Nonperturbative renormalization and $\mathrm{O}(a)$-improvement of the nonsinglet vector current with $N_f=2+1$ Wilson fermions and tree-level Symanzik improved gauge action

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 $\mathrm{O}(a)$ 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 $\mathrm{O}(a)$ improved. In addition, the local vector current, which has support on one lattice site, must be renormalized. At $\mathrm{O}(a)$, the breaking of the $\mathrm{SU}(N_{\mathrm{f}})$ symmetry by the quark mass matrix leads to a mixing between the local currents of different quark flavors. We present a nonperturbative calculation of all the required improvement and renormalization constants needed for the local and the conserved electromagnetic current in QCD with $N_{\mathrm{f}}=2+1$ $\mathrm{O}(a)$-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 initiative.

Figure 1

The chiral Ward identity in the continuum and in the limit $m_{12}=0$. Continuous horizontal lines indicate that the operator is projected onto vanishing spatial momentum.

Figure 2

Plateaus for the ratio $R(t=\frac{T}{2},t_1)$ and ${\widehat{c}}_{\mathrm{V}}$ defined through Eqs. (41) and (32) for the lattice ensemble N300.

Figure 3

Results of the fits used to determine the renormalization constant $Z_{\mathrm{V}}$ and improvement coefficients $b_{\mathrm{V}}$ and ${\overline{b}}_{\mathrm{V}}^{\mathrm{eff}}$ using the fit ansatz (42) for two different values of the bare coupling.

Figure 4

Left panel: Continuum limit of the ratio $Z_{\mathrm{V},\mathrm{sub}}^l/Z_{\mathrm{V}}$ where $Z_{\mathrm{V},\mathrm{sub}}^l$ was computed in [12] and where $Z_{\mathrm{V}}$ refers to our own determination. Middle panel: $b_{\mathrm{V}}^l$ and $b_{\mathrm{V}}^s$ correspond to our determinations with the light or strange spectator quarks, respectively, and $b_{\mathrm{V}}^F$ is the value computed in [23]. Right panel: ${\overline{b}}_{\mathrm{V}}^F$ is the value computed in [23]. We have defined the dimensionless parameter $\tilde{a}=a/0.5\mathrm{fm}$.

Figure 5

Chiral extrapolations of the improvement coefficients $c_{\mathrm{V}}^l$ and $c_{\mathrm{V}}^c$, respectively, for the local and conserved vector currents, for two different values of the bare coupling.

Figure 6

The dependence of the renormalization constant $Z_{\mathrm{V}}$ and improvement coefficients $b_{\mathrm{V}}$, ${\overline{b}}_{\mathrm{V}}^{\mathrm{eff}}$, and $c_{\mathrm{V}}$ on the bare coupling $g_0^2$. The blue and red points correspond to the local and conserved vector currents, respectively. The plain lines and error bands correspond to our Padé fits. For $b_{\mathrm{V}}$ and ${\overline{b}}_{\mathrm{V}}^{\mathrm{eff}}$ we also compare our results with previous lattice determinations [10, 23].

Figure 7

Left: Extrapolation of $c_{\mathrm{A}}$ for one value of the bare coupling $g_0^2$. Right: Improvement coefficient $c_{\mathrm{A}}$ as a function of the bare coupling $g_0^2$ with a (1,1)-Padé model. We also plot the results obtained by the ALPHA Collaboration in Ref. [11] using a different method.