Supersymmetry Inspired QCD Beta Function

TL;DR

This work introduces an NSVZ-inspired all-orders beta function for nonsupersymmetric SU($N$) gauge theories with arbitrary fermion representations, establishing a bound on the conformal window and recovering the two-loop limit at weak coupling. It demonstrates precise matching to the supersymmetric NSVZ result in the adjoint case and analyzes infrared fixed points, comparing with ladder approximations. The authors compare running couplings to lattice data in pure Yang-Mills and derive a universal ratio for conformal regions that is independent of the matter representation. The framework is extended to multiple representations and linked to phenomenological contexts such as walking technicolor and unparticle physics, with broader implications for the structure of gauge theory phase diagrams.

Abstract

We propose an all orders beta function for ordinary Yang-Mills theories with or without fermions inspired by the Novikov-Shifman-Vainshtein-Zakharov beta function of N=1 supersymmetric gauge theories. The beta function allows us to bound the conformal window. When restricting to one adjoint Weyl fermion we show how the proposed beta function matches the one of supersymmetric Yang-Mills theory. The running of the pure Yang-Mills coupling is computed and the deviation from the two loop result is presented. We then compare the deviation with the one obtained from lattice data also with respect to the two loop running.

Figures (3)

• Figure 1: Phase diagram for nonsupersymmetric theories with fermions in the: i) fundamental representation (black), ii) two-index antisymmetric representation (blue), iii) two-index symmetric representation (red), iv) adjoint representation (green) as a function of the number of flavors and the number of colors. The shaded areas depict the corresponding conformal windows. Above the upper solid curve the theories are no longer asymptotically free. Between the upper and the lower solid curves the theories are expected to develop an infrared fixed point according to the NSVZ inspired beta function. The dashed curve represents the change of sign in the second coefficient of the beta function.
• Figure 2: Phase diagram for nonsupersymmetric theories with fermions in the: i) fundamental representation (black), ii) two-index antisymmetric representation (blue), iii) two-index symmetric representation (red), iv) adjoint representation (green) as a function of the number of flavors and the number of colors. The shaded areas depict the corresponding conformal windows. Above the upper solid curve the theories are no longer asymptotically free. In between the upper and the lower solid curves the theories are expected to develop an infrared fixed point according to the NSVZ inspired beta function. The area between the upper solid curve and the dashed curve corresponds to the conformal window obtained in the ladder approximation.
• Figure 3: The evolution of the gauge coupling squared times the number of colors (i.e. the 't Hooft coupling) as a function of the energy scale for two, three and four colors. The solid curve is obtained using the susy inspired beta function, the dashed is obtained via the two loop beta function while the dotted curve is the one loop result. The green dots (biggest errorbars) correspond to lattice data for $SU(2)$Luscher:1992zx, the blue dots to $SU(3)$Luscher:1993gh and the red dots (smallest errorbars) to $SU(4)$Lucini:2007sa.