Unsafe but Calculable: Ratios of Angularities in Perturbative QCD

TL;DR

This paper introduces a method to calculate the cross section of ratio observables in perturbative QCD, which are not IRC safe but are Sudakov safe, using resummation techniques to handle their peculiar properties.

Contribution

It presents a novel resummation-based approach to compute ratios of IRC-safe observables that are themselves not IRC safe, expanding the scope of perturbative QCD calculations.

Findings

Ratios of angularities are Sudakov safe despite IRC unsafety.

Leading order distribution is finite with a square root dependence on the coupling.

Monte Carlo simulations reliably predict the ratio observable.

Abstract

Infrared- and collinear-safe (IRC-safe) observables have finite cross sections to each fixed-order in perturbative QCD. Generically, ratios of IRC-safe observables are themselves not IRC safe and do not have a valid fixed-order expansion. Nevertheless, in this paper we present an explicit method to calculate the cross section for a ratio observable in perturbative QCD with the help of resummation. We take the IRC-safe jet angularities as an example and consider the ratio formed from two angularities with different angular exponents. While the ratio observable is not IRC safe, it is "Sudakov safe", meaning that the perturbative Sudakov factor exponentially suppresses the singular region of phase space. At leading logarithmic (LL) order, the distribution is finite but has a peculiar expansion in the square root of the strong coupling constant, a consequence of IRC unsafety. The accuracy of the LL distribution can be further improved with higher-order resummation and fixed-order matching. Non-perturbative effects can sometimes give rise to order one changes in the distribution, but at sufficiently high energies Q, Sudakov safety leads to non-perturbative corrections that scale like a (fractional) power of 1/Q, as is familiar for IRC-safe observables. We demonstrate that Monte Carlo parton showers give reliable predictions for the ratio observable, and we discuss the prospects for computing other ratio observables using our method.