The potential was originally derived from QED to describe the muonium atom and accounts for all contributions to order α 5 . Since one loop corrections are included, the anomalous magnetic moment contributions of the muon can be identified and replaced by the proton anomalous magnetic moment to describe muonic hydrogen with a point-like proton. This serves as a convenient starting point to include the dominant electron vacuum polarization corrections to the spectrum and extract the proton's mean squared radius rp = r 2 . Our results are consistent with other theoretical calculations that find that the muonic hydrogen value for rp is smaller than the result obtained from electron scattering.
If proton structure corrections are included, the resulting values of r p from muonic hydrogen are systematically smaller than those generally obtained from electron scattering data [10], leading to a disparity between the two approaches. Some of disparity could be associated with uncertainties in the scattering data. The proton radius experiment (PRad) at the Jefferson Laboratory is designed to address this issue [11]. A recent spectroscopic measurement of the Rydberg constant [12] reports a smaller value of r p consistent with muonic hydrogen, but a new spectroscopic measurement of the 1S → 3S transition in hydrogen [13] supports a larger value as do most other spectroscopic measurements.
Here, we reexamine the theoretical calculation from a slightly different starting point. Our approach is to modify the two-body potential originally derived from QED to describe the muonium atom [14]. This potential contains all relativistic, recoil and one-loop terms that contribute to order α 5 . The inclusion of the one-loop corrections enables us to identify the muon anomalous magnetic moment and replace it by the proton's anomalous magnetic moment κ = 1.79285. The resulting potential can be used to calculate the fine structure, hyperfine structure, Lamb shift and recoil corrections for muonic hydrogen with a a point-like proton. It also serves as the starting point to include the dominant electron vacuum polarization contributions. The resulting hyperfine, spin-orbit, tensor and spin-independent potentials are [15] V HF = 4πα m 1 m 2 Here, µ is the reduced mass, Much of the reason for undertaking the following calculation is that Eq. (4) contains several terms that differ from those commonly used in determining the muonic hydrogen spectrum. In particular, the order α delta function term behaves as µ −2 whereas the corresponding term in [5] has an overall (1/m 2 1 + 1/m 2 2 ) factor. The difference in the s-state contribution is several meV in the order α 4 correction. Also, the one-loop term µ α 2 /(2m 1 m 2 r 2 ) contributes at order α 4 at the few meV level. The µ −2 dependence of the order α 2 one-loop ln(µ/λ IF ) term results in a recoil correction to the Lamb shift of order µ 3 α 5 /m 2 2 that is larger than the order α 6 correction to this contribution given in [5]. We undertook the calculation to determine the implication of these differences.

II.1. One loop correction to the Coulomb potential
The dominant contributions to the 2P − 2S splitting in muonic hydrogen are due to the electron vacuum polarization corrections to the photon propagator. These contributions can be included by using the dispersion representation for the photon propagator [17][18][19] where ∆(q 2 ) is For the eē intermediate state, If we take k 2 to be space-like, then the modified Coulomb interaction at the one-loop level is The explicit form of V V P ( k 2 ) in momentum space can be obtained by integrating over λ which results in with .Π (2) is the electron one-loop vacuum polarization correction in the spacelike region. Transforming to coordinate space To compute the effect of V V P , we need to calculate the difference between 21m|V V P |21m and 200|V V P |200 . For example, 21m|V V P |21m is with a = 1/µα and λ = m 2 e x. A similar calculation gives The difference between Eqs. (14) and (15) is This agrees with [5] . It is worth noting that the muon contribution from Eq. (16) is 0.0167 meV which is virtually the same as the 0.0168 meV result coming from m −2 1 muon contribution in last term of Eq. (4). The proton vacuum contribution from the m −2 2 term in Eq. (4) is indistinguishable from the result obtained using Eq. (16).

