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

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.