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Heavy-Heavy-Light Asymptotics from Thermal Correlators

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Pith's one-line read Thermal one-point functions on a circle times a sphere determine the large-dimension averages of heavy-heavy-light OPE coefficients in three-dimensional CFTs.

desk verdict A genuinely new method for HHL OPE asymptotics, well tested for a≤2, with the a≤4 claim resting on an observed pattern rather than a proof. read the letter →

arxiv 2506.21671 v1 pith:LKCUJUI6 submitted 2025-06-26 hep-th cond-mat.stat-mechmath-phmath.MP

Ilija Burić , Francesco Mangialardi , Francesco Russo , Volker Schomerus , Alessandro Vichi This is my paper
classification hep-thcond-mat.stat-mechmath-phmath.MP
keywords conformalfieldtheorythermalone-pointfunctionheavy-heavy-lightOPEcoefficientsblocksspectraldensitythree-dimensionalCFThigh-temperatureexpansiontensorstructures
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper shows that in three-dimensional CFTs that are gapped on the thermal circle, the large-dimension behavior of averaged heavy-heavy-light OPE coefficients is controlled by the high-temperature expansion of thermal one-point functions on $S^1 \times S^2$. The central result is an asymptotic formula in which each tensor structure of the three-point function is governed by a single dynamical coefficient, with a hierarchy that makes higher tensor structures increasingly suppressed as $\Delta \to \infty$. The same machinery yields a next-to-leading correction to the spectral density of primary operators. Checks in the free scalar theory agree closely with exact coefficients and improve on earlier leading-order estimates by up to three orders of magnitude even at moderate $\Delta$.

What carries the argument

The central object is a new inversion formula for thermal one-point functions on $S^1 \times S^2$, which expresses the product of the spectral density and the averaged OPE coefficient as an integral of a shadow thermal block against $Z(\beta,\mu)\langle \phi \rangle$. The formula rests on an orthonormal basis of one-point conformal blocks built from Jacobi polynomials, in which the blocks diagonalize the natural inner product. The blocks satisfy Casimir differential equations, and their large-$\Delta$ expansion has a simple leading term $F^{(0),a} = f^{\ell,a}_0(u,s)/((1-q)^3 - (1-q)qu)$. The inversion integral is then evaluated by saddle point in the variables $\beta_L,\beta_R$, using the high-temperature expansions of the partition function and the one-point function; a selection rule from Jacobi orthogonality ensures that only the single coefficient $b_{0,2a,a}$ survives at leading order.

What would settle it

Take a gapped 3D CFT, compute its thermal one-point function at high temperature with a nonzero rotation chemical potential to the order $\Omega^{2a} s^a$, extract $b_{0,2a,a}$, and compare with the large-$\Delta$ averaged OPE coefficient $\lambda^a_{\phi OO}$; formula (4.26) fails if the ratio $\lambda^a_{\phi OO} / [b_{0,2a,a} (\Delta/8\pi f)^{\Delta_\phi/3} N_{a,\ell} \Delta^{-2a}]$ does not tend to 1. A sharper falsifier is to find any CFT whose high-temperature one-point function contains an odd power of $\Omega$ or a non-polynomial dependence on $s$, since that violates the explicit assumption behind the derivation.

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Extended reading notes

Core claim

For a gapped three-dimensional CFT, the averaged OPE coefficient $\lambda^a_{\phi OO}(\Delta, \ell)$ for a light scalar $\phi$ and two identical heavy operators of dimension $\Delta$ and spin $\ell$ behaves as $b_{0,2a,a} (\Delta/(8\pi f))^{\Delta_\phi/3} N_{a,\ell} \Delta^{-2a} (1 + O(\Delta^{-1/3}))$ for tensor structures $a \le 4$, where $f$ is the free-energy density, $N_{a,\ell}$ is a kinematical normalization, and $b_{0,2a,a}$ is a coefficient in the high-temperature expansion of the one-point function. Consequently $\lambda^a/\lambda^b \to 0$ for $a > b$, so the tensor-structure label $a$ controls the large-$\Delta$ suppression. The paper also derives a next-to-leading correction to the density of primaries that depends only on the same thermal data as the leading term. In the free scalar theory, all-order high-temperature expansions for the partition function and $\langle \phi^2 \rangle$ produce asymptotic OPE coefficients that match exact data closely and far outperform previous leading-order estimates.

Load-bearing premise

The asymptotic formula rests on the assumption that the high-temperature one-point function of the light operator expands in only even powers of the angular chemical potential, with coefficients that are polynomials in the position variable $s$ of bounded order; a CFT with odd terms or non-polynomial dependence would not be captured by the single coefficient $b_{0,2a,a}$, and the hierarchy could fail.

