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The sphere free energy of the vector models to order $1/N$

Ludo Fraser-Taliente

TL;DR

This work computes the sphere free energy $F=-\log Z_{S^d}$ for the $O(N)$ vector $\phi^4$ and Gross-Neveu CFTs at large $N$ up to $\mathcal{O}(1/N)$, spanning both long-range and short-range versions. A key methodological advance is the analytic regularization scheme DREG$+\delta$, where the UV scaling of the auxiliary field is shifted by $\delta$ via a modified kinetic term, and counterterms are tuned to preserve finiteness, enabling precise matching to the $\varepsilon$-expansion. The authors obtain compact, dimension-dependent expressions for $F$ in terms of the field scaling dimensions and specific diagrams (tetrahedron and prism), resolving prior discrepancies and revealing structural simplicities across LR/SR and bosonic/fermionic cases. They show that SR fixed points correspond to extrema of the LR free-energy function, provide explicit $F$ formulas for several models (including Box$^k$ CFTs and SUSY variants), and demonstrate that anomalous dimensions naturally appear in $F$ through vacuum–self-energy relations of the $\sigma$ field. The results also include a neat derivation of the free energy for a free conformal field and a discussion of the UV completion of the LR theory, with implications for $\tilde{F}$-extremization and crossovers between LR and SR theories. Overall, the paper deepens the understanding of universal sphere free energies in large-$N$ CFTs and clarifies regularization subtleties that align with known perturbative results.

Abstract

We calculate the large-$N$ expansion of the sphere free energy $F=-\log Z_{S^d}$ of the O(N) $φ^4$ and the Gross-Neveu $(\barψ ψ)^2$ 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 in arXiv:1609.09113. These $F$s can be written compactly in terms of the anomalous dimensions, for both the short-range and the 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 $\tilde{F} =-\sin \tfrac{πd}{2} F$ is maximized as a function of the vector's scaling dimension.

The sphere free energy of the vector models to order $1/N$

TL;DR

This work computes the sphere free energy for the vector and Gross-Neveu CFTs at large up to , spanning both long-range and short-range versions. A key methodological advance is the analytic regularization scheme DREG, where the UV scaling of the auxiliary field is shifted by via a modified kinetic term, and counterterms are tuned to preserve finiteness, enabling precise matching to the -expansion. The authors obtain compact, dimension-dependent expressions for in terms of the field scaling dimensions and specific diagrams (tetrahedron and prism), resolving prior discrepancies and revealing structural simplicities across LR/SR and bosonic/fermionic cases. They show that SR fixed points correspond to extrema of the LR free-energy function, provide explicit formulas for several models (including Box CFTs and SUSY variants), and demonstrate that anomalous dimensions naturally appear in through vacuum–self-energy relations of the field. The results also include a neat derivation of the free energy for a free conformal field and a discussion of the UV completion of the LR theory, with implications for -extremization and crossovers between LR and SR theories. Overall, the paper deepens the understanding of universal sphere free energies in large- CFTs and clarifies regularization subtleties that align with known perturbative results.

Abstract

We calculate the large- expansion of the sphere free energy of the O(N) and the Gross-Neveu CFTs to order . 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 -expansion, resolving a puzzle raised by Tarnopolsky in arXiv:1609.09113. These s can be written compactly in terms of the anomalous dimensions, for both the short-range and the 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 is maximized as a function of the vector's scaling dimension.

Paper Structure

This paper contains 61 sections, 238 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Paper structure
  3. The large-N expansion and the epsilon expansion
  4. Which CFTs?
  5. The shortest summary of our results
  6. Proper regularization
  7. Discussion
  8. The sphere free energies
  9. O(N) vector model
  10. Long-range O(N)
  11. Short-range O(N)
  12. Gross-Neveu model
  13. Long-range GN
  14. Short-range GN
  15. Supersymmetric model
  16. ...and 46 more sections

Figures (2)

  • Figure 1: A schematic of our IR duality in theory space: there are two different QFTs that flow to the $\mathrm{O}(N)$ vector CFT. The standard UV completion of $N$ free scalars is on the left. The WF CFT found by perturbing this theory by $(\phi_i \phi_i)^2$, \ref{['eq:ONaction']}, can be solved for finite $N$ by using DREG, working perturbatively in $\epsilon=4-d$. In the large-$N$ limit, DREG no longer suffices: we need a QFT \ref{['eq:completeAction']} with a new regulator $\delta$ that improves the UV behaviour -- this is shown on the right. Now $\sigma \phi_i \phi_i$ is the perturbing relevant operator, and we can solve the IR CFT by working perturbatively in $1/N$. Computationally, we can solve for the $\mathrm{O}(N)$ theory by perturbing about either of these theories because in each case the ""*length of the flow (i.e. the size of the distance between the UV and IR CFTs, as measured by their conformal data) is proportional to that small parameter $\epsilon$ or $1/N$. We ignore the subtlety that in some $d$s the roles of free UV and interacting IR might be reversed Fei:2014yja.
  • Figure 2: A schematic of the long-range free energy $\tilde{F}^\mathrm{LR}(s)$ for the $\mathrm{O}(N)$ model. We have taken odd $k$, and assume $d>2k$, meaning that $\tilde{F}$ is maximized to leading order in $N$; this is the situation for the standard $\phi^4$$\mathrm{O}(N)$ CFT ($k=1$). The free CFT of $N$ scalars (black line) has $\tilde{F}=N\tilde{F}_b(\tfrac{d-s}{2})$, which is maximal at $s=2k$, marked with a box. The WF interacting CFT (blue line) has a maximum at $s=s_\mathrm{SR}=2k-2\hat{\gamma}_{\phi,1}/N + O(1/N^2)$, marked with a box. Both maxima are the locations of the short-range CFTs, and we have chosen a value of $d$ such that the free local theory has a larger $\tilde{F}$. Unless counterterms are tuned, for $s> s_\mathrm{SR}$ we flow to the short-range model on perturbing the free scalars by $\phi^4$, and so that regime of the interacting model is dashed.