Meson-Meson interaction in a non-perturbative chiral approach

E. Oset; J. A. Oller; J. R. Pelaez

REVIEW

Meson-Meson interaction in a non-perturbative chiral approach

J. A. Oller, E. Oset, J. R. Pelaez

High Energy Physics - Phenomenology · Submitted 2 Apr 1998 · 48 references

Pith's verdict: The full coupled-channel unitarized ChPT paper behind their PRL — the method holds up and earned its citations, but the fit is reported only as curves, and the σ and κ poles rest on a cutoff that is never varied.

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arXiv IDhep-ph/9804209 (version 1)

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CitationE. Oset, J. A. Oller, J. R. Pelaez. "Meson-Meson interaction in a non-perturbative chiral approach." arXiv:hep-ph/9804209 (1998). https://pith.science/paper/STNQMSNS

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Editor's note

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This is the long paper behind the Oller–Oset–Peláez PRL, and it's the one people actually cite for the method. The core is a coupled-channel generalization of the inverse amplitude method: T = T2 [T2 − T4^P − T2 G T2]^{-1} T2, with the seven O(p^4) coefficients fitted and a hard cutoff at 1.02 GeV. What's new here over the letter is the full machinery — all seven (I,J) channels, the explicit amplitudes in Appendix B, the pole positions, and the f0/a0 partial widths. The reader's novelty score is a bit harsh; a companion letter announcing a method doesn't make the full exposition derivative. Two things it does well. The toy model in §3 is clean reasoning: for a Breit-Wigner amplitude the expansion of T^{-1} truncates at O(p^4) and the method is exact, which explains why it works better than the power counting suggests. And the coupled-channel version genuinely fixes the single-channel IAM's Adler-zero pathology — a vanishing (T2)11 no longer produces a double zero as long as (T2)12 ≠ 0. That's a real technical improvement, stated plainly. Where I'd push: the fit is reported entirely as curves against data points. No chi-squared, no errors on the seven L̂_i, no error on any pole in Table III, and — the one that matters — no scan over qmax. The authors say themselves that the L̂_i depend on the cutoff, and Appendix A.2 ties it to µ = 2qmax/√e. The σ (442 − i227) and κ (770 − i250) sit far off the real axis, so their positions come entirely from the second-sheet continuation and are exactly the quantities most sensitive to a cutoff that is never varied. A refit at a different qmax would tell us whether those poles are pinned or just where this regularization puts them. The stress-test concern lands. The φ at 980 MeV with zero width is weaker still, but they flag it as a plausible identification rather than a result, which is the right call. The circularity worry is real but modest: the coefficients are fitted to the same phase shifts where the resonances appear, so the Eq. (31) widths are consistency checks. But the poles aren't fitted directly, and a coherent spectrum across seven channels from seven parameters is not nothing. Send it out. It's a methods paper that earned its citations, and the revision I'd ask for is a fit-quality table and a cutoff scan, not new physics.

Referee report

Significance

This is one of the founding papers of the unitarized chiral approach, and its importance is not in doubt. The step from the single-channel inverse-amplitude method to the coupled-channel form of Eq. (16) is what allows the f0 and a0 to appear at all, and the paper makes that mechanism explicit: the K̄K channel, absent from earlier single-channel treatments, is what generates those poles. The toy-model argument in §3 is a genuinely illuminating piece of reasoning rather than decoration, because it identifies the class of amplitudes (propagator-like, resonance-dominated) for which the truncation of T^{-1} converges faster than the truncation of T, and it explains in advance why the scheme works better than naive power counting would suggest. The second-sheet contour plots of Figs. 9 and 10, which show that the σ and κ poles are oriented nearly parallel to the real axis while the f0, ρ, K* and a0 poles are nearly perpendicular, are a real contribution to the phenomenology of broad scalars and anticipate later dispersive work. The paper also ships falsifiable content: the κ and σ pole positions are predictions, not fits, and the qualitative statement that a broad κ should exist in (1/2

Major objections (4)

