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  • Letter
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Scheme independent 3-sphere free energy is not a monotone F-function

Giacomo Santoni1,* and Francesco Scardino1,2,†

  • 1Physics Department, INFN Roma 1, Piazzale A. Moro 2, Roma, I-00185, Italy
  • 2Physics Department, Sapienza University, Piazzale A. Moro 2, Roma, I-00185, Italy

  • *Contact author: giacomo.santoni@alumni.sns.it
  • †Contact author: francesco.scardino@uniroma1.it
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Phys. Rev. D 113, L121701 – Published 4 June, 2026

DOI: https://doi.org/10.1103/bpyr-jzd8

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(g2) under any relevant scalar deformation of a three-dimensional Conformal Field Theory. However, an exact analysis of the free massive scalar on S3 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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References (15)

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Phys. Rev. D 113, L121701– Published 4 June, 2026
  • Received 11 March 2026
  • Accepted 12 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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