Three ways to solve critical $\phi^4$ theory on $4-\epsilon$ dimensional real projective space: perturbation, bootstrap, and Schwinger-Dyson equation

TL;DR

This paper computes the two-point function in critical $\,\phi^4$ theory on $4-\epsilon$ dimensional real projective space using perturbation, bootstrap, and Schwinger-Dyson methods, demonstrating their consistency and respective advantages.

Contribution

It introduces three complementary methods to analyze the critical $\,\phi^4$ theory on real projective space and shows their results agree.

Findings

All three methods yield consistent two-point functions.

Each method has unique advantages in analyzing the theory.

The approach advances understanding of conformal field theories on non-trivial manifolds.

Abstract

We solve the two-point function of the lowest dimensional scalar operator in the critical $\phi^4$ theory on $4-\epsilon$ dimensional real projective space in three different methods. The first is to use the conventional perturbation theory, and the second is to impose the crosscap bootstrap equation, and the third is to solve the Schwinger-Dyson equation under the assumption of conformal invariance. We find that the three methods lead to mutually consistent results but each has its own advantage.