II.2. Two-loop correction to the Coulomb potential
The two-loop contributions to ∆(λ), ∆ (4) (λ), have been calculated by Källen and Sabry [19]. They contain both the reducible double electron bubble diagram and the irreducible fourth order term Π (4) f ( k 2 ). These corrections can be expressed as a correction to the Coulomb potential in the form [20] where with By transforming this potential to momentum space and comparing it with Eq. (5), λ can be identified with 4t 2 and the second order energy shift expressed as where ∆ (4) (x) is The terms in the large square brackets result from evaluating the last integral in Eq. (18) and Li 2 (z) is the Spence function

II.3. Three-loop correction to the Coulomb potential
The three-loop contribution to ∆(λ), ∆ (6) (λ), is not available in the literature. These contributions have been calculated by Kinoshita and Nio [21]. They note that the three-loop correction is comprised of two reducible diagrams and one irreducible diagram that can be represented as Remembering the reducible fourth order dispersion result and it is possible to verify the contributions of the reducible terms. They are 0.000396 meV and 0.0029312 meV, respectively. The numerical evaluation of the Π (6) f ( k 2 ) contribution is [21] .001103 meV, giving a total three-loop correction to the Coulomb potential of 0.004431 meV.

II.4. α 4 Second order non-relativistic perturbation correction
The large size of the one-loop correction suggests that the contribution of V V P ( r) in second order nonrelativistic perturbation theory is not negligible. Evaluating this correction necessitates using the radial portions of the Coulomb Green's function for n = 2 and ℓ = 0, 1, expressed as µ 2 α g 2ℓ (x, x ′ ). General expressions for these Green's functions were derived by Hostler [23,24] and explicit expressions for small values of n are contained in [25,26]. Due to some typographical errors in the latter papers (g 20 in Ref. [25] and Eq. (2.18) in Ref. [26]), the expressions for g 20 (x, x ′ ) and g 21 (x, x ′ ) (x = r/a, x ′ = r ′ /a) are given here.
Here, Ei(x) is and The contributions take the form for n = 2 and ℓ = 0, 1. The integrations over r, r ′ can be evaluated exactly using Mathematica, and integrations over propagator parameters λ and λ ′ can then be calculated numerically. The results are EE (2) 2P V P = −0.0022671 meV and EE (2) 2S V P = −0.153164 meV for a net contribution of [5] EE II.5. α 5 Second order non-relativistic perturbation correction There is also a second order non-relativistic perturbative contribution from the combination of a one loop vacuum polarization correction and a two loop vacuum polarization correction . The calculation is similar to the one loop second order calculation and results in a 0.00215 meV contribution. In addition, there is a third order non-relativistic perturbative contribution from three one-loop vacuum polarization corrections [21,22] which gives 0.00007 meV.

III. PROTON SIZE CORRECTION
The effect of the proton size can be obtained in terms of its mean square radius r 2 by modifying the Coulomb potential with a charge form factor F ( k 2 ) defined as where ρ(r) is the proton charge density. In momentum space, the Coulomb potential becomes Expanding the exponential in Eq. (30) and integrating gives This gives The perturbative contribution due to a finite proton radius is then Numerically for n = 2, this is Since the reported energy difference is P 3/2 − S 1/2 , the sign of the contribution is negative. The electron vacuum polarization corrections to Eq. (36) are calculated in the Appendix. We do not directly address the order r 3 /a 3 proton size correction, but any single parameter functional form for F ( k 2 ) that satisfies k 2 dF ( k 2 )/d k 2 = − r 2 /6 can give an estimate of the size of this correction. For example, if F ( k 2 ) = (r 2 p k 2 /12 + 1) −2 the correction to the Coulomb interaction is and The O(r 3 p /a 3 ) coefficient is roughly the size obtained by more detailed calculations [27][28][29][30]. The 2P contribution is O(r 4 p /a 4 ). The remaining corrections to the energy levels come from the potentials in Eqs. (1)(2)(3)(4). In what follows, we use The terms in the first line of Eq. (4) contribute to the s and p levels in order α 4 . Their expectation values are (in meV) and it should be noted that the 1/r 2 term is part of the one-loop correction. For the n = 2 state, the contributions are The remainder of the terms in Eq. (4) are of order α 5 or α 5 ln(α). There are two issues to address when evaluating these terms. The first is the elimination of the photon mass dependence. This is accomplished by using the 'Bethe logarithm' technique, which amounts to the replacement of ln(µ/λ IF ) by [31] ln R ∞ α 2 k 0 (n, 0) The other is the matrix element of ∇ 2 [(ln(µr) + γ)/r]. For states with ℓ > 0, this reduces to −1/r 3 . When ℓ = 0, the result is Using and denoting the expectation values of the order α 5 as E 4 (nℓ), the results are The expectation value of V HF affects only s-states and is

