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Closed-form thermal threshold functions for the proper-time renormalisation group

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Pith's one-line read This paper derives closed-form thermal threshold functions for the proper-time renormalisation group, replacing numerical Matsubara evaluation with rapidly convergent Bessel winding-number series and reducing every higher threshold to a…

desk verdict The closed-form Bessel threshold functions and the cross-family identity are real, but the finite-temperature anomalous dimension (Eq. 44) is asserted without derivation and the LPA' scan rests on it. read the letter →

arxiv 2608.09803 v1 pith:UE5NDBBE submitted 2026-08-10 hep-th

Daniele Rizzo This is my paper · ORCID
classification hep-th
keywords proper-timerenormalisationgroupthermalthresholdfunctionsMatsubarasummationPoissonresummationmodifiedBesselfunctionalO(N)scalartheoryfinite-temperaturefixedpoints
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 aims to prove that every thermal threshold function of the proper-time renormalisation group, for the standard one-parameter regulator family, has a closed analytic form as a rapidly convergent series of modified Bessel functions, and that all higher threshold functions reduce to a single such form by an algebraic identity. If true, finite-temperature renormalisation-group computations for O(N)-symmetric scalar theories no longer need numerical evaluation mode by mode: the flow equations and their fixed points become analytic objects that can be scanned continuously across the regulator parameter. The paper builds the local-potential approximation and its refined version on these closed forms, recovers known results in every limiting case, and extends the zero-temperature anomalous-dimension construction to finite temperature, finding the refined fixed point smooth and bounded along the entire regulator line.

What carries the argument

The machinery is the modular inversion of the $\theta$ function that appears when each Matsubara term is written as a Mellin-Laplace transform: $\sum_{n\in\mathbb{Z}} e^{-4\pi^2\tau^2 t n^2} = \frac{1}{2\tau\sqrt{\pi t}} \sum_{\ell\in\mathbb{Z}} e^{-\ell^2/(4\tau^2 t)}$. This trades the slowly convergent sum over Matsubara modes for an exponentially convergent sum over thermal windings, each carrying a modified Bessel function $K_{m-2}(\ell\sqrt{1+w}/\tau)$. The central algebraic identity is established at the integrand level: differentiating the proper-time kernel $u^m e^{-u}/\Gamma(m)$ with respect to $w$ simply raises the kernel parameter $m$, reproducing the same kernel at parameter $m+n$ up to a binomial factor. The proper-time kernel $F(u;m) = 2u^m e^{-u}/\Gamma(m)$ is the named central object, and the closed forms (12) and (22) are what carry every subsequent flow calculation.

What would settle it

Evaluate the right-hand side of the Matsubara representation (11) by direct numerical summation at a non-special value of the regulator parameter, say m=3.7, w=0.5, tau=0.5, and compare with the Bessel closed form (12) to high precision; any disagreement beyond summation tolerance would contradict the central claim. Separately, compute the finite-temperature anomalous dimension by a direct derivative expansion of the proper-time two-point function without assuming uniform Z_k, and see whether the LPA' fixed point (48)-(50) shifts beyond the regulator spread.

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Extended reading notes

Core claim

The central claim is that the thermal threshold function $L_0^{(m)}(w;\tau)$ has the closed form $L_0^{(m)}(w;\tau) = \frac{1}{16\pi^2\Gamma(m)}\left[\frac{\Gamma(m-2)}{(1+w)^{m-2}} + 4\sum_{\ell\ge1}\left(\frac{\ell}{2\tau\sqrt{1+w}}\right)^{m-2} K_{m-2}\left(\frac{\ell\sqrt{1+w}}{\tau}\right)\right]$, obtained by Poisson-resumming the Matsubara sum into a sum over windings around the thermal circle. The companion identity $L_n^{(m)}(w;\tau) = \binom{m+n-1}{n} L_0^{(m+n)}(w;\tau)$ follows already from the kernel integrand and makes every derivative-level threshold a shifted copy of the same basic function. For the sharp proper-time regulator, the $m\to\infty$ endpoint, the closed form factorises exactly into $e^{-w}/(16\pi^2)$ times a thermal $\theta$ function, with the field and temperature dependences separating. At $m=5/2$ the threshold coincides identically with the exact-flow coth form at all temperatures, not only in the high-temperature or zero-temperature limits. The author presents these results as the complete analytic infrastructure of the finite-temperature PTRG at LPA and LPA'.

