Abstract
We perform the step-scaling investigation of the running coupling constant, using the gradient-flow scheme, in SU(3) gauge theory with twelve massless fermions in the fundamental representation. The Wilson plaquette gauge action and massless unimproved staggered fermions are used in the simulations. Our lattice data are prepared at high accuracy, such that the statistical error for the renormalised coupling, g_GF, is at the subpercentage level. To investigate the reliability of the continuum extrapolation, we employ two different lattice discretisations to obtain g_GF. For our simulation setting, the corresponding gauge-field averaging radius in the gradient flow has to be almost half of the lattice size, in order to have this extrapolation under control. We can determine the renormalisation group evolution of the coupling up to g^2_GF ~ 6, before the onset of the bulk phase structure. In this infrared regime, the running of the coupling is significantly slower than the two-loop perturbative prediction, although we cannot draw definite conclusion regarding possible infrared conformality of this theory. Furthermore, we comment on the issue regarding the continuum extrapolation near an infrared fixed point. In addition to adopting the fit ansatz a'la Symanzik for performing this task, we discuss a possible alternative procedure inspired by properties derived from low-energy scale invariance at strong coupling. Based on this procedure, we propose a finite-size scaling method for the renormalised coupling as a means to search for infrared fixed point. Using this method, it can be shown that the behaviour of the theory around g^2_GF ~ 6 is still not governed by possible infrared conformality.
Highlights
We present the details of our analysis and results in section 3. section 4 contains our comment on the continuum extrapolation, and the proposal of a finite-size scaling test of the renormalised coupling in the strong-coupling regime
We find that the systematic errors associate with the continuum extrapolation can be significant
We present our step-scaling analysis of the coupling constant in SU(3) gauge theory with 12 massless flavours, using the Gradient-Flow scheme
Summary
We make use of twisted boundary condition (TBC) [58], where the gauge field is periodic up to a gauge transformation (μ, ν = 1, 2, 3, 4 are the Lorentz indices). The different flavours (usually called “smells”) transform one in the other under translations of a full period of the torus according to ψαa (n + νLν /a) = eiπ/3Ωaνbψβb (n) (Ων )†βα ,. The factor eiπ/3 is introduced to lift the zero-momentum modes in these directions. In the weak-coupling regime, the power laws in the coupling in finite-volume perturbation theory are the same as those in the infinite-volume, continuum case [60,61,62] This boundary condition lifts the zero-momentum modes, making it possible to perform simulations at vanishing fermion mass
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