Nonperturbative renormalization of the quark chromoelectric dipole moment with the gradient flow: Power divergences

The $CP$-violating quark chromoelectric dipole moment (qCEDM) operator, contributing to the electric dipole moment (EDM), mixes under renormalization and—particularly on the lattice—with the pseudoscalar density. The mixing coefficient is power-divergent with the inverse lattice spacing squared, $1/a^2$, regardless of the lattice action used. We use the gradient flow to define a multiplicatively renormalized qCEDM operator and study its behavior at small flow time. We determine nonperturbatively the linearly divergent coefficient with the flow time, $1/t$, up to subleading logarithmic corrections, and compare it with the 1-loop perturbative expansion in the bare and renormalized strong coupling. We also discuss the $\mathrm{O}(a)$ improvement of the qCEDM defined at positive flow time.

Figure 1

${\overline{R}}_P(t)/t$ defined in Eq. (17) and determine on the ensemble $A_3$ (see Table 1) for values of the flow time $t/a^2=0.5$, 2.0, corresponding to a flow time radius $r_f=\sqrt{8t}=2a,4a$.

Figure 2

Flow time dependence of the ratio ${\overline{R}}_P$ for all our ensembles. For the x-axis we choose the standard definition of $t_0/a^2$, i.e., $t_0=t_0^{(0)}$ (see main text).

Figure 3

Tree-level lattice artifacts of the energy density defined in Eq. (45) for different orders $(\frac{a^2}{t}{)}^M$.

Figure 4

Simultaneous fit for chiral and continuum extrapolation of renormalized coupling ${\overline{g}}^2$ at $t/t_0=0.4$ choosing the standard definition of $t_0$ with $M=0$. See main text for more details. The empty point at the coarser lattice spacing is excluded from the continuum and chiral extrapolation.

Figure 5

Global fit of the ratio ${\overline{R}}_P$ for the definition of $t_0$, with $M=0$. See main text for more details. The empty point at the coarser lattice spacing is excluded from the continuum and chiral extrapolation.

Figure 6

Blue band corresponds to chiral, continuum limit of data. The error is obtained from the maximum difference between $t_0^{(M)}$, $M=0$, 1, 2, 3, 4. For plotting ${\overline{g}}^2$ for data points are obtained using $t_0^{(4)}$.

Figure 7

Error bands are obtained from the fitting of the chiral, continuum limit data. The error band is obtained from the maximum difference between the fitting results of extrapolated data using different $t_0^{(M)}$, $M=0$, 1, 2, 3, 4.

Figure 8

The ratio $R_P/t$ defined in Eqs. (11) and (43) and determine on the ensemble $A_3$ (see Table 1) for several values of the flow time $t/a^2=0$, 0.5, 2.0, corresponding to a flow time radius $r_f=\sqrt{8t}=0,2a,4a$. Different colors correspond to different sources.

Figure 9

Ratio $R_P$ in Eqs. (11) and (43) as a function of $t/t_0$. In the right plot, error bands are reconstructed based on the final fitting results and solid data points are belonged in the selected fitting ranges in Table 4.

Figure 10

Left plot: flow-time dependence of $c_{\chi }$ for the ensemble $A_3$. The different colored lines show the following contributions. Magenta: example of a single fit satisfying the p-value condition. The magenta band represents the statistical uncertainty for the particular fit range chosen. Orange: the contribution of the cutoff effect, parametrized by $B_{-1}$, to the single fit chosen in the plot. The orange band represents the associated statistical and systematic uncertainties stemming from the variation of the fit ranges. Green: the fit result obtained removing cutoff effects. The central value this time represents the median of the distribution obtained varying the fit ranges and satisfying our p-value condition (see Appendix pp2). The width of the band represents the associated statistical uncertainty. Red data point: the central value is $B_0$ obtained as the median of the distribution of values of $B_0$ varying the fit ranges and the error represents the sum in quadrature of the statistical and systematic uncertainties on $B_0$. Right plot: the blue data points represent the raw data after subtracting the cutoff effects and the higher dimensional operator contributions, linear in $t/t_0$, determined from the fit. The red band represents our estimate of $c_{\chi }$ including the statistical and systematic uncertainties added in quadrature.

Figure 11

Nonperturbative dependence of $c_{\chi }$ as a function of the bare coupling $g_0$. The blue band represents the Padé approximant in Eq. (56). The width of the band represents the statistical and systematic uncertainties added in quadrature.

Figure 12

Euclidean time, $x_4$, dependence of ${\tilde{\mathrm{\Gamma }}}_{PP}(x_4;t)$ for the ensemble $A_3$ at 3 values of the flow time.

Figure 13

Euclidean time, $x_4$, dependence of ${\tilde{\mathrm{\Gamma }}}_{CP}(x_4;t)$ for the ensemble $A_3$ at 3 values of the flow time.

Figure 14

p-values obtained from the ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}$. The darker regions correspond to the acceptable fit ranges. In the right plot, the blue and red rectangles correspond to blue and red peak in Fig. 15. See main text in this Appendix.

Figure 15

Distribution of the fit parameter $B_0$ for all the fit ranges satisfying $p>0.05$.

Figure 16

These results are $t_0/a^2$ dependence of each fitting parameters $B_{-1}$, $B_1$. With the exception of the $A_1$ and $A_3$ ensembles, the data points for each ensemble were computed from the fitting ranges from peak and splitting positions given in Table 4.

Figure 17

The only contribution to ${\mathrm{\Gamma }}_{PP}(x_4;t)$ at tree level.

Figure 18

Leading-order contributions to ${\mathrm{\Gamma }}_{CP}(x_4;t)$.