Non-perturbative determination of the $\Lambda$-parameter in the pure SU(3) gauge theory from the twisted gradient flow coupling

Abstract

We evaluate the $\Lambda$-parameter in the $\overline{\mathrm{MS}}$ scheme for the pure SU(3) gauge theory with the twisted gradient flow (TGF) method. A running coupling constant $g_{\mathrm{TGF}}^2(1/L)$ is defined in a finite volume box with size of $L^4$ with the twisted boundary condition. This defines the TGF scheme. Using the step scaling method for the TGF coupling with lattice simulations, we can evaluate the $\Lambda$-parameter non-perturbatively in the TGF scheme. In this paper we determine the dimensionless ratios, ${\Lambda }_{\mathrm{TGF}}/\sqrt{\sigma }$ and $r_0{\Lambda }_{\mathrm{TGF}}$ together with the $\Lambda$-parameter ratio ${\Lambda }_{\mathrm{SF}}/{\Lambda }_{\mathrm{TGF}}$ on the lattices numerically. Combined with the known ratio ${\Lambda }_{\overline{\mathrm{MS}}}/{\Lambda }_{\mathrm{SF}}$, we obtain ${\Lambda }_{\overline{\mathrm{MS}}}/\sqrt{\sigma }=0.517(10){(}_{-7}^{+8})$ and $r_0{\Lambda }_{\overline{\mathrm{MS}}}=0.593(12){(}_{-9}^{+12})$, where the first error is statistical one and the second is our estimate of systematic uncertainty.

Comments: 23 pages, 11 figures, LaTeX