The sphere free energy of the vector models to order $1/N$

TL;DR

This paper computes the large-N expansion of the sphere free energy for certain vector models, resolving regularization issues and connecting short-range and long-range conformal field theories.

Contribution

It provides a detailed calculation of the sphere free energy to order 1/N for O(N) models, addressing regularization ambiguities and establishing a link between different CFT regimes.

Findings

Matching results with epsilon-expansion resolves previous puzzles.

Sphere free energies expressed via anomalous dimensions.

Long-range CFT transitions to short-range CFT at a specific point.

Abstract

We calculate the large-$N$ expansion of the sphere free energy $F=-\log Z_{S^d}$ of the O(N) $\phi^4$ and the Gross-Neveu $(\bar{\psi} \psi)^2$ CFTs to order $1/N$. Analytic regularization of these theories requires consistently shifting the UV scaling dimension of the auxiliary field: this can only be done by modifying its kinetic term. This modification combines with the counterterms to give the result that matches the $\epsilon$-expansion, resolving a puzzle raised by Tarnopolsky in arXiv:1609.09113. These $F$s can be written compactly in terms of the anomalous dimensions, for both the short-range and the long-range versions of these CFTs. We also provide various technical results including a computation of the counterterms on the sphere and a neat derivation of the sphere free energy of a free conformal field. Finally, we observe that the long-range CFT becomes the short-range CFT at exactly the point where its $\tilde{F} =-\sin \tfrac{\pi d}{2} F$ is maximized as a function of the vector's scaling dimension.