Revisiting Roy-Steiner-equation analysis of pion-kaon scattering from lattice QCD data

Feng-Kun Guo; Qu-Zhi Li; Xiong-Hui Cao; Zhi-Hui Guo

REVIEW 3 major objections 5 minor 130 references

Revisiting Roy-Steiner-equation analysis of pion-kaon scattering from lattice QCD data

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Kappa meson is a broad resonance at m_pi=391 MeV, not a virtual state

desk verdict A technically strong Roy-Steiner reanalysis of HSC lattice data makes the κ at mπ = 391 MeV a broad resonance, but the claim is conditional on the σKK coupling fixed only by a no-cusp condition. read the letter →

arxiv 2506.10619 v2 pith:X6WG34SW submitted 2025-06-12 hep-ph

Xiong-Hui Cao , Feng-Kun Guo , Zhi-Hui Guo , Qu-Zhi Li This is my paper

classification hep-ph

keywords pion-kaonscatteringkapparesonanceRoy-SteinerequationslatticeQCDdispersionrelationsleft-handcutslengthsanalyticcontinuation

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 argues that the scalar strange meson $\kappa$ (also called $K_0^*(700)$) remains a broad resonance when the pion mass is raised to 391 MeV, rather than becoming the virtual-state pole that previous K-matrix analyses of the same lattice data reported. The authors solve Roy-Steiner-type dispersion equations for pion-kaon scattering, which enforce analyticity, unitarity, and crossing symmetry simultaneously, with the $\sigma$ and $K^*(892)$ treated as bound states at this heavy pion mass. Their result is a $\kappa$ pole at $\sqrt{s_\kappa}=(966_{-24}^{+41}-i198_{-17}^{+38})$ MeV, together with S-wave scattering lengths $m_\pi a_0^{1/2}=0.92_{-0.28}^{+0.06}$ and $m_\pi a_0^{3/2}=-(0.32_{-0.02}^{+0.05})$. If the analysis is right, the earlier virtual-state conclusion is an artifact of neglecting left-hand cuts, so extracting non-ordinary resonances from lattice data requires crossing-symmetric continuation methods.

What carries the argument

The machinery is the pair of s-channel and t-channel Roy-Steiner-type partial-wave equations built from twice-subtracted hyperbolic dispersion relations (with one subtraction for the isospin-odd amplitude), solved iteratively using Muskhelishvili-Omnès techniques in the t-channel and a Schenk-like phase-shift parametrization in the s-channel. The equations include explicit bound-state pole terms for $K^*(892)$ and $\sigma$, and their partial-wave projection generates the left-hand cuts whose positions are computed and displayed. The solution is made unique by demanding no-cusp behavior at the matching point, which pins the sign and value of the $\sigma K\bar K$ coupling ($g_{\sigma K\bar K}=-(296_{-44}^{+45})$ MeV) and fixes the two scattering lengths and the $K^*$ residue. A new general framework for the complex validity domain of the hyperbolic equations, for arbitrary hyperbola parameter $a$, is also established.

What would settle it

Re-run the Roy-Steiner equations with the sign of $g_{\sigma K\bar K}$ flipped to $+296$ MeV while refitting the lattice phase shifts; if the $I=1/2$ S-wave S-matrix still has a zero near $\sqrt{s}=966-i198$ MeV, the paper's conclusion is robust, and if the zero moves to the virtual-state sheet or disappears, the central claim is falsified.

Watch

Extended reading notes

Core claim

The central discovery is that at $m_\pi=391$ MeV the $I=1/2$ S-wave $\pi K$ amplitude, continued into the complex energy plane with Roy-Steiner-type equations derived from crossing-symmetric hyperbolic dispersion relations, has a second-sheet pole at $\sqrt{s_\kappa}=(966_{-24}^{+41}-i198_{-17}^{+38})$ MeV with coupling $g_{\kappa\pi K}=759_{-25}^{+63} e^{-i(1.05_{-0.06}^{+0.14})}$ MeV. This is a broad resonance above threshold, not the deep virtual-state pole near 600 to 750 MeV obtained from K-matrix fits of the same lattice phase shifts. The authors trace the discrepancy to left-hand cuts generated by the t-channel $\sigma$ bound state and the u-channel $K^*$ bound state, which the K-matrix parameterizations effectively discard. Crossing symmetry is therefore not a refinement but the deciding input for this pole.