. Spin-orbit and Tensor terms
The largest contribution of V LS is that associated with the L · S 1 term. It accounts for the fine structure splitting between the P 3/2 and the P 1/2 states. The contribution is The expectation value of r −3 is Since the eigenstates we are using are eigenstates of J 2 , Then, and The remaining spin dependent terms are the L · S 2 portion of Eq. (2) (call it V ′ LS ) and V T . Their matrix elements for a generic 2P state are 2P For V ′ LS , we can use the fact that F 2 and J 2 are diagonal, so F 2 = ( J + S 2 ) 2 and we can obtain the relation Both the 5 P 3/2 and 1 P 1/2 states are eigenstates of S 2 with eigenvalue 2, so 5 P 3/2 | L· S 2 | 5 P 3/2 = 1/2 and 1 P 1/2 | L· S 2 | 1 P 1/2 = −1.
The matrix elements of V T for these states are Combining these two contributions gives The 3 P 3/2 and 3 P 1/2 states are are mixed by the V ′ LS and V T potentials. This results in and 3 P 1/2 |3 S 1 ·r S 2 ·r − S 1 · S 2 | 3 P 1/2 = 1/3 , 3 P 3/2 |3 S 1 ·r S 2 ·r − S 1 · S 2 | 3 P 3/2 = 1/6 , The matrix elements of V ′ LS are and those of V T are The V ′ LS and V T contributions can be combined to give The expression for E MIX omits the a µ contribution and all these results agree with Ref. [5].
Diagonalizing the triplet P mixing matrix has the effect of shifting the 3 P 3/2 level up by ∆ = 0.1447 meV and the 3 P 1/2 level down by the same amount.
There are small electron vacuum polarization corrections to all of the terms in the potential that contribute to order α 4 . These are computed in the Appendix and included in the results that are compared with experiment.

