Renormalizing vector currents in lattice QCD using momentum-subtraction schemes

We examine the renormalization of flavor-diagonal vector currents in lattice QCD with the aim of understanding and quantifying the systematic errors from nonperturbative artifacts associated with the use of intermediate momentum-subtraction schemes. Our study uses the highly improved staggered quark action on gluon-field configurations that include $n_f=2+1+1$ flavors of sea quarks, but our results have applicability to other quark actions. Renormalization schemes that make use of the exact lattice vector Ward-Takahashi identity for the conserved current also have renormalization factors, $Z_V$, for nonconserved vector currents that are free of contamination by nonperturbative condensates. We show this by explicit comparison of two such schemes: that of the vector form factor at zero momentum transfer and the RI-SMOM momentum-subtraction scheme. The two determinations of $Z_V$ differ only by discretization effects (for any value of momentum transfer in the RI-SMOM case). The ${\mathrm{RI}}^{\prime }$-MOM scheme, although widely used, does not share this property. We show that $Z_V$ determined in the standard way in this scheme has $\mathcal{O}(1\%)$ nonperturbative contamination that limits its accuracy. Instead we define an ${\mathrm{RI}}^{\prime }$-MOM $Z_V$ from a ratio of local to conserved vector current vertex functions and show that this $Z_V$ is a safe one to use in lattice QCD calculations. We also perform a first study of vector current renormalization with the inclusion of quenched QED effects on the lattice using the RI-SMOM scheme.

Figure 1

Demonstration of the vector Ward-Takahashi identity in momentum space [Eq. (8)] for HISQ quarks on the lattice on a single gluon-field configuration from Set 2. The plot shows the ratio of the right-hand side of this equation to the amputated vertex function for the conserved vector current on the left-hand side for the SMOM kinematic configuration, Eq. (18). This is a matrix equation and this plots shows the result of averaging over all matrix components (which agree) and the two nonzero components of $q_{\mu }$. The solid line is the value of $2\sin (a{q}_{\mu }/2)$ for a nonzero component of $q_{\mu }$. The points correspond to lattice results for the ratio on a single configuration with crosses giving Coulomb gauge-fixed results and the circles Landau gauge-fixed results. Orange points correspond to a valence mass of $a{m}_{\mathrm{val}}=0.0306$ while purple points correspond to 0.0102. The Ward-Takahashi identity requires these points to lie on the line as they do.

Figure 2

A test of the expression for the difference of inverse propagators with momentum $p_1$ and $p_2$ in Eq. (14). We show results on coarse, fine and superfine lattices (Sets 2, 5 and 7) for a variety of $\mu$ values in lattice units where $|a{p}_1|=|a{p}_2|=|aq|=a\mu$.

Figure 3

The $Z_V$ value obtained for the conserved vector current in the RI-SMOM scheme on coarse, fine and superfine gluon-field configurations (Sets 2, 5 and 7). Values are given for a variety of $\mu$ values in lattice units where $|a{p}_1|=|a{p}_2|=|aq|=a\mu$.

Figure 4

Points labeled “MOM” show the renormalization factor for the HISQ conserved vector current which takes results from the lattice scheme to the $\overline{\mathrm{MS}}$ scheme obtained using a lattice calculation in the ${\mathrm{RI}}^{\prime }$-MOM scheme. These should be contrasted with results obtained using a lattice calculation in the RI-SMOM scheme which give a value of 1.0, shown as the black line labeled “SMOM.” Points are given for $\mu$ values of 2, 2.5, 3 and 4 GeV, indicated by different colors, and on coarse, fine and superfine lattices. The fit shown [see Eq. (26)] accounts for discretization errors and condensate contributions, which prove to be necessary for a good fit. The separation of the results for different $\mu$ values in the continuum limit (at $a=0$) is a result of the condensate contributions that appear in $Z_V$ when calculated in the ${\mathrm{RI}}^{\prime }$-MOM scheme on the lattice.

Figure 5

Valence mass dependence of $Z_V^{\mathrm{loc}}(\mathrm{SMOM})$ values obtained in the RI-SMOM scheme. Results and extrapolation are shown for $\mu =3\mathrm{GeV}$ on Set 2.