Load-bearing premise

The analysis stands on the assumption that the sign and magnitude of the $\sigma$ coupling to kaons, $g_{\sigma K\bar K}$, is correctly fixed by requiring no cusp in the t-channel amplitude at the matching point, even though the paper notes the sign cannot be definitively determined from data and the $\kappa$ pole sits near the cuts this coupling controls.

Editorial extensions

If this is right

The previously reported virtual-state $\kappa$ at $m_\pi=391$ MeV should be replaced by a broad resonance pole, so resonance classifications derived from K-matrix fits of these lattice data need revision.
The S-wave scattering lengths at this pion mass, $m_\pi a_0^{1/2}=0.92_{-0.28}^{+0.06}$ and $m_\pi a_0^{3/2}=-(0.32_{-0.02}^{+0.05})$, fall outside the NLO chiral perturbation theory band, indicating poor convergence of the chiral expansion at this pion mass.
The t-channel $\pi\pi\to K\bar K$ S-wave amplitude obtained from the Roy-Steiner solution satisfies the unitarity bound and agrees with lattice results near threshold, providing a consistency check for future simulations.
The no-cusp matching condition combined with the dispersion equations fixes the $\sigma K\bar K$ coupling to $g_{\sigma K\bar K}=-(296_{-44}^{+45})$ MeV, a quantity that lattice data alone had not pinned down.
The new complex validity-domain analysis shows that when the hyperbola parameter $a\neq 0$, the equations cannot be continued into the vicinity of $s=0$, so $a=0$ is the preferred setup for analytic continuation.

Reading between the lines

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

If confirmed, the same crossing-symmetric continuation method could be applied to other non-ordinary resonances extracted from lattice phase shifts, such as $D_0^*(2300)$, where K-matrix instabilities have also been reported.
The paper's mechanism suggests that at sufficiently heavy pion masses the distinction between resonance and virtual state can flip purely because crossing-symmetric cuts move poles across thresholds; mapping the $\kappa$ pole trajectory as a function of $m_\pi$ from 139 to 391 MeV would directly test that picture.
A direct lattice computation of the $\sigma K\bar K$ coupling, for example through $K\bar K$ scattering or three-point functions evaluated near the $\sigma$ pole, could convert the no-cusp assumption into a tested input and would strengthen or weaken the resonance conclusion.
The new complex-validity-domain criterion likely transfers to Roy-Steiner analyses of pion-nucleon scattering, where earlier treatments with nonzero hyperbola parameter may have continued into excluded regions near $s=0$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know up front. First, this is a serious technical paper: explicit kernels for the non-minimal Roy-Steiner equations with a nonzero hyperbolic parameter, corrected typos from the earlier literature, and a general complex validity-domain framework that fills a real gap—previous works with a ≠ 0 missed the exclusion region near s = 0. Second, the headline physics—κ is a broad resonance, √sκ = 966 − i198 MeV, not the virtual state HSC found—is plausible but not settled. The load-bearing coupling g_σKK = −296 ± 45 MeV is fixed only by a no-cusp condition at the t-channel matching point, and the paper says outright (Sec. VI.A) that the sign cannot be determined from data.

Credit where due. The numerical solution satisfies the RS equations to χ² ~ 10⁻³. The singularity analysis is honest: Sec. II.E states the κ pole sits at comparable distances from the unitarity cut and the LHCs and is sensitive to them, which is exactly why the σKK input matters. The physics claim itself was announced in the companion letter [70]; what this paper adds is the machinery, the validity-domain analysis, and the detailed account of the continuation.