VI. RESULTS AND CONCLUSIONS
Relative to the n = 2 Bohr level, the energies of the various n = 2 states, including the small corrections calculated in the Appendix, are Here, EE V P is the sum of the first seven rows in Table I, E 2 is the eight row, E 4 the ninth row, E F S the tenth row, the s-state hyperfine splitting the eleventh row and the p-state hyperfine splittings the twelfth row. The additional small corrections in the Appendix include the spin independent terms in subsection 1, the ℓ = 0 hyperfine splitting in subsection 4, the fine structure splittings in subsection 2, and the ℓ = 1 hyperfine splitting in subsections 3,4 and 5. Eq. (63g) gives our value for the Lamb shift in the absence of proton structure corrections.
Including all the contributions summarized in Tables I and II, the theoretical expression for the 2 3 S 1/2 ↔ where r p = r 2 denotes the proton mean square radius and the terms in parenthesis include an estimate of the proton polarizability correction [27][28][29][30]. The corresponding result for the 2 1 S 1/2 ↔ 2 3 P 3/2 interval is 229.5652 meV − 5.
These results are consistent with each other but low compared to other theoretical calculations. Reference [1] makes a well reasoned argument that theoretical value for the 2 3 S 1/2 ↔ 2 5 P 3/2 transition is 209.978 meV or about 0.1 meV greater than the present result. It gives the 0.841 fm result and one needs less than an additional 0.2 meV to agree with the electron scattering result whereas we need something closer to an additional 0.3 meV.  We have used nonrelativistic wavefunctions throughout because our potential contains the leading order relativistic, recoil and one-loop corrections. However, if we use the solutions to the Dirac equation given in Rose [32], the value of the electron one loop vacuum polarization changes from 205.007 meV to 205.028 meV for the 2P 1/2 − 2S 1/2 interval and from 205.007 meV to 205.033 meV for the 2P 3/2 − 2S 1/2 interval [4]. We have also calculated the relativistic corrections to the two-loop vacuum polarization contribution using the approach of [21]. In this order, the change in the 2P 1/2 − 2S 1/2 interval is 0.0001 meV and the corresponding change in the 2P 3/2 − 2S 1/2 interval is 0.0002 meV.
A more fundamental way to calculate the relativistic corrections to the dominant electron vacuum polarization contribution would be to use the solutions to the Salpeter equation with an instantaneous Coulomb kernel. Estimates of this correction using the scalar Salpeter equation have been made [33] and the results are small. Unfortunately, analytic solutions for the spin 1/2 Salpeter wave functions with unequal masses are not available.
The difference in the values of r p given in (64b) and (65b) suggest that a reasonable estimate of the error (deviation from the average) in r p would be ±0.005. This leaves a substantial difference from the CODATA value of r p = 0.8775(51) fm obtained from electron-proton scattering [10].
Finally, one might wonder how the relatively large contributions from the mass dependence of the delta function term and the one loop α 2 /r 2 term mentioned in the Introduction can still lead to a Lamb shift value that is in agreement with other calculations. The answer is that there are two versions of the spin-independent fine structure Hamiltonian that contribute order α 4 corrections to the 2P 1/2 − 2S 1/2 energy difference. One is the Breit-Pauli version [7] H ′ B−P given by The other is H ′ GRS of reference [14], which has the form where the last term arises in the calculation of the one-loop corrections. The last two terms of Eqs. (67) and (68) give identical contributions to the 2P 1/2 − 2S 1/2 splitting, namely All the spin-dependent fine structure terms of the two versions are the same so, apart from minor differences in some recoil terms, the Lamb shift values agree. This implies that there should be no α 2 /r 2 term associated with the one-loop corrections to Breit-Pauli Hamiltonian. In Appendix B this is shown to be the case. The electron vacuum polarization corrections to the leading order α 4 terms in the potential can be obtained from the momentum space representation of the potential, which is where the next to last term is a one-loop correction that contributes at order α 4 . The electron vacuum polarization correction is obtained by making the replacement In addition, there is a second order perturbative correction containing V V P (r) for each of these terms as well as for the relativistic kinetic energy terms. In these calculations, all integrals except those over λ can be evaluated analytically using Mathematica. The integrals over λ are performed using the Mathematica NIntegrate routine.
1. Spin independent terms e 2 k 2 /8µ 2 This leads to the expression and transforming to coordinate space gives The delta function only contributes to ℓ = 0 and gives The integral diverges, but the ℓ = 0 contribution of the remaining term cancels this divergence. Using Eq. (14) and the extra factor of λ = m 2 e x, the remaining contribution to the E(2P ) − E(2S) interval is The vacuum polarization correction to this term is The second order correction is where the integration over dλ has been suppressed. The Greens function g 20 (x, 0) is The final expression for the 2p − 2s splitting is where and β = m e a.
−e 2 p 2 /rm 1 m 2 Here, the expression is For the ℓ = 1 state, the result of integrating over r is The ℓ = 0 state, when integrated over r gives Integrating the difference of these two results over λ gives The expression for the second order correction is (again, suppressing the integral over dλ and including the factor of 2) This expression reduces to for ℓ = 1 and for ℓ = 0. The integrals are, respectively and Integrating the difference over dλ gives a 2p − 2s contribution of ∆E 2 (−e 2 p 2 /rm 1 m 2 ) = 0.01459 meV.
When integrated over dλ, the second order correction to the relativistic kinetic energy contribution is This correction is negligible for ℓ = 1. After completion of the integrals above, the contribution from the muon to the 2p − 2s splitting for ℓ = 0 is with Here, the momentum space integral is where ε is taken to 0 after the integral is evaluated. The dt integral can be evaluated at this point, but it is more convenient to first integrate over dr with the integrand multiplied by R 2 2ℓ (r). For the ℓ = 1, the result is The integration over dt then gives The calculation for ℓ = 0 is similar. Taking the difference and integrating over λ results in the expression where β = m e a.
The second order correction for µα 2 /2m 1 m 2 r 2 is obtained by integrating The integral to be evaluated is This leads to the spin-orbit contributions Only the p state is affected and we have For the L · S 1 term, L · S 1 is 1/2 for the P 3/2 states and −1 for the P 1/2 states. Thus, their (fine structure) contributions are The second order contribution can be obtained from the expression integrated over λ. The result is