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

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 $\mathcal{O}({\alpha }_s^3)$, 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 $\mathcal{O}({\Lambda }_{\mathrm{QCD}})$ renormalon ambiguity in the soft function, we switch from the $\overline{\mathrm{MS}}$ 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 $Q=m_Z$.

34 pages, 20 figures

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