Load-bearing premise

The load-bearing premise is that the finite-temperature anomalous-dimension formula (44) remains valid when the kinetic term is taken uniform in the field and when the finite-temperature difference between spatial and temporal wave-function renormalisation is neglected; the paper states this as an untested limitation.

Editorial extensions

If this is right

  • Every LPA and LPA' computation at finite temperature in the O(N) scalar theory can be evaluated with exponentially convergent Bessel sums instead of numerical Matsubara sums, at cost independent of the number of modes.
  • The sharp regulator endpoint factorises, so the field and temperature dependence are separately exact and the scheme has exponential rather than algebraic decoupling.
  • Because of the identity (22), one implementation of $L_0$ at real $m$ evaluates every threshold function of every family member, and derivative-expansion coefficients inherit the same convergence.
  • The LPA' fixed point can be tracked as a continuous function of the regulator parameter; the paper finds both the anomalous dimension and the correlation-length exponent smooth and bounded, with the exponent varying by about 1.3% while the anomalous dimension varies by about 30% across the family.
  • At $m=5/2$ the proper-time PTRG is identically, not just asymptotically, the thermal exact flow with the optimised regulator, so the two schemes coincide exactly at this point for all temperatures.

Reading between the lines

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

  • The same modular-resummation route suggests closed forms for any regulator kernel of Laplace type, since the only property used is that the kernel enters the integrand as an exponential; the paper does not explore broader kernel classes.
  • A natural test is to compute the finite-temperature anomalous dimension without the uniform-$Z_k$ assumption and ask whether the smoothness of the regulator scan survives; the paper flags this as its main open limitation.
  • Because the closed forms make regulator scans effectively free, they could be used to quantify scheme dependence of first-order-transition observables, including the non-convex region whose scheme dependence the paper explicitly notes.
  • The integrand-level proof of the cross-family identity hints that derivative-expansion coefficients in broader proper-time constructions may share the same shifted-parameter structure, not just the threshold functions computed here.
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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 / 4 minor

Summary. The paper develops closed-form expressions for the thermal threshold functions of the finite-temperature proper-time renormalisation group (PTRG). Starting from the Matsubara representation of L_0^{(m)}, it applies Poisson resummation and Mellin-Barnes techniques to obtain a winding-number series of modified Bessel functions (Eq. 12), proves the cross-family identity L_n^{(m)} = binom(m+n-1,n) L_0^{(m+n)} (Eq. 22), derives the m→∞ sharp-kernel endpoint in factorised form (Eq. 15), and checks these results against the zero-temperature limit, dimensional reduction, heavy-mode decoupling, and one-loop thermal perturbation theory. It then assembles LPA and LPA' flows for the O(N) scalar theory, states a finite-temperature anomalous-dimension formula (Eq. 44), and uses it to scan the LPA' quartic fixed point continuously in the regulator parameter m (Figs. 3 and 4).

Significance. The threshold-function calculus is a genuine technical advance if the main derivation is accepted. Appendix A is self-contained and complete: the Gaussian momentum integral, Mellin-Barnes representation, Poisson inversion, and Bessel integral are all exhibited, and Eq. (22) is proved at the level of the kernel integrand before any integration. The closed forms contain no fitted constants; the only free parameter is the regulator label m. The exact finite-τ equality with the Wetterich threshold at m=5/2, the sharp-regulator factorisation, and the frozen-curvature one-loop completeness check (Eq. 34) are strong internal consistency tests. The finite-temperature anomalous dimension and the continuous regulator scan, by contrast, are not yet on the same footing: Eq. (44) is asserted rather than derived, and the subsequent fixed-point results inherit that gap.

