Analytical results for the C-angularity soft function at NNLO

TL;DR

This work provides analytic NLO and NNLO soft-function results for a new class of event shapes, C-angularity, parameterized by $a$. By leveraging SCET factorization and a global-plus-correction strategy, the authors obtain a high-accuracy expansion in $a$ (up to $a^4$ for anomalous dimensions and $a^3$ for non-logarithmic NNLO terms), with cross-checks in known limits $a\to0$ (C-parameter) and $a\to2$ (threshold). The results enable NNLL′ resummation of C-angularities and bear on precise determinations of $\alpha_s$, while offering a computationally stable analytic framework that can be extended to N^3LO. The approach, including a publicly available implementation, also clarifies the structure of correlated soft emissions and their color-channel contributions in a broad class of dijet observables.

Abstract

We compute the soft function at NLO and NNLO for a one-parameter family of event shapes we call C-angularity. This family contains C-parameter as a specific choice of the parameter, in close analogy with how conventional angularity contains thrust as a special case. By construction, C-angularity and angularity coincide in the collinear limit such that the anomalous dimensions are equal. However, unlike angularity, C-angularity is a continuously differentiable function of the final state momenta, which makes the analytic calculation of the C-angularity soft function simpler. We obtain analytical results for the C-angularity soft function and anomalous dimension as an expansion in the C-angularity parameter $a$, to third and fourth order in $a$ respectively. These expansions yield results that are accurate at the few per mille level for $-1\le a < 1$.

Figures (4)

• Figure 1: A comparison of the function, $f_e(\eta)$ for several of the dijet event shapes in the class defined by \ref{['eq:taue']}. In particular, we compare $f_e$ for broadening ($e=b$), angularity ($e=\tau,a$) and C-angularity ($e=C,a$). For angularity, we plot $f_{\tau,a}$ for $a=0$ (i.e. thrust) and $a=0.5$. For C-angularity, we plot $f_{C,a}$ for $a=0$ (i.e. C-parameter) and $a=0.5$.
• Figure 2: Example two loop soft factor diagrams containing the colour factors (a) $C_R^2$, (b) $C_R C_A$ and (c) $C_R n_f T_F$.
• Figure 3: A comparison of the exact $a$-dependence of $\Delta \hat{\gamma}_{a;S}^{(1,c)}$, computed numerically, with a Laurent expansion in $a$ to different orders in $a$, the coefficients of which were computed analytically and are tabulated in \ref{['tab:Icnm']}.
• Figure 4: A comparison of the exact $a$-dependence of $S^{(2,c)}_{a}$, where the $\Delta S^{(2,c)}_{a}$ component has been computed numerically, to a fully analytic calculation, where $\Delta S^{(2,c)}_{a}$ has been computed as a Laurent expansion in $a$ to ${\mathcal{O}}(a^3)$. Here we have set $\mu={\mathcal{T}}_{\mathrm{cut}}$ so only the non-logarithmic terms contribute.