[§5 and Table II] The fit is never quantified. Seven parameters are adjusted against phase shifts and inelasticities in seven channels, yet the paper reports no χ², no number of data points, no list of which data sets and energy ranges entered the fit, and no uncertainties on the L̂_i in Table II. The abstract's claim of 'good agreement with a vast amount of experimental analyses' (abstract:s3) therefore rests entirely on visual comparison of curves in Figs. 1–7. This matters for the second half of Table II as well: the comparison with χPT at µ = 1.2 GeV is presented as a consistency check, but L̂5 = 1.7 against L5 = 0.4 ± 0.5 and L̂7 = 0.2 against L7 = −0.4 ± 0.2 are two- to three-sigma displacements, not agreement, and the text's 'of the same order' understates this. A table of fitted parameters with errors, a χ² per channel, and an explicit statement of the data sets used would make the central claim checkable rather than asserted.
[§5.2 and Table III] The cutoff qmax = 1.02 GeV is fixed by hand and never varied, although §2 and Appendix A.2 state explicitly that the fitted L̂_i depend on the regularization scale, which 'appears through the cut-off'. Table III then quotes pole positions to the MeV with no uncertainty attached. For the narrow states this is harmless, since the ρ and K* poles are pinned by the phase-shift data near δ = 90°. For the two states the abstract singles out as 'very broad' (abstract:s4), the σ at 442 − i227 MeV and the κ at 770 − i250 MeV, the pole position is fixed by analytic continuation far from the real axis and is precisely where cutoff sensitivity would show up. A stability study — refit the L̂_i at, say, qmax = 0.8 and 1.2 GeV and report how the σ and κ poles move — is needed before the second-sheet positions of these two states can be regarded as determined by the data rather than by the choice of regulator.
[§5.1.7 and abstract:s4] The φ identification is the weakest of the seven and the abstract's flat listing of φ among the poles 'corresponding to' resonances overstates it. The pole sits at 980 MeV, about 30 MeV below K̄K threshold, with zero width, in a single-channel treatment that omits the three-pion channel carrying the physical φ width of about 4 MeV. The authors discuss this candidly in §5.1.7 and argue that the small OZI-suppressed 3π coupling could shift the mass and generate the width, which is reasonable, but the abstract does not carry that qualification. Either the φ should be flagged in the abstract as a marginal identification, or the 3π channel should be included; as written, the reader is invited to count seven successes where six are secure.
[Eq. (31) and abstract:s5] The partial decay widths of the f0 and a0 are computed from Eq. (31) using the same fitted amplitudes that produced the poles, so the statement that they are 'well reproduced' (abstract:s5) is a consistency check on the fit rather than an independent prediction. This is not a circularity in the vicious sense — the widths are compared to measured values and the comparison could have failed — but the framing should say so. The caveat already noted in the text, that the σ background rotates the f0 → ππ amplitude by a factor i so that Im T12 must be replaced by Re T12, is a good illustration of how easily such a check can be misread, and it deserves to be stated in the abstract-level summary of results rather than buried in §5.2.

Minor objections (7)

[§5.1.1 vs Table III] The f0 pole is quoted as 'around 980 − i14 MeV' in §5.1.1 but as 994 − i14 MeV in Table III and §5.2; the text appears to conflate the pole mass with the effective mass of about 980 MeV. The σ pole is likewise given as 442 − i225 in §5.1.1 and 442 − i227 in Table III and §5.2. These should be made consistent.
[Fig. 4 caption vs §5.1.4] The panel labels in the caption and in the text disagree: §5.1.4 places the K̄K → πη phase shifts in panel (b) and the πη mass distribution in panel (d), while the caption assigns the mass distribution to (b), the K̄K → πη phase shifts to (c) and the inelasticity to (d). One of the two should be corrected.
[Appendix A.1, Eq. (32)] The second logarithmic term in Eq. (32) appears to contain the factor (1 + √(1 + M_2^2/q_max^2)) twice in the denominator, and the overall bracket structure is difficult to parse as printed. This should be checked against the equal-mass limit of Eq. (33), which is clean.
[Appendix B, Eq. (47)] There is a stray comma in the symbol 'T ,+−+−4' in Eq. (47), and the same superscript/subscript pattern is used inconsistently for the other amplitudes in this appendix.
[§2 and throughout] The distinction between the standard chiral coefficients L_i and the fitted coefficients L̂_i is not maintained consistently; for instance the sentence in §2 beginning 'Thus all these contributions will make the L_i differ from the L_i' renders both with the same symbol. A single consistent notation would help the reader track which quantities are fitted.
[§1] The a0 is called a0(890) in the introduction and a0(980) everywhere else, including Table III. The introduction should use the same label as the rest of the paper.
[Table I] The table is difficult to read as typeset, since the states for adjacent (I,J) entries run together without clear column separation (for example 'KπKη' and 'πηK̄K'). A clearer layout, or a separate row per channel, would make the channel content unambiguous.

