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  • Open Access

Sphere free energy of vector models to order 1/N

Ludo Fraser-Taliente*

  • Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, OX1 3PU, United Kingdom

  • *Contact author: ludovic.fraser-taliente@physics.ox.ac.uk
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Phys. Rev. D 113, 125024 – Published 17 June, 2026

DOI: https://doi.org/10.1103/2981-lgcg

Abstract

We calculate the large-N expansion of the sphere free energy F=−logZSd of the O(N)ϕ4 and the Gross-Neveu (ψ¯ψ)2 conformal field theories (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 ε expansion, resolving a puzzle raised by Tarnopolsky. These F’s can be written compactly in terms of the anomalous dimensions, for both the short- and 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 F˜=−sinπd2F is maximized as a function of the vector’s scaling dimension.

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Phys. Rev. D 113, 125024– Published 17 June, 2026
  • Received 26 January 2026
  • Accepted 27 April 2026
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