The Gluon Beam Function at Two Loops

TL;DR

This paper calculates the two-loop gluon beam function, completing the set of NNLO virtuality-dependent beam functions, which are essential for precise resummation of certain collider observables involving initial-state radiation.

Contribution

It provides the first complete two-loop calculation of the gluon beam function, extending previous work on quark beam functions, enabling more accurate theoretical predictions at NNLO and N^3LL orders.

Findings

NNLO matching corrections significantly reduce scale dependence.

Results enable improved resummation for N-jettiness and beam thrust.

Numerical results demonstrate the importance of NNLO corrections.

Abstract

The virtuality-dependent beam function is a universal ingredient in the resummation for observables probing the virtuality of incoming partons, including N-jettiness and beam thrust. We compute the gluon beam function at two-loop order. Together with our previous results for the two-loop quark beam function, this completes the full set of virtuality-dependent beam functions at next-to-next-to-leading order (NNLO). Our results are required to account for all collinear ISR effects to the N-jettiness event shape through N^3LL order. We present numerical results for both the quark and gluon beam functions up to NNLO and N^3LL order. Numerically, the NNLO matching corrections are important. They reduce the residual matching scale dependence in the resummed beam function by about a factor of two.