Manifestly Soft Gauge Invariant Formulation of vNRQCD

TL;DR

This paper develops a gauge-invariant formulation of vNRQCD using Wilson lines, simplifying operator bases and clarifying the role of soft gluons, ghosts, and color structures in heavy quark interactions.

Contribution

It introduces a manifestly gauge-invariant operator formulation in vNRQCD with Wilson lines, reducing complexity and improving the understanding of soft gluon interactions.

Findings

Operators can be expressed in a gauge-invariant form using Wilson lines.

Soft ghosts do not couple to heavy quarks until two loops.

Color structures follow non-Abelian exponentiation.

Abstract

Homogeneous power counting in Non-Relativistic QCD (NRQCD) implies the simultaneous existence of both soft and ultrasoft gluons. In the velocity renormalization group (vNRQCD) formalism we show that operators involving soft fields interacting with heavy potential quarks can be put in a manifestly gauge invariant form by utilizing gluon and quark building blocks which contain Wilson lines, and are analogous to those used in the soft collinear effective theory. This leads to several simplifications, in particular significantly reducing the size of the operator basis, which stream-lines matching and anomalous dimension calculations at subleading order in the velocity expansion. Also, soft ghosts no longer couple via potential like interactions to the heavy quark fields, and hence do not appear until two loops. Furthermore the the color structures that appear at each order in $\alpha_s$ in the static potential are only those which should arise according to non-Abelian exponentiation. We also discuss how zero-bin subtractions clarify the role of $i\varepsilon$ terms in the heavy quark propagator poles carrying soft momenta. Even though the choice of the direction of the soft Wilson lines in our building blocks does not matter, there is still a limited set of consistent choices, and the predicted form of the eikonal poles does influence results at two-loops and beyond.