Abstract
In calculating hadronic contributions to precision observables for tests of the Standard Model in lattice QCD, the electromagnetic current plays a central role. Using a Wilson action with O($a$) improvement in QCD with $N_{\mathrm{f}}$ flavors, a counterterm must be added to the vector current in order for its on-shell matrix elements to be O($a$) improved. In addition, the local vector current, which has support on one lattice site, must be renormalized. At O($a$), the breaking of the SU($N_{\mathrm{f}}$) symmetry by the quark mass matrix leads to a mixing between the local currents of different quark flavors. We present a non-perturbative calculation of all the required improvement and renormalization constants needed for the local and the conserved electromagnetic current in QCD with $N_{\mathrm{f}}=2+1$ O($a$)-improved Wilson fermions and tree-level Symanzik improved gauge action, with the exception of one coefficient, which we show to be order $g_0^6$ in lattice perturbation theory. The method is based on the vector and axial Ward identities imposed at finite lattice spacing and in the chiral limit. We make use of lattice ensembles generated as part of the Coordinated Lattice Simulations (CLS) initiative.
Highlights
Precision tests of the Standard Model typically require reliable theory input from first-principles calculations
We present a nonperturbative calculation of all the required improvement and renormalization constants needed for the local and the conserved electromagnetic current in QCD with Nf 1⁄4 2 þ 1 OðaÞ-improved Wilson fermions and tree-level Symanzik improved gauge action, with the exception of one coefficient, which we show to be order g60 in lattice perturbation theory
Appendix A presents a determination of the improvement coefficient cA of the axial current, and Appendix B contains some results on the employed correlation functions in lattice perturbation theory
Summary
Precision tests of the Standard Model typically require reliable theory input from first-principles calculations. Our main motivation for the present calculation is to determine the two-point function of the electromagnetic current with only Oða2Þ discretization effects This will in particular allow for a shorter continuum extrapolation of the leading hadronic contribution to the anomalous magnetic moment of the muon, and a more cost-effective set of lattice QCD simulations. Given that phenomenologically the πþπ− channel, which is described by the timelike electromagnetic form factor of the pion, accounts for more than two-thirds of the total hadronic contributions, it is very natural to impose the renormalization condition on the local vector current that the electric charge of the pion be unity at every lattice spacing. Appendix A presents a determination of the improvement coefficient cA of the axial current, and Appendix B contains some results on the employed correlation functions in lattice perturbation theory
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