Collins-Soper Equation for the Energy Evolution of Transverse-Momentum and Spin Dependent Parton Distributions

TL;DR

This work derives the Collins-Soper energy-evolution equation for all leading-twist TMD quark distributions in impact-parameter space, showing that soft and hard contributions factorize into $K(b,\mu,\rho)$ and $G(x\zeta,\mu,\rho)$, with the sum being $\rho$-independent. The one-loop analysis re-derives the unpolarized evolution kernel and demonstrates that the same evolution applies to all spin-dependent leading-twist distributions, implying universality and no mixing among these distributions. Using these kernels, the authors develop a resummation framework for polarized SIDIS structure functions, expressing observables in $b$-space with a Sudakov form factor ${\cal S}$ and perturbative functions $A$ and $B$, and they show DL-level universality across observables. The approach also extends to Drell-Yan and provides a path toward precise predictions of $P_{h\perp}$ spectra and spin asymmetries in SIDIS across varying $Q^2$. Overall, the paper offers a unified, gauge-invariant method for resumming transverse-momentum logarithms in spin phenomena within QCD.

Abstract

The hadron-energy evolution (Collins and Soper) equation for all the leading-twist transverse-momentum and spin dependent parton distributions is derived in the impact parameter space. Based on the result, we present resummation formulas for the spin structure functions in the semi-inclusive deep inelastic scattering.

Figures (4)

• Figure 1: Feynman rules for the eikonal vertex in the TMD parton distribution (a) and its derivative (b).
• Figure 2: Factorization for the derivative $\partial/\partial \ln\zeta$ on the un-subtracted TMD parton distribution.
• Figure 3: One-loop diagrams for the soft part $K$. The mirror diagrams are not shown, but included in the result.
• Figure 4: One-loop diagrams for the Collins-Soper evolution for the un-subtracted TMD parton distributions.