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We investigate possible renormalization-group fixed points at nonzerocoupling in $\phi^3$ theories in six spacetime dimensions, using beta functionscalculated to the four-loop level. We analyze three theories of this type, with(a) a one-component scalar, (b) a scalar transforming as the fundamentalrepresentation of a global ${\rm SU}(N)$ symmetry group, and (c) a scalartransforming 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 intheories (a) or (b). Theory (c) has the special feature that the one-loop termin the beta function is zero; implications of this are discussed.

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