Renormalization-group behavior of φ3 theories in d=6 dimensions

We investigate possible renormalization-group fixed points at nonzero coupling in φ3 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 SU(N) symmetry group, and (c) a scalar transforming as a bi-adjoint representation of a global SU(N)âŠ-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.