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

TL;DR

The paper develops a complete SCET-based framework for the $e^+e^-$ C-parameter distribution, achieving $N^3$LL$^\\prime$ resummation with fixed-order input up to ${\cal O}(\alpha_s^3)$ and leading nonperturbative power corrections via a renormalon-free shape function. The soft function is determined to one loop analytically and to two loops numerically, with three-loop logarithms fixed by thrust universality, while the leading power correction $\Omega_1^C$ is implemented in the renormalon-free $R$-gap scheme and extended to include hadron-mass running. The framework yields strong perturbative convergence and a cross-section uncertainty of about $2$–$3\%$ at $Q=m_Z$, and it enables direct tests of thrust–C parameter universality in the tail region. A companion paper uses these results to extract $\alpha_s(m_Z)$ and to perform a universality test with thrust, including hadron-mass effects.

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 ${\cal O} (α_s^3)$, 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 $Ω_n$. In order to eliminate an ${\cal O} (Λ_{\rm QCD})$ 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 $Ω_1$. We show how to simultaneously account for running effects in $Ω_1$ 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 $α_s(m_Z)$ and $Ω_1$, the perturbative uncertainty in our cross section is $\simeq 3\%$ at $Q=m_Z$.

Figures (20)

• Figure 1: Plot of $\mathrm{d}/\mathrm{d} C\, \ln[ (1/\sigma)\mathrm{d}\sigma/\mathrm{d} C]\simeq h'(C)/h(C)$, computed from experimental data at $Q=m_Z$. The derivative is calculated using the central difference with neighboring points.
• Figure 2: $\mathcal{O}(\alpha_s)$ nonsingular C-parameter distribution, corresponding to Eq. (\ref{['eq:1NS']}).
• Figure 3: $\mathcal{O}(\alpha_s^2)$ nonsingular C-parameter distribution. The solid line shows our reconstruction, whereas dots with error bars correspond to the EVENT2 output with the singular terms subtracted. Our reconstruction consists of fit functions to the left of the red dashed line at $C = 0.15$ and between the two red dashed lines at $C = 0.75$ and $C = 0.8$ and interpolation functions elsewhere.
• Figure 4: $\mathcal{O}(\alpha_s^3)$ nonsingular C-parameter distribution. The solid line shows our reconstruction, whereas dots with error bars correspond to the EERAD3 output with the singular terms subtracted. Our reconstruction consists of two fit functions, one for $C < 0.75$ and another one for $0.75 < C < 0.835$, and an interpolation for $C>0.835$ (to the right of the red dashed vertical line).
• Figure 5: Singular and nonsingular components of the fixed-order C-parameter cross section, including up to ${\cal O}(\alpha_s^3)$ terms, with $\Omega_1=0.25\,$GeV and $\alpha_s(m_Z) = 0.1141$.