Gauge dependence and matching procedure of a nonrelativistic QED/QCD boundstate formalism

TL;DR

This paper investigates gauge dependence in nonrelativistic QED/QCD boundstate calculations, providing a formalism to ensure gauge invariance and highlighting potential pitfalls in current matching procedures.

Contribution

It introduces a transformation charge Q for gauge changes, formal arguments based on BRST symmetry, and demonstrates how to avoid gauge ambiguities in boundstate computations.

Findings

Calculated Q up to O(1/c) to analyze gauge dependence.

Showed gauge cancellations can be bypassed by calculating Q directly.

Highlighted gauge dependence issues in top quark momentum distribution calculations.

Abstract

A nonrelativistic boundstate formalism used in contemporary calculations is investigated. It is known that the effective Hamiltonian of the boundstate system depends on the choice of gauge. We obtain the transformation charge Q of the Hamiltonian for an arbitrary infinitesimal change of gauge, by which gauge independence of the mass spectrum and gauge dependences of the boundstate wave functions are dictated. We give formal arguments based on the BRST symmetry supplemented by power countings of Coulomb singularities of diagrams. For illustration: (1)we calculate Q up to O(1/c), (2)we examine gauge dependences of diagrams for a decay of a qqbar boundstate up to O(1/c) and show that cumbersome gauge cancellations can be circumvented by directly calculating Q. As an application we point out that the present calculations of top quark momentum distribution in the ttbar threshold region are gauge dependent. We also show possibilities for incorrect calculations of physical quantities of boundstates when the on-shell matching procedure is employed. We give a proof of a justification for the use of the equation of motion to simplify the form of a local NRQCD Lagrangian. The formalism developed in this work will provide useful cross checks in computations involving NRQED/NRQCD boundstates.