The soft spots, in proportion. The g_σKK issue is the main one, but the authors are transparent about it, and the no-cusp condition is a defensible way to fix a free parameter. The real gap is that ±45 MeV only covers the bootstrap spread under that constraint, not the allowed 0–900 MeV range; a different sign or magnitude could move the pole or turn it back into a virtual state. That keeps the κ claim conditional. I would push back gently on the reader's circularity point: the reliance on the group's earlier Roy analysis [38] for √sσ and g_σππ is fine—that is a published, separate ππ study. The circularity that actually bites is internal: g_σKK is fixed by smoothness at an artificial matching point (the authors note it cannot be pinned down from the s-channel physical region), and it controls the continuation that produces the headline pole. The hand-tuned Regge λ and the P-wave extrapolation above 1.4 GeV are minor; the paper shows they barely affect the S-wave.

Bottom line: this deserves a serious referee and will be useful to dispersion-relation practitioners and to anyone extracting resonances from lattice phase shifts. If I were refereeing, my one demand would be the pole's dependence on g_σKK over its full allowed range, or a code release, before the resonance claim is treated as settled.

Referee Report

3 major / 5 minor

Summary. The paper constructs Roy-Steiner-type dispersion relations for πK→πK and ππ→KKbar scattering at the unphysical pion mass mπ=391 MeV, where both the K*(892) and the σ become bound states. Using HSC lattice phase shifts and moduli as inputs, together with Regge asymptotics and a t-channel Omnès solution, the authors solve the coupled s- and t-channel systems, determine the S-wave scattering lengths, and analytically continue the I=1/2 S-wave amplitude to extract the κ pole. Their central result is that the κ remains a broad resonance, sqrt(s_kappa)=966^{+41}_{-24}-i198^{+38}_{-17} MeV, in contrast to the virtual-state pole found in the HSC K-matrix analyses. The paper also develops a general method for the complex validity domain of hyperbolic dispersion relations for arbitrary hyperbola parameter a, and gives explicit kernel functions corrected and extended relative to earlier literature.

Significance. If the central claim survives scrutiny, this is an important result: it would show that a dispersive treatment respecting crossing symmetry and including left-hand cuts can change the interpretation of lattice phase shifts for a non-ordinary resonance, and it would provide a concrete warning against unconstrained K-matrix continuations in the presence of bound-state poles in crossed channels. The paper is technically substantial: it derives many kernel functions, gives a systematic validity-domain framework, and reports a numerical solution satisfying the RS equations to chi^2 ~ 10^-3. The extracted phase shifts, scattering lengths, and pole parameters provide concrete predictions for future lattice studies. However, the main physical conclusion depends on a coupling, g_{sigma KK}, that is not determined by the lattice data and is fixed only through a smoothness condition at an artificial matching point; this makes the central claim currently less robust than the error bars in Eq. (114) suggest.

major comments (3)

