Nonperturbative determination of improvement coefficients using coordinate space correlators in $N_f=2+1$ lattice QCD

We determine quark mass dependent order $a$ improvement terms of the form $b_{J}am$ for nonsinglet scalar, pseudoscalar, vector and axialvector currents using correlators in coordinate space on a set of Coordinated Lattice Simulations ensembles. These have been generated employing nonperturbatively improved Wilson fermions and the tree-level Lüscher-Weisz gauge action at $\beta =3.4$, 3.46, 3.55 and 3.7, corresponding to lattice spacings ranging from $a\approx 0.085\mathrm{fm}$ down to 0.05 fm. In the $N_f=2+1$ flavor theory two types of improvement coefficients exist: $b_J$, proportional to nonsinglet quark mass combinations, and ${\overline{b}}_J$ (or ${\tilde{b}}_J$), proportional to the trace of the quark mass matrix. Combining our nonperturbative determinations with perturbative results, we quote Padé approximants parametrizing the $b_J$ improvement coefficients within the above window of lattice spacings. We also give preliminary results for ${\tilde{b}}_J$ at $\beta =3.4$.

Figure 1

Dependence of the ratio $R_P-1$ at the separation $x=(0,1,1,1)a$ as a function of the inverse hopping parameter difference $1/{\kappa }_s-1/{\kappa }_{\ell }$, see Eq. (20). For an ensemble list, see Table 2.

Figure 2

Dependence of the ratio ${\tilde{R}}_P-1$ at the separation $x=(0,1,2,2)a$ as a function of the inverse hopping parameter difference $1/{\kappa }^{(\sigma )}-1/{\kappa }^{(\rho )}$.

Figure 3

Tree-level improvement coefficients. The data correspond to the largest mass difference that we can encounter. The band indicates the region with cutoff effects smaller than 15%. All separations $n=x/a$ within this band are accepted.

Figure 4

$B_J(x)$ for the ensembles H102, S400 and N203 which share similar pion and kaon masses, see Table 2. Solid lines denote the $x^6$ fit, Eq. (46), to the data in the shaded region. The $n_{0}a\approx x_0$ points that are used to define the $b_J$ coefficients are plotted as large circles. The difference between a fit function at this position from its value at $x=0$ constitutes one of our systematic errors.

Figure 5

Numerical results for $b_J$ for all the mass nondegenerate ensembles. The outer (blue) error bars denote the total uncertainties while the inner (red) error bars indicate the statistical errors.

Figure 6

$b_J$ as functions of $g^2$. The error bands correspond to Eqs. (47)–(48) with the parameter values of Table 4. The dashed lines are the perturbative one-loop expectations.

Figure 7

The improvement coefficients $b_J$, extracted for different choices of $x_0$. In addition to $x_0=N_{0}a$ (blue central point within each group of three points), the results obtained for $x_{0<}$ (left points where available) and $x_{0>}$ (right points) are shown, see Table 5. Inner error bars are statistical only. Outer error bars include systematics. The “H” and “C” ensembles correspond to $\beta =3.4$ ($a\approx 0.085\mathrm{fm}$), S400 to $\beta =3.46$, N203 to $\beta =3.55$ and J303 to $\beta =3.7$ ($a\approx 0.050\mathrm{fm}$).