Deeply virtual Compton scattering amplitude at the tree level: Transversality, twist-3, and factorization

We study the virtual Compton amplitude in the generalized Bjorken region ${(q}^2\to \infty ,$ t small) in QCD by means of a light-cone expansion of the product of EM currents in string operators in coordinate space. Electromagnetic gauge invariance (transversality) is maintained by including in addition to the twist-2 operators “kinematical” twist-3 operators which appear as total derivatives of twist-2 operators. The non-forward matrix elements of the elementary twist-2 operators are parametrized in terms of two-variable spectral functions (double distributions), from which twist-2 and 3 skewed distributions are obtained through reduction formulas. Our approach is equivalent to a Wandzura-Wilczek–type approximation for the twist-3 skewed distributions. The resulting Compton amplitude is manifestly transverse up to terms of order ${t/q}^2.$ We find that in this approximation the tensor amplitude for longitudinal polarization of the virtual photon is finite, while the one for transverse polarization contains a divergence already at the tree level. However, this divergence has zero projection on the polarization vector of the final photon, so that the physical helicity amplitudes are finite.