Editorial extensions

If this is right

  • In any gapped 3D CFT whose thermal one-point function admits the assumed high-temperature expansion, the leading averaged HHL OPE coefficient for each tensor structure is fixed by one dynamical coefficient per structure.
  • The hierarchy $\lambda^a/\lambda^b \to 0$ for $a > b$ explains why the $a=0$ tensor structure dominates at large $\Delta$ and why subdominant structures are increasingly suppressed.
  • The spectral density of primaries receives a next-to-leading correction of relative order $\Delta^{-1/3}$ that depends only on the same EFT parameters as the leading term, not on new dynamical data.
  • Subleading corrections in $\Delta^{-1/3}$ can be computed systematically; in the free scalar theory including them reduces the error in the spectral density at $\Delta = 210$ from about 40 percent to about 0.06 percent.
  • The methods apply to any theory where the high-temperature partition function and one-point function are available, including generalized free theories, vector models on $S^1 \times S^2$, and holographic CFTs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: if the tensor-structure hierarchy extends beyond $a \le 4$, then for sufficiently large $\Delta$ only the dominant $a=0$ structure contributes to thermal one-point functions, making the leading HHL data effectively spin-independent.
  • Inference: the inversion formula could be run in reverse: given high-precision OPE coefficients, one could reconstruct the high-temperature coefficients $b_{0,2a,a}$, offering a new route to extract thermal effective data from spectra.
  • Inference: in the free theory the extra $\log \Delta$ term in the OPE asymptotics is tied to the $\log \beta/\beta$ divergence of $\langle \phi^2 \rangle$; for weakly coupled or nearly free theories, the relative weight of $\log \Delta$ and $\Delta^{-1/3}$ terms could serve as a diagnostic of closeness to the free limit.
  • Inference: extending the external operator to nonzero spin would allow the same Casimir-driven inversion to constrain stress-tensor and conserved-current OPE coefficients, which are not fixed by Ward identities; the paper notes this extension should be straightforward.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper revisits the extraction of CFT data from thermal one-point functions on S1×S2. It constructs an orthonormal basis of one-point thermal blocks (2.33), derives an inversion formula (2.44) from the Casimir symmetry, and develops both low-temperature recursion relations and a systematic large-Δ expansion for the blocks. On this basis the authors derive an asymptotic formula for the density of primaries (4.16), with a new next-to-leading correction, and an asymptotic formula (4.26) for averaged heavy-heavy-light OPE coefficients, including a hierarchy in the tensor-structure label a. The general results are tested extensively in the free scalar theory, where all-order high-temperature expansions of Z and ⟨ϕ2⟩ are obtained with the help of Zagier's Mellin-transformation method. The reported agreement includes a 0.06% accuracy of the seven-term spectral-density asymptotics at Δ=210 and several subleading orders for OPE coefficients.

Significance. If correct, Eq. (4.26) is a genuinely new universal statement: it determines the large-Δ, fixed-ℓ behaviour of HHL OPE coefficients separately for each tensor structure and explains the dominance hierarchy observed numerically in [44]. The inversion formula and the recursive block construction are methodological contributions likely to be reused. The paper deserves credit for shipping reproducible code and for validating analytic predictions against exact free-field data with explicit error quotes. The main caveat is that the derivation of the a≤4 result relies on an observed, not proven, block-expansion structure, and the universality claim is conditional on an assumed high-temperature form of the one-point function.

