Leptonic widths of heavy quarkonia: QCD/NRQCD matching for the electromagnetic current at O(α_s v²)

We construct the S-wave part of the electromagnetic vector annihilation current to $O(\alpha_s v^2)$, where $v$ is the non-relativistic quark velocity, for heavy quarks whose dynamics are described by the NRQCD action on the lattice. The NRQCD vector current for $Q\bar{Q}$ 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 to the corresponding continuum current in QCD to$O(v^2)$ at one-loop. The annihilation channel gives a complex amplitude with Coulomb-exchange and infrared singularities, making a careful choice for the contours of integration and infrared subtraction functions in the numerical integration necessary. 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 simulations of electromagnetic decays of heavy quarkonia, notably the $\Upsilon$ meson.