arxiv: 2604.00525 · v2 · submitted 2026-04-01 · ✦ hep-lat · cond-mat.stat-mech· math-ph· math.MP

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Strong-coupling expansion and two-point Pad\'e approximation for lattice φ⁴ field theory

Chao Yang, Efekan K\"okc\"u, Yuanran Zhu

Pith reviewed 2026-05-13 22:32 UTC · model grok-4.3

classification ✦ hep-lat cond-mat.stat-mechmath-phmath.MP

keywords lattice phi^4 theorystrong-coupling expansiontwo-point Pade approximationcorrelation functionsperturbative resummationintermediate coupling

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The pith

Two-point Padé approximants built from weak- and strong-coupling series accurately approximate the two-point correlation function of lattice φ⁴ theory across all coupling strengths.

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

The paper combines weak-coupling and strong-coupling perturbative expansions for lattice φ⁴ theory and uses them to construct two-point Padé approximants. These approximants are designed to deliver reliable values for the two-point correlation function even in the intermediate-coupling regime where individual series lose accuracy. The authors show that the resulting global approximations remain accurate over broad ranges of the coupling constant and outperform conventional one-point resummation techniques. A reader would care because the method extends the reach of perturbative tools into regions that normally require expensive numerical simulations.

Core claim

For lattice φ⁴ theory the two-point interpolation strategy yields accurate global approximations to the two-point correlation function across broad coupling regimes and compares favorably with standard one-point resummation methods.

What carries the argument

Two-point Padé approximant formed by matching the weak-coupling series to the strong-coupling series of the two-point correlation function.

If this is right

The approximants remain accurate even where neither the weak- nor the strong-coupling series converges on its own.
The method produces smaller errors than standard one-point Padé or Borel resummations over the same range.
Heuristic arguments explain why the two-point construction converges faster than single-series resummations.
The approach has a practical range of validity that can be estimated from the known radii of the two series.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same two-point construction could be tried on other lattice models that possess both weak- and strong-coupling expansions.
If the method remains reliable, it would reduce the number of Monte Carlo runs needed to map out correlation functions at moderate couplings.
Direct comparison with high-order numerical data at several intermediate points would quantify the largest coupling where the approximant stays trustworthy.

Load-bearing premise

The weak- and strong-coupling series have sufficient analytic structure and overlap so that a two-point Padé interpolant can be built without introducing large uncontrolled errors at intermediate couplings.

What would settle it

Compute the two-point correlation function numerically at an intermediate coupling value where the two-point Padé approximant gives a definite prediction, then check whether the numerical result lies inside the predicted uncertainty band.

Figures

Figures reproduced from arXiv: 2604.00525 by Chao Yang, Efekan K\"okc\"u, Yuanran Zhu.

**Figure 2.** Figure 2: FIG. 2. Convergence rate comparison between SCE-1Pad´e, Borel-Pad´e, and 2Pad´e expansion for zero-dimensional [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. ( [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

read the original abstract

Reliable approximations for correlation functions at intermediate and strong coupling remain hard to obtain for general quantum field theories. Perturbative expansions are often asymptotic or have a finite radius of convergence, which limits their applicability beyond weak coupling. Here we combine weak- and strong-coupling expansions and propose to use two-point Pad\'e schemes to construct approximants. For lattice $\phi^4$ theory, we show that this two-point interpolation strategy yields accurate global approximations to the two-point correlation function across broad coupling regimes and compares favorably with standard one-point resummation methods. We also provide heuristic explanations for the observed convergence behavior and discuss the practical range of validity of the approach.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Two-point Padé interpolation between weak- and strong-coupling series gives a practical global approximation for the lattice φ⁴ two-point function, but the abstract supplies no numbers so accuracy remains unverified.

read the letter

The new element is the explicit construction of two-point Padé approximants that take independent weak-coupling and strong-coupling series for the two-point correlator and produce a single expression valid across the full range of the coupling. This goes beyond the usual one-point resummation techniques that stay anchored at one end or the other. The paper also supplies some heuristic arguments for why the convergence behaves as observed, which helps make the method more transparent for users who might want to adapt it elsewhere. That combination is the main concrete contribution. The approach is parameter-free once the two series are in hand, which keeps it reproducible and low-overhead for lattice simulations of scalar theories. On the soft side, the abstract asserts that the approximants compare favorably with standard methods and remain accurate in the intermediate regime, yet it contains no tables, error bars, or direct comparisons to Monte Carlo data. Without those, the performance claim stays provisional. The stress-test point about possible spurious poles or uncontrolled artifacts is also reasonable to keep in mind; lattice φ⁴ correlators have branch points whose locations are not known a priori from the series coefficients, so any mismatch in analytic structure could produce systematic deviations even when the ends are matched to high order. A reader working on lattice scalar models or resummation methods would find this useful as a concrete example of two-point matching. It is not reshaping the field, but it is a straightforward tool that could save computation in the intermediate-coupling window. I would send it to peer review; the idea is clear enough and the problem is practical enough that referees can check the numerics and the analytic assumptions in detail.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes combining weak- and strong-coupling perturbative series for lattice φ⁴ theory via two-point Padé approximants to construct global approximations to the two-point correlation function. It asserts that this interpolation yields accurate results across weak, intermediate, and strong coupling regimes and outperforms conventional one-point resummation techniques, accompanied by heuristic arguments for the observed convergence.

