Non-perturbative determination of the $Λ$-parameter in the pure SU(3) gauge theory from the twisted gradient flow coupling
We evaluate the $Λ$-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 $Λ$-parameter non-perturbatively in the TGF scheme. In this paper we determine the dimensionless ratios, $Λ_{\mathrm{TGF}}/\sqrtσ$ and $r_{0}Λ_{\mathrm{TGF}}$ together with the $Λ$-parameter ratio $Λ_{\mathrm{SF}}/Λ_{\mathrm{TGF}}$ on the lattices numerically. Combined with the known ratio $Λ_{\overline{\mathrm{MS}}}/Λ_{\mathrm{SF}}$, we obtain $Λ_{\overline{\mathrm{MS}}}/\sqrtσ = 0.517(10)(^{+8}_{-7})$ and $r_{0}Λ_{\overline{\mathrm{MS}}}=0.593(12)(^{+12}_{-9})$, where the first error is statistical one and the second is our estimate of systematic uncertainty.