major comments (2)
  1. [§4.2, Eqs. (4.20), (4.26); Appendix C, Eqs. (C.2), (C.4), Table 1] The central formula (4.26) for a ≤ 4 is derived from a 1/Δ block expansion whose structure is only observed, not proven. Eq. (4.20) is introduced with the words 'inspecting these terms for the cases a ≤ 4, n ≤ 7, we observe', and Eq. (C.2) is explicitly called an 'ansatz based on observations up to a = 4'. The subsequent cancellation analysis, including the statement that only the single pair {f^(0,0), b^(2a)} survives the θ-integral, is therefore an empirical pattern. This is load-bearing because (4.26) and the hierarchy (4.28) are claimed for a ≤ 4, while the validation of the asymptotic formula does not cover a = 3,4: Appendix D.2.2 gives asymptotic coefficients only up to a = 2, and Figure 8, while showing exact data with a ≤ 4, plots the asymptotic curve only for a = 1. Table 1 also lists terms only up to n = 5, although (4.20) is used for n ≤ 7. Please either prove, or at least justify to all orders in n,k, the structure (4.20)/(C.2) and the single-pair cancellation, or restrict the stated domain of validity of (4.26)/(4.28) to the checked cases; if the unrestricted claim is retained, add free-field comparisons for a = 3 and a = 4.
  2. [§4.2, Eqs. (4.23)–(4.24); Abstract; §6] Equation (4.26) is conditional on the assumed high-temperature form of the one-point function, Eq. (4.23)-(4.24), with only even powers of Ω and polynomial coefficients b_{i,2j}(s) of degree j. The manuscript states this as an assumption, but the abstract and Section 6 describe the results as valid for generic CFTs ('any conformal field theory'). If a CFT has odd-Ω terms in ⟨ϕ⟩ (e.g., a parity-violating theory without the symmetry that forces even Ω), then the single coefficient b_{0,2a,a} would not control the leading OPE contribution and the hierarchy (4.28) could fail. The scope should be stated explicitly as gapped CFTs satisfying (4.23)-(4.24), or the assumption should be derived from the block decomposition and the assumed convergence of the high-temperature expansion.
minor comments (3)
  1. [§3.1, Eq. (3.6)] There is a typo in the displayed value of A(1,1,0): the first expression has (2Δ - Δφ) / (2Δ) while the second has (2Δ - Δφ^2) / (2Δ); the first should be Δφ^2.
  2. [General prose] Small language issues: '40% form the numerical value' should be 'from' in Section 1; 'concerened' should be 'concerned' in Section 5; and 'T able' appears in the Table 1 caption.
  3. [§4.2 vs. Appendix D.2.2] Eq. (4.30) and Eq. (D.12) use different prefactors, (Δ/(8π f))^{1/3} and (Δ/(π f))^{1/3}, with the same f. The two series are consistent only after the constants in I^{⟨ϕ2⟩} are adjusted; a one-sentence remark making this conversion explicit would help readers reproduce the plotted curves.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the HHL asymptotic formula is derived from an independent inversion formula and high-temperature data, not from the OPE coefficients it predicts.

full rationale

The derivation chain is self-contained against circularity. The inversion formula (2.44) follows from orthogonality of thermal blocks, which is established via Jacobi-polynomial orthogonality and the shadow transform, not assumed from the target OPE data. The central asymptotic (4.26) is obtained by saddle-point evaluation of this inversion formula using the large-Delta block expansion from the Casimir equation and the high-temperature expansions of Z and <phi>. The dynamical coefficient b_{0,2a,a} is an independent input coming from the assumed high-temperature expansion (4.23)-(4.24) of the one-point function, not a parameter fitted to the OPE coefficients being predicted. In the free-field validation, exact low-temperature block decompositions are compared with all-order high-temperature asymptotic expansions of the same explicit two-point/one-point expressions; no fitted quantity is relabeled as a prediction. Self-citations to [44] provide conventions, prior numerical observations, and starting points, but the load-bearing steps (Casimir recursion, block orthogonality, inversion, large-Delta expansion, saddle point) are re-derived here or draw on external references such as [24] and [46]. The only fragile element is the observed, unproven 1/Delta block structure (4.20) and the ansatz (C.2), which is explicitly labelled as based on observations up to a=4; this is an inductive verification gap and a correctness risk, not a reduction of the result to its own inputs. No equation is defined in terms of the quantity it purports to derive, and no fitted parameter is renamed as a prediction. Therefore no significant circularity is present.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The derivation rests on the assumed high-temperature forms of the partition function and one-point function, and on an observed structural ansatz for the 1/Delta expansion of blocks. The undetermined coefficients b_{0,2a,a} and the higher b's are external inputs; the paper's contribution is the derivation of the asymptotic formulas in terms of these inputs.

free parameters (2)
  • b_{0,2a,a} (a=0,...,4) = not determined for general CFT; computed in free scalar theory (e.g. eq. 5.25 and App. D.2.2)
    Coefficients in the assumed high-temperature expansion of the one-point function (4.23)-(4.24). They set the scale of each tensor-structure OPE coefficient in (4.26). The paper does not derive them from first principles.
  • higher coefficients b_{i,2j,k} of (4.24) = not determined
    Control subleading O(Delta^{-1/3}) and higher corrections to (4.26) and (4.16); they are inputs from the thermal one-point function expansion.
assumptions (4)
  • domain assumption High-temperature partition function has the form Z = A(beta,Omega) exp(4 pi f / (beta^2 (1+Omega^2))) with A admitting an expansion in non-negative powers of beta and Omega^2 (eqs. 4.3-4.5).
    Used to evaluate the spectral density inversion integral via saddle point; for gapped theories log A = -8 pi c1 - (32 pi c2 Omega^2)/(3(1+Omega^2)) + O(beta^2).
  • domain assumption The thermal one-point function at high temperature expands as eq. (4.23), with only even powers of Omega and coefficients b_{i,2j}(s) that are polynomials in s of order j (eq. 4.24).
    Motivated by the block expansion property (3.11), but assumed for general CFTs. It determines which coefficient b_{0,2a,a} controls the leading OPE coefficient.
  • ad hoc to paper The 1/Delta expansion of thermal one-point blocks has the structure (4.20), with exponents involving min(floor(n/3), a), and the selection rule (C.4).
    Observed for a <= 4, n <= 7 by direct inspection and summarized in Appendix C; not proven in general. Used to derive the scaling Delta^{-2a} in (4.26).
  • standard math Thermal one-point blocks are orthonormal under the inner product (2.15) with the shadow transform, eq. (2.22).
    Central for the inversion formula (2.44). Established at q -> 0 by Jacobi polynomial orthogonality with a contour shift; Appendix A notes the harmonic-analysis derivation does not cover the full Lorentzian continuation.

