Pith. sign in

REVIEW 3 major objections 4 minor 6 cited by

The paper claims the first lattice-QCD determination of the π(1300) resonance pole, with mass and width consistent with experiment.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-04 10:25 UTC pith:GS7A52GQ

load-bearing objection The heavy-pion three-body analysis is a solid first; take the physical-point pole with a grain of salt until the Mπ-dependence of the three-body force is tamed. the 3 major comments →

arxiv 2510.09476 v2 pith:GS7A52GQ submitted 2025-10-10 hep-lat hep-phnucl-th

Emergence of the π(1300) Resonance from Lattice QCD

Haobo Yan , Maxim Mai , Marco Garofalo , Yuchuan Feng , Michael D\"oring , Chuan Liu , Liuming Liu , Ulf-G. Mei{\ss}ner
show 1 more author
Carsten Urbach
This is my paper
classification hep-lat hep-phnucl-th PACS 12.38.Gc14.40.Be13.75.Lb
keywords lattice QCDπ(1300) resonancethree-body scatteringfinite-volume formalismchiral perturbation theoryresonance polehadron spectroscopymodel averaging
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper reports the first extraction of the π(1300) resonance parameters directly from lattice QCD. Using finite-volume spectra of two- and three-pion systems at heavy pion masses, combined with chiral perturbation theory for the two-body interactions, the authors map the spectra to infinite-volume scattering amplitudes and analytically continue them to complex energies. After model averaging, they obtain a pole position of (1169±46) − i(62−62+168) MeV, consistent with the experimental mass and width. This establishes that the π(1300) is a three-pion resonance whose properties can be predicted from the fundamental theory of quarks and gluons.

Core claim

The central claim is that the π(1300) exists as a resonance in QCD and its pole position can be computed from lattice QCD. The calculation uses three-pion operators and the finite-volume three-body quantization condition to constrain the three-body scattering amplitude at a pion mass of about 305 MeV, where the resonance lies within reach (around 4.2 Mπ). Heavy-pion fits all produce a π(1300) pole—even when no explicit pole term is included in the three-body force—indicating that the resonance is generated dynamically by the coupled σπ and ρπ interactions. Extrapolating to the physical pion mass under the assumption that the three-body force has negligible pion-mass dependence, with two-body

What carries the argument

The key machinery is the finite-volume three-body quantization condition det(B+C−τ−1EL)=0, which relates the discrete energy levels in a finite box to the infinite-volume three-body scattering amplitude. The matrices B and EL encode the momentum kinematics; τ−1 encodes the two-body sub-amplitudes, which are constrained by the modified Inverse Amplitude Method (mIAM) built from Chiral Perturbation Theory; and C is a volume-independent three-body force parameterized by a few low-energy constants. Solving this condition at two volumes and two pion masses, then analytically continuing the resulting amplitude to complex energies, gives the resonance pole.

Load-bearing premise

The physical-point pole position assumes that the three-body force parameters fitted at a pion mass of about 305 MeV do not change significantly when the pion mass is lowered to its physical value; if they do, the extrapolated mass and width could shift outside the quoted errors.

What would settle it

A second lattice calculation at an intermediate pion mass (e.g., Mπ≈250 MeV) with three-body spectra would test the assumed pion-mass independence: if the three-body force parameters fitted there differ significantly from those at 305 MeV, the physical-point extrapolation collapses. Alternatively, a direct simulation at the physical pion mass, or a precise experimental measurement of the π(1300) pole from a Dalitz-plot analysis that lies outside 1169±46 MeV and the quoted width range, would falsify the central claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • π(1300) is a genuine three-pion resonance whose mass and width are computable from QCD, not just phenomenologically inferred.
  • The appearance of the pole even without a pole term in the three-body force suggests the resonance is dynamically generated by three-body interactions, not an explicit quark-antiquark state.
  • The same workflow can be applied to other three-body resonances (e.g., ω(782), a1(1260), and excited baryons) with similar quantum numbers.
  • The two-body results (ρ(770), f0(500)) extracted from the same fits agree with experiment within the quoted uncertainties, cross-validating the three-body framework.
  • The light-pion three-body data alone cannot pin down the π(1300), so future heavier-pion or physical-point simulations are needed to sharpen the extrapolation.

