Abstract
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 $\Upsilon(1S)$ 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 $\Gamma(J/\Psi\to \ell^+\ell^-)$ and compare to the experimental data.
Highlights
Bottomonia, the bound states of a bottom and an antibottom quark, are excellent systems to investigate the dynamics of bound states in QCD
Starting from QCD, we first integrate out the hard modes to arrive at nonrelativistic QCD (NRQCD)
We integrate out the soft modes and potential gluons with ultrasoft energies and soft momenta to arrive at potential NRQCD (PNRQCD)
Summary
Bottomonia, the bound states of a bottom and an antibottom quark, are excellent systems to investigate the dynamics of bound states in QCD. Together with the N3LO results for the energy levels and the wave function at the origin [2,3,7], this made it possible to evaluate the decay rate at N3LO in Ref. The aim of this paper is to go beyond this approximation and include the corrections due to the charm-quark mass at next-to-next-to-leading order (NNLO). The analytic result for this contribution completes the two-loop evaluation of cv and together with its application to ΓðΥð1SÞ → lþl−Þ constitutes the main result of our paper Another possibility to include the charm-quark mass effects is to consider mc to be soft. In the Appendix, analytic results for all matching coefficients up to two loops, which are not presented in the main part of the paper, are provided.
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