Soft-Collinear Factorization in Effective Field Theory

TL;DR

This paper develops soft-collinear effective theory (SCET) to describe high-energy hadronic processes by separating collinear, soft, and ultrasoft degrees of freedom through a power counting in \lambda. It shows how usoft and soft gluons factor from collinear jets via Wilson lines (Y and S for usoft and soft, with W for collinear), yielding universal factorization at leading order and enabling operator-level constructions that respect gauge invariance. The authors illustrate the framework with exclusive $B\to D\pi$ decays and inclusive $B\to X_s\gamma$ spectra, deriving a convolution structure that combines hard, jet, and soft (shape) functions. The formalism provides a systematic approach to power corrections and a coherent, gauge-invariant picture of factorization in heavy-quark processes.

Abstract

The factorization of soft and ultrasoft gluons from collinear particles is shown at the level of operators in an effective field theory. Exclusive hadronic factorization and inclusive partonic factorization follow as special cases. The leading order Lagrangian is derived using power counting and gauge invariance in the effective theory. Several species of gluons are required, and softer gluons appear as background fields to gluons with harder momenta. Two examples are given: the factorization of soft gluons in B->D pi, and the soft-collinear convolution for the B->Xs gamma spectrum.

Figures (11)

• Figure 1: A matching calculation which shows how $W$ appears. On the left collinear $\bar{n}\!\cdot\! A_{n,q}$ gluons hit an incoming soft quark. Integrating out the offshell quark propagators gives the effective theory operator on the right which contains a factor of $W$.
• Figure 2: Collinear gluon propagator with label momentum $q$ and residual momentum $k$, and the order $\lambda^0$ interactions of collinear gluons with the usoft gluon field. Here usoft gluons are springs, collinear gluons are springs with a line, and $\alpha$ is the covariant gauge fixing parameter in Eq. (\ref{['Lcg']}).
• Figure 3: The attachments of usoft gluons to a collinear quark line which are summed up into a path-ordered exponential.
• Figure 4: The attachments of usoft gluons to a collinear gluon which are also summed up into a path-ordered exponential.
• Figure 5: The interaction of a soft and collinear gluon with momenta $k\sim Q(\lambda,\lambda,\lambda)$ and $q\sim Q(\lambda^2,1,\lambda)$ respectively, to produce an offshell gluon with momentum $k+q\sim Q(\lambda,1,\lambda)$.