Drell-Yan production at small q_T, transverse parton distributions and the collinear anomaly

TL;DR

This paper derives an all-order expression for Drell-Yan cross sections at small transverse momentum using effective field theory, addressing the collinear anomaly and providing explicit NNLL resummation details.

Contribution

It introduces a factorization theorem accounting for the collinear anomaly and calculates the three-loop coefficient A^(3), advancing the understanding of transverse parton distributions.

Findings

Derived an exact all-order expression for the Drell-Yan cross section in q_T space.

Explicitly provided anomalous dimensions and matching coefficients at NNLL order.

Identified and addressed the collinear anomaly affecting factorization at small transverse momentum.

Abstract

Using methods from effective field theory, an exact all-order expression for the Drell-Yan cross section at small transverse momentum is derived directly in q_T space, in which all large logarithms are resummed. The anomalous dimensions and matching coefficients necessary for resummation at NNLL order are given explicitly. The precise relation between our result and the Collins-Soper-Sterman formula is discussed, and as a by-product the previously unknown three-loop coefficient A^(3) is obtained. The naive factorization of the cross section at small transverse momentum is broken by a collinear anomaly, which prevents a process-independent definition of x_T-dependent parton distribution functions. A factorization theorem is derived for the product of two such functions, in which the dependence on the hard momentum transfer is separated out. The remainder factors into a product of two functions of longitudinal momentum variables and x_T^2, whose renormalization-group evolution is derived and solved in closed form. The matching of these functions at small x_T onto standard parton distributions is calculated at O(alpha_s), while their anomalous dimensions are known to three loops.