Where Pith is reading between the lines

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

  • If the three-body force does vary with pion mass (the main assumption), the quoted pole position could shift; a direct calculation at an intermediate pion mass (e.g., Mπ≈250 MeV) would quantify this systematic and likely tighten the width.
  • The near-consistency with the experimental width, despite the very broad and asymmetric error, hints that the π(1300) may be a broad, dynamically generated state rather than a conventional quark-antiquark excitation; this can be tested by comparing lineshapes and Dalitz plots against future decay experiments.
  • The method's reliance on chiral perturbation theory for the two-body channels suggests that a complementary approach using fully lattice-determined two-body phase shifts (without CHPT input) could be used at heavier pion masses to isolate the three-body force uncertainties.
  • The model-averaging procedure with Akaike weights provides a template for how systematic model choices (cutoffs, parameterizations, data subsets) can be folded into a single uncertainty, a practice that could be adopted in other lattice QCD resonance studies.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript presents a lattice QCD determination of the π(1300) resonance parameters using four CLQCD ensembles at two pion masses (Mπ≈305 and 208 MeV) and two volumes. It combines GEVP spectra for ππ (I=0,1,2) and πππ (I=1) with the Mai–Döring finite-volume three-body formalism, mIAM parameterization of two-body interactions, and a generic three-body force linear in (s−m0²). Fitting about 2000 model variants at the heavy pion mass yields a clear π(1300) pole; extrapolating to the physical pion mass under the stated assumption that all terms in Eq. (2) are pion-mass independent, the AIC model average gives Mπ(1300)=(1169±46)−i(62−62+168) MeV, consistent with PDG. The paper also quotes ρ(770) and f0(500) poles from the same framework as cross-checks.

Significance. If the physical-point result holds, this is the first direct lattice QCD extraction of the π(1300) resonance parameters and a nontrivial demonstration that a three-body resonance emerges dynamically from QCD at Mπ≈305 MeV. The heavy-mass signal is credible: the best fit has χ²/dof=1.08, all acceptable heavy-mass fits produce a pole, and the pole appears even without an explicit pole term in the three-body force. The two-body ρ and f0 results serve as useful consistency checks of the framework, though they are not independent validations because they share the same two-body input. The main result, however, rests on an unquantified assumption about the pion-mass dependence of the three-body force; the paper is transparent about this, but the claim 'supporting values from phenomenology' is conditional on that assumption.

major comments (3)
  1. [Analysis and results, Eq. (2) and Eq. (4)] The physical-point pole is obtained by taking parameters fitted at Mπ≈305 MeV and evaluating the amplitude at Mπ=139 MeV with 'assuming negligible pion-mass dependence of all terms in Eq. (2) evaluated in physical/lattice units.' This assumption is neither tested by the light-pion data (the authors state that only a few light-only fits lead to a π(1300) pole, because the pole lies at ≳6Mπ) nor assigned a systematic error. At the physical point the pole lies near 8.4Mπ, far outside the energy region constrained by the lattice spectra. Since c_c, c_p, and m0 in Eq. (2) are undetermined functions of Mπ, an O(1) variation over the factor-of-two Mπ range could shift the pole by more than the quoted 46 MeV or remove it entirely. This is the central claim of the Letter, so the error budget in Eq. (4) is incomplete. I request either an estimate of this systematic (e.g., by including a linear Mπ
  2. [Fig. 3 and 'Lattice QCD spectrum'] Several energy levels in Fig. 3 are shown faded and excluded from the analysis without a stated criterion. In the πππ channel the number of levels is small, and the resonance signal at Mπ≈305 MeV comes from an aggregation around 4Mπ; excluding levels by hand can bias the fit. The Supplemental Material documents operator construction and fit-range choices but not the exclusion rule. Please state the selection criterion (e.g., unstable plateaus, poor overlap) and check the stability of the extracted π(1300) pole under including or excluding borderline levels.
  3. [Analysis and results, 'For the fits to light pion mass data only'] The claim 'all fits in 68%, 95%, and 99% confidence intervals ... lead to a π(1300) pole' applies only to fits that include heavy three-body input. The light-only fits mostly do not produce a pole, and the physical-point extrapolation is made from heavy-mass parameters. Since the light-pion spectra do not constrain the pole, the no-resonance scenario is effectively ruled out only at Mπ≈305 MeV, not at the physical point. The conclusion should state this distinction explicitly, because the current phrasing in the conclusion ('At heavy pion mass, all fits lead to ... effectively ruling out the no-resonance scenario') could be read as applying to the physical point.
minor comments (4)
  1. [Eq. (4) and abstract] The uncertainty on Im Mπ(1300) is written inconsistently: Eq. (4) gives (62±169) MeV, while the abstract and Fig. 5 caption give (62−62+168) MeV. Also, the central value 62 is not the midpoint of the asymmetric range; please use a consistent asymmetric notation.
  2. [Fig. 5] The axis labels use 'Mω' in all panels, including the π(1300) panel and the f0(500) panel. This appears to be a typo for Mπ or a mass scale label; please correct to avoid confusion.
  3. [Supplemental Material, Table S2] Several entries in Table S2 appear garbled (e.g., '61720.70.78', '59071.6 69.', '18028.–'). The table should be checked and reformatted so that all fitted parameters and uncertainties are legible.
  4. [References] Reference [8] (d'Argent et al.) is missing the year and journal volume/page details; the entry currently reads 'JHEP05, 143, arXiv:1703.08505' without a year. Please complete the citation.

