C-parameter Distribution at N${}^3$LL$^\prime$ including Power Corrections

TL;DR

This paper presents a high-precision calculation of the C-parameter distribution in electron-positron collisions using effective field theory, including resummation, fixed-order results, and nonperturbative corrections, to improve the extraction of the strong coupling constant.

Contribution

It provides the first N³LL' resummed prediction for the C-parameter distribution including power corrections and renormalon subtractions, enabling more accurate QCD tests.

Findings

Achieves approximately 3% perturbative uncertainty at the Z boson mass.

Demonstrates universality of power corrections between C-parameter and thrust.

Provides a framework for testing nonperturbative effects in event shape distributions.

Abstract

We compute the $e^+ e^-$ C-parameter distribution using the Soft-Collinear Effective Theory with a resummation to N${}^3$LL$^\prime$ accuracy of the most singular partonic terms. This includes the known fixed-order QCD results up to ${\cal O} (\alpha_s^3)$, a numerical determination of the two loop non-logarithmic term of the soft function, and all logarithmic terms in the jet and soft functions up to three loops. Our result holds for $C$ in the peak, tail, and far tail regions. Additionally, we treat hadronization effects using a field theoretic nonperturbative soft function, with moments $\Omega_n$. In order to eliminate an ${\cal O} (\Lambda_{\rm QCD})$ renormalon ambiguity in the soft function, we switch from the $\overline {\rm MS}$ to a short distance "Rgap" scheme to define the leading power correction parameter $\Omega_1$. We show how to simultaneously account for running effects in $\Omega_1$ due to renormalon subtractions and hadron mass effects, enabling power correction universality between C-parameter and thrust to be tested in our setup. We discuss in detail the impact of resummation and renormalon subtractions on the convergence. In the relevant fit region for $\alpha_s(m_Z)$ and $\Omega_1$, the perturbative uncertainty in our cross section is $\simeq 3\%$ at $Q=m_Z$.