On critical exponents without Feynman diagrams

TL;DR

This paper revisits Polyakov's unitarity-based, diagram-free method to compute anomalous dimensions at the Wilson–Fisher fixed point in $4-\epsilon$ dimensions, extending the approach to $O(\epsilon^2)$ and validating it against known diagrammatic results for $O(n)$ models. By enforcing consistency between the algebraic OPE data and unitarity-inspired momentum-space amplitudes, the authors derive leading and subleading anomalous dimensions for external scalars $\phi_i$ and scalar composites $\phi^2\phi_i$, including mixed correlators, and demonstrate how large-spin data emerges from bootstrap considerations. The work highlights a complementary analytical route to the conformal bootstrap, capable of yielding subleading corrections without Feynman diagrams and offering insights into the role of channel mixing and higher-scalar exchanges. Its findings strengthen the bridge between Polyakov’s program and modern bootstrap techniques, with potential extensions to other dimensions and nonperturbative regimes via Mellin-space formulations.

Abstract

In order to achieve a better analytic handle on the modern conformal bootstrap program, we re-examine and extend the pioneering 1974 work of Polyakov's, which was based on consistency between the operator product expansion and unitarity. As in the bootstrap approach, this method does not depend on evaluating Feynman diagrams. We show how this approach can be used to compute the anomalous dimensions of certain operators in the $O(n)$ model at the Wilson-Fisher fixed point in $4-ε$ dimensions up to $O(ε^2)$.

Figures (1)

• Figure 1: A generic time-ordered four point function in momentum space. The left hand side represents the discontinuity in the four point function which can be written as a product of the three point functions with the imaginary part of the exchange propagator.