An improved study of the kappa resonance and the non-exotic $s$ wave $\pi K$ scatterings up to $\sqrt{s}=2.1$GeV of LASS data

H. Q. Zheng; Z. Y. Zhou

REVIEW

An improved study of the kappa resonance and the non-exotic $s$ wave $\pi K$ scatterings up to $\sqrt{s}=2.1$GeV of LASS data

Z. Y. Zhou, H. Q. Zheng

High Energy Physics - Phenomenology · Submitted 8 Mar 2006 · 21 references

Pith's verdict: The formal correction to the dispersion relation is real; the kappa pole claim is plausible but leans on a chiral left-hand-cut input the paper itself concedes is doubtful.

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arXiv IDhep-ph/0603062 (version 2)

Pith IDJLPWJWUW, Pith's permanent record of this paper and its review

CitationH. Q. Zheng, Z. Y. Zhou. "An improved study of the kappa resonance and the non-exotic $s$ wave $\pi K$ scatterings up to $\sqrt{s}=2.1$GeV of LASS data." arXiv:hep-ph/0603062 (2006). https://pith.science/paper/JLPWJWUW

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Editor's note

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Dear [--],

Here is my read on Zhou and Zheng (hep-ph/0603062).

The genuinely new piece is the observation that the left-hand-cut dispersion relation has no subtraction constant even for unequal-mass scatterings. The argument via polynomial boundedness of the amplitude and f(0)=0 is a meaningful correction to their earlier paper, and it removes a free parameter from the fit. That is a real formal result, and it is the strongest part of the paper.

What the paper does well beyond that: the fits are presented with enough detail to follow, and the chi2 comparison with and without the kappa pole (80/54 vs 1055/56, and the analogous high-energy version) shows that within their parametrization the pole is not optional. The authors are also unusually straightforward about what could undermine the claim. They say plainly that if the I=1/2 background were estimated to be positive and convex, the existence of the kappa would be severely doubted. That sentence is the key to the whole paper.

Where the soft spots are, in order of importance:

First, the left-hand-cut background is computed from one-loop chiral perturbation theory, and the sign and concavity of that background do the heavy lifting. The robustness argument—pomeron dominance gives |S|>1, so the left-cut phase is negative and concave—is a plausible heuristic, but it does not prove that the chiral estimate has the correct sign and concavity in the integration region that controls the fit. The large cutoff Lambda_{1/2} with its huge error shows only that the overall size is not pinned down by LASS data, not that the sign is.

Second, the threshold parameters are outputs of a fit whose left-hand-cut input is itself chiral. So the agreement with Roy–Steiner and chiral results is partly inherited. The authors acknowledge this implicitly by saying the scattering lengths are no longer free, but they do not fully address the circularity worry.

Third, the data handling: some LASS points are excluded on unitarity grounds, and solution B is dropped. This is defensible, but it would be worth a robustness test where the excluded points are kept and the fits re-done.

The K0*(1430) and K0*(1950) results are fine as far as they go, and the no-twin-pole conclusion is clearly stated as a statement about LASS data in this parametrization, not a general theorem.

Net: this is a serious, careful paper with a real formal correction and an interesting but conditional claim about the kappa. It should go to peer review. The referee should push on the sensitivity of the pole to the sign and shape of the left-hand-cut background, and ask for an explicit test varying the chiral input, if that is possible.

I would probably cite the formal dispersion-relation point if I work on scalar mesons, and it is a decent reading-group paper for hadron spectroscopy.

Referee report

Significance

If the central claim is accepted, the paper provides a significant addition to the long-standing debate on the κ meson: it uses a manifestly analytic and unitary parametrization to show that a broad low-mass pole is required to describe LASS data. The strengths of the work include the internal consistency of the fits, explicit robustness checks against variations of the cutoff and the fitted energy range, the dramatic deterioration of χ² when the κ pole is removed, and the authors' frank discussion of the assumptions and limitations of their scheme. The main weakness is that the existence of the κ pole is established only within a framework where the left-hand-cut background is computed from one-loop chiral perturbation theory; the authors themselves acknowledge that if that background had the opposite sign and curvature, the κ pole would be severely doubted. The circularity in using the same chiral framework both to build the background and to validate the threshold parameters is an additional concern. These issues make the claim of 'firm evidence' conditional on an input that is not fully tested.

