A non-perturbative determination of $b_{\mathrm{g}}$

Abstract

Close to the continuum limit, lattice QCD with mass-degenerate Wilson quarks can be described by Symanzik's effective continuum action, which contains the dimension 5 operator, $m\mathrm{tr}(F_{\hat{I}¼\hat{I}½}{F}_{\hat{I}¼\hat{I}½})$. Its effect can be eliminated by an O($a{m}_{\mathrm{q}}$) rescaling of the bare lattice coupling constant. Until recently, the corresponding improvement coefficient, $b_{\mathrm{g}}$, was only known perturbatively to 1-loop order and an estimate of the remaining uncertainty is the dominant systematic error in the ALPHA collaboration's recent determination of $\hat{I}{\pm }_s(m_Z)$ with the decoupling method. To remove this error we have determined $b_{\mathrm{g}}$ non-perturbatively for the corresponding parameter range. We here briefly review improvement conditions for $b_{\mathrm{g}}$, perform a perturbative test and report on our non-perturbative results for $b_{\mathrm{g}}$.

8 pages, 1 figure, contribution to "The 40th International Symposium on Lattice Field Theory (Lattice 2023)", CERN-TH-2024-004