I will show how the use of standard perturbative RG in dimensional regularization allows for a renormalization group based computation of both the spectrum and a family of coefficients of the operator product expansion (OPE) for the whole family of scalar multi-critical universality classes.

The task is greatly simplified by a straightforward generalization of perturbation theory to a functional perturbative RG approach. I illustrate the procedure in the e-expansion by obtaining the next-to-leading corrections for the spectrum and the leading corrections for the OPE coefficients of Ising and Lee-Yang universality classes and then give several results for the whole family of renormalizable multi-critical models.

Whenever comparison is possible our RG results explicitly match the ones recently derived using combination of CFT constraints, Schwinger-Dyson equation and the free theory behavior at the upper critical dimension.