Power Counting in the Soft-Collinear Effective Theory

TL;DR

The paper derives a gauge-invariant, vertex-based power counting formula for SCET that expresses the $\lambda$-dimension $\delta$ of any diagram solely in terms of vertex scalings and four types of vertices, enabling order-by-order identification of required operators in the $\lambda$ expansion. It shows equivalence with direct power counting and links operator scaling to the power of $Q$ and to dynamical twist in the OPE, demonstrated through explicit examples in DIS, semileptonic $B$ decays, and hadronic $B$ decays. The key result, $\delta = 4u + 4 + \sum_k (k-4)(V_k^C+V_k^S+V_k^{SC}) + (k-8)V_k^U$, together with the definitions of $V_k^i$, provides a simple and universal framework for leading and subleading power corrections in soft and collinear interactions. This approach significantly streamlines the construction of complete operator bases and the assessment of power-suppressed contributions across inclusive and exclusive processes.

Abstract

We describe in some detail the derivation of a power counting formula for the soft-collinear effective theory (SCET). This formula constrains which operators are required to correctly describe the infrared at any order in the Lambda_QCD/Q expansion (lambda expansion). The result assigns a unique lambda-dimension to graphs in SCET solely from vertices, is gauge independent, and can be applied independent of the process. For processes with an OPE the lambda-dimension has a correspondence with dynamical twist.

Figures (1)

• Figure 1: Graphs used for the power counting examples in the text. (a), (b), and (c) correspond to the processes DIS, $B\to X_u \ell \bar{\nu}_\ell$, and $B\to D\pi$ respectively. Dashed lines are collinear quarks, double solid lines are usoft or soft heavy quarks, and the single solid lines are soft light quarks. Gluons with a line through them are collinear, while those without a line are soft or usoft.