Higher-loop calculations of the ultraviolet to infrared evolution of a vectorial gauge theory in the limitNc→∞,Nf→∞withNf/Ncfixed

Abstract

We consider an asymptotically free vectorial $\mathrm{SU}({N}_{c})$ gauge theory with ${N}_{f}$ fermions in the fundamental representation and analyze higher-loop contributions to the evolution of the theory from the ultraviolet to the infrared in the limit where ${N}_{c}\ensuremath{\rightarrow}\ensuremath{\infty}$ and ${N}_{f}\ensuremath{\rightarrow}\ensuremath{\infty}$ with $r={N}_{f}/{N}_{c}$ a fixed, finite constant. We focus on the case where the $n$-loop beta function has an infrared zero, at $\ensuremath{\xi}={\ensuremath{\xi}}_{\mathrm{IR},n\ensuremath{\ell}}$, where $\ensuremath{\xi}=\ensuremath{\alpha}{N}_{c}$. We give results on ${\ensuremath{\xi}}_{\mathrm{IR},n\ensuremath{\ell}}$, the anomalous dimension of the fermion bilinear evaluated at ${\ensuremath{\xi}}_{\mathrm{IR},n\ensuremath{\ell}}$, denoted ${\ensuremath{\gamma}}_{\mathrm{IR},n\ensuremath{\ell}}$, and certain structural properties of the beta function, ${\ensuremath{\beta}}_{\ensuremath{\xi}}$. The approach to this limit is investigated, and it is shown that the leading correction terms are strongly suppressed, by the factor $1/{N}_{c}^{2}$. This provides an understanding of a type of approximate universality in calculations for moderate values of ${N}_{c}$ and ${N}_{f}$, namely, that ${\ensuremath{\alpha}}_{\mathrm{IR},n\ensuremath{\ell}}{N}_{c}$, ${\ensuremath{\gamma}}_{\mathrm{IR},n\ensuremath{\ell}}$, and structural properties of the beta function are similar in theories with different values of ${N}_{c}$ and ${N}_{f}$ provided that they have similar values of ${N}_{f}/{N}_{c}$. We give results up to four loops for nonsupersymmetric theories and up to three loops for supersymmetric theories.