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Large-$N$ $\pi^+\pi^+$ scattering from twisted Eguchi--Kawai reduction

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Claudio Bonanno, Margarita García Pérez, Antonio González-Arroyo, Ken-Ichi Ishikawa, Masanori Okawa, Fernando Romero-López

High Energy Physics - Lattice · Submitted 26 Oct 2026 · 134 references

Pith's verdict: First TEK extraction of the π⁺π⁺ scattering length at large N — clean, well-controlled, and the main soft spot is openly handled.

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arXiv ID2610.35176 (version 1)
Pith ID4FUY7R7V, Pith's permanent record of this paper and its review
CitationAntonio Gonz\'alez-Arroyo, Claudio Bonanno, Fernando Romero-L\'opez, Ken-Ichi Ishikawa, Margarita Garc\'ia P\'erez, Masanori Okawa. "Large-$N$ $\pi^+\pi^+$ scattering from twisted Eguchi--Kawai reduction." arXiv:2610.35176 (2026). https://pith.science/paper/4FUY7R7V
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The paper

The paper claims

Using the Twisted Eguchi–Kawai reduction, the paper computes the large-N π⁺π⁺ scattering length and extracts the low-energy constant ℓππ/N=0.345(21), consistent with S-matrix bootstrap bounds.

The reading

The paper aims to demonstrate that scattering amplitudes can be computed directly in large-N QCD using the twisted Eguchi–Kawai (TEK) reduction. By computing the π⁺π⁺ scattering length for values of N up to 1156 and performing controlled continuum, chiral, and large-N limits, they extract the large-N limit of the low-energy constant ℓππ/N. This quantity governs the subleading quark-mass dependence of the scattering length in chiral perturbation theory. The result is consistent with bounds from the S-matrix bootstrap program, and when combined with previous finite-N lattice data, it yields the full 1/N expansion of ℓππ. The work establishes TEK as a viable tool for hadronic scattering observables in the 't Hooft limit.

What carries the argument

The central object is the rescaled energy shift ∆ε = N ℓ³ ∆E of the two-pion system, extracted from the linear growth of the ratio of correlation functions R₂(τ) = ⟨C(τ)⟩/⟨D(τ)⟩² in Euclidean time. The ratio avoids large-N and infinite-volume divergences. The Lüscher formula relates ∆ε to the scattering length. The TEK reduction maps color degrees of freedom to spatial extent ℓ ∝ √N, so the large-N limit simultaneously takes the infinite-volume limit. The calculation uses stochastic estimators for fermion traces and relies on previous TEK determinations of the pion decay constant and the rho meson mass.

Extended reading notes

Core claim

The paper provides the first extraction of a scattering observable from the twisted Eguchi–Kawai reduction of large-N QCD. The central numerical result is the large-N low-energy constant ℓππ/N = 0.345(21), obtained from the chiral behavior of the π⁺π⁺ scattering length. Combining this with prior finite-N lattice results gives the 1/N expansion ℓππ/N = 0.345(21) − (1/N)[N_f 1.43(50) + 3.6(1.4)] + O(1/N²). The chiral limit of the scattering length matches the leading-order chiral perturbation theory prediction, and the coupling ℓππ is consistent with recent S-matrix bootstrap bounds.

Load-bearing premise

The extrapolation to infinite N relies on the assumption that leading finite-N corrections in the TEK model scale as 1/$N^{{3/2}}$, based on an analogy with finite-volume corrections (since ℓ ∝ √N). The data cannot distinguish this from a pure 1/$N^{{3/2}}$ correction without the volume factor, and for the finest lattice spacings only N=841 is available, making the correction model-dependent.

Editorial extensions

If this is right

  • The methodology extends to other scattering channels, such as isospin-0 ππ scattering, where the fate of the σ resonance at large N can be studied.
  • The extracted ℓππ/N provides a direct lattice input for the S-matrix bootstrap program and for effective field theory analyses at large N.
  • The full 1/N expansion of ℓππ (including the N_f-dependent term) constrains the subleading structure of the chiral Lagrangian beyond the leading order.
  • The success of TEK for scattering opens the door to calculating other dynamical quantities, such as ρπ scattering, trilinear couplings, and form factors in the large-N limit.