major comments (2)
  1. [§4.2, Eq. (44)] Equation (44), the finite-temperature anomalous dimension, is introduced with the phrase 'Expanding ... gives' and no derivation is shown. This is the load-bearing new ingredient of the LPA' section: the quadratic fixed-point equation (48), the eigenvalue formulas (49)-(50), and the continuous regulator scan in Figs. 3 and 4 all presuppose it. At finite temperature the O(p^2) projection of the proper-time heat-kernel trace is not automatically the T=0 formula with τ-dependent L_3: the discrete p_0 direction, the Z_k^{-3/2} prefactor produced by the Gaussian spatial-momentum integral when Z_k ≠ 1, and the O(4)-breaking split Z_spatial ≠ Z_temporal can all generate additional contributions. The paper's own Sec. 5 limitation statement concedes that the split is untested. Please provide the projection calculation from Eqs. (2) and (43), keeping all η-dependent and Z-dependent terms, or explicitly label Eq. (44) and the fixed-point results that depend on it as conjectural.
  2. [§4.2, Eqs. (46)-(50)] The claim that the LPA' fixed point varies smoothly with 'no special or pathological point anywhere on the line' is a property of the specific two-coupling, uniform-Z_k truncated system, not of the PTRG itself. Conditional on Eq. (44) and the high-temperature power laws (41), the algebra from (46) to (50) is internally consistent, but if Eq. (44) is modified by O(4)-splitting or Z-dependent terms, the cancellation that produces the closed quadratic (48) will not necessarily survive. The discussion should therefore separate the robust threshold-function calculus from the truncation-dependent fixed-point scan, and the comparison with the Ising values in Sec. 4.2 should be presented as an assessment of the two-coupling truncation, not as a regulator-dependence result of the full theory.
minor comments (4)
  1. [§3.1, Eq. (16)] The claim that the m→∞ limit at fixed physical scales approaches the sharp-kernel threshold with O(1/m) corrections is stated without proof for finite τ; because the limit is taken after the ℓ-sum in Eq. (12), please provide the uniform large-order Bessel asymptotics or a suitable reference.
  2. [§3.1, Eq. (13)] As printed, the Poisson inversion factor (2τ√π t)^{-1} is the standard one; a one-line derivation or an explicit normalisation convention would remove any ambiguity.
  3. [§3.1, notation] The symbol τ denotes both the imaginary-time coordinate in Eq. (9) and the dimensionless temperature T/k from Eq. (10) onward; the re-use is flagged in the text, but a distinct symbol for one of the two quantities would improve readability.
  4. [§5] The phrases 'complete analytic infrastructure' and 'complete map of the regulator dependence' overstate the status of the LPA' results given the unresolved status of Eq. (44); I suggest softening these summary statements.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the closed-form threshold derivation is self-contained, and the finite-temperature anomalous-dimension formula is an asserted extension rather than a circular step.

full rationale

The central results, Eqs. (12) and (22), are derived directly from the defining proper-time flow (2) by explicit Gaussian momentum integration, Mellin–Barnes representation, Poisson resummation of the Matsubara sum, and a standard Bessel integral; the cross-family identity (22) is proven from the linearity of the defining integral at the level of the integrand, Eq. (25), and is not assumed. No parameter is fitted to any target quantity: m is a scheme label, and the fixed-point results (47)–(50) are solved algebraically from the derived threshold functions rather than adjusted to benchmarks. The comparisons against the exact Wetterich flow, one-loop thermal perturbation theory, and known T=0 thresholds are genuinely external checks, not inputs. The only in-scope caveat is Eq. (44), the finite-temperature anomalous dimension, which is introduced by analogy to the T=0 construction of [22] rather than re-derived in the paper; this is a derivation-gap/correctness concern, not a case of a prediction being equivalent to its inputs by construction or by self-citation. Likewise, the Section 5 limitation about Z_spatial ≠ Z_temporal is an acknowledged boundary on the LPA′ application, not a circular step. Because the paper's central derivation reduces neither to a fitted quantity nor to a load-bearing self-citation chain, the appropriate finding is no significant circularity.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The closed-form threshold derivation introduces no fitted parameters: m labels the regulator family but is not adjusted to data. The physical framework is standard thermal field theory plus the PTRG approximation. The finite-temperature anomalous dimension relies on an unproven extension of a known T=0 construction, which is the main ledger item beyond standard assumptions.