[Sec. VI.A, Sec. VI.D, Eq. (114)] The central claim that κ is a broad resonance rather than a virtual state rests on the σ-induced left-hand cut, whose strength is controlled by g_{σKK}. The paper states in Sec. VI.A that the sign of g_{σKK}=±(0~900) MeV 'cannot even be definitively determined' from data, and Sec. VI.D fixes g_{σKK}=-296±45 MeV only by the no-cusp condition for g_0^0 at the t-channel matching point. Section II.E explicitly says that the κ pole sits at similar distances from the unitarity cut and the LHCs, especially those induced by the σ bound state, and Eqs. (A59)-(A60) show that the σ-pole term enters directly into the S-wave partial-wave projection. The quoted uncertainty in Sec. VI.D is the bootstrap spread under the no-cusp constraint; it does not include the alternate sign or couplings outside the fitted range. I request an explicit scan over the sign of g_{σKK} and over its magnitude, with the no-cusp condition re-imposed where possible, with the resulting κ pole reported. Unless the broad-resonance conclusion is stable under that scan, Eq. (114) is not supported.
[Sec. VI.A-D] The uniqueness of the RS solution is established by imposing no-cusp conditions at the s- and t-channel matching points s_m and t_m. These matching points are introduced for numerical convenience, and the no-cusp requirement is an external smoothness assumption rather than a consequence of unitarity, analyticity, or crossing symmetry. The paper explains in Sec. VI.A that without this condition the s-channel system has a one-parameter family of solutions. Because the choice of matching point and the smoothness criterion can move the solution within that family, the paper should demonstrate that the κ pole is stable under (i) replacing the no-cusp condition by a different smoothness or minimal-variation criterion at s_m, and (ii) varying √t_m over the full range allowed by the unitarity bound, not only 1.18-1.22 GeV as in Sec. VI.D. Without such a check, the selection of the physical solution, and therefore the pole position in Eq. (114), is not uniquely determined.
[Sec. VII, Fig. 13] The conclusion that the HSC K-matrix analyses 'almost completely neglect the influence of LHCs, resulting in an inaccurate description' is stronger than what the comparison actually shows. Figure 13 demonstrates that the RS solution reproduces the HSC phase shifts within uncertainties on the real axis; the difference in the κ interpretation is generated by the analytic continuation, not by a direct disagreement in the physical region. The paper should either provide a test showing that a K-matrix parametrization augmented by the σ LHC cannot reproduce the RS pole, or soften the wording in Sec. VII to state that the RS-type analytic continuation favors a broad resonance. This distinction matters because the real-axis phase shifts alone do not select between the two pole interpretations.

minor comments (5)

[Around Figs. 5 and 6] The text contains a Chinese-language paragraph and duplicated figure captions that appear to be remnants of an earlier draft; these must be removed or translated before publication.
[Abstract and Sec. IV] The abstract says 'all important partial waves' are analyzed, but the numerical solution explicitly solves only the (I,J)=(1/2,0), (1/2,1), and (3/2,0) channels; the wording should be adjusted to match the actual scope.
[Sec. IV.B, Eq. (88)] The Regge/Pomeron parameters b_P and σ_P are taken from the physical pion mass with an ad hoc scaling σ_P~0.7σ_phys. Since the Regge coupling λ is the only parameter varied in the bootstrap, a sentence quantifying the isolated effect of the Regge-model choice on the κ pole would help the reader judge the sensitivity.
[Fig. 12] The legend entry 'HadSpec, 18' is unclear; it should be labeled consistently with the reference list, e.g., 'HSC Ref. [37]', and the same applies to the data source in Fig. 13.
[Sec. VI.C, Eq. (109)] The parameterization imposes continuity of the phase shift and its derivative at s_m, but the relation between the parameters C_J and D_J and the input derivative δ'_{m,J} should be checked for sign conventions; a brief derivation or reference would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the κ pole is an output of the coupled Roy-Steiner equations, and the acknowledged g_{σKK} uncertainty is a robustness concern rather than a circular reduction.

full rationale

The paper's central claim—that the κ state at m_π = 391 MeV is a broad resonance rather than the virtual state found by HSC—is not equivalent to its inputs by construction. The pole is obtained by solving the coupled s- and t-channel RS-type equations (Eqs. 38, 39, 42) with inputs that do not include the pole position: lattice phase shifts above the matching point s_m, Regge amplitudes matched near 1.8 GeV, and t-channel phases and σ parameters taken from the earlier Roy-equation analysis of ππ scattering [38]. The zero of S_0^{1/2}(s) below threshold is then searched for in the complex plane and emerges as an output of the integral equations, not as a fitted parameter. The scattering lengths and g_{K*πK} are free parameters of the equations fixed by requiring a smooth physical solution at the matching point, and the quoted κ pole is not used to determine them. The σKK coupling is fixed by a no-cusp condition at the t-channel matching point, which is an internal consistency requirement and not a surrogate for the κ pole; the paper explicitly states that its sign 'cannot even be definitively determined' and that its precise value cannot be determined from the physical s-channel region. That is a genuine robustness and uncertainty-coverage problem, but it is not a circular reduction of the kind defined here. The self-citations [38] and [70] overlap with the authors, but [38] is an independent, earlier dispersive analysis of ππ lattice data whose assumptions do not include the κ pole, and [70] is the companion letter reporting the same calculation; neither introduces the κ result as an input. The central derivation is therefore self-contained against the external lattice and Roy-equation benchmarks, and the most serious limitation is the underdetermined σKK sign, which should be assessed as a systematic risk rather than as circularity.