Major objections (3)

[§2, Eq. (11) and the following paragraph] The existence of the κ pole is not established by the LASS data alone but by the combination of the data and a specific chiral estimate of the left-hand-cut background. The authors concede in the paragraph after Eq. (11) that if the I=1/2, J=0 background contribution 'were estimated to be positive and convex,' the existence of the κ pole 'would be severely doubted.' The two arguments offered for the robustness of the negative and concave behavior — pomeron dominance giving |Sphy|>1, and the absence of unitarity constraints on the left — are heuristic and are not derivations for the integration region that controls the fit. Because the no-κ fits (χ²/dof=1055/56 and 3726/66) use the same chiral background, they cannot test this input. Please provide a quantitative sensitivity test, for example by perturbing the sign or curvature of the left-hand-cut amplitude within a class of admissible forms, and showing that the κ pole remains required by the data.
[§2, Eqs. (12)–(13) and the comparison with Refs. [13,14]] The threshold parameters a0 and b0 are presented as outputs that agree with chiral and Roy–Steiner predictions, but the same chiral perturbation theory is used to compute the left-hand-cut integrals that enter the fit. The agreement is therefore partly inherited from the input rather than being an independent consequence of analyticity and the data. This weakens the statement that 'the correct use of analyticity also put strong constraints on threshold parameters.' Please quantify the sensitivity by, for instance, redoing the fit with the left-hand cut obtained from the Roy–Steiner or dispersive representation, or by removing the chiral input and checking how much of the threshold-parameter agreement survives.
[Appendix, Eqs. (19)–(22)] The proof of f(0)=0 is sketched rather than fully rigorous. The argument that T^I_J(s)=O(s^{-n}) from polynomial boundedness is plausible, but the step to Eq. (9) also requires control of the pole sum on the right-hand side of Eq. (8) uniformly as s→0; this is not explicitly shown. In addition, the theorem invoked for Eq. (21) is stated without a reference and its hypotheses are not verified for the isospin amplitude. Since the entire subtraction-free scheme relies on f(0)=0, please either provide a complete proof, with explicit bounds on the pole term, or state the precise assumptions under which Eq. (9) is expected to hold.

Minor objections (5)

[Eq. (12) and Fig. 2] The sign convention for the cutoff Λ² is confusing: Eq. (12) reports Λ²_{1/2} and Λ²_{3/2} as negative, while the fixed-cutoff tests and the Fig. 2 caption write Λ²_L=1.5 GeV² with a positive sign. Please clarify whether the plotted and fitted quantities differ by a sign convention or a typo.
[Eq. (17)] The expression for a0 e^{iφ0} contains a stray semicolon in the middle of the parentheses; this should be removed for readability.
[Abstract and text] There are several typographical errors: 'one of our previous paper' should be 'one of our previous papers,' 'Gev' in Eq. (12) should be 'GeV,' and 'cutoﬀ' appears with a nonstandard character. These should be corrected in a final revision.
[Notation, second and third sheet poles] The notation K0^II(1430) and K0^III(1950) is used without an explicit definition of the Riemann-sheet convention. Please define these terms at first use so the distinction between second-sheet and third-sheet poles is unambiguous.
[Fig. 2 caption] The caption says 'solid line: two–pole fit, dashed line: one pole fit,' but it is not stated which channel the dashed line corresponds to; specify the I=3/2 curves and the meaning of 'one pole fit' in the caption.

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The paper

The paper claims

A subtraction-free dispersion analysis of LASS data forces the kappa resonance into existence.

The reading

This paper argues that when the analytic structure of pion–kaon scattering is treated correctly, the LASS phase-shift data force the existence of a broad, low-mass scalar meson, the $\kappa$, with mass about $694$ MeV and width about $606$ MeV. The key step is to show that the dispersion integral for the left-hand cut requires no subtraction constant, so the threshold parameters are no longer free inputs but consequences of the fit. With that constraint, dropping the $\kappa$ pole degrades the fit from $\chi^2/\mathrm{d.o.f.}=80.3/54$ to $1055/56$, which the authors read as firm evidence for the resonance. If the analysis is right, the $\kappa$ joins the $\sigma$ as a firmly established light scalar, and the same LASS data become consistent with chiral symmetry predictions for the threshold parameters.