Circularity Check

0 steps flagged

No significant circularity: the π(1300) pole is not an input and emerges from fitted amplitudes; the physical-point extrapolation rests on an explicit assumption, not a definitional reduction.

full rationale

The derivation chain is: (1) lattice correlators are reduced to finite-volume energies via GEVP; (2) the FVU quantization condition Eq. (1) maps those spectra to two-/three-body amplitudes; (3) two-body amplitudes are parameterized by mIAM/CHPT and the three-body force by the generic form Eq. (2), with parameters fitted to the spectra; (4) analytic continuation yields pole positions; (5) the physical point is reached by setting M_π to its physical value, keeping the fitted three-body parameters fixed. No step defines the π(1300) pole as an input. The heavy-pion resonance signal is directly visible in the spectra and is reproduced even by fits with no explicit pole term in C, so the pole is a consequence of the amplitude, not a self-definitional output. The ρ(770) and f0(500) results in Eq. (4) are derived from the same two-body fits and are presented as cross-checks, not independent predictions; that is a consistency check, not circularity. The physical-point extrapolation relies on the explicitly stated ansatz 'assuming negligible pion-mass dependence of all terms in Eq. (2)' — this is an unquantified model assumption and a systematic-uncertainty concern, but it is not a fitted parameter renamed as a prediction, nor does any equation reduce the pole to the input by construction. Methodological self-citations (FVU formalism of Ref. [33], prior ω study Ref. [16], mIAM experience Refs. [41-43]) supply established tools and prior benchmarks; no load-bearing argument reduces to an unverified self-citation chain. The result is benchmarked against external PDG values and previous lattice studies, and the central claim has independent content. Therefore no significant circularity is found.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The analysis introduces no new particles or forces. Its free parameters are the chiral LECs and three-body contact/pole terms fitted to lattice spectra; the most consequential input is the ad hoc assumption of pion-mass independence for the three-body force, which drives the physical-point extrapolation. The quantization-condition and mIAM machinery is imported from prior work by the same and other authors.

free parameters (3)
  • mIAM chiral LECs l1^r, l2^r, l3^r (l4^r from FLAG in mIAM3) = See Table S2; varies across fits, e.g., l1^r ~ -4.6e-3, l2^r ~ 4.2e-3, l3^r ~ 8.5e-3 (fit 203)
    These enter the two-body phase shifts through the modified Inverse Amplitude Method, and are fitted to the ππ finite-volume spectra.
  • Three-body force constants c_c^σπ, c_p^σπ, c_c^ρπ, c_p^ρπ, m0 = See Table S2; e.g., fit 203 has c_c^σπ = 8.5, others varied or zero
    These parameterize the short-range three-body force C(s,p,p') in Eq. (2); they are fitted to the combined two- and three-body spectra.
  • Cutoffs σ_MP, |k_max|, |l_max| = CT = {1 Mπ², √3·5·2·2π/aL, √3·2π/aL} (chosen, varied)
    The matching point and momentum cutoffs define the truncated three-body formalism; the paper states that variation has no significant impact, but they are free choices.
axioms (5)
  • domain assumption The finite-volume quantization condition det[(B+C)−τ^{-1}E_L]=0 of Refs. [33] correctly maps finite-volume spectra to infinite-volume amplitudes.
    Invoked in Eq. (1) and used to extract scattering amplitudes; relies on prior literature on three-body quantization conditions.
  • domain assumption The two-body amplitude is well described by the modified Inverse Amplitude Method (mIAM) with chiral LECs, including behavior below threshold down to the matching point σ_MP.
    Used for K^{-1}_{I2} in Eq. (3); standard but an approximation to full QCD.
  • ad hoc to paper The three-body force has the generic form c_αβ = c_c^{αβ} + c_p^{αβ}(s−m0²), with only σπ and ρπ channels.
    Eq. (2): 'Little is known about the form of the three-body force for this system, forcing us to pick a generic form.'
  • ad hoc to paper The three-body force parameters extracted at Mπ≈305 MeV are unchanged when extrapolating to Mπ≈139 MeV.
    Stated before Fig. 5: 'assuming negligible pion-mass dependence of all terms in Eq. (2).' This is the key extrapolation assumption.
  • domain assumption Higher partial waves and inelastic channels (e.g., K Kbar, ηη) are negligible in the energy region of interest.
    The quantization condition uses a truncated basis (σπ, ρπ, Gπ) and the operator list is restricted below inelastic thresholds (Supplemental section 1).

pith-pipeline@v1.3.0-alltime-deepseek · 31250 in / 7393 out tokens · 73683 ms · 2026-08-04T10:25:46.117881+00:00 · methodology

