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High Energy Physics - Lattice · arXiv:hep-lat/0503025 v2 · submitted 18 Mar 2005 · 28 references

I=2 Pion Scattering Length from Two-Pion Wave Functions

CP-PACS Collaboration : S. Aoki, M. Fukugita, K-I. Ishikawa, N. Ishizuka, Y. Iwasaki, K. Kanaya, T. Kaneko, Y. Kuramashi

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M. Okawa, A. Ukawa, T. Yamazaki, T. Yoshié

PDF · arXiv · The abstract, sentence by sentence · Cite

The paper says

“We calculate the two-pion wave function in the ground state of the I=2 S-wave system and find the interaction range between two pions, which allows us to examine the validity of the necessary condition for the finite-volume method for the scattering length proposed by Lüscher.”

more of the abstract

We work in the quenched approximation employing a renormalization group improved gauge action for gluons and an improved Wilson action for quarks at $1/a=1.207(12) {\rm GeV}$ on 163 × 80, 203 × 80 and 243 × 80 lattices. We conclude that the necessary condition is satisfied within the statistical errors for the lattice sizes L ≥ 24 ($3.92 {\rm fm}$) when the quark mass is in the range that corresponds to $m_\pi^2 = 0.273-0.736 {\rm GeV}^2$. We obtain the scattering length with a smaller statistical error from the wave function than from the two-pion time correlator.

from the abstract

Pith says

Machine review · deepseek-v4.1-flash · 13 Nov 2026 · what was read was not recorded · not checked by a person

Reviewed in 2026 with later literature in view.

It can be wrong. Challenge this review or reply to it.

Holds up. A solid CP-PACS methods paper: the first direct 4D check that the two-pion range fits inside L/2, plus a cheaper wave-function route to a0 — but the R < L/2 claim is resolution-dependent and the fit window starts inside R. This line is wrong

2 major objections · 7 minor

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The paper says, at Sec. IV A, Eq. (29), footnote 1

The review did not quote this passage.

Pith says: Objection 1 of 2 · major

The operational definition of R as the scale where U(x;k) is buried in the statistical error is acknowledged by the authors (footnote 1 and Appendix A) to have no direct physical meaning and to grow with statistical accuracy, so the abstract's 'satisfied within the statistical errors' is honest.

more of objection 1

What remains unresolved is the interpretation of the observed mass dependence: R is largest (~10, 1.6 fm) for the heaviest quark mass m_π² = 0.736 GeV² and smaller for the lighter masses, whereas a physical interaction range should scale roughly as 1/m_π and therefore be largest for the lightest pion. This pattern is what one expects if R tracks the signal-to-noise of φ at large x rather than a physical range. A concrete test would be to repeat the R extraction on a half-statistics subsample, or with a fixed and tighter error criterion, and show how R moves; if R shrinks as statistics improve, the abstract should be restated as 'no violation resolvable at current precision,' which is weaker but still a useful and defensible statement.

The paper says, at Sec. IV B, Table I

The review did not quote this passage.

Pith says: Objection 2 of 2 · major

For the heaviest mass the fit range quoted in Table I is x_m = 9.8, while the interaction range extracted in Sec. IV A is R ≈ 10 for the same mass. The 'from V' and 'from φ' fits therefore include points at or just inside the region where the Helmholtz/Green-function ansatz is not assumed to hold, and the paper does not discuss the sensitivity of the extracted k² and a_0 to the choice of x_m.

more of objection 2

Because these two methods supply the headline smaller-error scattering lengths, the authors should show a_0 versus x_m (for example x_m = 10, 11, 12) for at least the heaviest and lightest masses, or justify the choice of x_m independently of the measured R.

7 minor objections

The paper says, at Eq. (13) and Eq. (A24)

Pith says: minor objection 1 of 7

As printed, Eq. (13) gives 1/tan δ_0 = g_00 k/(4π) = 4πk (1/L³)Σ 1/(p²−k²), and Eq. (A24) gives 1/tan δ_0 = 4πk·g_00; these two right-hand sides differ by (4π)², and neither matches the ratio of coefficients obtained from Eqs. (A22)–(A23), which yields 1/tan δ_0 = 4π g_00/k. Please check the placement of k and 4π in both equations.