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The paper

The paper claims

A coupled-channel unitarized chiral Lagrangian with seven fitted fourth-order coefficients describes meson-meson scattering up to about 1.2 GeV and generates poles for the $f_0$, $a_0$, $rho$, $K^*$, $phi$, $sigma$ and $kappa$ resonances.

The reading

The paper asks whether chiral symmetry plus exact coupled-channel unitarity can describe the strong interactions of light mesons beyond the low-energy region where ordinary chiral perturbation theory converges. It builds the scattering amplitude by expanding the inverse amplitude rather than the amplitude itself, keeping the $O(p^2)$ and $O(p^4)$ chiral Lagrangians and fitting seven fourth-order coefficients to data. With a cutoff of $1.02$ GeV, the resulting amplitudes reproduce phase shifts and inelasticities in seven isospin and angular-momentum channels up to about $1.2$ GeV. The same amplitudes develop poles in the complex energy plane for the known $f_0$, $a_0$, $rho$, $K^*$ and $phi$ resonances and for the broad $sigma$ and $kappa$. If correct, one symmetry-constrained scheme accounts for the light-meson resonance spectrum below $1.2$ GeV without adding those resonances as elementary fields.

What carries the argument

The load-bearing formula is Eq. (16): $$T = T_2 cdot [T_2 - T_4^P - T_2 G T_2]^{-1} cdot T_2,$$ where $T_2$ is the $O(p^2)$ chiral amplitude, $T_4^P$ is the polynomial part of the $O(p^4)$ amplitude, and $G$ is the two-meson loop function whose imaginary part gives the phase space. This is a coupled-channel generalization of the inverse amplitude method: expanding $T^{-1}$ instead of $T$ moves the resonance poles into zeros of the inverse, and the matrix structure keeps unitarity when several channels couple. The toy-model section shows that for an exact Breit-Wigner resonance the same construction reproduces the propagator, which is the paper's explanation for why the method works well in resonant channels.

Extended reading notes

Core claim

The paper claims that a coupled-channel amplitude built from the $O(p^2)$ and $O(p^4)$ chiral Lagrangians, with seven fourth-order coefficients fitted to data and a fixed cutoff $q_{max}=1.02$ GeV, describes meson-meson scattering in seven isospin and angular-momentum channels up to about $1.2$ GeV. The amplitude is not a polynomial fit: it imposes coupled-channel unitarity through the loop function $G$, and its analytic continuation develops poles in the complex energy plane for the $f_0$, $a_0$, $rho$, $K^*$, $phi$, $sigma$ and $kappa$. The paper reports pole positions such as $sigma$ at $442-i227$ MeV, $f_0(980)$ at $994-i14$ MeV, $rho(770)$ at $759-i70$ MeV, $K^*(890)$ at $892-i21$ MeV and $a_0(980)$ at $1055-i21$ MeV, and it reproduces total and partial decay widths, including $f_0 to pi pi$ or $K bar K$ and $a_0 to pi eta$ or $K bar K$. The authors acknowledge that crossed-channel loops and tadpoles are not explicit and are absorbed into the fitted coefficients.

Load-bearing premise

The load-bearing premise is that the fourth-order chiral amplitude can be split as a polynomial plus a loop term, Eq. (15), and that all crossed-channel loops and tadpoles omitted by the calculation can be absorbed into the seven fitted coefficients without changing the amplitude's form.