What carries the argument

The central object is the logarithmic cut function $f(s)$ that arises when the partial-wave $S$-matrix is written as a product of second-sheet pole factors times $e^{2i\rho(s)f(s)}$. Its dispersion integral over the left-hand, circular, and right-hand cuts is evaluated with the subtraction point at $s=0$, and the paper proves $f(0)=0$ using $\rho(s)\sim s^{-1}$ and $T^I_J(s)\sim O(s^{-n})$ as $s\to0$, both consequences of the assumed analyticity of the amplitude. This single identity removes the two subtraction constants from the fit, so the scattering lengths are no longer adjustable, and it converts the negative, concave background phase from chiral estimates into a condition that the data must meet; the $\kappa$ pole is then needed to reproduce the measured rise of the phase shift.

Extended reading notes

Core claim

The central discovery is that the left-hand-cut dispersion relation for unequal-mass elastic scattering is free of subtraction constants: in the $s\to0$ limit the logarithmic cut function satisfies $f(0)=0$, provided the full amplitude satisfies the standard analyticity and polynomial-boundedness assumptions. Feeding this condition into the unitarity-and-analyticity respecting parametrization of the $\pi K$ $S$-matrix and fitting the LASS data up to $2.1$ GeV yields a second-sheet $\kappa$ pole at $M_\kappa=694\pm53$ MeV, $\Gamma_\kappa=606\pm59$ MeV, with an extended fit giving $M_\kappa=649\pm63$ MeV, $\Gamma_\kappa=602\pm22$ MeV. Removing the $\kappa$ pole worsens $\chi^2/\mathrm{d.o.f.}$ from $80.3/54$ to $1055/56$. The fit also produces threshold scattering lengths $a_0^{1/2}=0.219\pm0.034$ and $a_0^{3/2}=-0.042\pm0.002$ that agree with chiral and dispersive determinations, a second-sheet $K_0^*(1430)$ pole near $1443$ MeV, and a third-sheet $K_0^*(1950)$ pole near $1917$ MeV, without requiring the twin-pole structure of a typical resonance shape.

Load-bearing premise

The entire kappa conclusion rests on the chiral estimate that the non-resonant background phase shift in the isospin-1/2, spin-0 channel is negative and concave; if that background were instead positive and convex, the data would no longer force a kappa pole.

Editorial extensions

If this is right

Once analyticity is imposed, the LASS data cannot be described without a $\kappa$ pole: omitting it changes $\chi^2/\mathrm{d.o.f.}$ from $80.3/54$ to $1055/56$.
The same fit yields threshold lengths $a_0^{1/2}=0.219\pm0.034$ and $a_0^{3/2}=-0.042\pm0.002$, in line with chiral and dispersive predictions, so LASS data and chiral symmetry can coexist.
$K_0^*(1430)$ emerges as a second-sheet pole near $1443$ MeV (about $1435$ MeV in the full fit), with no evidence for a third-sheet partner.
$K_0^*(1950)$ is confirmed as a third-sheet pole at about $1917$ MeV, with total width $145\pm38$ MeV and partial $K\pi$ width $87\pm14$ MeV.
Neither $K_0^*(1430)$ nor $K_0^*(1950)$ needs the twin-pole structure of a conventional resonance shape.

Reading between the lines

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

If the subtraction-free identity holds generally, the same dispersion analysis can be applied to other unequal-mass channels to turn chiral predictions into sharp constraints on threshold parameters and on broad low-lying poles.
The result strengthens the case that the light scalar nonet contains a broad $\kappa$; a high-statistics production experiment or a lattice QCD determination of the $\pi K$ $I=1/2$ amplitude could independently confirm or revise the pole position.
Because the authors themselves point to the sign and curvature of the background as the decisive input, a model-independent dispersive evaluation of that background would settle the $\kappa$ question more cleanly than any additional fit to the same LASS data.