Figure 6

The top plot shows $Z_V^{\mathrm{loc}}(\mathrm{SMOM})$ for $\mu$ values between 1 and 4 GeV, plotted as a difference to the corresponding $Z_V$ at that lattice spacing obtained from the vector form factor at zero momentum transfer. The fit shown (see text) accounts for discretization errors and condensate contributions, but no condensate contributions are seen and they are strongly constrained to zero by the fit. The lower plot shows the same difference but for a $Z_V^{\mathrm{loc}}(\mathrm{SMOM})$ derived from results at $\mu =2$ and 3 GeV in such a way as to reduce discretization effects [see Eq. (30)]. The fit here is to a simple ($\mathrm{constant}+a^4$) form as described in the text.

Figure 7

Valence mass dependence of our raw results for $Z_V^{\mathrm{loc}}$ calculated in the ${\mathrm{RI}}^{\prime }$-MOM scheme, before multiplication by the additional renormalization factor needed to match to $\overline{\mathrm{MS}}$. Results and extrapolation are shown for $\mu =3\mathrm{GeV}$ on Set 2.

Figure 8

$Z_V^{\mathrm{loc}}(\mathrm{MOM})$ for $\mu$ values between 2 and 4 GeV, plotted as a difference to the corresponding $Z_V$ at that lattice spacing obtained from the vector form factor at zero momentum transfer. The fit shown (see text) accounts for discretization errors and condensate contributions, with condensate contributions being necessary to obtain a good fit.

Figure 9

$Z_V^{\mathrm{loc}}({\mathrm{MOM}}_{\mathrm{Rc}})$ from the modified ${\mathrm{RI}}^{\prime }$-MOM scheme of Eq. (32), plotted as a difference to the corresponding $Z_V$ at that lattice spacing obtained from the vector form factor at zero momentum transfer. Results are shown for $\mu$ values from 2 to 4 GeV. The fit shown (see text) accounts for discretization errors and condensate contributions, but condensate contributions are strongly constrained to be zero.

Figure 10

$Z_V^{\mathrm{loc}}({\mathrm{SMOM}}_{{\gamma }_{\mu },\mathrm{Rc}})$ from the modified RI-${\mathrm{SMOM}}_{{\gamma }_{\mu }}$ scheme, plotted as a difference to the corresponding $Z_V$ at that lattice spacing obtained from the vector form factor at zero momentum transfer. Results are shown for $\mu$ values from 2 to 4 GeV. The fit shown (see text) accounts for discretization errors and condensate contributions, but condensate contributions are strongly constrained to be zero.

Figure 11

The difference between the vertex functions for the local vector and local axial-vector currents as a function of $\mu$ in the ${\mathrm{RI}}^{\prime }$-MOM and RI-SMOM schemes. The RI-SMOM scheme gives a much smaller nonperturbative contribution to $Z_A-Z_V$ than the ${\mathrm{RI}}^{\prime }$-MOM scheme, but neither scheme gives $Z_V=Z_A$ which is true to all orders in perturbation theory. The values plotted result from an extrapolation to zero valence quark mass and are shown for the finest lattice we use (Set 7 in Table 1).

Figure 12

Results for the renormalization factor, $Z_V$ for the $\mathrm{QCD}+\mathrm{QED}$ conserved current for the HISQ action, calculated using the RI-SMOM scheme. Results are given for coarse, fine and superfine gluon-field configurations for quark electric charge, $Q=2e/3$ and a variety of momenta with magnitude $a\mu$ in lattice units.

Figure 13

The impact of quenched QED (with quark charge $2e/3$) on the determination of $Z_V^{\mathrm{loc}}$ and $Z_q$ using the RI-SMOM scheme as a function of the lattice spatial extent, $L_s$ in lattice units. Results are for coarse lattices (Sets 3, 4 and 5), and $\mu =2\mathrm{GeV}$. The volume dependence is negligible.

Figure 14

The ratio of $Z_V^{\mathrm{loc}}$ values for $\mathrm{QCD}+\mathrm{QED}$ to QCD calculated in the RI-SMOM scheme. Results are given for coarse to superfine lattices at $\mu$ values from 2 to 4 GeV and plotted against the square of the lattice spacing. The dashed lines give the result of a fit described in the text that shows that the results are fully described by a perturbative series (of which the leading coefficient is known) up to discretization effects. The dips in the fit functions close to $a=0$ are the result of the fact that the argument of ${\alpha }_s$ in the fit function [Eq. (34)] is inversely related to $a$.

Figure 15

$Z_V^{1\mathrm{link}}(\mathrm{SMOM})$ for $\mu$ values between 1 and 4 GeV, plotted as a difference to the corresponding $Z_V$ at that lattice spacing obtained from the vector form factor at zero momentum transfer. The fit shown [see Eq. (b7)] accounts for discretization effects only.