Renormalization-Group Behavior of ${\varphi }^3$ Theories in $d=6$ Dimensions

Abstract

We investigate possible renormalization-group fixed points at nonzero coupling in ${\varphi }^3$ 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 $\mathrm{SU}(N)$ symmetry group, and (c) a scalar transforming as a bi-adjoint representation of a global $\mathrm{SU}(N)\otimes \mathrm{SU}(N)$ 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.

9 pages

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