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

Abstract

Abstract:We evaluate the Λ-parameter in the MS scheme for the pure SU(3) gauge theory with the twisted gradient flow (TGF) method. A running coupling constant g 2 TGF (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 TGF / √ σ and r 0 Λ TGF together with the Λ-parameter ratio Λ SF /Λ TGF on the lattices numerically. Combined with the known ratio Λ MS /Λ SF , we obtain Λ MS / √ σ = 0.5315(81)( +269 −48 ) and r 0 Λ MS = 0.6062(92)( +309 −52 ), where the first error is statistical one and the second is our estimate of systematic uncertainty.

SU(N ) gauge theories in 2+1 dimensions: glueball spectra and k-string tensions

“…= 263(4) [9]MeV, (22) where the first error is statistical and the second is systematic. This value is consistent with the values recently obtained in the dedicated calculations [47,48] that use very different methods, and our value has a similar accuracy. This adds confidence that our calculations of Λ M S for the other SU(N) groups are also reliable.…”

“…The lattice coupling, g 2 L , satisfies the β-function in eqn (47) with s = L, and the mean-field coupling will satisfy a β-function with the same b 0 and b 1 coefficients but with a different coefficient, b I 2 , of the g 7 power. To determine b I 2 one can use the expression for TrU p as a power series in g 2 L to write g 2 I as…”

The twisted gradient flow running coupling in SU(3): a non-perturbative determination

“…Our scheme is based on a modification to the finite volume scheme introduced in [6], and was already used in [7] to compute the running coupling at NLO in continuum perturbation theory for several SU(N) groups. Similar schemes have also been used in recent years in SU(3) pure gauge theory to non-perturbatively determine the ratio of Λ parameters with the MS scheme [8,9], or, along with ideas of volume reduction [10,11,12,13,14], in the obtention of the SU(∞) running coupling on a single-site lattice, through the use of step scaling procedures based on rescaling the rank of the gauge group instead of the size of the lattice [15], among other interesting computations [16,17,18,19].…”

The twisted gradient flow coupling at one loop

“…One of its most common applications has been, in combination with finite-size scaling techniques, the determination of the non-perturbative scale dependence of the gauge coupling constant. Examples of the usefulness of this approach range from precise, non-perturbative determinations of the QCD coupling constant and Λ parameter [4][5][6][7][8] to the study of Yang-Mills theories with near conformal behavior [9][10][11], or with large number of colors [12]. Several coupling renormalization schemes based on gradient flow techniques have been proposed to that end [3,[13][14][15][16].…”