Momentum Broadening in the Opacity Expansion: All-Path-Length Corrections and Improved Regge Kinematics

Abstract

We present a detailed study of momentum broadening and the jet transport coefficient $\hat{q}$ for high-energy partons traversing the Quark-Gluon Plasma (QGP), extending the Gyulassy-Levai-Vitev (GLV) formalism to include both all-path-length (APL) and sub-Regge kinematical corrections. Traditional GLV calculations rely on the large separation distance and large formation time approximations, which are valid for large systems but whose applicability to small systems, such as $pp$ and $p/d$A collisions, may fail. We derive analytic expressions for the momentum broadening distributions and $\hat{q}$ to first order in the opacity expansion, and perform a detailed numerical investigation to quantify their impact. The APL correction suppresses momentum broadening at low $p_{\perp }$, with a correction scaling as $\propto 1/(L{p}_{\perp })$ that dominates for small systems, while converging to the standard GLV result at large $L$. The sub-Regge kinematical correction enhances momentum broadening at high $p_{\perp }$, becoming significant when the transverse momentum transfer approaches the magnitude of the parton's large light-cone momentum component, and vanishing in the Regge limit where this ratio is small. When both corrections are combined, the sub-Regge kinematical correction partially mitigates the suppression induced by the APL term; in the case of $\hat{q}$, this mitigation is essentially complete, with ${\hat{q}}_{(\mathrm{APL}+\mathrm{SUB})}$ found to coincide closely with the standard GLV result. These findings demonstrate that sub-Regge kinematical corrections can resolve the long-standing problem of large negative energy-loss contributions at high energies identified in earlier studies.

Comments: 41 pages, 8 figures