The Running Coupling from SU(3) Potentials

A non-perturbative finite-size scaling technique is used to study the evolution of the running coupling (in a certain adapted scheme) in the SU(3) Yang-Mills theory. At low energies contact is made with the fundamental dynamical scales, such as the string tension K, while at larger energies the coupling is shown to evolve according to perturbation theory. In that regime the coupling in the MS-bar scheme of dimensional regularization is obtained with an estimated total error o...

I discuss some calculations of the running coupling in SU($N$) gauge theories from lattice simulations, centering on the work of the UKQCD collaboration. This talk is introductory in nature; full details have been published elsewhere.

Interpreting the way that the SU(3) bare lattice coupling runs with the lattice spacing is complicated by the fact that there is a smooth cross-over region in which the strong coupling expansion transforms into a weak-coupling one. For N > 4, however, there is a first order bulk transition that cleanly separates the strong and weak coupling regimes. We find that in this case the calculated string tension can be readily fitted throughout the weak coupling region by a standard ...

The running of the coupling is studied in SU(4) gauge theory using the Schr\"odinger functional technique. Up to energies of the order of the square root of the string tension $\sigma$, the running is found to agree with the two-loop perturbative formula. Relating the perturbative to the non-perturbative regime of the running and converting to the $\msbar$ scheme allows one to extract the ratio $\Lambda_{\msbar}/\sqrt{\sigma}$. The result is then used in combination with simi...

We present a simple parameterization of a running coupling constant, defined via the static potential, that interpolates between 2-loop QCD in the UV and the string prediction in the IR. Besides the usual $\Lam$-parameter and the string tension, the coupling depends on one dimensionless parameter, determining how fast the crossover from UV to IR behavior occurs (in principle we know how to take into account any number of loops by adding more parameters). Using a new Ansatz fo...

A recent lattice calculation of the QCD running coupling is presented. The coupling is extracted from the force between two static quarks in the framework of the valence quark approximation. A value of the Lambda-parameter for zero quark flavours is determined: Lambda_msbar=0.630(38)\sqrt{\sigma} =293(18)^{+25}_{-63} MeV. The first error is statistical, the second stems from the overall scale uncertainty in the string tension sigma. Combining this value with results from full...

Nonperturbative determinations of the renormalization group $\beta$ function are essential to connect lattice results to perturbative predictions of strongly coupled gauge theories and to determine the $\Lambda$ parameter or the strong coupling constant. The continuous $\beta$ function is very well suited for this task because it is applicable both in the weakly coupled deconfined regime as well as the strongly coupled confined regime. Here we report on our results for the $\...

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. ...

Using continuum extrapolated lattice data we trace a family of running couplings in three-flavour QCD over a large range of scales from about 4 to 128 GeV. The scale is set by the finite space time volume so that recursive finite size techniques can be applied, and Schr\"odinger functional (SF) boundary conditions enable direct simulations in the chiral limit. Compared to earlier studies we have improved on both statistical and systematic errors. Using the SF coupling to impl...

For N>4 there is a first order bulk transition that cleanly separates the strong and weak coupling regimes of SU(N) lattice gauge theories with the plaquette action. We find that in this case the calculated string tension can be readily fitted throughout the weak coupling region by a standard 3-loop perturbative expression modified by lattice spacing corrections of the expected form. While our fits demand the presence of the latter, they are not constraining enough to tell us...