Leading CFT constraints on multi-critical models in d>2

We consider the family of renormalizable scalar QFTs with self-interacting potentials of highest monomial ϕ^m below their upper critical dimensions d_c=((2m)/(m−2)), and study them using a combination of CFT constraints, Schwinger-Dyson equation and the free theory behavior at the upper critical dimension. For even integers m≥4 these theories coincide with the Landau-Ginzburg description of multi-critical phenomena and interpolate with the unitary minimal models in d=2, while for odd m the theories are non-unitary and start at m=3 with the Lee-Yang universality class. For all the even potentials and for the Lee-Yang universality class, we show how the assumption of conformal invariance is enough to compute the scaling dimensions of the local operators ϕ^k and of some families of structure constants in either the coupling's or the ϵ-expansion. For all other odd potentials we express some scaling dimensions and structure constants in the coupling's expansion.