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Subtractions for SCET Soft Functions

Christian W. Bauer, Nicholas Daniel Dunn, Andrew Hornig

TL;DR

The paper addresses the challenge of computing soft functions at NLO for N-jet events with arbitrary observables/algorithms in SCET. It introduces a subtraction-based framework that categorizes jets into Type I and Type II by UV behavior and leverages analytically known components to obtain finite, general results. The approach is validated against known results and illustrated with examples such as eta–phi jets with cuts and 1-jettiness, enabling numerical soft-function calculations across many observables. This work broadens the set of accessible NLO predictions for jet cross sections at hadron colliders by providing a practical, general subtraction method.

Abstract

We present a method to calculate the soft function in Soft-Collinear Effective Theory to NLO for N-jet events, defined with respect to arbitrarily complicated observables and algorithms, using a subtraction-based method. We show that at one loop the singularity structure of all observable/algorithm combinations can be classified as one of two types. Type I jets include jets defined with inclusive algorithms for which a jet shape is measured. Type II jets include jets found with exclusive algorithms, as well as jets for which only the direction and energy are measured. Cross sections that are inclusive over a certain region of phase space, such as the forward region at a hadron collider, are examples of Type II jets. We show that for a large class of measurements the required subtractions are already known analytically, including traditional jet shape measurements at hadron colliders. We demonstrate our method by calculating the soft functions for the case of jets defined in eta-phi space with an out-of-jet pT cut and a rapidity cut on the jets, as well as for the case of 1-jettiness.

Subtractions for SCET Soft Functions

TL;DR

The paper addresses the challenge of computing soft functions at NLO for N-jet events with arbitrary observables/algorithms in SCET. It introduces a subtraction-based framework that categorizes jets into Type I and Type II by UV behavior and leverages analytically known components to obtain finite, general results. The approach is validated against known results and illustrated with examples such as eta–phi jets with cuts and 1-jettiness, enabling numerical soft-function calculations across many observables. This work broadens the set of accessible NLO predictions for jet cross sections at hadron colliders by providing a practical, general subtraction method.

Abstract

We present a method to calculate the soft function in Soft-Collinear Effective Theory to NLO for N-jet events, defined with respect to arbitrarily complicated observables and algorithms, using a subtraction-based method. We show that at one loop the singularity structure of all observable/algorithm combinations can be classified as one of two types. Type I jets include jets defined with inclusive algorithms for which a jet shape is measured. Type II jets include jets found with exclusive algorithms, as well as jets for which only the direction and energy are measured. Cross sections that are inclusive over a certain region of phase space, such as the forward region at a hadron collider, are examples of Type II jets. We show that for a large class of measurements the required subtractions are already known analytically, including traditional jet shape measurements at hadron colliders. We demonstrate our method by calculating the soft functions for the case of jets defined in eta-phi space with an out-of-jet pT cut and a rapidity cut on the jets, as well as for the case of 1-jettiness.

Paper Structure

This paper contains 1 section, 21 equations, 3 figures.

Table of Contents

  1. Introduction

Figures (3)

  • Figure 1: The coefficient $[ \mathcal{D} ^0]_{ij}$ normalized to $\frac{\alpha_s}{\pi} \mathbf{T}_i\!\cdot\!\mathbf{T}_j$ for three equally-spaced jets as a function of the subtraction jet size $R_S$.
  • Figure 2: The soft function for an $\eta$-$\phi$ jet algorithm as a function of the angle of the measured jet with different $R$ for the case when the emitters $\langle ij \rangle$ are the two beam directions.
  • Figure 3: The coefficient $\mathcal{D} ^0$ for one-jettiness, shown as a function of the angle of the measured jet. The labels refer to the three $\langle ij \rangle$ emitters, with 1 and 2 referring to the beams and 3 to the jet.