Phase Space and Jet Definitions in SCET

TL;DR

This paper clarifies phase space integration in SCET for various jet algorithms, highlighting how different definitions affect infrared safety and the cancellation of divergences at one loop.

Contribution

It introduces a consistent power counting method in SCET for phase space, addressing double-counting and divergence cancellation across different jet algorithms.

Findings

Phase space integrals are UV divergent but cancel in sum.

JADE and Sterman-Weinberg are IR safe at one loop.

k_T algorithm's IR safety depends on the regularization scheme.

Abstract

We discuss consistent power counting for integrating soft and collinear degrees of freedom over arbitrary regions of phase space in the soft-collinear effective theory (SCET), and illustrate our results at one loop with several jet algorithms: JADE, Sterman-Weinberg and k_T. Consistently applying SCET power-counting in phase space, along with non-trivial zero-bin subtractions, prevents double-counting of final states. The resulting phase-space integrals over soft and collinear regions are individually ultraviolet divergent, but the phase-space ultraviolet divergences cancel in the sum. Whether the soft and collinear contributions are individually infrared safe depends on the jet definition. We show that while this is true at one loop for JADE and Sterman-Weinberg, the k_T algorithm does not factorize into individually infrared safe soft and collinear pieces in dimensional regularization. We point out that this statement depends on the ultraviolet regulator, and that in a cutoff scheme the soft functions are infrared safe.