The paper says, at Eq. (28)

Pith says: minor objection 2 of 7

The lattice Laplacian as written, Σ_μ [f(x+μ̂) − f(x−μ̂) + 2f(x)], is not the standard second-difference operator; the usual form is Σ_μ [f(x+μ̂) + f(x−μ̂) − 2f(x)]. Please verify the signs and the constant term.

show 5 more

The paper says, at Abstract and Sec. V

Pith says: minor objection 3 of 7

The phrase 'for the lattice sizes L ≥ 24' is supported by a single lattice at L = 24; it would be more accurate to say 'for L = 24' and to note that no larger volume was simulated.

The paper says, at Abstract

Pith says: minor objection 4 of 7

The abstract reports the smaller statistical error from the wave function but does not mention that the chiral extrapolation of a_0/m_π spans −1.29 to −2.39 (1/GeV²) depending on the fitting form (Eq. 31). A sentence in the abstract noting that no continuum- or chirally-controlled physical-mass number is claimed would prevent over-reading of the title.

The paper says, at Sec. III B

Pith says: minor objection 5 of 7

The quenched approximation and the single lattice spacing (a = 0.163 fm, no continuum extrapolation) are stated in the methods but not flagged as limitations in the conclusions; a brief explicit caveat in Sec. V would help the reader weigh the result.

The paper says, at Figs. 5, 9, 10

Pith says: minor objection 6 of 7

The panels show several quark masses but the figure captions do not identify which mass corresponds to which curve or panel; please label them explicitly so the mass dependence of R can be read directly from the figures.

The paper says, at Tables II and III

Pith says: minor objection 7 of 7

The 'from V' and 'from φ' rows are populated only for the lighter masses, consistent with the text, but the tables do not state which columns these entries belong to; please annotate the correspondence between the blank entries and the masses for which no plateau region was found.

What would settle it

Split the 24³ ensemble into two halves and redo the U(x;k) analysis on each half. If the inferred R is smaller for the half-statistics subsample than for the full sample (as footnote 1 predicts), quantify the trend and extrapolate to, say, 4× statistics. If R would reach or exceed L/2 = 12 with a plausible statistics increase, the 'satisfied for L ≥ 24' claim is fragile and should be softened. Also redo the 'from V' and 'from φ' fits with xm raised above the measured R for the heaviest mass and check whether k² and a0 shift within errors.

from the skeptic's reading

How the review went

4 readings by deepseek-v4.1-flash · 13 Nov 2026

  1. Reader holds up confidence moderate
  2. Skeptic agrees 1 min 13 s · 20,245 tokens read, 3,942 written
  3. Editor wrote the verdict above 36 s · 21,241 tokens read, 1,941 written
  4. Referee minor revision: 2 major objections, 7 minor 3 min 5 s · 20,549 tokens read, 10,149 written

Not run on this paper: the simulated authors' defence.

Pith did not check this review's objections against the paper.

