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

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.

Figures (1)

• Figure 1: (a) Thermodynamic $F$-function $F_{\cal E}(x)$ for a free massive scalar on $S^3$ ($x=mR$). $F_{\cal E}$ starts at $F_{\rm UV}\approx 0.064$ (dashed line), overshoots below $F_{\rm IR}=0$, reaching a minimum at $x_\ast\approx 1.58$ (dot), then returns to $0^-$ from below. (b) Flow derivative $dF_{\cal E}/d\log R$. Negative for $x<x_\ast$, positive for $x>x_\ast$ (shaded regions). The dashed curve on the top right corner is the analytic asymptotic $\pi/(96\,x)$ in Eq. \ref{['eq:large_x']}, which controls the large-$x$ tail. The sign change demonstrates that $F_{\cal E}$ is not a monotone $F$-function.