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Light-by-Light Scattering Nonlogarithmic Corrections to Hyperfine Splitting in Muonium

We consider three-loop corrections to hyperfine splitting in muonium generated by the gauge invariant set of diagrams with virtual light-by-light scattering block. These diagrams produce both recoil and nonrecoil contributions to hyperfine splitting. Recoil corrections are enhanced by large logarithms of the muon-electron mass ratio. Both nonrecoil and logarithmically enhanced radiative-recoil corrections where calculated some time ago. Here we calculate nonlogarithmic radiative-recoil corrections generated by the insertions of the light-by-light scattering block.


I. INTRODUCTION
Theoretical and experimental research on hyperfine splitting (HFS) in the ground state of muonium has a long history, see e.g., [1][2][3]. Measurement of the HFS in muonium is currently the best way to determine the value of the electron-muon mass ratio. Nowadays the HFS in the ground state of muonium is measured [4,5] with error bars in the ballpark of 16-51 Hz, and a new higher accuracy measurement is now planned at J-PARC, Japan [6].
The value of α 2 (m µ /m e ) is obtained from comparison of the HFS theory and experiment with the uncertainty that is dominated by the 2.3 × 10 −8 relative uncertainty of the HFS theory [3]. Improvement of the HFS theory would allow further reduction of the uncertainty of the electron-muon mass ratio. The current theoretical uncertainty of the HFS interval is estimated to be about 70-100 Hz, respective relative error does not exceed 2.3×10 −8 (see discussions in [1][2][3]). Reduction of the theoretical error of the HFS theory in muonium to about 10 Hz is a realistic goal [1,2]. Still unknown contributions include three-loop purely radiative corrections, three-loop radiative-recoil corrections, and nonlogarithmic recoil corrections (see detailed discussion in [2,3]) which are the main sources of the theoretical uncertainty. Below we consider three-loop radiative-recoil contributions to HFS generated by the light-by-light (LBL) scattering diagrams in Fig. 1 (and by three more diagrams with the crossed photon lines). These radiative-recoil corrections are additionally enhanced by the large logarithm of the electron-muon mass ratio. The logarithm squared and single-logarithmic terms are already calculated [7,8]. Here we calculate the nonlogarithmic contribution. We will follow the general approach to calculation of the three-loop radiative-recoil corrections to HFS developed in [7,[9][10][11][12][13][14][15] and start with the general expression for the LBL scattering contribution in Fig. 1 (see, e.g., [1,2]) where k µ is the four-momentum carried by the upper photon lines, q µ is the four-momentum carried by the lower photon lines, m is the electron mass, M is the muon mass, Z = 1 is the muon charge in terms of the electron charged used for classification of different contributions, and S αβµν is the light-by-light scattering tensor. The Fermi energy is defined as where m r is the reduced mass. The angle brackets in Eq. (2) denote the projection of the γ-matrix structures on the HFS interval (difference between the states with the total spin one and zero).
The integral in Eq. (1) contains both nonrecoil and recoil corrections to HFS that are partially already calculated (see [1,2,8] for a collection of these results) where a µ is the muon anomalous magnetic moment.
The leading nonrecoil term in Eq. (4) is generated by the nonrelativistic pole in the muon propagator 1 and was calculated in [16,17]. This is a numerically dominant contribution and it should be extracted analytically from the expression in Eq. (1) before calculation of the radiative-recoil corrections.
Recoil corrections generated by the diagrams in Fig. 1 contain three loop integrations and each of them could in principle generate a large logarithm of the electron-muon mass ratio.
The strongly ordered region of integration momenta m ≪ k ≪ p ≪ q ≪ M would produce logarithm cubed contribution but it turns into zero due to the tensor structure of the LBL block and fermion factors in this region [9]. The large logarithm squared, calculated in [7], Calculation of the single-logarithmic contributions is more involved and requires knowledge of the leading terms in the large momentum expansion of the function T (q 2 , q 0 ) in Eq. (2). In [8] after integration over the photon momenta k and q we obtained an integral representation for this function written as a sum of the ladder and crossed diagrams contributions in Fig. 1 T The ladder contribution is represented as a sum of nine multidimensional integrals where The crossed diagram contribution is represented as a sum of three multidimensional integrals where In Eq. (8)-Eq. (20) The large momentum expansions of the ladder and crossed functions T L,C (q 2 , q 0 ) as well as the large momentum expansion of the total function T were calculated in [8]. Both the already known double-logarithmic and the new singlelogarithmic radiative-recoil contributions to HFS were obtained in [8] from these large momentum expansions. Below we will use the exact explicit expressions for the function T (q 2 , q 0 ) to calculate a nonlogarithmic radiative-recoil contribution generated by the diagrams in Fig. 1.

