$ε$-Expansion in Critical $φ^3$-Theory on Real Projective Space from Conformal Field Theory

TL;DR

This work extends the ε-expansion approach to the non-unitary critical φ^3 theory at d=6−ε by analyzing conformal field theory data on real projective space RP^d. By enforcing conformal symmetry and the equations of motion, the authors analytically determine the leading behavior of the one-point function ⟨φ⟩ and the anomalous dimension γ_φ, showing consistency with conventional perturbation theory and numerical truncated conformal bootstrap results. They derive g^2 = -(2·4^3π^3/3)ε + O(ε^2) and γ_φ = -ε/18 + O(ε^2), and obtain a perturbative constraint relating C_{φφ}^{φ}A_φ to ε. The results demonstrate that critical data on RP^d can be fixed by CFT consistency, suggesting extensions to other critical theories and deeper exploration of RP^d/CFT data.

Abstract

We use a compatibility between the conformal symmetry and the equations of motion to solve the one-point function in the critical $φ^3$-theory (a.k.a the critical Lee-Yang model) on the $d = 6 - ε$ dimensional real projective space to the first non-trivial order in the $ε$-expansion. It reproduces the conventional perturbation theory and agrees with the numerical conformal bootstrap result.