Editorial extensions

If this is right

In the $(0,0)$ channel the $f_0(980)$ pole appears only because the $K bar K$ channel is included; a single-channel treatment misses it. The same coupled-channel requirement brings out the $a_0(980)$ in the $(1,0)$ channel.
The $rho$ and $K^*$ come out as nearly elastic Breit-Wigner resonances, so single-channel treatments of those channels are justified and the coupled-channel corrections are small there.
The fitted coefficients $hat L_i$ are of the same order as the standard chiral perturbation theory $L_i$ at $mu simeq 1.2$ GeV, giving a quantitative bridge between the low-energy and resonance regions.
The scheme predicts broad $sigma$ and $kappa$ poles and gives partial widths, for example $f_0$ about $65%$ $pi pi$ and $35%$ $K bar K$, and $a_0$ about $50%$ $pi eta$ and $50%$ $K bar K$.
The same amplitudes can be inserted into photoproduction, meson-nucleon, nuclear-medium and strongly interacting symmetry-breaking problems.

Reading between the lines

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

Because the $sigma$ and $kappa$ poles lie far from the real axis, their exact positions are the least robust output; a full crossed-channel calculation could move them appreciably while still fitting the same data, so their quoted masses and widths should be treated as scheme-dependent.
The $phi$ pole appears about $30$ MeV below its physical mass with zero width because three-pion coupling is omitted; adding that small coupling is a sharp test of whether the fitted parameters survive when a new channel is opened.
The breakdown above $1.2$ GeV is a natural place to look for the omitted $eta eta$ and multi-pion channels; if adding them leaves the seven coefficients nearly unchanged, the two-meson truncation is better justified than the paper's own caveat suggests.

Original abstract

A non-perturbative method which combines constraints from chiral symmetry breaking and coupled channel unitarity is used to describe the meson-meson interaction up to about 1.2 GeV. The approach uses the O(p^2) and O(p^4) chiral Lagrangians. The seven free parameters of the O(p^4) Lagrangian are fitted to the data. The results are in good agreement with a vast amount of experimental analyses. The amplitudes develop poles in the complex plane corresponding to the f0, a0, rho, K*, phi, sigma and kappa resonances; the latter two, very broad. The total and partial decay widths of the resonances are also well reproduced. Further extensions and applications of this chiral non-perturbative scheme are also discussed.

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

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,460 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.44 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.83 · compares with prior results 0.57 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

A non-perturbative method which combines constraints from chiral symmetry breaking and coupled channel unitarity is used to describe the meson-meson interaction up to about 1.2 GeV. The approach uses the O(p^2) and O(p^4) chiral Lagrangians. The seven free parameters of the O(p^4) Lagrangian are fitted to the data. The results are in good agreement with a vast amount of experimental analyses. The amplitudes develop poles in the complex plane corresponding to the f0, a0, rho, K*, phi, sigma and kappa resonances; the latter two, very broad. The total and partial decay widths of the resonances are also well reproduced. Further extensions and applications of this chiral non-perturbative scheme are also discussed.

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	A non-perturbative method which combines constraints from chiral symmetry breaking and coupled channel unitarity is used to describe the meson-meson interaction up to about 1.2 GeV.	method 0.80	0.90	0.22	asserted 1.00
s1	The approach uses the O(p^2) and O(p^4) chiral Lagrangians.	method 1.00	0.64	0.03	asserted 1.00
s2	The seven free parameters of the O(p^4) Lagrangian are fitted to the data.	method 0.99	0.88	0.05	asserted 1.00
s3	The results are in good agreement with a vast amount of experimental analyses.	result 0.92	0.75	0.35	asserted 1.00
s4	The amplitudes develop poles in the complex plane corresponding to the f0, a0, rho, K*, phi, sigma and kappa resonances; the latter two, very broad.	result 1.00	0.89	0.24	asserted 1.00
s5	The total and partial decay widths of the resonances are also well reproduced.	result 1.00	0.87	0.13	asserted 1.00
s6	Further extensions and applications of this chiral non-perturbative scheme are also discussed.	future work 1.00	0.53	0.04	asserted 0.85

What kind of paper the classifier reports this is: method or tool 0.69 · evidence from analysis of existing data 0.62.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

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

Figures

**Figure 2.** Results in the I = J = 1 channel. (a) phase shifts for ππ → ππ. Data: open circle [24], black square [30]. (b), (c) same as in fig. 1. (d) inelasticity

**Figure 3.** Phase shifts for ππ → ππ in the I = 2, J = 0 channel. Data: cross [31], empty square [32], full triangle [33]. 24

**Figure 4.** Results in the I = 1, J = 0 channel. (a) phase shifts for πη → πη. (b) Invariant mass distribution for πη data from [34]. (c) Phase shifts for KK¯ → πη. (d) inelasticity

