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Infrared Safe but Infinite: Soft-Gluon Divergences Inside the Physical Region

S. Catani, B. R. Webber

TL;DR

The paper shows that infrared- and collinear-safe QCD observables can diverge at interior points of phase space due to non-smooth behavior, even though fixed-order cancellations exist. It classifies singularities into boundary and interior (critical-point) types and demonstrates that interior divergences arise from steps in the distribution at C0, leading to double-logarithmic instabilities at higher orders. By performing leading double-log resummation, the authors show these divergences are tamed into a finite, smooth structure called a Sudakov shoulder, extending the familiar Sudakov resummation near exclusive boundaries to interior phase-space points. The work highlights the need to identify critical points when using safe observables, and shows that resummed predictions can provide new, precise tests of QCD beyond fixed-order calculations.

Abstract

We show that QCD observables defined as infrared- and collinear-safe, according to the usual Sterman-Weinberg criteria, can nevertheless be infinite at accessible points inside phase space, to any finite order of perturbation theory. Infrared finiteness is restored after resummation of divergent terms to all orders. The resulting characteristic structure, which we call a Sudakov shoulder, represents an interesting new class of QCD predictions.

Infrared Safe but Infinite: Soft-Gluon Divergences Inside the Physical Region

TL;DR

The paper shows that infrared- and collinear-safe QCD observables can diverge at interior points of phase space due to non-smooth behavior, even though fixed-order cancellations exist. It classifies singularities into boundary and interior (critical-point) types and demonstrates that interior divergences arise from steps in the distribution at C0, leading to double-logarithmic instabilities at higher orders. By performing leading double-log resummation, the authors show these divergences are tamed into a finite, smooth structure called a Sudakov shoulder, extending the familiar Sudakov resummation near exclusive boundaries to interior phase-space points. The work highlights the need to identify critical points when using safe observables, and shows that resummed predictions can provide new, precise tests of QCD beyond fixed-order calculations.

Abstract

We show that QCD observables defined as infrared- and collinear-safe, according to the usual Sterman-Weinberg criteria, can nevertheless be infinite at accessible points inside phase space, to any finite order of perturbation theory. Infrared finiteness is restored after resummation of divergent terms to all orders. The resulting characteristic structure, which we call a Sudakov shoulder, represents an interesting new class of QCD predictions.

Paper Structure

This paper contains 7 sections, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Singularities in fixed order
  3. Examples
  4. The $C$-parameter in $e^+e^-$ annihilation
  5. Jet shapes in hadron collisions
  6. Resummation and the Sudakov shoulder
  7. Discussion and conclusions

Figures (2)

  • Figure 1: Predictions of the $C$-parameter distribution for $\alpha_S = 0.12$. Dashed: ${\cal O}(\alpha_S)$. Solid: ${\cal O}(\alpha_S^2)$. Dot-dashed: resummed.
  • Figure 2: Second-order prediction of the $C$-parameter distribution for $C\to \hbox{\small ${{3} \over {4} }$}^+$. Points: EVENT Monte Carlo. Curve: Eqs. (\ref{['Cto34p']}-\ref{['d0']}).