On the Two $R$-Factors in the Small-$x$ Shockwave Formalism

There are two $R$-factors frequently used in the phenomenology of exclusive processes at small values of the Bjorken $x$ variable. One $R$-factor takes into account the effects of non-zero longitudinal momentum transfer, which is assumed to be zero in the dipole scattering amplitude. Another $R$-factor accounts for the real part of the elastic scattering amplitude which is often neglected, with the standard dipole scattering amplitude giving only the imaginary part of the elastic amplitude. In this work we present two new theoretical developments aimed at eliminating the need for the two $R$-factors. We argue that the $R$-factors can be replaced by (i) modifying the argument of the dipole scattering amplitude and by (ii) augmenting the initial conditions for its non-linear small-$x$ evolution. Specifically, we show that to account for the effects of non-zero skewness $\xi$, one has to replace the rapidity argument $Y = \ln (1/x)$ of the eikonal dipole amplitude $N$ and the odderon dipole amplitude $\cal O$ by $Y = \ln \min \left\{ 1/|x|, 1/|\xi|\right\}$. The prescription applies to the elastic scattering cross sections, as well as for calculations of the Generalized Parton Distributions and Generalized Transverse Momentum Dependent parton distributions at small $x$ and at small but non-zero skewness $\xi$. We also show that the real part of the scattering amplitude, proportional to Im~$N$, which is intimately connected to the signature factor of the amplitude, can be accounted for by a more careful evaluation of the initial condition for the evolution and by writing the non-linear evolution equation in an integral form. One can similarly construct Im~$\cal O$ for the odd-signature odderon amplitude. We hope that future implementation of our prescriptions presented here will eliminate the need for both phenomenological $R$-factors.