Renormalization-Group Behavior of $\phi^3$ Theories in $d=6$ Dimensions

TL;DR

This paper explores the existence of nonzero coupling fixed points in six-dimensional $\,\phi^3$ theories using four-loop beta functions, finding limited evidence for such fixed points across different symmetry representations.

Contribution

It provides a detailed four-loop analysis of renormalization-group fixed points in various six-dimensional $\,\phi^3$ theories, including cases with different global symmetries.

Findings

No robust fixed points in scalar and fundamental representation theories.

Potential fixed point in bi-adjoint scalar theory with zero one-loop beta function.

Implications of the zero one-loop term in the bi-adjoint case are discussed.

Abstract

We investigate possible renormalization-group fixed points at nonzero coupling in $\phi^3$ theories in six spacetime dimensions, using beta functions calculated to the four-loop level. We analyze three theories of this type, with (a) a one-component scalar, (b) a scalar transforming as the fundamental representation of a global ${\rm SU}(N)$ symmetry group, and (c) a scalar transforming as a bi-adjoint representation of a global ${\rm SU}(N) \otimes {\rm SU}(N)$ symmetry. We do not find robust evidence for such fixed points in theories (a) or (b). Theory (c) has the special feature that the one-loop term in the beta function is zero; implications of this are discussed.