Original abstract

We point out that the dispersion relation for the left hand cut integral presented in one of our previous paper (Nucl. Phys. {\bf A}733(2004)235) is actually free of subtraction constant, even for unequal mass elastic scatterings. A new fit to the LASS data~\cite{Aston} is performed and firm evidence for the existence of $\kappa$ pole is found. The correct use of analyticity also put strong constraints on threshold parameters -- which are found to be in good agreement with those obtained from chiral theories. We also determined the pole parameters of $K_0^*(1430)$ on the second sheet, and reconfirm the existence of $K_0^*(1950)$ on the third sheet. We stress that the LASS data do not require them to have the twin pole structure of a typical Breit--Wigner resonance.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.24 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.22 · compares with prior results 0.91 · mentions code or data availability 0.02.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We point out that the dispersion relation for the left hand cut integral presented in one of our previous paper (Nucl. Phys. {\bf A}733(2004)235) is actually free of subtraction constant, even for unequal mass elastic scatterings. A new fit to the LASS data~\cite{Aston} is performed and firm evidence for the existence of $\kappa$ pole is found. The correct use of analyticity also put strong constraints on threshold parameters -- which are found to be in good agreement with those obtained from chiral theories. We also determined the pole parameters of $K_0^*(1430)$ on the second sheet, and reconfirm the existence of $K_0^*(1950)$ on the third sheet. We stress that the LASS data do not require them to have the twin pole structure of a typical Breit--Wigner resonance.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We point out that the dispersion relation for the left hand cut integral presented in one of our previous paper (Nucl.	contribution 0.63	0.41	0.04	asserted 0.78
s1	Phys. {\bf A}733(2004)235) is actually free of subtraction constant, even for unequal mass elastic scatterings.	result 0.57	0.54	0.07	asserted 1.00
s2	A new fit to the LASS data~\cite{Aston} is performed and firm evidence for the existence of $\kappa$ pole is found.	result 0.91	0.96	0.64	asserted 1.00
s3	The correct use of analyticity also put strong constraints on threshold parameters -- which are found to be in good agreement with those obtained from chiral theories.	result 0.87	0.88	0.11	asserted 0.99
s4	We also determined the pole parameters of $K_0^(1430)$ on the second sheet, and reconfirm the existence of $K_0^(1950)$ on the third sheet.	result 0.99	0.96	0.12	asserted 1.00
s5	We stress that the LASS data do not require them to have the twin pole structure of a typical Breit--Wigner resonance.	interpretation 0.88	0.76	0.21	asserted 0.97

What kind of paper the classifier reports this is: theoretical result 0.71 · evidence from analysis of existing data 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

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.** The left hand cut, circular cut and the right hand cut of pa

**Figure 2.** Phase shifts obtained from the fit to LASS data up to 1510 /

**Figure 3.** Fit to a0 of LASS data up to 2.1GeV (data points encircled violate unitarity and are not used in the fit). In the J=0, I=1/2 channel there are contributions from the left hand cut; 3 second sheet poles, κ, KII 0 (1430) and one to be determined; two third sheet poles KIII 0 (1430) and KIII 0 (1950) through the right hand cut integrals. The fit results up to 2.1GeV are shown below, χ 2 d.o.f. = 229.5/(96 − 14) ; Mκ = 649 ± 63MeV , Γκ = 602 ± 22MeV ; MKII 0 (1430) = 1435 ± 6MeV , ΓKII 0 (1430) = 288 ± 22MeV MK∗ 0 (1950) = 1917 ± 12MeV , Γ tot K∗ 0 (1950) = 145 ± 38MeV

**Figure 4.** Fit to φ0 of LASS data up to 2.1GeV (data points marked by red circles violate unitarity) χ 2 d.o.f. = 777/(96 − 12), and the κ pole’s mass and width are both further decreased by roughly 100MeV comparing with Eq. (18) but with ridiculously small error bars. Hence the results are un-trustworthy. For the case without κ the output is rather similar at low energies to the situation of the one pole fit as depicted by

Figures reproduced from arXiv:hep-ph/0603062 by the authors.

References

Works cited (21)

[1]

D. Aston et al. (LASS Collaboration), Nucl. Phys. B296(1988)493

View on Pith 1988
[2]

Here we are only able to provide a few previous publications on this is sue: E. van Beveren et al. , Z. Phys. C30(1986)615; N. A. Tornqvist, Z. Phys. C68(1995)647; S. Ishida et al. , Prog. Theor. Phys. 98(1997)621; D. Black et al. , Phys. Rev. D58(1998)054012. F. K. Guo et al. , hep-ph/0509050

View on Pith Pith review arXiv 1986
[3]

S. N. Cherry and M. R. Pennington, Nucl. Phys. A688(2001)823

View on Pith 2001
[4]

H. Q. Zheng et al. , Nucl. Phys. A733(2004)235

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[5]

