$ε$-Expansions Near Three Dimensions from Conformal Field Theory
Pallab Basu, Chethan Krishnan
TL;DR
The authors formulate a conformal-field-theory based $\epsilon$-expansion for scalar theories with potential $\phi^{\frac{2d_0}{d_0-2}}$ in $d=d_0-\epsilon$, unifying the $d_0=3$ and $d_0=4$ cases and deriving anomalous dimensions via CFT axioms and OPE constraints. They extend the framework to the $O(N)$ model in three dimensions, introducing a recursive contraction-calculation method (cow-pies) to obtain $f$ and $\rho$ coefficients and then compute anomalous dimensions for both primary towers and mixed operators, finding results that agree with known perturbative loop outcomes. The approach highlights a universal structure for $d_0=3,4$ through a telescoping recursion that fixes leading anomalous dimensions such as $\gamma_{\phi}$ and $\gamma_{\phi^n}$, and extends to general $N$ with explicit formulas for several operator classes. The paper also clarifies why $d_0=6$ (the $\phi^3$ theory) cannot be treated in the same way due to a non-integer $r$ and qualitatively different OPE behavior, indicating a distinct $\epsilon$-expansion regime.
Abstract
We formally extend the CFT techniques introduced in arXiv:1505.00963, to $φ^{\frac{2d_0}{d_0-2}}$ 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 $φ^3$ theory in $d_0=6$ is qualitatively different.