0 comments
read the original abstract

The mass of the lightest hadron in nature, the pion, is one seventh of that of the nucleon and one tenth of the mass of its first excited state, the $\pi(1300)$. This enormous energy difference opens an interesting window into the confinement of quarks and the structure of the lightest hadrons. In this Letter, we provide the first calculation of resonance parameters of the $\pi(1300)$ from lattice quantum chromodynamics (QCD). For this purpose, recently derived state-of-the-art tools are adapted and applied both in the construction of three-hadron operators and for mapping finite-volume spectra to infinite-volume amplitudes, subsequently analytically continuing these to complex energies. For our heavy pion mass ensembles, we find a clear signal of the resonance. Making a simple assumption of vanishing pion mass dependence for the three-body force, but incorporating constraints from Chiral Perturbation Theory for all the two-body channels, enables a robust extrapolation to the physical point. Applying model averaging, we extract a pole position of $M_{\pi(1300)}=(1169\pm46)-i(62_{-62}^{+168})\,\MeV$ supporting values from phenomenology.

Figures

Figures reproduced from arXiv: 2510.09476 by Carsten Urbach, Chuan Liu, Haobo Yan, Liuming Liu, Marco Garofalo, Maxim Mai, Michael D\"oring, Ulf-G. Mei{\ss}ner, Yuchuan Feng.

Figure 1
Figure 1. Figure 1: FIG. 1. Workflow of the pole position determination for two [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Diagram topologies for [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Finite-volume spectra for [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Quality of the 47 central fit results. The used LQCD [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Pole positions on the second Riemann sheet of the [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The $a_1(1420)$ in a Unitary Coupled-Channel Three-Body Approach

    hep-ph 2026-06 conditional novelty 6.5

    Nine-channel unitary three-body fits to COMPASS lineshapes reproduce the a1(1420) enhancement by triangle singularity without requiring a genuine a1(1420) pole.

  2. Production Effects and Final-state Interactions in $\pi_1 \to 3\pi$

    hep-ph 2026-07 conditional novelty 6.0

    A Khuri-Treiman model with contact and one-pion-exchange production terms fits COMPASS's π1→3π freed-isobar data and produces a smooth 1.6 GeV structure in the extracted production strengths.

  3. Three-body unitary determination of the $f_1(1285)$ and $f_1(1420)$ pole positions

    hep-ph 2026-06 unverdicted novelty 6.0

    Fitting a spectator-isobar three-body unitary amplitude to BESIII K0S K0S pi0 data yields poles at (1277±2±1)-i(12±1±0) MeV for f1(1285) and (1435±2±7)-i(40±2±1) MeV for f1(1420), with the latter traced to a K Kbar* q...

  4. Coupled-channel approach to isotensor $\pi\pi\pi$ scattering from lattice QCD

    hep-lat 2026-01 conditional novelty 6.0

    The I=2 three-pion spectrum from lattice QCD is described by a repulsive rho-pi S-wave interaction, consistent with a leading-order effective Lagrangian.

  5. The $a_1(1420)$ in a Unitary Coupled-Channel Three-Body Approach

    hep-ph 2026-06 unverdicted novelty 5.0

    Unitary coupled-channel three-body model fitted to COMPASS data reproduces the a1(1420) enhancement via triangle singularity, indicating no genuine resonance pole is required.

  6. Lattice QCD study of the $K^*(892)$ resonance at the physical point

    hep-lat 2026-03 unverdicted novelty 5.0

    Lattice QCD yields the K*(892) resonance pole at physical pion mass and continuum limit as 883(22) - i20(13) MeV, in agreement with experiment.

Reference graph

Works this paper leans on

66 extracted references · 47 linked inside Pith · cited by 5 Pith papers

  1. [1]

    Primary Color (Soft Blue): # 4A90E2 (or RGB: 74, 144, 226)

  2. [2]

    Secondary Color (Light Gray): # D3D3D3 (or RGB: 211, 211, 211)

  3. [3]

    Accent Color (Dark Blue): #1D3557 (or RGB: 29, 53, 87)

  4. [4]

    Accent Color (Dark Green): #2C6B5A (or RGB: 44, 107, 90) FIG. 1. Workflow of the pole position determination for two- and three-body systems from LQCD. K(1460), and as heavy as the excitedN(1440) state of the nucleon, which is only 1.5 times heavier than the pro- ton. Unraveling this pattern is, therefore, fundamental to understanding the emergence of the...

