Effects of light-by-light scattering in the Lamb shift and hy- perfine structure of muonic hydrogen

Abstract. We calculate the meson exchange contribution to the interaction operator of muon and proton, which is determined by the meson coupling with two photon state. For the construction of transition form factor Meson → γγ we use the existing parametrizations based on experimental data including the monopole parametrization over photon four-momenta. For an estimate of the form factor value at zero photon four-momenta squared we use experimental data on the decay width ΓMeson→γγ . It is shown that scalar, pseudoscalar, axial vector and tensor mesons exchanges give significant contribution to the Lamb shift (LS) and hyperfine splitting (HFS) in muonic hydrogen which should be taken into account for a comparison with precise experimental data.


Introduction
The discrepancy between results for the proton charge radius obtained by different methods got the name "proton radius puzzle". In particular, measurements using electronic hydrogen lead to a different proton charge radius compared with those using muonic hydrogen. This problem arose after the CREMA experiments on Lamb shift measurement in muonic hydrogen [1,2]. Increasingly accurate experiments require corresponding theoretical calculations. To achieve high accuracy it is necessary to take into account the problem of a more accurate construction of the particle interaction operator in muonic atoms and the inclusion of new contributions to this operator. Emerging new experimental data on the Lamb shift in electron hydrogen, as well as a new analysis of experiments already performed on the scattering of leptons by nucleons, show that the values of the proton charge radius obtained from electron and muon systems are approaching [3,4]. The problem of the proton charge radius is gradually beginning to be solved. However, the analysis of new interactions between the proton and the muon is important for future more accurate experiments. Among the interactions of the proton and the muon there are those in which two virtual photons turn into a meson, which leads to an effective one-meson potential. The calculation of the transition form factor (TFF) of two photons into a meson can be performed within the framework of nonperturbative quantum chromodynamics. In this work we continue our investigations [5][6][7][8][9] (see also [10][11][12][13][14]) of the role of one-meson exchange interactions in muonic hydrogen (µp).

Scalar meson exchange
Our approach to the calculation of one-meson exchange contributions to the energy levels of (µp) is based on quasipotential approach in quantum field theory [15][16][17][18]. The general parametrization of scalar meson → γ * + γ * vertex function takes the form [14]: where A(t 2 , k 2 1 , k 2 2 ), B(t 2 , k 2 1 , k 2 2 ) are two scalar functions, k 1,2 are four momenta of virtual photons, t is the four momentum of scalar meson. Then the muon-proton interaction amplitude via the scalar meson exchange can be written as follows: where p 1 , p 2 are four momenta of particles in initial state, q 1 , q 2 are four momenta of particles in final state. We set t = q 1 − p 1 = 0 because this momentum is small and we consider further the leading order in fine structure constant α contribution to the interaction operator (|t| ∼ µα). This leads to the cancelation of the term with the function B(p 2 , k 2 1 , k 2 2 ). Using projection operator on muon-proton states with spin S=0, S=1 [15] we can construct the interaction operator for these states. In the case of triplet state we find: where m 1 , m 2 are the masses of a muon and proton, M S is the mass of scalar meson. After the trace calculation using the package Form [19] we obtain in the leading order: So in the leading order the scalar meson exchange doesn't contribute to hyperfine structure. The typical momentum integral contributing to the shift has the following form: where a 1 = 2m 1 /Λ. An analytical value I 1 is presented after an expansion over 2m 1 /Λ up to terms of the second order. Such integral appears if we suppose that the parametrization of function A(t 2 , k 2 , k 2 ) for scalar mesons has monopole form for variables k 2 1 and k 2 2 : where A S = A(0, 0, 0). Then the interaction potential takes the form : Averaging (7) over wave functions of 2S-state we obtain the shift of 2S-level in the form: The contribution to 2P-state is suppressed by additional degrees of α:

Pseudoscalar meson exchange
The general parametrization of pseudoscalar meson (π 0 , η, η ′ ) → two virtual photon vertex function is expressed through transition form factor in the form [20]: Transition form factor has normalization F π 0 γ * γ * (0, 0, 0) = 1. At first we consider construction of the hyperfine part of interaction potential in the case of S-states. Using projection operators technique we present interaction amplitude via pseudoscalar meson exchange as follows: Introduction total and relative momenta of particles in the initial and final states p = (0, p), q = (0, q) instead p 1,2 , q 1,2 and taking into account that relative momenta are small (|p| ∼ µα, |q| ∼ µα) we obtain for the numerator of the amplitude the result in the leading order (proportional to t 2 ): As a result the hyperfine part of muon proton interaction potential takes the form: where For A(t 2 ) there is a dispersion relation with one subtraction, which has the form: Imaginary part of A(t 2 ) doesn't depend on specific form of transition form factor parametrization and is well known [21]: Numerical value of the A(0) for an electron is equal A e (0) = −21.9 ± 0.3 [22] and for a muon A µ (0) = −6.1 ± 0.3. Going in (13) to coordinate representation by means of the Fourier transform we obtain: For numerical calculations we also use the Goldberg-Treiman relation for the pion-nucleon interaction constant g p = g πNN = m p g A F π with g A = 1.27, F π = 0.0924 GeV. Averaging (17) over wave functions of 1S and 2S-state we obtain the hyperfine splitting of 1S and 2S states: In a similar way we obtain the potential of 2P 1/2 -state hyperfine splitting: Averaging (19) over wave function of 2P-state we obtain the numerical value of the hyperfine splitting of the level 2P 1/2 : ∆E h f s,P 2P 1/2 = 0.0004 µeV.

Axial vector meson exchange
The general parametrization of axial vector meson → γ * γ * vertex function has the form [23]: Interaction amplitude via axial vector meson exchange has the following general structure: where the vertex function where G A (t 2 ) , G P (t 2 ) are axial and induced form factors respectively. Axial vector meson propagator is D αβ (t) = Using the Lorentz transformation for wave function of muon and proton and formalism of projection operators we obtain for numerator of the amplitude in the case of state with F = 1: T r (q 1 + m 1 )γ ν (p 1 −k + m 1 )γ µp 1 + m 1 ) For transition form factor F AVγ * γ * (k 2 , k 2 ) we use the following representation [24]: where R ′ (0) is the value of derivative of radial wave function at zero, < e 2 q > is squared effective charge of a light quark in a bound state in units of electron charge. As a result the potential contributing to hyperfine structure has the form of a Yukawa potential: Equation (26) contains a characteristic loop momentum integral, that can be calculated analytically: After the Fourier transform of the potential (26) and averaging it over the wave functions, we obtain the contribution to the HFS spectrum:

Tensor meson exchange
Following [25] we will assume further that hadronic light-by-light amplitude for tensor mesons is dominated be helicity Λ = 2 exchange. The amplitude of the process γ * + γ * → T can be parameterized as follows [26,27]: where F T γ * γ * (k 2 1 , k 2 2 ) is a transition form factor (TFF) of tensor meson,

Conclusion
The obtained numerical values of the meson contributions to the Lamb shift and the hyperfine structure show that they are significant and must be taken into account in a more accurate comparison with experimental data. The contribution of the tensor meson to the lamb shift is comparable to the contribution of the scalar σ-meson. Other tensor mesons apparently make a significantly smaller contribution, since their constant of interaction with the nucleon is much smaller. Experimental data [30] show that all tensor mesons have a significant decay width into a pair of pions, which interact well with the nucleon, therefore, such processes need to be investigated additionally. Work in this direction is in progress.