The review in full

The review in brief, machine-written

The useful thing here isn't the scattering length number, it's the check. Everyone who uses Lüscher's formula assumes R < L/2, and nobody had verified it in four-dimensional QCD. CP-PACS computes the Bethe-Salpeter wave function on 16³, 20³ and 24³ at five quark masses, plots U(x;k) = (Δ+k²)φ/(k²φ), and shows it's buried in the noise beyond x ≈ 8–10 on the 24³. That's a real, if modest, contribution, and it's scoped honestly to 'within the statistical errors' (abstract:s2). The second contribution is the wave-function extraction of a0 at a single time slice, with errors two to three times smaller than from the time correlator. Tables I–III show the three methods agreeing, and that agreement is the strongest evidence in the paper that the machinery is sound. The Lüscher derivation in Sec. II and Appendix A is a review, not new, and they say so; Lin et al. and Balog et al. are cited properly for the wave-function idea. Credit where due: this is careful, well-organized work with real error bars and explicit residual plots (Figs. 5, 9, 10). Now the soft spots. The R definition is the one that matters. They state it themselves in footnote 1 and Appendix A: R is where U is buried in statistics, not a physical range, and it grows with precision. So the headline is 'no resolvable violation,' which is what they wrote, but the margin is thin — R ≈ 10 against L/2 = 12 on the heaviest mass, about two lattice spacings. The stress-test's sharper point lands: the fit window xm = 9.8 for that mass starts at or inside R ≈ 10, so the 'from V' and 'from φ' fits include points where the Helmholtz equation isn't assumed to hold. The paper doesn't discuss this. It probably doesn't move a0 much, since the tail is small by construction, but varying xm upward would have cost nothing and should have been shown. Minor: 'L ≥ 24' really means 'L = 24 works'; 20³ and 16³ fail for heavier masses, and they say so. The chiral extrapolation is form-dependent (Eq. 31, spread from −1.29 to −2.39), acknowledged, and they're right that it can't be fixed without lighter masses. Quenched, acknowledged. This is for lattice people doing scattering and anyone wanting a cheaper route to a0. It deserves a referee.

The strongest objection

The central claim (abstract:s2) is that the necessary condition R < L/2 is satisfied. R is defined operationally in Eq. (29) as the scale where U(x;k) = (Δ+k²)φ/(k²φ) is buried in statistical error. The authors themselves state in footnote 1 that this R 'has no direct relevance to the physical scale' and 'becomes larger as the statistical accuracy of k2 and φ(x;k) increases,' and in Appendix A that 'we cannot rigorously define the two-pion interaction range.' So the claim is a statement about current statistical resolution, not a physical bound. On the 24³ lattice the margin is thin: R ≈ 10 (1.6 fm) for the heaviest mass vs L/2 = 12 (1.96 fm), only about two lattice spacings. A modest increase in statistics would push R outward and could erase the margin. Separately, the fit ranges xm in Table I (9.8 for the heaviest mass) start at or inside R ≈ 10, so the 'from V' and 'from φ' extractions include points where the Helmholtz equation is not yet assumed valid — a tension the paper does not discuss. The claim as stated ('within the statistical errors') is honest, but it should be read as 'no violation resolvable at current precision,' not 'the condition is verified.'

The reader's weakest_assumption identifies exactly the load-bearing soft spot: the operational definition of R in Eq. (29) and footnote 1. I agree and sharpen it with the specific numbers. The paper's own text (footnote 1, Appendix A) concedes that R cannot be rigorously defined and grows with statistical precision, so the headline claim (abstract:s2) is a statement about current resolution rather than a physical bound. The margin on the 24³ lattice is thin (R ≈ 10 vs L/2 = 12), and the fit ranges xm in Table I begin at or inside R for the heaviest mass, which the paper does not discuss. None of this overturns the paper: the claim is explicitly scoped to 'within the statistical errors,' the three extraction methods agree within errors (Tables I–III), and the authors flag the limitation themselves. The chiral-extrapolation form dependence (Eq. 31) and the quenched approximation are stated scope limitations, not new findings. The paper is a legitimate methods-and-measurement contribution with an honest, hedged headline. I therefore keep the verdict at ACCEPT/UNCHANGED and recommend only that the claim be read as 'no resolvable violation at current precision.'

The referee's report

If correct, this is the first direct, four-dimensional QCD check of the premise on which Lüscher's finite-volume relation for the scattering length rests, namely that the two-pion interaction range stays below half the box. The paper is careful in its error treatment: three independent extraction methods are cross-checked, the Helmholtz residual is shown explicitly (Figs. 5, 9, 10), and the derivation of Lüscher's formula is reviewed in Sec. II and Appendix A rather than merely cited. The demonstration that the wave function at a single time slice yields a smaller statistical error than the time correlator is a useful methodological result that plausibly extends to phase-shift and multi-hadron problems. The authors are also candid about the operational, resolution-dependent character of R (footnote 1, Appendix A) and about the spread in the chiral extrapolation (Eq. 31).