Assumptions & free parameters 10 free parameters · 8 assumptions · 0 invented entities

The central claim rests on a large body of prior formalism and inputs. The free parameters are the Regge coupling λ (fitted to lattice amplitudes), the σ pole parameters (from the same group's earlier Roy analysis), the σKK coupling (fixed by a no-cusp condition), the K* coupling and scattering lengths (tuned to make the RS equations self-consistent), plus a number of hand-chosen matching points and Schenk coefficients. The axioms include standard S-matrix analyticity, unitarity, and crossing symmetry, the Mandelstam double-spectral representation, and more ad hoc assumptions about Regge asymptotics, elastic extrapolation, and the no-cusp uniqueness condition. No new particles are introduced.

free parameters (10)

Regge coupling λ = 10^{+10}_{-5}
Sec. IV.B; matched to LQCD amplitudes at sqrt(s_c) = 1.8 GeV; enters Regge contributions to driving terms.
σ pole position sqrt(s_σ) = 759^{+7}_{-16} MeV
Input from Ref. [38] (same group's Roy analysis); controls σ pole term and LHC.
σππ coupling g_σππ = 493^{+27}_{-46} MeV
Input from Ref. [38]; used in pole terms.
σKK coupling g_σKK = -296^{+45}_{-44} MeV
Determined by no-cusp condition at t-channel matching point, Sec. VI; sign not fixed by data.
K*πK coupling g_K*πK = 35^{+5}_{-7} MeV (final)
Tuned parameter in solving RS equations; initial value 51 MeV from HSC.
S-wave scattering length m_pi a0^{1/2} = 0.92^{+0.06}_{-0.28}
Output of the RS solution, but a free parameter in the numerical optimization; correlated with a0^{3/2}.
S-wave scattering length m_pi a0^{3/2} = -0.32^{+0.05}_{-0.02}
Same as above; output and free parameter.
Schenk parameterization coefficients B_I^J = not quoted
Free parameters in the phase-shift parameterization optimized to satisfy the RS equations, Sec. VI.C.
t-channel matching points sqrt(t_m) = 1.2 GeV (S-wave), 1.7 GeV (P-wave)
Chosen by hand; S-wave value restricted to 1.18-1.22 GeV to maintain unitarity bound, Sec. VI.D.
s-channel matching point sqrt(s_m) = 1.3 GeV
Chosen slightly above ηK threshold and below three-body threshold; phase shifts and derivatives there are inputs.

assumptions (8)

domain assumption Mandelstam double spectral representation holds for πK scattering at m_pi = 391 MeV
Sec. III.C assumes it to locate Lehmann-Martin ellipse boundaries and the complex validity domain; standard but not proven at unphysical masses.
standard math s ↔ u crossing symmetry of the isospin-even and odd amplitudes F±
Used to construct hyperbolic DRs and relate s- and t-channel partial waves (Sec. II.B).
standard math Unitarity and Watson's theorem allow a single-channel Muskhelishvili-Omnes form for t-channel g0_0 and g1_1 up to the matching point
Sec. V relies on this to solve the t-channel equations.
ad hoc to paper Regge/Veneziano model with linear degenerate trajectories and a fitted Pomeron term describes the high-energy asymptotics at this unphysical pion mass
Sec. IV.B; no cross-section data at m_pi = 391 MeV, so the asymptotic amplitudes are modeled.
domain assumption The I = 1/2 P-wave πK phase shift remains elastic and can be linearly extrapolated from 1.4 to 1.8 GeV
Sec. IV.A; K*(1410) coupling assumed negligible.
domain assumption ηK inelasticity in the I = 1/2 S-wave is weak enough to be neglected
Sec. IV.A, following HSC observation of smooth phase behavior.
domain assumption The σ and K* bound-state pole terms plus their generated left-hand cuts are the only new singular structures needed at m_pi = 391 MeV
Sec. II.E; motivated by HSC spectra and previous Roy analysis, but the completeness of this singularity set is assumed.
ad hoc to paper The no-cusp condition at the s-channel matching point selects the unique physical solution among the one-parameter family
Sec. VI.A-B; used to fix the underdetermined system; mathematical necessity not independently established.

