Improving the kinetic couplings in lattice nonrelativistic QCD

We improve the nonrelativistic QCD (NRQCD) action by comparing the dispersion relation to that of the continuum through $\mathcal{O}(p^6)$ in perturbation theory. The one-loop matching coefficients of the $\mathcal{O}(p^4)$ kinetic operators are determined, as well as the scale at which to evaluate ${\alpha }_s$ in the $V$-scheme for each quantity. We utilize automated lattice perturbation theory using twisted boundary conditions as an infrared regulator. The one-loop radiative corrections to the mass renormalization, zero-point energy and overall energy-shift of an NRQCD $b$-quark are also found. We also explore how a Fat3-smeared NRQCD action and changes of the stability parameter $n$ affect the coefficients. Finally, we use gluon field ensembles at multiple lattice spacing values, all of which include $u$, $d$, $s$ and $c$ quark vacuum polarization, to test how the improvements affect the nonperturbatively determined $\mathrm{\Upsilon }(1S)$ and ${\eta }_b(1S)$ kinetic masses, and the tuning of the $b$ quark mass.

Figure 1

Contributions to the one-loop self energy $\mathrm{\Sigma }$ of the heavy quark, showing the rainbow diagram (left) and the tadpole diagram (right). The straight lines represent heavy quarks, while the curly lines represent gluons.

Figure 2

The raw data for $W_1$ at multiple values of $1/L$ overlaid with our fit curve.

Figure 3

Numerical values for $W_1$ (top) and $W_2$ (bottom) for different NRQCD actions without mean-field improvement as described in Sec. 2.

Figure 4

Numerical values for ${\tilde{c}}_1^{(1)}$ (top) and $c_5^{(1)}$ (bottom) for different NRQCD actions without mean-field improvement.

Figure 5

As in Fig. 4 but with the mean-field improved data for ${\tilde{c}}_1^{(1)}$ (top) and $c_5^{(1)}$ (bottom). Note the change in vertical scale. The mean-field corrections are given in Appendix pp2. Note that the fat3 smeared data are the same as in Fig. 4, since no mean-field correction is applied in this case.

Figure 6

The one-loop radiative correction ${\tilde{c}}_1^{(1)}{\alpha }_s(q^{*})$ (top) and $c_5^{(1)}{\alpha }_s(q^{*})$ (bottom) with ${\alpha }_s$ defined in the $V$-scheme, for different NRQCD actions and different values of the lattice spacings. We give a subset of the numerical values relevant for nonperturbative calculations in Appendix pp3.

Figure 7

Numerical values for $Z_m^{(1)}$ (top) and the one-loop radiative correction $Z_m^{(1)}{\alpha }_s(q^{*})$ (bottom) with ${\alpha }_s$ defined in the $V$-scheme, for different NRQCD actions and different values of the lattice spacings. Note that the unsmeared data are mean-field improved as described Sec. 2b, and the fat3 smeared results are not mean-field improved.

Figure 8

The same as in Fig. 7 but for $W_0$ in place of $Z_m^{(1)}$.

Figure 9

The same as in Fig. 7 but with $\delta C$ in place of $Z_m^{(1)}$.

Figure 10

Nonperturbatively obtained values for $M_{\mathrm{kin}}$ for the ${\eta }_b$ and the $\mathrm{\Upsilon }$ plotted against momentum squared, together with the spin-averaged kinetic mass, on set 5. The solid lines show experimental values and errors are statistical only.

Figure 11

Nonperturbatively obtained spin-averaged kinetic masses, given by Eq. (33), on very coarse set 1 (above) and coarse set 3 (below) for $p^4$ and $p^6$ actions with both tree-level and $\mathcal{O}({\alpha }_s)c_1,c_6$ and $c_5$. The errors shown are statistical only, excluding lattice spacing uncertainty, and are correlated. The data points at each value of $P^2{a}^2$ have been offset symmetrically for clarity. The larger energy with $a|\mathbf{P}|=9$ is from the (2, 2, 1) ground state. The solid line is the experimental value.

Figure 12

Same as Fig. 11 but on fine set 5.

Figure 13

Nonperturbatively obtained values for $\mathrm{\Delta }{E}_{221–300}$ for the ${\eta }_b$ for each action plotted against the square of the lattice spacing.

Figure 14

The speed of light squared, $c^2{P}^2=(\mathrm{\Delta }E+M_{\mathrm{kin}}{)}^2-M_{\mathrm{kin}}^2$, on set 5 with the $\mathcal{O}({\alpha }_s{p}^4,p^6)$ NRQCD action. The errors shown are statistical only, and we use the value of $M_{\mathrm{kin}}$ computed using momentum (1, 1, 1). The dashed line corresponds to $c^2=1$.