Pith's reading of the paper (a machine's summary, not the authors')

Lattice QCD calculations of how two pions scatter rely on a formula derived by Martin Lüscher in the 1980s. The formula converts the energy of two pions sitting in a finite periodic box into the scattering length, the low-energy parameter that controls the interaction. The derivation assumes that the range of the pion-pion interaction is smaller than half the box size, so that the box boundary does not distort the interaction. Until this paper, that assumption had not been checked in QCD. The CP-PACS collaboration computed the two-pion wave function, the amplitude for finding the two pions at a given separation, on three lattice volumes with a quenched gluon sector. From the wave function they extracted the quantity (Laplacian + k^2) phi / (k^2 phi), which measures how far the wave function is from satisfying the free Helmholtz equation. Away from the interaction region this quantity should vanish. They find it consistent with zero beyond about 1.6 fm at the heaviest quark mass, which is the largest range they observe. On the 24^3 lattice (3.92 fm) this leaves a margin, so the condition R < L/2 holds within statistical errors for all five quark masses. They also extract the scattering length directly from the wave function, by fitting the asymptotic value of the Laplacian ratio to a constant, or by fitting the wave function itself to a periodic Green function. Both agree with the conventional method based on the two-pion time correlator, but with smaller statistical errors. The chiral extrapolation to the physical pion mass remains ambiguous: three fitting forms give different limits, and the authors say so.

Core claim. The paper's load-bearing assertion is abstract:s2: 'We conclude that the necessary condition is satisfied within the statistical errors for the lattice sizes L >= 24 (3.92 fm) when the quark mass is in the range that corresponds to m_pi^2 = 0.273 - 0.736 GeV^2.' If correct, this is the first direct verification in four-dimensional QCD that the two-pion interaction range R stays below L/2 on 3.92 fm lattices, which is the premise on which Lüscher's finite-volume formula for the scattering length rests.

The premise it rests on. The operational definition of the interaction range R in Eq. (29) and footnote 1: R is defined as the scale where U(x;k) = (Delta + k^2) phi / (k^2 phi) is buried in the statistical error. Footnote 1 states that 'The radius R thus defined has no direct relevance to the physical scale... It becomes larger as the statistical accuracy of k2 and phi(x;k) increases.' The claim that R < L/2 is therefore partly a statement about current statistical precision rather than a fixed physical range, and better statistics would push R outward. The authors acknowledge this, but the headline conclusion inherits the dependence.

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The paper

Abstract

We calculate the two-pion wave function in the ground state of the I=2 S-wave system and find the interaction range between two pions, which allows us to examine the validity of the necessary condition for the finite-volume method for the scattering length proposed by Lüscher. We work in the quenched approximation employing a renormalization group improved gauge action for gluons and an improved Wilson action for quarks at $1/a=1.207(12) {\rm GeV}$ on 163 × 80, 203 × 80 and 243 × 80 lattices. We conclude that the necessary condition is satisfied within the statistical errors for the lattice sizes L ≥ 24 ($3.92 {\rm fm}$) when the quark mass is in the range that corresponds to $m_\pi^2 = 0.273-0.736 {\rm GeV}^2$. We obtain the scattering length with a smaller statistical error from the wave function than from the two-pion time correlator.

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. No sentence in this record was written by a model.

abstract compiled · 2026-09-23

The authors' main claim, as the classifier picked it out 0.98

“We conclude that the necessary condition is satisfied within the statistical errors for the lattice sizes $L\ge 24$ ($3.92 {\rm fm}$) when the quark mass is in the range that corresponds to $m_\pi^2 = 0.273-0.736 {\rm GeV}^2$.”

