$ϵ$-Expansions Near Three Dimensions from Conformal Field Theory

We formally extend the CFT techniques introduced in arXiv:1505.00963, to ${\varphi }^{\frac{2d_0}{d_0-2}}$ theory in $d=d_0-ϵ$ dimensions and use it to compute anomalous dimensions near $d_0=3,4$ in a unified manner. We also do a similar analysis of the $O(N)$ model in three dimensions by developing a recursive combinatorial approach for OPE contractions. Our results match precisely with low loop perturbative computations. Finally, using 3-point correlators in the CFT, we comment on why the ${\varphi }^3$ theory in $d_0=6$ is qualitatively different.