Gauge dependence and matching procedure of a nonrelativistic QED-QCD bound state formalism

A nonrelativistic bound state formalism used in contemporary calculations is investigated. It is known that the effective Hamiltonian of the bound state 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 bound state 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 (Formula presented) and (2) we examine the gauge dependences of diagrams for a decay of a (Formula presented) bound state up to (Formula presented) and show that cumbersome gauge cancellations can be circumvented by directly calculating Q. As an application we point out that the present calculations of the top quark momentum distribution in the (Formula presented) threshold region are gauge dependent. We also show the possibilities for incorrect calculations of physical quantities of bound states 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 or NRQCD bound states.