Renormalization of Dijet Operators at Order $1/Q^2$ in Soft-Collinear Effective Theory

TL;DR

The paper tackles the challenge of resumming power-suppressed logarithms in dijet event shapes such as thrust by using a novel Soft-Collinear Effective Theory (SCET) formalism that avoids explicit $\lambda$-scaling. It systematically matches QCD onto a tower of subleading dijet operators $O_2^{(0)}$, $O_2^{(1i)}$, and $O_2^{(2i)}$ with corresponding Wilson coefficients $C_2^{(j)}$, and computes the renormalization group evolution through anomalous dimensions $\gamma_2^{(j)}(u,v)$, including the first full results for $\gamma_2^{(2a_1)}(u,v)$ and $\gamma_2^{(2a_2)}(u,v)$. The analysis builds on overlap subtraction between collinear sectors and a position-space to momentum-space translation via a Fourier transform in $t \to u$, with known results for $\gamma_{(1a)}$ and $\gamma_{(1c)}$ reproduced and new subleading contributions derived. While full resummation also requires matching to observable-dependent soft functions, these anomalous dimensions enable the leading power-suppressed logarithm resummation in the cumulative thrust distribution and advance precision QCD applications, including improved extractions of $\alpha_s$.

Abstract

We make progress towards resummation of power-suppressed logarithms in dijet event shapes such as thrust, which have the potential to improve high-precision fits for the value of the strong coupling constant. Using a newly developed formalism for Soft-Collinear Effective Theory (SCET), we identify and compute the anomalous dimensions of all the operators that contribute to event shapes at order $1/Q^2$. These anomalous dimensions are necessary to resum power-suppressed logarithms in dijet event shape distributions, although an additional matching step and running of observable-dependent soft functions will be necessary to complete the resummation. In contrast to standard SCET, the new formalism does not make reference to modes or $λ$-scaling. Since the formalism does not distinguish between collinear and ultrasoft degrees of freedom at the matching scale, fewer subleading operators are required when compared to recent similar work. We demonstrate how the overlap subtraction prescription extends to these subleading operators.

Figures (3)

• Figure 1: QCD graphs contributing to three-body final states.
• Figure 2: The Feynman diagrams for any operator with the $q^n\bar{q}^{{\bar{n}}}g^{n}$ configuration. The Feynman rules for the effective vertex are determined by the structure of each operator. Diagram (g) is the overlap amplitude, and must be subtracted.
• Figure 3: The Feynman diagrams for any operator with the $q^n\bar{q}^{n}g^{{\bar{n}}}$ configuration. The Feynman rules for the effective vertex are determined by the structure of each operator. Diagram (e) is the overlap amplitude, and must be subtracted.