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

Abstract

We determine quark mass dependent order $a$ improvement terms of the form $b_Jam$ for non-singlet scalar, pseudoscalar, vector and axialvector currents using correlators in coordinate space on a set of CLS ensembles. These have been generated employing non-perturbatively improved Wilson Fermions and the tree-level L\"uscher-Weisz gauge action at $\beta = 3.4, 3.46, 3.55$ and $3.7$, corresponding to lattice spacings ranging from $a \approx 0.085$ fm down to $0.05$ fm. In the $N_f=2+1$ flavour theory two types of improvement coefficients exist: $b_J$, proportional to non-singlet quark mass combinations, and $\bar{b}_J$ (or $\tilde{b}_J$), proportional to the trace of the quark mass matrix. Combining our non-perturbative determinations with perturbative results, we quote Pad\'e approximants parameterizing the $b_J$ improvement coefficients within the above window of lattice spacings. We also give preliminary results for $\tilde{b}_J$ at $\beta=3.4$.