Doubly virtual Compton scattering and the beam normal spin asymmetry

We construct an invariant basis for Compton scattering with two virtual photons (VVCS). The basis tensors are chosen to be gauge invariant and orthogonal to each other. The properties of the corresponding 18 invariant amplitudes are studied in detail. We consider the special case of elastic VVCS with the virtualities of the initial and final photons equal. The invariant basis for VVCS in this orthogonal form does not exist in the literature. We furthermore use this VVCS tensor for a calculation of the beam normal spin asymmetry in the forward kinematics. For this, we relate the invariant amplitudes to the helicity amplitudes of the VVCS reaction. The imaginary parts of the latter are related to the inclusive cross section by means of the optical theorem. We use the phenomenological value of the transverse cross section σT~0.1 mb and the Callan-Gross relation which relates the longitudinal cross section σL to the transverse one. The result of the calculation agrees with an existing calculation, and predicts negative values of the asymmetry Bₙ of order of 5 ppm in the energy range from 3 to 45 GeV and for very forward angles.