Analytical Solution of the Sudakov--BFKL Interpolation Equation for Small-$x$ Gluon TMDs

Abstract

We analytically solve the evolution equation for small-$x$ gluon transverse-momentum-dependent distributions, which describes the interpolation between the Sudakov and BFKL regimes. We first derive its two limiting forms: the BFKL equation for $\xi = \alpha\sigma s{\bm z}^2/4 \ll 1$ and the Sudakov equation for $\xi \gg 1$. The analytical solutions in these limits are obtained through Mellin-space diagonalization of the BFKL kernel and direct integration of the Sudakov evolution equation, respectively. We then solve the full interpolation equation using a Mellin-space diagonalization ansatz, in which the evolution factor $F(Y,\xi)$ describes the nontrivial $\xi$-dependent modification of a Mellin eigenfunction of the BFKL kernel. This procedure reduces the original two-dimensional integral to a one-dimensional form and permits an analytical evaluation of the resulting evolution kernel. The obtained solution interpolates consistently between the BFKL and Sudakov regimes through an exponential factor $\exp[H(\xi,\gamma)]$. An analysis of the structure of $H(\xi,\gamma)$ allows a quantitative estimation of the transition region between the two dynamical regimes. Our calculation implies a potential matching point in the range of $\xi^{*}\simeq 0.04-0.15$, which is substantially smaller than the naive evaluation value.