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arXiv · 2603.09799

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

Giacomo Santoni·Francesco Scardino
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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.

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Giacomo Santoni, Francesco Scardino. 2026-03-10. The scheme independent 3-sphere free energy is not a monotone F-function. https://arxiv.org/abs/2603.09799

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