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Double logarithmical corrections to beam asymmetry in polarized electron-proton scattering

The up-down asymmetry in transversally polarized electron proton scattering is induced by the interference between one and two photon exchange amplitudes. Inelastic intermediate hadronic states (different from one-proton state) of the two photon exchange amplitude give rise to contributions containing the square of"large logarithm"(logarithm of the ratio of the transferred momentum to the electron mass). We investigate the presence of such contributions in higher orders of perturbation theory. The relation with the case of zero transfer momentum is explicitly given. The mechanism of cancellation of infrared singularities is discussed.

The beam asymmetry due to transversal polarization of an electron beam scattered on (unpolarized) protons is a pure quantum effect arising from the interference of the Born amplitude (with one photon exchange) and the imaginary part of the two-photon exchange amplitude (Fig. 1).
The corresponding contribution to the differential cross section as well as to the beam asymmetry is proportional to the electron mass. Therefore, the presence of this contribution does not contradict the Kinoshita-Lee-Nauenberg theorem [1], about cancellation of mass singularities, since the corresponding cross sections are suppressed by the lepton mass.
We show that the main contribution arises from the kinematical region of loop momenta when the energy of the electron in the intermediate state, being on mass shell, is small in the reference frame of the center of mass of the initial particles.
The kinematics is determined by the conservation laws where a is electron spin. In general, the emission of real photons must be considered, in order to avoid infrared divergences, when higher orders of perturbation theory (PT) are taken into where c = cos θ and θ is the scattering angle in the center of mass frame. The absolute values of the photon momenta squared, in TPE amplitude, can reach zero: where c 1,2 are the cosines of the angles between the initial and intermediate electron momenta The main contribution arises from two regions −t 1 ≪ t 2 = t and −t 2 ≪ t 1 = t.
Moreover, we will show that the energy of the intermediate electron is much lower than the electron energy corresponding to the elastic case ǫ ′′ ≪ ǫ.
Neglecting the dependence on p ′′ 1 from the remaining part of the amplitude we find the main (double-logarithmical, or DL) asymptotic behavior of the amplitude: We do not distinguish here the two kinds of "large logarithms" The result of exact calculation consists in the replacement

II. CALCULATION OF DOUBLE LOGARITHM CORRECTION
Let us investigate the question about the size of radiative corrections to the cross section for the scattering of transversally polarized electron and to the relevant beam asymmetry.
Let us consider first the radiative corrections (RC) at the lowest order. The contribution from the emission of virtual photons is twofold. Firstly, it is due to the vertex functions The main contribution arises from the Dirac form factor. In the scattering channel we have: The second class of contributions arise from emission and absorption of real photons as an intermediate state of the leptonic block.
Let firstly restrict our considerations to the emission of soft real intermediate photons (see Fig. 3).
For the corresponding contribution we have where λ is a fictitious "photon mass", the resulting contribution suffers from infrared divergences. To remove these divergencies, we must take into account the inelastic process of electron-proton scattering with emission of additional soft (or hard) real photons by initial and final electrons (see Fig 4): The total sum δ = δ vs + δ inel s is free from infrared divergences: As the dominant (DL) contribution arises from the region of intermediate state with a soft lepton in presence of photons, we can generalize the above result including the emission of "hard" internal and external photons. This can be realized by the choice In this case the hadronic block remains the same as for the lowest order of PT.
In DL approximation we have Up to now we calculate the contribution of box diagram to the matrix element. For the calculation of the asymmetry we must to take into account imaginary pairs of the contribution.
By general arguments of analiticity [7] the matrix diagram with two-photon exchange and insertion of vertex function with correction of α n order in DL approximation is proportional It is naturally to expect (keeping in mind the arguments in favor of exponentiation of soft photon emission contributions proved by Yenne, Frautchi and Suura [2]) that this result can be generalized to all orders of PT: We can use here The calculation in the lowest order of PT leads to The numerical results for R in the lowest order, R LO , and in higher orders, R ∞ , are presented in Fig. 5. One can see they are sizable and should be taken into account.
The asymmetry at the lowest order of PT has been calculated in previous papers, (see for instance [3]).
Mass suppressed amplitudes connected with Higgs production and decay in DL approximation were calculated in the paper [8]. The contribution to the imaginary part of the amplitude from the square of the box diagram ( Fig. 6) is of order (α/π) 2 L, and can be omitted in DL approximation. This holds also for the interference of the born diagram with the two loop box diagram (Fig. 7) [6].
We note that the limiting case t → 0 can not be obtained using the approach described above. Contrary to the hadronic block, where the t → 0 limit can be put smoothly, the leptonic block drastically depends on the parameter (−t/m 2 ). For the t = 0 case, the exact as it was shown in Appendix D of [4].
The presence of infrared singularities in RC to the impact factor of the electron was previously noted in the paper of one of us [5]. In the present paper, the way to eliminate such singularities is explicitly shown.