Small-x evolution of 2n-tuple Wilson line correlator revisited: The nonsingular kernels

The Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner equation tells how gauge invariant higher order Wilson line correlators would evolve at high energy. In this article, we have revisited the equation and presented a convenient integrodifferential form of this equation that is carrying identical-looking generalized kernels for all explicit real and virtual terms. In the equation, the "real" terms correspond to splitting (say at position z) of this 2n-tuple correlator to various pairs of 2m-tuple and (2n + 2 - 2m)-tuple correlators, whereas "virtual" terms correspond to splitting into pairs of 2m-tuple and (2n - 2m)-tuple correlators. The generalized kernels of virtual terms with m = 0 (no splitting) and of real terms with m = 1 (splitting with at least one dipole) have poles, and when integrated over z, they do generate ultraviolet logarithmic divergences, separately for real and virtual terms. However, we have shown that, except these two cases in all other terms, the corresponding kernels, separately for real and virtual terms, have rather softened ultraviolet singularity and when integrated over z do not generate ultraviolet logarithmic divergences. We have also studied implication of this in the strong scattering regime where only virtual terms are effective.