The Fate of Ultra-Collinear Modes in On-Shell Massive Sudakov Form Factors

TL;DR

This paper demonstrates the cancellation of ultra-collinear contributions in on-shell massive Sudakov form factors in QCD, confirming the robustness of the leading-power factorization formula and providing detailed two-loop calculations and resummation.

Contribution

It shows that ultra-collinear modes cancel to all orders in the on-shell form factor, and provides explicit two-loop calculations and resummation including mass hierarchies.

Findings

Ultra-collinear contributions cancel to all orders due to gauge invariance.

Two-loop calculations of soft and jet functions for massive Sudakov form factors.

Resummation of logarithms at NNLL accuracy including fermion mass hierarchies.

Abstract

Individual multi-loop diagrams for the massive Sudakov form factor contain an infinite tower of ultra-collinear momentum regions. We show that, for the on-shell form factor in QCD, these contributions cancel to all orders as a consequence of gauge invariance, so the leading-power SCET$_{\rm II}$ factorization formula is unchanged. Using the $\eta$ rapidity regulator, we compute the soft function and the massive jet function of the quark and gluon Sudakov form factors through two loops and resum logarithms at NNLL accuracy, including hierarchies of fermion masses. We also show that with a gauge-boson mass regulator, the infinite tower of modes is truncated and ultra-collinear and ultra-soft modes become manifest and factorize explicitly, providing a direct EFT derivation of the regulated infrared dependence.