V. Bernard, N. Kaiser and Ulf-G. Meissner, Nucl. Phys. B357(1991)129

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Similar opinion is also emphasized by David Bugg: private communica- tions

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J. Kennedy and T. D Spearman, Phys. Rev. 126(1962)1956

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Z. Y. Zhou et al., JHEP0502(2005)043

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H.-P. Jakob and F. Steiner, Z. Physik 228(1969)353

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P. D. B. Collins, An Introduction to Regge Theory and High Energy Physics, Cambridge University Press, 1977

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Z. G. Xiao and H. Q. Zheng, Nucl. Phys. A695(2001)273; Z. G. Xiao and H. Q. Zheng, talk given at International Conference on Flavor Physics (ICFP 2001), Zhang-Jia-Jie City, Hunan, China, 31 May - 6 Jun 2001, hep-ph/0107188

View on Pith Pith review arXiv 2001
[12]

J. Y. He, Z. G. Xiao and H. Q. Zheng, Phys. Lett. B536(2002)59; Erratum-ibid. B549(2002)362

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[13]

P. B¨ uttiker, S. Descotes-Genon and B. Moussallam, Eur. Phy s. J. C33(2004)409

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M. Krivoruchenko, Z. Phys. A350(1995)343

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P. Estabrooks et al. , Nucl. Phys. B133(1978)490

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[16]

M. Jamin, J. A. Oller and A. Pich, Nucl. Phys. B587(2000)331. 12

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[17]

M. Ablikim et al. (BES Collaboration), Phys. Lett. B633(2006)681; D. Bugg, Eur. Phys. J. A25(2005)107, Erratum-ibid. A26(2005)151; E. M. Aitala et al. (E791 Collaboration), Phys. Rev. Lett. 89(2002)121801

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[18]

J. J. Wang, Z. Y. Zhou and H. Q. Zheng, JHEP 0512(2005)019

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[19]

A. V. Anisovich and A. V. Sarantsev, Phys. Lett. B413(1997) 137

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[20]

I. Caprini, G. Colangelo and H. Leutwyler, Phys. Rev. Lett. 96(2006)132001

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[21]

C. B. Lang, Fortschritte der Physik, 26(1978)509. Appendix Here we present the proof of the statement that if the full scatt ering ampli- tude is polynomial bounded then f (0) = 0. In order to achieve this, according to the previous text in this paper, one only needs to prove T I J (s → 0) is no more singular than O(s−n) where n is an arbitrary but ﬁnite ...

View on Pith 1978

Forward citations

Cited by 4 Pith papers

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

Revisiting Roy-Steiner-equation analysis of pion-kaon scattering from lattice QCD data
hep-ph 2025-06 conditional novelty 6.0 of 10

A Roy-Steiner dispersive analysis of lattice QCD pion-kaon data at m_pi = 391 MeV finds the κ resonance at (966-i198) MeV, a broad resonance rather than the virtual-state pole reported in K-matrix analyses.
Unitary $KK$ and $K\pi$ inputs for an amplitude analysis of the $D^+ \to K^-K^+\pi^+$ decay
hep-ph 2026-11 conditional novelty 5.0 of 10

A unitarized KK and Kpi FSI model with parametrized weak vertices fits the toy S-wave of D+ -> K-K+pi+ and is proposed as the basis for a CPV-sensitive amplitude analysis of LHCb data.
Lattice QCD study of the $K^*(892)$ resonance at the physical point
hep-lat 2026-03 unverdicted novelty 5.0 of 10

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.
Dispersive $\pi K\to \pi K$ and $\pi \pi \to K \bar{K}$ amplitudes from scattering data, threshold parameters and the lightest strange resonance $\kappa$ or $K^*_0(700)$
hep-ph 2020-10 conditional novelty 5.0 of 10

Simultaneous dispersive fits to pion-kaon and pion-pion-to-kaon-antikaon data yield constrained amplitude parameterizations, precise threshold parameters, and a kappa/K*0(700) pole consistent with prior dispersive det...

How the literature uses it

7 arXiv papers cite this in 9 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 6 sentences without a confident label

Compares with it 1 paper

Using a BW ansatz with constant width, improved by incorporating some of the left- and circular-cut contributions via a dispersion integral, Ref. [ref] finds a $$ mass and width similar to Ref. [ref] , which should probably be expected given that the left-hand cut contribution should not influence the $1.9$ mass region significantly.