free parameters (1)
  • m = not fitted; scanned over (2, infinity)
    Regulator shape parameter of the proper-time kernel family. The central closed forms hold for arbitrary m>2; the fixed-point scan treats m as an external scheme knob, not as a fitted constant.
assumptions (6)
  • standard math Poisson summation formula (Jacobi theta modular inversion), Eq. (13)
    Used in App. A to resum the Matsubara sum into a winding-number series; a standard identity from analytic number theory and thermal field theory.
  • standard math Mellin-Barnes integral representation (53) and tabulated Bessel integral (55)
    Core steps in the closed-form derivation of (12), with domains of validity stated (m>3/2 for the Mellin form, m>2 for the final integral).
  • domain assumption Thermal field theory on R^3 x S^1 with Matsubara frequencies omega_n = 2 pi n T
    Defines the finite-temperature setup in Eq. (1) and (9); standard framework inherited from the literature.
  • domain assumption PTRG flow equation (2) as the defining approximation
    The paper's subject; not an exact ERG flow, and the author explicitly notes beyond-one-loop differences. All threshold statements are statements about this defined scheme.
  • domain assumption Uniform-Z_k LPA' truncation ignoring field dependence and O(4)-breaking Z_spatial != Z_temporal
    Needed for the eta formula (44) and the LPA' fixed-point scan; the paper itself lists this as a limitation in Sec. 5.
  • ad hoc to paper The T=0 heat-kernel construction of eta [22] extends to finite T by inserting tau-dependence through L_3, with no additional eta-dependent or O(4)-splitting terms
    The formula (44) is asserted by analogy to [22] without a derivation in the present text; this is the weakest load-bearing premise in the LPA' part.

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Pith. "Pith review of Closed-form thermal threshold functions for the proper-time renormalisation group." pith.science (2026). https://pith.science/paper/UE5NDBBE

@misc{pith2026260809803,
  author       = {Pith},
  title        = {Pith review of: Closed-form thermal threshold functions for the proper-time renormalisation group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UE5NDBBE}},
  note         = {Machine review of arXiv:2608.09803}
}
abstract

We derive closed-form expressions for the thermal threshold functions of the finite-temperature proper-time renormalisation group (PTRG): until now these have been evaluated numerically, Matsubara mode by Matsubara mode. For the standard one-parameter regulator family, Poisson resummation of the Matsubara sum yields a rapidly convergent winding-number series of modified Bessel functions, and a single algebraic identity reduces every higher threshold function to this same closed form at a shifted kernel parameter. The sharp proper-time regulator, recovered as the exact $m\to\infty$ endpoint of the family with a controlled $O(1/m)$ approach, factorises into a field-dependent and a purely thermal piece. Built on them, the local potential approximation (LPA) and its refinement to a running anomalous dimension (LPA$'$) for the $O(N)$-symmetric theory reduce to established results, all cross-checked against the exact Wetterich equation with the optimised regulator. Two findings go beyond reduction. First, the known zero-temperature anomalous-dimension construction is extended here to finite temperature. Second, because the closed form holds for any real regulator parameter, the refined truncation's fixed point can for the first time be tracked as a continuous function of the regulator, rather than at a handful of isolated points, and is found to vary smoothly and remain bounded, with no special or pathological point anywhere on the line.

Figures

Figures reproduced from arXiv: 2608.09803 by the authors.

Figure 1
Figure 1. Thermal threshold functions at vanishing curvature, normalised to their [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Higher thermal threshold functions Ln(0; τ ), n = 1, 2, 3, 4, normalised to their T = 0 values, 1/(32π 2 ) for every n (m = 3, solid blue), 1/(16π 2n!) (sharp, dashed orange) and Γ(n + 1 2 )/(12π 2n! Γ( 1 2 )) (ERG, dot-dashed green). As in [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. The η ≡ 0 quartic fixed point’s correlation-length exponent ν(m) (47), as a continuous function of the regulator parameter m (solid blue), together with the exact Wetterich flow’s known closed-form value ν = 1/2 [13] (dot-dashed green). The two curves cross exactly at m = 5/2 (marker), the point where the proper-time and Wetterich threshold functions coincide identically. The Wilson–Fisher fixed point. Substituting … view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The LPA′ quartic fixed point, η ∗ (left) and ν (right), as a continuous function of the regulator parameter m, evaluated from the closed form (48)–(49) at each m and plotted out to m ∼ 103 , where the curve has visibly converged to the sharp-kernel value. Reference lin…

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