Soft-collinear factorization in effective field theory

TLDR

Factorization separates hard, collinear, and soft dynamics in QCD, with exclusive hadronic and inclusive partonic factorization as special cases. The authors derive the leading‑order Lagrangian via power counting and gauge invariance, introduce multiple gluon species with softer gluons acting as background fields, and illustrate the formalism with soft‑gluon factorization in \(B\to D\pi\) and the soft‑collinear convolution for the \(B\to X_s\gamma\) spectrum. They demonstrate that soft and ultrasoft gluons can be factorized from collinear particles at the operator level within the effective field theory.

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 $\stackrel{\ensuremath{\rightarrow}}{B}D\ensuremath{\pi}$ and the soft-collinear convolution for the $\stackrel{\ensuremath{\rightarrow}}{B}{X}_{s}\ensuremath{\gamma}$ spectrum.