II. CALCULATION OF NONLOGARITHMIC CONTRIBUTIONS
In terms of the function T (q 2 , q 0 ) the total contribution to HFS of the diagrams in Fig. 1 in Eq. (1) can be written in the form where We calculate this integral in Euclidean space and parameterize Euclidean four-vectors q 0 = q cos θ, |q| = q sin θ. After the Wick rotation d 4 q → (4πi/2) ∞ 0 q 2 dq 2 π 0 dθ sin 2 θ, and the integrand simplifies Only the even in q 0 terms in the function T (q 2 , q 0 ) contribute to the integral in Eq. (26).
The numerators on the LHS in Eq. (28) and Eq. (29) can be multiplied by polynomials in q 2 and q 2 0 . These polynomials on the RHS turn into polynomials in (−q 2 ) and (−q 2 cos 2 θ). The function J in Eq. (26) depends on µ = m/(2M) only due to the integrals containing in the integrand the first term on the RHS in Eq. (28). We call these integrals µ-integrals, and the general methods of their calculation are developed and described in [18,19]. These We are looking for the µ-independent terms in the small µ (large M) expansion of the integral in Eq. (32). It is tempting to substitute 4M 2 /(m 2 q 2 + 4M 2 cos 2 θ) → 1/ cos 2 θ directly inside the integrand in Eq. (32). Obviously this is not safe since the integral over θ can become divergent at cos θ = 0 if an extra factor cos 2 θ is not supplied by the function T (q 2 , cos 2 θ). Just by inspection we see that there are entries in the function T (q 2 , cos 2 θ) that do not contain such a compensating factor. The reason for this spurious divergence at cos θ = 0, or, what is the same, at q 0 = 0 is pretty obvious: q 0 = 0 corresponds to the nonrecoil contribution to HFS, and this spurious divergence is cutoff by 1/M in the original integral. This is the mechanism how an apparently recoil integral in Eq. (32) produces a nonrecoil correction of order 1/µ. Hence, in case of such spurious divergence we cannot make the substitution 4M 2 /(m 2 q 2 + 4M 2 cos 2 θ) → 1/ cos 2 θ inside the integral, and we need to calculate the integral over angles more accurately. By inspection we see that the integrals over angles in Eq. (32) have the form where n = 0, 1, 2, 3, . . ., and explicitly for n = 0, 1, 2, 3 (see [19]) Considering the integrand in Eq. (33) and/or the small µ expansions of the functions in we observe that only the integrals with n = 0 generate singular at µ → 0 contributions and do not admit the naive substitution 1/(µ 2 q 2 + cos 2 θ) → 1/ cos 2 θ in the integrand. Using the explicit expansions in Eq. (35) it is easy to check now that to separate the nonrecoil (1/µ) contributions in the integrals and simplify the calculation of µ-independent terms in Eq. (32) in the small µ case it is sufficient to make the substitution Here we have introduced a new "principal value" prescription for integration over θ 1 π π 0 dθP 1 cos 2 θ = 0.
As usual with the principal value ℘(1/cos 2 θ) cos 2 θ = 1. Using this rule we can easily derive the rules for integration of the products P(1/cos 2 θ) with arbitrary polynomials of cos 2 θ and sin 2 θ, for example 1 π π 0 dθ sin 2 θP 1 After calculations we obtain nonlogarithmic contributions to HFS produced by the ladder and by the crossed diagrams The total nonlogarithmic recoil contribution to HFS generated by the diagrams in Fig. 1 is This result makes us one step closer to calculation of all nonlogarithmic three-loop radiativerecoil corrections to HFS. Only two gauge invariant sets of diagrams with two radiative photon insertions either in the electron or the muon line remain uncalculated. We hope to report on the respective results in not so far future.

ACKNOWLEDGMENTS
This work was supported by the NSF grant PHY-1066054. The work of V. S. was also supported in part by the RFBR grant 12-02-00313 and by the DFG grant GZ: HA 1457/7-2.