小x胶子TMD的Sudakov-BFKL插值方程的解析解
Analytical Solution of the Sudakov--BFKL Interpolation Equation for Small-$x$ Gluon TMDs
中文总结 AI 辅助
本文对描述小x胶子TMD的Sudakov-BFKL插值方程进行解析求解，推导其极限形式并通过Mellin空间方法得到完整解，定量估计两动力学区域过渡的ξ匹配点。
中文摘要 AI 辅助
我们对描述Sudakov regime与BFKL regime之间插值的小x胶子横动量依赖分布的演化方程进行了解析求解。首先推导其两个极限形式：当ξ = ασs z²/4 ≪1时为BFKL方程，当ξ≫1时为Sudakov方程。这两个极限下的解析解分别通过BFKL核的Mellin空间对角化和Sudakov演化方程的直接积分得到。随后，我们采用Mellin空间对角化假设求解完整的插值方程，其中演化因子F(Y,ξ)描述BFKL核的Mellin本征函数随ξ的非平凡修正。该过程将原二维积分简化为一维形式，可对所得演化核进行解析计算。所得解通过指数因子exp[H(ξ,γ)]在BFKL和Sudakov regime之间实现一致插值。对H(ξ,γ)结构的分析可定量估计两种动力学 regime之间的过渡区域。我们的计算表明，ξ*≈0.04-0.15范围内存在一个潜在匹配点，该值远小于朴素评估值。
英文摘要
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 $ξ= ασs{\bm z}^2/4 \ll 1$ and the Sudakov equation for $ξ\gg 1$. 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 $F(Y,ξ)$ describes the nontrivial $ξ$-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 $\exp[H(ξ,γ)]$. An analysis of the structure of $H(ξ,γ)$ allows a quantitative estimation of the transition region between the two dynamical regimes. Our calculation implies a potential matching point in the range of $ξ^{*}\simeq 0.04-0.15$, which is substantially smaller than the naive evaluation value.