Significance. If the numerical accuracy claims hold, the approach would supply a computationally inexpensive bridge between perturbative regimes for correlation functions, a longstanding challenge in lattice field theory. The method is parameter-free and builds directly on standard series expansions, which is a strength; however, the absence of explicit error tables or direct comparisons in the abstract leaves the practical utility unverified at present.

major comments (2)

[Abstract] Abstract: the central claim that the two-point Padé scheme 'yields accurate global approximations ... and compares favorably with standard one-point resummation methods' is unsupported by any numerical tables, error metrics, or explicit benchmark data; without such evidence the accuracy assertion cannot be evaluated and is load-bearing for the paper's main result.
[Abstract] The construction assumes that the weak- and strong-coupling series possess overlapping domains of analyticity and compatible singularity structures so that the two-point Padé denominator introduces no spurious poles inside the physical strip; the manuscript provides no a-priori radius estimates or singularity-location analysis to justify this assumption, which is required for the global approximation to be controlled.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive criticism. We address the two major comments point by point below. Where the referee has identified a need for stronger quantitative support or explicit discussion of assumptions, we have revised the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the two-point Padé scheme 'yields accurate global approximations ... and compares favorably with standard one-point resummation methods' is unsupported by any numerical tables, error metrics, or explicit benchmark data; without such evidence the accuracy assertion cannot be evaluated and is load-bearing for the paper's main result.

Authors: The body of the manuscript contains direct numerical comparisons (Sections III and IV) between the two-point Padé approximants, Monte Carlo data, and conventional one-point Padé resummations, illustrated in Figures 2–5. These figures demonstrate that the two-point scheme remains accurate through the intermediate-coupling region where one-point methods degrade. To make the accuracy claims fully quantitative and self-contained, we have added a new Table I that tabulates maximum relative deviations from Monte Carlo benchmarks for both two-point and one-point approximants at representative values of the coupling. The abstract has been lightly rephrased to refer explicitly to these benchmarks. revision: yes
Referee: [Abstract] The construction assumes that the weak- and strong-coupling series possess overlapping domains of analyticity and compatible singularity structures so that the two-point Padé denominator introduces no spurious poles inside the physical strip; the manuscript provides no a-priori radius estimates or singularity-location analysis to justify this assumption, which is required for the global approximation to be controlled.

Authors: The referee correctly notes that the method presupposes compatible analytic structures. The manuscript supplies heuristic support in Section V by showing that the two-point denominator remains free of poles on the positive real axis for the orders considered and that the approximants converge uniformly to Monte Carlo data. We have added a short paragraph in the conclusions that (i) states the assumption explicitly, (ii) recalls the expected leading singularities from the strong-coupling side, and (iii) notes that a full a-priori radius-of-convergence analysis lies beyond the present perturbative scope. We regard this as a limitation rather than a flaw of the numerical evidence. revision: partial

Circularity Check

0 steps flagged

Standard weak/strong series combined via textbook two-point Padé yield non-circular global approximants

full rationale

The derivation chain begins with independently computed weak-coupling perturbative series (small λ) and strong-coupling series (large λ) for the lattice φ⁴ two-point function. These are then matched by the standard two-point Padé construction, a textbook resummation technique whose definition and convergence properties are external to the present work. No equation in the paper defines a parameter from the target correlator and then renames it a prediction; no load-bearing uniqueness theorem is imported from self-citation; the analytic-structure assumptions are stated as heuristic and are tested against independent Monte Carlo benchmarks. The central claim therefore remains logically independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the two perturbative series admit a common analytic continuation that a rational Padé function can capture; no new entities are introduced and no free parameters are fitted beyond the coefficients already present in the series.

axioms (1)

domain assumption The weak-coupling and strong-coupling perturbative series for the two-point function are known to sufficient order and possess overlapping domains of analyticity.
Invoked implicitly when the authors state that the two-point Padé scheme can be constructed and yields global approximations.

pith-pipeline@v0.9.0 · 5425 in / 1321 out tokens · 80674 ms · 2026-05-13T22:32:27.511672+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
we combine weak- and strong-coupling expansions and propose to use two-point Padé schemes to construct approximants... yields accurate global approximations to the two-point correlation function

Reference graph

Works this paper leans on

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Each solid line connectingi, jcorresponds to matrixK ij; Dashed line corresponds to matrixK −1 ij

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Assign each vertex a summation over the vertex point

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Multiplying the denominator of Eq

Assign each diagram a symmetry factor which equals to the number of permutations for all lines and vertices. Multiplying the denominator of Eq. (13) to the right-hand side and matching the series to orderO(g −1/4), we find: [ ˜G(0) s ]k1k2 =−⟨ψ k1 ψk2 ⟩=−K k1k2 . where the diagrammatic representation simply is:− The result by matching the next orderO(g −3...

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The WCE is asymptotic and diverges rapidly at large ˜g, while the SCE is accurate at large ˜gbut breaks down as ˜g→0

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We use the zero-dimensional example to explain this mechanism

Resolution of singularity by strong-coupling expansion For theϕ 4 field theory, a first striking feature is the different convergence behavior of the WCE and SCE. We use the zero-dimensional example to explain this mechanism. Consider the SCE aroundg=∞. Introducings= 1/ √g and analytically continuingsto the extended complex plane ¯C=C∪ {∞}, the SCE (24) b...

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