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Pith. "Pith review of Heavy-Heavy-Light Asymptotics from Thermal Correlators." pith.science (2026). https://pith.science/paper/LKCUJUI6

@misc{pith2026250621671,
  author       = {Pith},
  title        = {Pith review of: Heavy-Heavy-Light Asymptotics from Thermal Correlators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LKCUJUI6}},
  note         = {Machine review of arXiv:2506.21671}
}
abstract

We revisit the calculation of spectral densities and heavy-heavy-light (HHL) operator product expansion (OPE) coefficients in three-dimensional conformal field theories using thermal one-point functions on $S^1 \times S^2$. A central element of our analysis is a new inversion formula for one-point functions which is derived via Casimir differential equations. We develop systematic expansions of the spectral density and HHL OPE coefficients in the regime of large $\Delta_H$. We validate our analytic tools by comparing the results with the partial wave expansions of thermal one-point functions in free field theories. The algorithms developed for these expansions make full use of Casimir recursion relations, thereby extending their applicability into the heavy exchange regime. In the end, we observe excellent agreement with our analytic predictions and an improvement of up to three orders of magnitude compared to all previous leading order estimates of the CFT data even for moderate values of $\Delta_H$.

Figures

Figures reproduced from arXiv: 2506.21671 by the authors.

Figure 1
Figure 1. Contributions to the recursion relations of the type A(n1+y,n2+x,·) . We have only displayed a few of the summands that appear. In total there are 44 terms involving shifts of the indices ni that can go to values as high as 6. By definition, in order for A(n1,n2,n3) to be the coefficients of a thermal block, the sum in the previous equation has to vanish which implies that A(n1,n2,n3) = −2 [PITH_FULL_IMAGE:figures/… view at source ↗
Figure 2
Figure 2. The order in which we proceed to solve the recursion relations. Each red dots stands for a linear recursion in which we determine the n3 dependence at fixed n1 and n2. In the recursion we start with the largest value n3 = n2 and go down until we reach n3 = 0. Remarks 1) In Section 5, we compute blocks with generic parameters (∆ϕ, ∆). This is slower that the calculation of blocks for fixed values of (∆ϕ, ∆). But it t… view at source ↗
Figure 3
Figure 3. Log of the ratio of exact multiplicities and the asymptotic density of scalars expanded to different orders. The notation NnL here stands for n terms in the asymptotic series after the leading order. On the right, a zoomed version of the same plot shows the difference of the asymptotics with five or seven terms in the regime ∆ > 100. 50 100 150 200 0.0 0.1 0.2 0.3 0.4 100 120 140 160 180 200 0.000 0.005 0.010 0.015 … view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p031_4.png]
Figure 5
Figure 5. Figure 5: Averaged OPE coefficients for dominant and subdominant tensor structures with internal spin ℓ = 2. Here, ψq(z) denotes the q-digamma function. In the second expression, we performed the expansion around β = 0. As mentioned before, the leading divergence is more singula…
Figure 6
Figure 6. Figure 6: Scalar exchange dominant OPE coefficients against asymptotic formulas with a varying number of terms: leading (L), next-to-leading (NL), etc. The darker the asymptotic curve, the more subleading terms were considered. To begin with, [PITH_FULL_IMAGE:figures/full_fig_p…
Figure 7
Figure 7. Figure 7: Dominant OPE coefficients in different spin sectors against the asymptotic formula with four terms. Different sectors are indicated by different colours, with blue corresponding to spin zero, orange to spin one and green to spin two. The curve represents the fourth ord…
Figure 8
Figure 8. Figure 8: Averaged OPE coefficients for some (a ≤ 4) of the subdominant tensor struc￾tures, for internal spin six. The other subdominant OPE coefficients with a = 5, 6, 7 are too small to be visualised properly, and have been therefore excluded from the plot. The curve is the as…

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