Resummation of the C-Parameter Sudakov Shoulder Using Effective Field Theory

TL;DR

The paper resolves the Sudakov shoulder in the C-parameter distribution at $C=3/4$ in $e^+e^-$ annihilation by formulating a complete SCET factorization theorem that introduces new C-parameter–specific jet and soft functions. The observable’s quadratic sensitivity near the symmetric Mercedes trijet configuration leads to a finite LO shoulder coefficient and necessitates resummation of $ ext{ln}^2 c$ and $ ext{ln} c$ terms at NLO, which the authors achieve at NLL with a momentum-space framework, avoiding Sudakov--Landau poles. They validate the singular structure against EVENT2 and present matched NLL+NLO predictions with profile scales, demonstrating a smooth shoulder and quantifiable uncertainties. The work offers a robust EFT foundation for Sudakov shoulder resummation in multi-jet configurations, with potential NNLL extensions and implications for precision QCD and future $e^+e^-$ colliders.

Abstract

The C-parameter distribution in $e^+e^-$ annihilation exhibits a kinematic shoulder at $C = 3/4$, where three-parton final states reach their maximum and a fourth parton is required to exceed it. This boundary generates large logarithms that must be resummed. Using soft-collinear effective theory, we derive a factorization theorem involving new jet and soft functions specific to the C-parameter measurement, in which soft radiation contributes quadratically in transverse momentum. This quadratic structure explains the step discontinuity at leading order. We compute all ingredients at one loop, validate against Monte Carlo, and present matched NLL+NLO results. Unlike thrust and heavy jet mass, the C-parameter has no Sudakov--Landau pole, making momentum-space resummation straightforward. All calculations, numerical analysis, and manuscript preparation were performed by Claude, an AI assistant developed by Anthropic, working under physicist supervision.

Figures (5)

• Figure 1: C-parameter distribution at LO and NLO from EVENT2 Monte Carlo (with $\alpha_s = 0.118$). Left: full distribution showing the characteristic $1/C$ divergence at small $C$ and the step discontinuity at the shoulder $C = 3/4$. The LO distribution $(\alpha_s/2\pi)A(C)$ from Eq. \ref{['eq:LO-dist']} agrees precisely with the exact analytical formula (black curve) from Eqs. \ref{['eq:A-from-F0']}--\ref{['eq:p0']}. Right: zoom on the shoulder region, showing the NLO spike just above $C = 3/4$ from unresummed Sudakov logarithms, while the LO distribution vanishes exactly for $C > 3/4$.
• Figure 2: Non-singular distribution $B(c) - B_{\rm sing}^{\rm logs}(c)$ for each color channel in the shoulder region, where $B_{\rm sing}^{\rm logs}$ contains only the $\ln c$ and $\ln^2 c$ terms predicted by SCET. The smooth extrapolation to $c \to 0$ validates the singular coefficients. Quadratic fits (dashed) give the non-singular intercepts $B_0$ for each channel.
• Figure 3: Comparison of the NLO distribution $B(C)$ (red), the NLL singular prediction $B_{\rm sing}^{\rm NLL}(C)$ (green), and the non-singular distribution $B_{\rm NS}(C) = B(C) - B_{\rm sing}^{\rm NLL}(C)$ (blue) in the shoulder region, where $c = (8/3)(C - 3/4)$. The singular coefficients are fixed by SCET (see Eq. \ref{['eq:SCET-full-expansion']}). The crossing points where $B_{\rm sing}^{\rm NLL} = 0$ and where $B_{\rm sing}^{\rm NLL} = B_{\rm NS}$ are indicated; beyond the latter, non-singular corrections dominate over resummed contributions.
• Figure 4: Comparison of the resummed contribution $(\alpha_s/2\pi) A(3/4)(1 - R_{\rm NLL})$ with the singular distributions $(\alpha_s/2\pi)^2 B_{\rm sing}^{\rm logs}$ (dashed) and $(\alpha_s/2\pi)^2 B_{\rm sing}^{\rm NLL}$ (solid) for different scale exponents $t$ defined in Eq. \ref{['eq:t-scales']}. The canonical choice $t = 1$ gives maximal Sudakov suppression, while $t = 0$ corresponds to fixed scales with no resummation. For numerical stability, the $t = 0$ curve uses $t = 0.01$. Left: both contributions to the cross section. Right: difference from $B_{\rm sing}^{\rm NLL}$. The agreement at small $t$ validates that the resummation correctly reproduces the singular structure.
• Figure 5: C-parameter distribution at LO (blue), LO+NLO (orange), and NLL+NLO matched (purple) with $\alpha_s(M_Z) = 0.118$. The shaded bands show theoretical uncertainties from scale variations. Left: full distribution. Right: shoulder region $0.6 < C < 1.0$. The matched distribution smoothly crosses the shoulder, with the Sudakov factor ensuring $R(c) \to 0$ as $c \to 0$ so the NLL piece approaches $(\alpha_s/2\pi)A(3/4)$ from above.