Leptonic widths of heavy quarkonia: S-Wave QCD/NRQCD matching coefficients for the electromagnetic vector annihilation current at O(alpha_s v^2)

TL;DR

This paper develops a lattice QCD/NRQCD method to accurately compute the S-wave electromagnetic vector annihilation current for heavy quarkonia, including one-loop corrections up to order v^2, aiding precise decay rate predictions.

Contribution

It introduces a novel lattice matching procedure for the electromagnetic current in heavy quarkonia, incorporating one-loop corrections and complex amplitude treatment with automated vertex generation.

Findings

Matching coefficients as a function of heavy quark mass Ma.

Successful treatment of Coulomb-exchange and infrared singularities.

Applicable to electromagnetic decays of heavy quarkonia like the Upsilon meson.

Abstract

We construct the S-wave part of the electromagnetic vector annihilation current to O(alpha_s v^2)$ on the lattice for heavy quarks whose dynamics are described by the NRQCD action, where v is the non-relativistic quark velocity. The lattice vector current for QQbar annihilation is expressed as a linear combination of lattice operators with quantum numbers L=0, J^P=1^-, and the coefficients are determined by matching this lattice current to the corresponding continuum current in QCD to O(v^2) to one-loop. The annihilation channel gives a complex amplitude and a proper choice for the contours of integration is needed; a simple Wick rotation is not possible. In this way, and with a careful choice of subtraction functions in the numerical integration, the Coulomb-exchange and infrared singularities appearing in the amplitudes are successfully treated. The matching coefficients are given as a function of the heavy quark mass Ma in lattice units. An automated vertex generation program written in Python is employed, allowing us to use a realistic NRQCD action and an improved gluon lattice action. A change in the definition of either action is easily accommodated in this procedure. The final result is applicable to the electromagnetic decays of heavy quarkonia, notably the Upsilon meson.