Restoration of $k_T$ factorization for low $p_T$ hadron hadroproduction

TL;DR

This paper demonstrates that the $k_T$ factorization theorem, which is broken by soft gluons at high transverse momentum, can be restored at low $p_T$ in hadron-hadron collisions through contour deformation and factorization of residual divergences.

Contribution

It introduces a method to restore $k_T$ factorization at low $p_T$ by deforming loop contours and factoring out Glauber gluon effects, maintaining universality of TMD parton densities.

Findings

$k_T$ factorization is broken at high $p_T$ by Glauber gluons.

Contour deformation restores $k_T$ factorization at low $p_T$.

Glauber factor is flavor-independent, enabling experimental constraints.

Abstract

We discuss the applicability of the $k_T$ factorization theorem to low-$p_T$ hadron production in hadron-hadron collision in a simple toy model, which involves only scalar particles and gluons. It has been shown that the $k_T$ factorization for high-$p_T$ hadron hadroproduction is broken by soft gluons in the Glauber region, which are exchanged among a transverse-momentum-dependent (TMD) parton density and other subprocesses of the collision. We explain that the contour of a loop momentum can be deformed away from the Glauber region at low $p_T$, so the above residual infrared divergence is factorized by means of the standard eikonal approximation. The $k_T$ factorization is then restored in the sense that a TMD parton density maintains its universality. Because the resultant Glauber factor is independent of hadron flavors, experimental constraints on its behavior are possible. The $k_T$ factorization can also be restored for the transverse single-spin asymmetry in hadron-hadron collision at low $p_T$ in a similar way, with the residual infrared divergence being factorized into the same Glauber factor.