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) ⊗ 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.
Gracey, J. A., Ryttov, T. A., & Shrock, R. (2020). Renormalization-Group Behavior of φ3 Theories in d=6 Dimensions. Physical Review D. Particles and fields, 102(4), . https://doi.org/10.1103/PhysRevD.102.045016