How the main claim is stated: asserted 0.96

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.53 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.90 · compares with prior results 0.83 · mentions code or data availability 0.01

The abstract, sentence by sentence

This table labels the abstract only (arXiv listing abstract v2). The number is the probability the classifier gave the label.

SentenceRoleThe authors' own workStates the main conclusionStrength
We calculate the two-pion wave function in the ground state of the I=2 $S$-wave system and find the interaction range between two pions, which allows us to examine the validity of the necessary condition for the finite-volume method for the scattering length proposed by L\"uscher. contribution 0.96 0.97 0.15 asserted 1.00
We work in the quenched approximation employing a renormalization group improved gauge action for gluons and an improved Wilson action for quarks at $1/a=1.207(12) {\rm GeV}$ on $16^3 \times 80$, $20^3 \times 80$ and $24^3 \times 80$ lattices. method 1.00 0.93 0.02 asserted 0.95
We conclude that the necessary condition is satisfied within the statistical errors for the lattice sizes $L\ge 24$ ($3.92 {\rm fm}$) when the quark mass is in the range that corresponds to $m_\pi^2 = 0.273-0.736 {\rm GeV}^2$. result 0.89 0.95 0.50 asserted 0.96
We obtain the scattering length with a smaller statistical error from the wave function than from the two-pion time correlator. result 0.99 0.96 0.21 asserted 1.00

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

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Sources read

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Figures (11)
Figure 1 of the paper
Figure 1. FIG. 1: Two-pion wave function
Figure 2 of the paper
Figure 2. FIG. 2: Two-pion wave function
Figure 4 of the paper
Figure 4. FIG. 4: Time dependence of
Figure 6 of the paper
Figure 6. FIG. 6: Normalized two-pion correlator
Figure 7 of the paper
Figure 7. FIG. 7: Effective mass for
Figure 8 of the paper
Figure 8. FIG. 8: Pion scattering length
Figure 11 of the paper
Figure 11. FIG. 11: Volume dependence of pion scattering length

Showing 7 of the paper's 11 figures. Figures are the authors', reproduced from the paper.

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  13. Generalized L\"uscher Formula in Multi-channel Baryon-Meson Scattering hep-lat · Sep 2012
  14. Nucleon-Nucleon Potential and its Non-locality in Lattice QCD hep-lat · Mar 2011
  15. Extracting Scattering Phase-Shifts in Higher Partial-Waves from Lattice QCD Calculations hep-lat · Jan 2011
  16. Precise Determination of the I=2 pipi Scattering Length from Mixed-Action Lattice QCD hep-lat · Jun 2007
  17. Nuclear Physics Review hep-lat · Jan 2014
  18. Towards grounding nuclear physics in QCD nucl-th · Oct 2019
  19. Nuclear Physics from Lattice QCD hep-lat · Apr 2010
  20. Multi-particle systems on the lattice and chiral extrapolations: a brief review hep-lat · Feb 2021
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  • The wavefunction given in Eq. ([ref]), $ϕ(|x|;k)∼Z_ππ(|x|;k) ψ_ππ(|x|;k)$ satisfies the Schrodinger equation [ref] [equation] where $U_k(x,y)$ is the Fourier transform of the modified Bethe-Salpeter kernel for the $π^+π^+$ interaction, which is generally nonlocal and energy-dependent [ref] .

    Nuclear Physics from Lattice QCD · Scattering Parameters from Wavefunctions. · 0.83

  • where we use Eqs. ([ref]) and ([ref]) in the arrow. $G(x;k)$ is evaluated using the formula in Appendix B of Ref. [ref] .

    Scattering amplitude from Bethe-Salpeter wave function inside the interaction range · Introduction · 0.99

  • We follow the strategy proposed by CP-PACS group [ref] , where asymptotic Bethe-Salpeter wavefunctions on the Euclidean lattice are adopted to evaluate pion scattering length.

    Hadron-hadron interaction from SU(2) lattice QCD · Formulation · 1.00

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