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 ϕm below their upper critical dimensions dc=2mm−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 ϕk 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.