The scheme independent 3-sphere free energy is not a monotone F-function

TL;DR

The paper investigates a scheme-independent quantity derived from the 3-sphere partition function in 3D QFTs, revealing it is not a monotone F-function along RG flows despite conformal invariance at fixed points.

Contribution

It demonstrates that the scheme-independent sphere free energy is not a monotone F-function along the entire RG flow, challenging previous assumptions.

Findings

Conformal perturbation theory shows local decrease at second order under relevant deformations.

Exact analysis of a free massive scalar reveals non-monotonic behavior along RG flow.

Obstruction identified in the differential operator used to remove local ambiguities.

Abstract

We study the natural scheme-independent quantity obtained from the three-sphere partition function of a $(2+1)$-dimensional quantum field theory by removing all local counterterm ambiguities. At conformal fixed points this quantity equals the standard $F$-theorem invariant. Conformal perturbation theory shows that it locally decreases at $O(g^2)$ under any relevant scalar deformation of a three-dimensional CFT. However, an exact analysis of the free massive scalar on $S^3$ shows that this sphere-free-energy interpolant is not monotone along the full renormalization-group flow: it dips below its infrared value and then returns to it. Thus the natural counterterm-subtracted quantity built from sphere thermodynamics is not, by itself, a monotone $F$-function. We trace the obstruction to the second-order differential operator required to eliminate the local ambiguities.