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Cite this review

Pith. "Pith review of Revisiting Roy-Steiner-equation analysis of pion-kaon scattering from lattice QCD data." pith.science (2026). https://pith.science/paper/X6WG34SW

@misc{pith2026250610619,
  author       = {Pith},
  title        = {Pith review of: Revisiting Roy-Steiner-equation analysis of pion-kaon scattering from lattice QCD data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X6WG34SW}},
  note         = {Machine review of arXiv:2506.10619}
}

abstract

A comprehensive analysis of $\pi K\rightarrow \pi K$ and $\pi\pi\rightarrow K\bar K$ amplitudes at large unphysical pion mass for all important partial waves is presented. A set of crossing-symmetric partial-wave hyperbolic dispersion relations is used to describe lattice QCD data at $m_\pi=391$ MeV. In the present analysis, the amplitudes for the $S$- and $P$-waves are formulated by combining the constraints of analyticity, unitarity, and crossing symmetry, fulfilling Roy-Steiner-type equations. We use these results to investigate the low-lying strange-meson resonances and resolve the instability problem tied to analytic continuation in prior lattice QCD studies based on the $K$-matrix formalism. At $m_\pi=391$ MeV, the rigorous Roy-Steiner-type equation approach allows us to determine the $S$-wave scattering lengths, $m_\pi a_0^{1/2}=\left(0.92_{-0.28}^{+0.06}\right)$, $m_\pi a_0^{3/2}=-\left(0.32_{-0.02}^{+0.05}\right)$, and the $\kappa$ (also known as $K_0^*(700)$) pole position, $\sqrt{s_\kappa}=\left(966_{-24}^{+41}-i 198_{-17}^{+38}\right)$ MeV. We also provide a detailed analysis of the complex validity domain of the Roy-Steiner-type equations.

Figures

Figures reproduced from arXiv: 2506.10619 by the authors.

**Figure 1.** The cut structure of the πK partial wave amplitudes in the complex s plane for the physical mπ = 139 MeV case (left) and the unphysical mπ = 391 MeV case (right). As the masses of the u and d quarks are gradually increased (while keeping the s quark mass fixed) until mπ ∼ 391 MeV (and mK ∼ 549 MeV), the scattering amplitude exhibits two new features. First, the I = 1/2 P-wave of πK scattering shows the emergence of … view at source ↗

**Figure 2.** FIG. 2. Two LM ellipses: the red one encompasses the origin, while the blue one does not. Lines extending [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

**Figure 3.** FIG. 3. Complex validity domain for the [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗

Figures from the paper (16 more)

**Figure 4.** Figure 4: FIG. 4. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]

**Figure 5.** Figure 5: FIG. 5. Unitarity box diagrams that are used to calculate the double spectral regions of [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]

**Figure 6.** Figure 6: FIG. 6. Complex validity domain of the RS-type equation for [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]

**Figure 7.** Figure 7: FIG. 7. Comparison of Im [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]

**Figure 8.** Figure 8: FIG. 8. Decomposition of the DTs in terms of LQCD input, extrapolated part between 1.4 and 1.8 GeV [PITH_FULL_IMAGE:figures/full_fig_p030_8.png]

**Figure 9.** Figure 9: FIG. 9. The moduli of the Omn`es functions for ( [PITH_FULL_IMAGE:figures/full_fig_p032_9.png]

**Figure 10.** Figure 10: FIG. 10. Comparison of the left-hand sides (lines) and right-hand sides (crosses) of the RS-type equations [PITH_FULL_IMAGE:figures/full_fig_p038_10.png]

