Charm-quark mass effects in NRQCD matching coefficients and the leptonic decay of the $\mathrm{{\rm Y}}(1S)$ meson

We compute two-loop corrections to the vector current matching coefficient involving two heavy quark masses. The result is applied to the computation of the $\mathrm{{\rm Y}}(1S)$ decay width into an electron or muon pair. We complement the next-to-next-to-next-to-leading order corrections of Ref. arXiv:1401.3005 by charm quark mass effects up to second order in perturbation theory. Furthermore, we apply the formalism to $\mathrm{\Gamma }(J\mathrm{/}\mathrm{\Psi }\to {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-})$ and compare to the experimental data.