Invariant Operators in Collinear Effective Theory

TL;DR

This work develops a collinear effective theory for processes with highly energetic hadrons, introducing a label-space operator and Wilson lines to build gauge-invariant, invariant operators whose Wilson coefficients depend on label operators. It provides a practical set of construction rules and illustrates the framework with a gauge-invariant collinear quark kinetic term and a heavy-to-light current, clarifying how label dependence enters physical amplitudes. The authors apply the formalism to nonleptonic B decays to a D meson and a pion, reproducing the known two-loop generalized factorization results and showing that collinear-gluon contributions yield a convolution of short-distance coefficients with the pion light-cone wavefunction at all orders. The approach offers a predictive, all-orders pathway to generalized factorization in heavy-to-light decays and underpins the structure of factorization theorems in collinear QCD.

Abstract

We consider processes which produce final state hadrons whose energy is much greater than their mass. In this limit interactions involving collinear fermions and gluons are constrained by a symmetry, and we give a general set of rules for constructing leading and subleading invariant operators. Wilson coefficients C(mu,P) are functions of a label operator P, and do not commute with collinear fields. The symmetry is used to reproduce a two-loop result for factorization in B -> D pi in a simple way.

Figures (2)

• Figure 1: Typical momenta/positions for soft particles (shaded bins), and collinear particles (white bins).
• Figure 2: The collinear graphs contributing at one loop in the effective theory.