**Figure 11.** Figure 11: FIG. 11. Decomposition of the right-hand sides of RS-type equations into different contributions. Black [PITH_FULL_IMAGE:figures/full_fig_p039_11.png]

**Figure 12.** Figure 12: FIG. 12. Comparison of the absolute values of [PITH_FULL_IMAGE:figures/full_fig_p041_12.png]

**Figure 13.** Figure 13: FIG. 13. Phase shifts for the ( [PITH_FULL_IMAGE:figures/full_fig_p042_13.png]

**Figure 14.** Figure 14: FIG. 14. Comparison of the scattering lengths between determinations from solving the RS-type equations [PITH_FULL_IMAGE:figures/full_fig_p043_14.png]

**Figure 15.** Figure 15: FIG. 15. The [PITH_FULL_IMAGE:figures/full_fig_p056_15.png]

**Figure 16.** Figure 16: FIG. 16. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p057_16.png]

**Figure 17.** Figure 17: FIG. 17. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p058_17.png]

**Figure 18.** Figure 18: FIG. 18. The NLO ChPT predictions of the [PITH_FULL_IMAGE:figures/full_fig_p058_18.png]

**Figure 19.** Figure 19: FIG. 19. Qualitative evolution of [PITH_FULL_IMAGE:figures/full_fig_p059_19.png]

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Reference graph

Works this paper leans on

130 extracted references · 16 canonical work pages

[70]

Hoferichter, J

M. Hoferichter, J. Ruiz de Elvira, B. Kubis, and U.-G. Meißner, Roy–Steiner-equation analysis of pion–nucleon scattering, Phys. Rept. 625, 1 (2016), arXiv:1510.06039 [hep-ph]

arXiv 2016
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D¨ oring and U.-G

M. D¨ oring and U.-G. Meißner, Finite volume effects in pion-kaon scattering and reconstruction of the κ(800) resonance, JHEP 01, 009, arXiv:1111.0616 [hep-lat]

arXiv
[1]

We derived the kernel functions for the equations with a ̸= 0, many of which are presented for the first time in this work

We established a framework to construct the non-minimal subtracted Roy-Steiner-type equa- tions for πK scattering at unphysical large pion masses when σ/f0(500) and K∗(892) both become bound state poles. We derived the kernel functions for the equations with a ̸= 0, many of which are presented for the first time in this work. In addition, we introduced a ...
[2]

Consequently, thet-channel equations can be solved, which in turn reformulates the s-channel equations into Roy-like equations

In the t-channel system, unitarity, as implemented through Muskhelishvili–Omn` es tech- niques, provides the necessary connection at least up to the onset of inelastic contributions, and even beyond if inelasticities are sufficiently well constrained. Consequently, thet-channel equations can be solved, which in turn reformulates the s-channel equations in...
[3]

We have demonstrated that the restrictions from crossing symmetry play a crucial role in πK scattering at large pion masses, especially in the presence of bound state poles

The most important finding is that the κ state at mπ = 391 MeV is not a deep virtual- state pole below the πK threshold [21], but rather remains a broad resonance [70]. We have demonstrated that the restrictions from crossing symmetry play a crucial role in πK scattering at large pion masses, especially in the presence of bound state poles. Anticipated im...
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R. A. Brice˜ no, J. J. Dudek, and R. D. Young, Scattering processes and resonances from lattice QCD, Rev. Mod. Phys. 90, 025001 (2018), arXiv:1706.06223 [hep-lat]. 60

arXiv 2018
[5]

Partial-wave projection for the t-channel amplitudes We start by enumerating all kernels that appear in the equation of g1
[6]

3 − 4qK(t)qπ(t) 2s′ + t − 2Σ 2! A t, s′ − 3 4qK(t)qπ(t) 2s′ + t − 2Σ # , (A32) G+ 2,1 t, s′ = 3 √ 3(2s′ + t − 2Σ)2 32(qK(t)qπ(t))5 P1 zs s′, t

According to Eq. (15), the partial-wave projection of the t-channel for F − (we only need the P -wave) can be written as g1 1(t) = √ 2 16πqπqK Z 1 0 dzt ztF −(s, t) = √ 2 4π Z 1 0 dzt z2 t F −(s, t) s − u . (A1) First, we perform the t-channel P -wave partial-wave projection using the above equation on both sides of Eq. (32). The left-hand side of Eq. (32...