    Pith/arXiv arXiv 2025

  5. [5]

    Ketzer, B

    B. Ketzer, B. Grube, and D. Ryabchikov, Light-Meson Spectroscopy with COMPASS, Prog. Part. Nucl. Phys. 113, 103755 (2020), [Erratum: Prog.Part.Nucl.Phys. 128, 104000 (2023)], arXiv:1909.06366 [hep-ex]

    Pith/arXiv arXiv 2020

  6. [6]

    Hyodo and M

    T. Hyodo and M. Niiyama, QCD and the strange baryon spectrum, Prog. Part. Nucl. Phys.120, 103868 (2021), arXiv:2010.07592 [hep-ph]

    Pith/arXiv arXiv 2021

  7. [7]

    H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu, and S.-L. Zhu, An updated review of the new hadron states, Rept. Prog. Phys.86, 026201 (2023), arXiv:2204.02649 [hep-ph]

    Pith/arXiv arXiv 2023

  8. [8]

    Mai, U.-G

    M. Mai, U.-G. Meißner, and C. Urbach, Towards a the- ory of hadron resonances, Phys. Rept.1001, 1 (2023), arXiv:2206.01477 [hep-ph]

    Pith/arXiv arXiv 2023

  9. [9]

    J. R. Pelaez, Light Meson Resonances, (2025), arXiv:2509.08648 [hep-ph]

    arXiv 2025

  10. [10]

    Navaset al.(Particle Data Group), Review of particle physics, Phys

    S. Navaset al.(Particle Data Group), Review of particle physics, Phys. Rev. D110, 030001 (2024)

    2024

  11. [11]

    M. A. Ananevaet al., OBSER V ATION OF PION EX- CITED STATE: A NEW PSEUDOSCALAR MESON, JETP Lett.34, 488 (1981)

    1981

  12. [12]

    d’Argent, N

    P. d’Argent, N. Skidmore, J. Benton, J. Dalseno, E. Gersabeck, S. Harnew, P. Naik, C. Prouve, and J. Rademacker, Amplitude Analyses ofD 0 → π+π−π+π− andD 0 →K +K −π+π− Decays, JHEP05, 143, arXiv:1703.08505 [hep-ex]

    Pith/arXiv arXiv

  13. [13]

    M. T. Hansen and S. R. Sharpe, Lattice QCD and Three- particle Decays of Resonances, Ann. Rev. Nucl. Part. Sci. 69, 65 (2019), arXiv:1901.00483 [hep-lat]

    Pith/arXiv arXiv 2019

  14. [14]

    M. Mai, M. D¨ oring, and A. Rusetsky, Multi-particle systems on the lattice and chiral extrapolations: a brief review, Eur. Phys. J. ST230, 1623 (2021), arXiv:2103.00577 [hep-lat]

    Pith/arXiv arXiv 2021

  15. [15]

    Romero-L´ opez, Multi-hadron interactions from lattice QCD, PoSLA TTICE2022, 235 (2023), arXiv:2212.13793 [hep-lat]

    F. Romero-L´ opez, Multi-hadron interactions from lattice QCD, PoSLA TTICE2022, 235 (2023), arXiv:2212.13793 [hep-lat]

    Pith/arXiv arXiv 2023

  16. [16]

    M¨ uller and A

    F. M¨ uller and A. Rusetsky, On the three-particle ana- log of the Lellouch-L¨ uscher formula, JHEP03, 152, arXiv:2012.13957 [hep-lat]

    Pith/arXiv arXiv 2012

  17. [17]

    M. T. Hansen, F. Romero-L´ opez, and S. R. Sharpe, De- cay amplitudes to three hadrons from finite-volume ma- trix elements, JHEP04, 113, arXiv:2101.10246 [hep-lat]

    Pith/arXiv arXiv

  18. [18]

    J.-Y. Pang, R. Bubna, F. M¨ uller, A. Rusetsky, and J.- J. Wu, Lellouch-L¨ uscher factor for the K→3πdecays, JHEP05, 269, arXiv:2312.04391 [hep-lat]

    Pith/arXiv arXiv

  19. [19]

    M. Mai, A. Alexandru, R. Brett, C. Culver, M. D¨ oring, F. X. Lee, and D. Sadasivan (GWQCD), Three-Body Dy- namics of the a1(1260) Resonance from Lattice QCD, Phys. Rev. Lett.127, 222001 (2021), arXiv:2107.03973 [hep-lat]

    Pith/arXiv arXiv 2021

  20. [20]

    H. Yan, M. Mai, M. Garofalo, U.-G. Meißner, C. Liu, L. Liu, and C. Urbach,ωMeson from Lattice QCD, Phys. Rev. Lett.133, 211906 (2024), arXiv:2407.16659 [hep- lat]

    Pith/arXiv arXiv 2024

  21. [21]

    J. J. Dudek, R. G. Edwards, M. J. Peardon, D. G. Richards, and C. E. Thomas, Toward the excited meson spectrum of dynamical QCD, Phys. Rev. D82, 034508 (2010), arXiv:1004.4930 [hep-ph]

    Pith/arXiv arXiv 2010

  22. [22]

    McNeile and C

    C. McNeile and C. Michael (UKQCD), The Decay con- stant of the first excited pion from lattice QCD, Phys. Lett. B642, 244 (2006), arXiv:hep-lat/0607032. 6

