Six dimensional QCD at two loops

TL;DR

This paper constructs and renormalizes six-dimensional QCD at two loops, revealing a rich fixed point structure and universality classes, and compares it to lower-dimensional theories.

Contribution

It provides the first two-loop renormalization of six-dimensional QCD and explores its fixed points and universality classes, extending the understanding of gauge theories in higher dimensions.

Findings

Gauge coupling is asymptotically free for all Nf.

The conformal window for SU(3) is at Nf=16.

Six-dimensional QCD shares universality with lower-dimensional models.

Abstract

We construct the six dimensional Quantum Chromodynamics (QCD) Lagrangian in a linear covariant gauge and subsequently renormalize it at two loops in the MSbar scheme. The coupling constant corresponding to the gauge interaction is asymptotically free for all numbers of quark fields, Nf. Analysing the beta-functions yields a rich spectrum of fixed points. For instance, the conformal window in the six dimensional theory is at Nf = 16 for the SU(3) colour group. The critical theory structure is similar to that of an O(N) scalar theory in eight dimensions. Using the large N expansion the latter is shown to be in the same universality class as the Heisenberg ferromagnet. Similarly using the large Nf expansion, six dimensional QCD is shown to be in the same class as the two dimensional non-abelian Thirring model and four dimensional QCD. Abelian gauge theories are also renormalized at high loops in six and eight dimensions. It is shown that the gauge parameter only appears in the electron anomalous dimension at one loop, similar to four dimensions.