Reading between the lines

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

  • The TEK reduction may be applied to three-meson amplitudes and resonant scattering, where finite-N effects are more subtle, providing a new window into large-N meson dynamics.
  • The good agreement with bootstrap bounds suggests that the bootstrap inequalities are already tight at large N, potentially allowing the extraction of low-energy constants from bootstrap alone.
  • The inability to discriminate between Lüscher-type and pure 1/N^{3/2} corrections implies that future simulations at larger N (e.g., N > 2000) are needed to settle the nature of finite-N effects in TEK for scattering.
Original abstract

We present the first computation of the $\pi^+\pi^+$ scattering length in large-$N$ QCD in the 't Hooft limit, $N_{\scriptscriptstyle{\rm f}}/N\to 0$, using the twisted Eguchi--Kawai formulation of the lattice theory. We employ values of $N$ up to 1156 and consider several quark masses and lattice spacings to achieve controlled large-$N$, continuum, and chiral limits. The chiral behavior of the scattering length is compared with next-to-leading-order Chiral Perturbation Theory predictions. Our result for the low-energy coupling describing the sub-leading quark-mass dependence is consistent with existing S-matrix bootstrap bounds. Combining TEK results with previous lattice QCD data obtained using standard methods for $3\le N\le 6$, we are also able to extract sub-leading $\mathcal{O}(1/N)$ corrections to the large-$N$ scattering length.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-27: 1 request, 4,430 input tokens.

What the abstract itself states, as probabilities: claims first 0.97 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.11 · randomized assignment 0.01 · states quantitative result 0.22 · compares with prior results 0.97 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We present the first computation of the $\pi^+\pi^+$ scattering length in large-$N$ QCD in the 't Hooft limit, $N_{\scriptscriptstyle{\rm f}}/N\to 0$, using the twisted Eguchi--Kawai formulation of the lattice theory. We employ values of $N$ up to 1156 and consider several quark masses and lattice spacings to achieve controlled large-$N$, continuum, and chiral limits. The chiral behavior of the scattering length is compared with next-to-leading-order Chiral Perturbation Theory predictions. Our result for the low-energy coupling describing the sub-leading quark-mass dependence is consistent with existing S-matrix bootstrap bounds. Combining TEK results with previous lattice QCD data obtained using standard methods for $3\le N\le 6$, we are also able to extract sub-leading $\mathcal{O}(1/N)$ corrections to the large-$N$ scattering length.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We present the first computation of the $\pi^+\pi^+$ scattering length in large-$N$ QCD in the 't Hooft limit, $N_{\scriptscriptstyle{\rm f}}/N\to 0$, using the twisted Eguchi--Kawai formulation of the lattice theory. contribution 0.99 0.97 0.34 asserted 1.00
s1 We employ values of $N$ up to 1156 and consider several quark masses and lattice spacings to achieve controlled large-$N$, continuum, and chiral limits. method 0.99 0.96 0.04 asserted 1.00
s2 The chiral behavior of the scattering length is compared with next-to-leading-order Chiral Perturbation Theory predictions. contribution 0.38 0.84 0.07 asserted 0.91
s3 Our result for the low-energy coupling describing the sub-leading quark-mass dependence is consistent with existing S-matrix bootstrap bounds. result 0.70 0.89 0.14 hedged possibility 0.86
s4 Combining TEK results with previous lattice QCD data obtained using standard methods for $3\le N\le 6$, we are also able to extract sub-leading $\mathcal{O}(1/N)$ corrections to the large-$N$ scattering length. result 0.73 0.94 0.13 asserted 0.96

