Infrared Safety in Factorized Hard Scattering Cross-Sections

TL;DR

Factorization in QCD via SCET expresses hard, jet, and soft contributions to observables, but its validity depends on the infrared safety of the jet and soft functions. The authors propose a regulator-independent, one-loop test by analyzing the regions of integration in soft and jet functions for angularities, showing IR safety for $a<1$ and breakdown for $a\ge1$. They demonstrate the crucial role of zero-bin subtraction and highlight how scaleless integrals in pure dimensional regularization can reclassify IR vs UV divergences without an explicit regulator. The work clarifies the limits of naive SCET_I factorization and points toward potential SCET_II approaches for observables with overlapping soft/collinear dynamics.

Abstract

The rules of soft-collinear effective theory can be used naively to write hard scattering cross-sections as convolutions of separate hard, jet, and soft functions. One condition required to guarantee the validity of such a factorization is the infrared safety of these functions in perturbation theory. Using e+e- angularity distributions as an example, we propose and illustrate an intuitive method to test this infrared safety at one loop. We look for regions of integration in the sum of Feynman diagrams contributing to the jet and soft functions where the integrals become infrared divergent. Our analysis is independent of an explicit infrared regulator, clarifies how to distinguish infrared and ultraviolet singularities in pure dimensional regularization, and demonstrates the necessity of taking zero-bins into account to obtain infrared-safe jet functions.

Figures (7)

• Figure 1: The (A), (B) virtual and (C), (D) real gluon contributions to the soft function. The gluons all have momentum $k$.
• Figure 2: Regions of integration in the $k^-,k^+$ plane for the coefficient of $\delta(\tau_0^s)$ in the $a=0$ soft function $S_0(\tau_0^s)$. (A) The region of integration for the virtual diagrams is the entire first quadrant. (B) For the real diagrams the region is $\widetilde{\mathcal{S}}$, which contains IR divergent regions. (C) These are converted in the sum of virtual and real graphs into the purely UV region $\mathcal{S}$.
• Figure 3: Region of integration in the $k^-,k^+$ plane for the coefficient of $\delta(\tau_a^s)$ in the soft function $S_a(\tau_a^s)$ for $a=-1$, $a=0.5$, and $a=1$. The region $\mathcal{S}$ is formed by summing real and virtual diagram regions as in Fig. \ref{['softregionsa0']}. For $a<1$, $\mathcal{S}$ always remains above the line $k^+ k^- = Q^2$, which is the boundary of the $a=1$ region and divides the infrared and ultraviolet regions of the soft loop integration.
• Figure 4: Diagrams contributing to the angularity jet function $J_a^n(\tau_a^n)$. The total momentum through each graph is $Qn/2 + l$, and each gluon momentum is $q$. (A) Wilson line emission diagram and (B) its mirror; (C) sunset and (D) tadpole QCD-like diagrams.
• Figure 5: Regions of integration in the $q^-,\mathbf{q}_\perp^2$ plane for virtual gluon diagram contributions to the coefficient of $\delta(\tau_a^n)$ in the jet function $J_a^n(\tau_a^n)$. $\widetilde{\mathcal{V}}$ is the region for the naı̈ve integral, $\widetilde{\mathcal{V}}_0$ for the zero-bin subtraction, and $\mathcal{V}$ for the sum of these two contributions.