    Pith/arXiv arXiv 2006

  23. [23]

    A. S. Kronfeld, Twenty-first Century Lattice Gauge The- ory: Results from the QCD Lagrangian, Ann. Rev. Nucl. Part. Sci.62, 265 (2012), arXiv:1203.1204 [hep-lat]

    Pith/arXiv arXiv 2012

  24. [24]

    T. G. Steele, J. C. Breckenridge, M. Benmerrouche, V. Elias, and A. H. Fariborz, QCD sum rules and the pi (1300) resonance, Nucl. Phys. A624, 517 (1997), arXiv:hep-ph/9708454

    Pith/arXiv arXiv 1997

  25. [25]

    A. Holl, A. Krassnigg, and C. D. Roberts, Pseudoscalar meson radial excitations, Phys. Rev. C70, 042203 (2004), arXiv:nucl-th/0406030

    Pith/arXiv arXiv 2004

  26. [26]

    Wang and J.-L

    L. Wang and J.-L. Ping, Mesons in the constituent quark model, Chin. Phys. Lett.24, 1195 (2007)

    2007

  27. [27]

    Martinez Torres, K

    A. Martinez Torres, K. P. Khemchandani, D. Jido, and A. Hosaka, Theoretical support for theπ(1300) and the recently claimedf 0(1790) as molecular resonances, Phys. Rev. D84, 074027 (2011), arXiv:1106.6101 [nucl-th]

    Pith/arXiv arXiv 2011

  28. [28]

    Williams, C

    R. Williams, C. S. Fischer, and W. Heupel, Light mesons in QCD and unquenching effects from the 3PI effective action, Phys. Rev. D93, 034026 (2016), arXiv:1512.00455 [hep-ph]

    Pith/arXiv arXiv 2016

  29. [29]

    Huet al.(CLQCD), Quark masses and low-energy constants in the continuum from the tadpole-improved clover ensembles, Phys

    Z.-C. Huet al.(CLQCD), Quark masses and low-energy constants in the continuum from the tadpole-improved clover ensembles, Phys. Rev. D109, 054507 (2024), arXiv:2310.00814 [hep-lat]

    Pith/arXiv arXiv 2024

  30. [30]

    Michael and I

    C. Michael and I. Teasdale, Extracting Glueball Masses From Lattice QCD, Nucl. Phys. B215, 433 (1983)

    1983

  31. [31]

    Luscher and U

    M. Luscher and U. Wolff, How to Calculate the Elastic Scattering Matrix in Two-dimensional Quantum Field Theories by Numerical Simulation, Nucl. Phys. B339, 222 (1990)

    1990

  32. [32]

    Blossier, M

    B. Blossier, M. Della Morte, G. von Hippel, T. Mendes, and R. Sommer, On the generalized eigenvalue method for energies and matrix elements in lattice field theory, JHEP04, 094, arXiv:0902.1265 [hep-lat]

    Pith/arXiv arXiv

  33. [33]

    Fischer, B

    M. Fischer, B. Kostrzewa, J. Ostmeyer, K. Ottnad, M. Ueding, and C. Urbach, On the generalised eigenvalue method and its relation to Prony and generalised pencil of function methods, Eur. Phys. J. A56, 206 (2020), arXiv:2004.10472 [hep-lat]

    Pith/arXiv arXiv 2020

  34. [34]

    See Supplemental Material for the method of operator construction, full topologies of the contraction diagrams, technical details on extracting energy spectra and corre- lator fittings, two- and three-body finite-volume analysis and fits, and the model averaging procedure

  35. [35]

    H. Yan, C. Liu, L. Liu, and Y. Meng, Construction of gen- eralN-body lattice operators with arbitrary momenta, (2025), arXiv:2507.16070 [hep-lat]

    arXiv 2025

  36. [36]

    Peardon, J

    M. Peardon, J. Bulava, J. Foley, C. Morningstar, J. Dudek, R. G. Edwards, B. Joo, H.-W. Lin, D. G. Richards, and K. J. Juge (Hadron Spectrum), A Novel quark-field creation operator construction for hadronic physics in lattice QCD, Phys. Rev. D80, 054506 (2009), arXiv:0905.2160 [hep-lat]

    Pith/arXiv arXiv 2009

  37. [37]

    Mai and M

    M. Mai and M. D¨ oring, Three-body Unitarity in the Finite Volume, Eur. Phys. J. A53, 240 (2017), arXiv:1709.08222 [hep-lat]

    Pith/arXiv arXiv 2017

  38. [38]

    Mai and M

    M. Mai and M. D¨ oring, Finite-Volume Spectrum ofπ+π+ andπ +π+π+ Systems, Phys. Rev. Lett.122, 062503 (2019), arXiv:1807.04746 [hep-lat]

    Pith/arXiv arXiv 2019

  39. [39]