On the scalar $\pi K$ form factor beyond the elastic region · Pole extraction · 0.96

Cites it as background 1 paper

For the case of physical quark masses, $κ$ also had remained a subject of considerable debates for several decades until it was established as the lowest-lying strange hadronic resonance in QCD in the past twenty years based on rigorous dispersive analyses of $πK$ scattering [ref] ; see, e.g., Refs. [ref] for reviews.

Revisiting Roy-Steiner-equation analysis of pion-kaon scattering from lattice QCD data · Introduction · 0.91

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

D. Aston et al. (LASS Collaboration), Nucl. Phys. B296(1988)493

View on Pith 1988

[2] [2]

Here we are only able to provide a few previous publications on this is sue: E. van Beveren et al. , Z. Phys. C30(1986)615; N. A. Tornqvist, Z. Phys. C68(1995)647; S. Ishida et al. , Prog. Theor. Phys. 98(1997)621; D. Black et al. , Phys. Rev. D58(1998)054012. F. K. Guo et al. , hep-ph/0509050

View on Pith Pith review arXiv 1986

[3] [3]

S. N. Cherry and M. R. Pennington, Nucl. Phys. A688(2001)823

View on Pith 2001

[4] [4]

H. Q. Zheng et al. , Nucl. Phys. A733(2004)235

View on Pith 2004

[5] [5]

V. Bernard, N. Kaiser and Ulf-G. Meissner, Nucl. Phys. B357(1991)129

View on Pith 1991

[6] [6]

Similar opinion is also emphasized by David Bugg: private communica- tions

View on Pith

[7] [7]

J. Kennedy and T. D Spearman, Phys. Rev. 126(1962)1956

View on Pith 1962

[8] [8]

Z. Y. Zhou et al., JHEP0502(2005)043

View on Pith 2005

[9] [9]

H.-P. Jakob and F. Steiner, Z. Physik 228(1969)353

View on Pith 1969

[10] [10]

P. D. B. Collins, An Introduction to Regge Theory and High Energy Physics, Cambridge University Press, 1977

View on Pith 1977

[11] [11]

Z. G. Xiao and H. Q. Zheng, Nucl. Phys. A695(2001)273; Z. G. Xiao and H. Q. Zheng, talk given at International Conference on Flavor Physics (ICFP 2001), Zhang-Jia-Jie City, Hunan, China, 31 May - 6 Jun 2001, hep-ph/0107188

View on Pith Pith review arXiv 2001

[12] [12]

J. Y. He, Z. G. Xiao and H. Q. Zheng, Phys. Lett. B536(2002)59; Erratum-ibid. B549(2002)362

View on Pith 2002

[13] [13]

P. B¨ uttiker, S. Descotes-Genon and B. Moussallam, Eur. Phy s. J. C33(2004)409

View on Pith 2004

[14] [14]

M. Krivoruchenko, Z. Phys. A350(1995)343

View on Pith 1995

[15] [15]

P. Estabrooks et al. , Nucl. Phys. B133(1978)490

View on Pith 1978

[16] [16]

M. Jamin, J. A. Oller and A. Pich, Nucl. Phys. B587(2000)331. 12

View on Pith 2000

[17] [17]

M. Ablikim et al. (BES Collaboration), Phys. Lett. B633(2006)681; D. Bugg, Eur. Phys. J. A25(2005)107, Erratum-ibid. A26(2005)151; E. M. Aitala et al. (E791 Collaboration), Phys. Rev. Lett. 89(2002)121801

View on Pith 2006

[18] [18]

J. J. Wang, Z. Y. Zhou and H. Q. Zheng, JHEP 0512(2005)019

View on Pith 2005

[19] [19]

A. V. Anisovich and A. V. Sarantsev, Phys. Lett. B413(1997) 137

View on Pith 1997

[20] [20]

I. Caprini, G. Colangelo and H. Leutwyler, Phys. Rev. Lett. 96(2006)132001

View on Pith 2006

[21] [21]

C. B. Lang, Fortschritte der Physik, 26(1978)509. Appendix Here we present the proof of the statement that if the full scatt ering ampli- tude is polynomial bounded then f (0) = 0. In order to achieve this, according to the previous text in this paper, one only needs to prove T I J (s → 0) is no more singular than O(s−n) where n is an arbitrary but ﬁnite ...

View on Pith 1978