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Direct contraction D(τ ) and crossed contraction C(τ ) contributing to the π + and π +π + correlators in Eqs. (3.8) and (3.9). Figure adapted from Ref. [39]. Note that chiral logs do not appear in these expressions, as they are sub-leading in the 1/N expansion, and that all the ¯ℓi denote scheme- and scale-independent quantities. Since F 2 ∼ O(N), one recovers a0 ∼ O(1/N) from Eq. (3.4), in agreement with general theoretical expectations about large-N QCD. Moreover, the large-N scaling of F also implies that the expansion parameter ξ ∼ O(1/N), and since ¯ℓi = 1,2,3,4 ∼ O(N), the…
Figure 2. Example of energy shift calculation from the slope of the linear large-time behavior of R2(τ ). The linear fit to extract the slope is always performed in the range τmin/a = 9 and τmax/a = √ N − 6 (vertical dashed lines), and errors are estimated via a binned jack-knife analysis. In all cases, this interval includes the range where the pion effective mass computed from ⟨D(τ )⟩ exhibits a plateau. This effective mass plateau is compared with the result obtained for mπ from the more extensive GEVP analysis performed in [112] (shown as a horizontal shaded band). end to compare with…
Figure 3. Top panels: Large-N extrapolations of the rescaled energy shift ∆ε assuming linear corrections in 1/N3/2 (inspired by Lüscher formula). Obtained best fits have reduced chi-squared ranging from 0.48 to 1.37, corresponding to p-values ranging from 25% to 70%. Bottom panels: pion mass behavior of the pre-factor C of the observed 1/N3/2 finite-N dependence according to two different models. These best fits have a reduced chi-squared of, respectively, 1.59 (left) and 1.39 (right), corresponding to p-values of 13% and 20%. The extraction of the scattering length from the energy shift…
Figure 4. Top panel: pion-mass dependence of the scattering length in the continuum limit. Bottom panel: lattice artifacts affecting the chiral limit of the scattering length. The shaded band represents in both cases the LO χPT prediction for N a0σ/mπ, expected to hold in the continuum and chiral limit. The chiral-continuum fit has a reduced chi-squared of 1.32 (p-value ∼ 23.5%). As it can be seen, both limits are under control: lattice artifacts in the chiral limit amount to a ∼ 15% effect at most (estimated as the relative difference between the values at the coarsest lattice spacing and in…
Figure 5. Study of possible systematic effects affecting our determination of ℓππ/N. The slope term −A1Nξ thus allows to determine the combination of LECs: A1 = ℓππ − ¯ℓ3/2 N = 0.357(18). (5.7) In order to extract ℓππ, one needs a determination of ¯ℓ3, parameterizing the sub-leading quark-mass dependence of the pion mass at large N. In the previous TEK study [112], m2 π is observed to be perfectly described by a linear behavior in the whole explored range of quark masses, thus pointing out that ¯ℓ3 is much smaller than our current error bars. This is consistent with previous large-N lattice…
Figure 6. Calculation of the 1/N expansion of the combination of LECs given by ℓππ−ℓ¯3/2−2ℓ¯4 N , obtained combining the large-N TEK determination with the finite-N results of [39]. Let us conclude by commenting that, using directly the string tension to set the scale (without passing through the gradient flow scale t1) would lead to the perfectly compatible results: A0 = 1.310(82) and A1 = 0.300(27) (differing by about 1 standard deviation with the ones obtained via t1). 5.3 The 1/N expansion of ℓππ The goal of this section is to combine the large-N TEK result for ℓππ/N obtained in this stud
Figure 7. Bounds on g˜2 and g˜ ′ 2 obtained in [48, 51], compared with the constraint provided by the determination of ℓππ/N obtained in this study, cf. Eq. (5.16). convert our result in the same notation adopted in [48, 51].3 In particular, let us introduce the following quantities: g˜2 = 1 48π 2
Figure 8. Further examples of energy shift calculations. – 26 –
Figure 9. Chiral behavior of the squared pion mass as a function of the renormalized quark mass in the continuum limit. The shaded line represents the best fit to Eq. (C.1), minus lattice artifact terms, which have also been subtracted from the data. C Large-N estimation of ¯ℓ3/N Using the data reported here and in Ref. [112] for the pion mass as a function of b and κ, the results for the critical hopping κc(b) and for the scalar renormalization constant ZS(b), one is able to determine m2 π as a function of the renormalized quark mass amR = Z −1 S  1 2κ − 1 2κc  , with κc the critical…
Figure 10. Estimation of the 1/N expansion of the NLO LEC ¯ℓ3/N, obtained combining the large￾N TEK determination of this quantity with the Nf = 2 finite-N results of [71]. N ¯ℓ3/N SU(2) χPT ¯ℓ3/N U(2) χPT 3 1.07(50) 4 0.90(33) -0.74(24) 5 0.58(26) -0.81(23) ∞ -0.022(22)

Figures reproduced from arXiv:2610.35176 by the authors.

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