    M. Mai, M. D¨ oring, C. Culver, and A. Alexandru, Three- body unitarity versus finite-volumeπ +π+π+ spectrum from lattice QCD, Phys. Rev. D101, 054510 (2020), arXiv:1909.05749 [hep-lat]

    Pith/arXiv arXiv 2020

  40. [40]

    Alexandru, R

    A. Alexandru, R. Brett, C. Culver, M. D¨ oring, D. Guo, F. X. Lee, and M. Mai, Finite-volume energy spectrum of theK −K −K − system, Phys. Rev. D102, 114523 (2020), arXiv:2009.12358 [hep-lat]

    Pith/arXiv arXiv 2020

  41. [41]

    Brett, C

    R. Brett, C. Culver, M. Mai, A. Alexandru, M. D¨ oring, and F. X. Lee, Three-body interactions from the finite- volume QCD spectrum, Phys. Rev. D104, 014501 (2021), arXiv:2101.06144 [hep-lat]

    Pith/arXiv arXiv 2021

  42. [42]

    Garofalo, M

    M. Garofalo, M. Mai, F. Romero-L´ opez, A. Rusetsky, and C. Urbach, Three-body resonances in theφ 4 theory, JHEP02, 252, arXiv:2211.05605 [hep-lat]

    Pith/arXiv arXiv

  43. [43]

    Y. Feng, F. Gil, M. D¨ oring, R. Molina, M. Mai, V. Shas- try, and A. Szczepaniak, A unitary coupled-channel three-body amplitude with pions and kaons, (2024), arXiv:2407.08721 [nucl-th]

    Pith/arXiv arXiv 2024

  44. [44]

    S. U. Chung, SPIN FORMALISMS 10.5170/CERN- 1971-008 (1971)

    doi:10.5170/cern- 1971

  45. [45]

    M. Mai, C. Culver, A. Alexandru, M. D¨ oring, and F. X. Lee, Cross-channel study of pion scattering from lattice QCD, Phys. Rev. D100, 114514 (2019), arXiv:1908.01847 [hep-lat]

    Pith/arXiv arXiv 2019

  46. [46]

    D¨ oring, B

    M. D¨ oring, B. Hu, and M. Mai, Chiral Extrapolation of the Sigma Resonance, Phys. Lett. B782, 785 (2018), arXiv:1610.10070 [hep-lat]

    Pith/arXiv arXiv 2018

  47. [47]

    N. R. Acharya, F.-K. Guo, M. Mai, and U.-G. Meißner, θ-dependence of the lightest meson resonances in QCD, Phys. Rev. D92, 054023 (2015), arXiv:1507.08570 [hep- ph]

    Pith/arXiv arXiv 2015

  48. [48]

    Hanhart, J

    C. Hanhart, J. R. Pelaez, and G. Rios, Quark mass de- pendence of the rho and sigma from dispersion relations and Chiral Perturbation Theory, Phys. Rev. Lett.100, 152001 (2008), arXiv:0801.2871 [hep-ph]

    Pith/arXiv arXiv 2008

  49. [49]

    Aokiet al.(Flavour Lattice Averaging Group (FLAG)), FLAG Review 2024, (2024), arXiv:2411.04268 [hep-lat]

    Y. Aokiet al.(Flavour Lattice Averaging Group (FLAG)), FLAG Review 2024, (2024), arXiv:2411.04268 [hep-lat]

    Pith/arXiv arXiv 2024

  50. [50]

    Sadasivan, A

    D. Sadasivan, A. Alexandru, H. Akdag, F. Amorim, R. Brett, C. Culver, M. D¨ oring, F. X. Lee, and M. Mai, Pole position of the a1(1260) resonance in a three-body unitary framework, Phys. Rev. D105, 054020 (2022), arXiv:2112.03355 [hep-ph]

    Pith/arXiv arXiv 2022

  51. [51]

    A. S. Sakthivasan, M. Mai, A. Rusetsky, and M. D¨ oring, Effects of final state interactions on Landau singularities, JHEP10, 246, arXiv:2407.17969 [hep-ph]

    Pith/arXiv arXiv

  52. [52]

    D¨ oring, J

    M. D¨ oring, J. Haidenbauer, M. Mai, and T. Sato, Dy- namical coupled-channel models for hadron dynamics, (2025), arXiv:2505.02745 [nucl-th]

    arXiv 2025

  53. [53]

    Akaike, A new look at the statistical model identifica- tion, IEEE Transactions on Automatic Control19, 716 (1974)

    H. Akaike, A new look at the statistical model identifica- tion, IEEE Transactions on Automatic Control19, 716 (1974)

    1974

  54. [54]

    Z. Wang, D. B. Leinweber, C. Liu, L. Liu, P. Sun, A. W. Thomas, J.-j. Wu, H. Xing, and K. Yu (CLQCD), Spec- tral parameters of theρresonance from lattice QCD, JHEP08, 064, arXiv:2502.03700 [hep-lat]