Show all 130 references

[7]

Partial-wave projection for the s-channel amplitudes We will continue to start withF −. In Eq. (32) (where both sides of the expression are previously multiplied by the factor s − u), the first subtraction terms read [46] ST− J (s) = δJ,0 m+a− 0 2(m2 + − m2 −) 3s2 − 2Σs − ∆2 2...
[8]

In the ( I, J) = (1/2, 0) channel, there are two Adler zeros [113], which at leading order (LO) of ChPT are given by s1/2 A± = 1 5 m2 K + m2 π ± 2 q 4m4 K − 7m2 Km2π + 4m4π

(I, J) = (1/2, 0) channel First, let us recall the prediction of ChPT. In the ( I, J) = (1/2, 0) channel, there are two Adler zeros [113], which at leading order (LO) of ChPT are given by s1/2 A± = 1 5 m2 K + m2 π ± 2 q 4m4 K − 7m2 Km2π + 4m4π . (B1) Note that only one of them...
[9]

Furthermore, as illustrated in Fig

(I, J) = (1/2, 1) channel In the ( I, J) = (1 /2, 1) channel, there is a P -wave BS below the threshold, which is always accompanied by a VS pole at nearly the same position on the second sheet. Furthermore, as illustrated in Fig. 16, the RS-type equation analysis reveals thre...
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However, this is not the case

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Reviewed August 7, 2026 · model on record in the stance chip above.

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We derived the kernel functions for the equations with a ̸= 0, many of which are presented for the first time in this work

We established a framework to construct the non-minimal subtracted Roy-Steiner-type equa- tions for πK scattering at unphysical large pion masses when σ/f0(500) and K∗(892) both become bound state poles. We derived the kernel functions for the equations with a ̸= 0, many of which are presented for the first time in this work. In addition, we introduced a ...

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Consequently, thet-channel equations can be solved, which in turn reformulates the s-channel equations into Roy-like equations

In the t-channel system, unitarity, as implemented through Muskhelishvili–Omn` es tech- niques, provides the necessary connection at least up to the onset of inelastic contributions, and even beyond if inelasticities are sufficiently well constrained. Consequently, thet-channel equations can be solved, which in turn reformulates the s-channel equations in...

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We have demonstrated that the restrictions from crossing symmetry play a crucial role in πK scattering at large pion masses, especially in the presence of bound state poles

The most important finding is that the κ state at mπ = 391 MeV is not a deep virtual- state pole below the πK threshold [21], but rather remains a broad resonance [70]. We have demonstrated that the restrictions from crossing symmetry play a crucial role in πK scattering at large pion masses, especially in the presence of bound state poles. Anticipated im...

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Partial-wave projection for the t-channel amplitudes We start by enumerating all kernels that appear in the equation of g1

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3 − 4qK(t)qπ(t) 2s′ + t − 2Σ 2! A t, s′ − 3 4qK(t)qπ(t) 2s′ + t − 2Σ # , (A32) G+ 2,1 t, s′ = 3 √ 3(2s′ + t − 2Σ)2 32(qK(t)qπ(t))5 P1 zs s′, t

According to Eq. (15), the partial-wave projection of the t-channel for F − (we only need the P -wave) can be written as g1 1(t) = √ 2 16πqπqK Z 1 0 dzt ztF −(s, t) = √ 2 4π Z 1 0 dzt z2 t F −(s, t) s − u . (A1) First, we perform the t-channel P -wave partial-wave projection using the above equation on both sides of Eq. (32). The left-hand side of Eq. (32...