    Pith/arXiv arXiv

  55. [55]

    M. A. Clark, R. Babich, K. Barros, R. C. Brower, and C. Rebbi (QUDA), Solving Lattice QCD systems of equa- tions using mixed precision solvers on GPUs, Comput. Phys. Commun.181, 1517 (2010), arXiv:0911.3191 [hep- lat]

    Pith/arXiv arXiv 2010

  56. [56]

    Babich, M

    R. Babich, M. A. Clark, B. Joo, G. Shi, R. C. Brower, and S. Gottlieb (QUDA), Scaling lattice QCD beyond 100 GPUs, inInternational Conference for High Per- formance Computing, Networking, Storage and Analysis (2011) arXiv:1109.2935 [hep-lat]

    Pith/arXiv arXiv 2011

  57. [57]

    M. A. Clark, B. Jo´ o, A. Strelchenko, M. Cheng, A. Gamb- hir, and R. C. Brower (QUDA), Accelerating lattice 7 QCD multigrid on GPUs using fine-grained paralleliza- tion, inInternational Conference for High Performance Computing, Networking, Storage and Analysis(2016) arXiv:1612.07873 [hep-lat]

    Pith/arXiv arXiv 2016

  58. [58]

    H. Yan, C. Liu, L. Liu, Y. Meng, and H. Xing, Pion mass dependence in Dπscattering and the D0*(2300) resonance from lattice QCD, Phys. Rev. D111, 014503 (2025), arXiv:2404.13479 [hep-lat]

    Pith/arXiv arXiv 2025

  59. [59]

    H. Yan, C. Liu, L. Liu, Y. Meng, and H. Xing, Isospin- 1 2 Dπscattering and theD ∗ 0 resonance, PoSLA T- TICE2023, 055 (2024), arXiv:2312.01078 [hep-lat]

    Pith/arXiv arXiv 2024

  60. [60]

    J. J. Dudek, R. G. Edwards, and C. E. Thomas, S and D- wave phase shifts in isospin-2 pi pi scattering from lattice QCD, Phys. Rev. D86, 034031 (2012), arXiv:1203.6041 [hep-ph]

    Pith/arXiv arXiv 2012

  61. [61]

    Carrascoet al.(European Twisted Mass), Up, down, strange and charm quark masses with N f = 2+1+1 twisted mass lattice QCD, Nucl

    N. Carrascoet al.(European Twisted Mass), Up, down, strange and charm quark masses with N f = 2+1+1 twisted mass lattice QCD, Nucl. Phys. B887, 19 (2014), arXiv:1403.4504 [hep-lat]. 8 SUPPLEMENT AL MA TERIAL

    Pith/arXiv arXiv 2014

  62. [62]

    We employ only local, light-flavor operators of the formO= ¯l′Γl, wherel, l′ ∈u, d and Γ determines the Dirac quantum number

    Operator constructions The operators used to interpolate the single-hadron, two-body, and three-body systems are constructed from linear combinations of quark bilinears. We employ only local, light-flavor operators of the formO= ¯l′Γl, wherel, l′ ∈u, d and Γ determines the Dirac quantum number. We define the elementary building blocksσ l = ¯l1l,ρ l i = ¯l...

  63. [63]

    Instead, we display all topologies of the contraction diagrams

    T opologies of the contraction diagrams Since the number of contraction diagrams is on the order of several hundred, it is impractical to present explicit expressions for all correlation functions. Instead, we display all topologies of the contraction diagrams. For theI= 1 πππchannel, the topologies for all operator types are shown in Figs. S1, S2, S3, S4...

  64. [64]

    Finite-volume spectra We provide technical details and fitting plots for the extraction of finite-volume energies. The generalized eigenvalue problem (GEVP) of the correlation matrixC ij reads C(t)v n(t, t0) =λ n(t, t0)C(t 0)vn(t, t0),(S8) whereλ n(t, t0) andv n(t, t0) are the eigenvalues and eigenvectors, respectively. The reference timet 0 is chosen as ...

  65. [65]

    Two- and three-body input Best fit parameters of the combined fits to two- and three-body finite-volume spectra are provided in Table S2. These include fits to either heavy, light, or both ensembles using the modified Inverse Amplitude method (mIAM) [44] for the two-body part and three-body force of the general form cαβ =c αβ c + cαβ p s−m 2 0 , α=β∈ {σπ,...

  66. [66]

    Model average procedure To find the average across the different results of the lattice data analyses, we employ the procedure developed in Ref. [57]. Namely, starting fromNcomputations with mean valuesx k and uncertaintiesσ x,k (k= 1,· · ·, N) their averagexand uncertaintyσ x are given by x= NX k=1 ωk xk , σ 2 x =σ 2 x,stat +σ 2 x,syst , σ 2 x,stat = NX ...