**Figure 5.** Results in the I = 1/2, J = 0 channel. (a) phase shifts for Kπ → Kπ. Data: full circle [36], cross [37] , open square [38], full triangle [39]. (b) phase shifts for Kπ → Kη. (c) phase shifts for Kη → Kη. (d) inelasticity. 25

**Figure 6.** Results in the I = 1/2, J = 1 channel. (a) phase shifts for Kπ → Kπ. Data: full triangle [36], open circle [39]. (b) phase shifts for Kπ → Kη. (c) phase shifts for Kη → Kη. (d) inelasticity

**Figure 7.** Phase shifts for Kπ → Kπ in the I = 3/2, J = 0 channel. Data: open triangle [39], open circle [42]

**Figure 8.** (|TIJ=01|) 2 for KK¯ → KK¯ showing the singularity corresponding to the φ resonance. 26

**Figure 9.** Imaginary part of the ππ amplitude in the (I,J)=(0,0) channel in the second Riemann sheet. On the left we show a three dimensional plot were we can observe the different structure of the σ and f0 poles. On the right we show a contour plot of the lower half plane of the second sheet. The σ pole is very far away from the real (physical) axis and its lines of maximum gradient are parallel to it, in contrast with the f0. That is why the effect of both poles in the phase shifts (

**Figure 10.** The poles associated to the ρ (left) and a0 (right) are oriented differently. The ρ mass seen on the (I,J)=(1,1) phase shifts is slightly bigger than the real part of the position of the ρ pole, whereas the peak of the mass distribution where the a0 is observed (see

Figures reproduced from arXiv:hep-ph/9804209 by the authors.

References

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, open square [38], full triangle [39]. (b) phase shifts for Kπ → Kη . (c) phase shifts for Kη → Kη . (d) inelasticity. 25 Fig. 6: Results in the I = 1/2, J = 1 channel. (a) phase shifts for Kπ → Kπ . Data: full triangle [36], open circle [39]. (b) phase shifts for Kπ → Kη . (c) phase shifts for Kη → Kη . (d) inelasticity. Fig. 7: Phase shifts for Kπ → Kπ...

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How later papers used it

134 later papers cite it

Evidence: substantial. 110 of the citing papers share no author with it, and 44 of those carry a confident label from Pith's citation classifier.

Pith read 17 of the 258 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

3 later papers by other authors are listed below: 2 used its method or tool and 1 used its data or values.

Uses its method or tool 2 of 48 papers with this label

Briefly introduced the inverse amplitude method of this paper and showed its fit to meson-meson partial waves with special weight on the (ηK, πK) system.

Finite volume effects in pion-kaon scattering and reconstruction of the kappa(800) resonance (2011) · 0.97 · paraphrased: the licence does not allow quoting
Extracted resonance parameters using a unitary chiral perturbation theory model based on this paper.

Rho resonance parameters from lattice QCD (2016) · 1.00 · paraphrased: the licence does not allow quoting

Uses its data or values 1 of 5 papers with this label

Used the two-pion decay width Γ(f0→ππ)=19.5 MeV derived in this paper within a chiral unitarity approach to estimate the effective coupling gKK̅ππ around 10 GeV.

f0(980) meson as a K bar K molecule in a phenomenological Lagrangian approach (2007) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

117 arXiv papers cite this in 216 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 · 122 sentences without a confident label

Builds on it 1 paper

In the further extension of Ref. [ref] , the next-order $𝒪(p^4)$ Lagrangian of chPT is taken into account, where additional parameters entering by this procedure are also fitted to data.

Dynamical generation of hadronic resonances in effective models with derivative interactions · Coupled-channel unitarity approach by Oller, Oset, and Peláez · 0.86

Uses its method or tool 44 papers

To parametrize the scattering amplitude we use unitarized chiral perturbation theory (U$χ$PT) [ref] , which is derived from standard $χ$PT [ref] by applying the Inverse Amplitude Method.

From QCD to Physical Resonances · METHOD AND RESULTS · 0.99
The squares in the diagrams mean the full meson–meson scattering amplitudes involving the loop resummation through the BS equation [ref] .

Final state interactions in the decays J/psi->VPP · Formalism · 0.99
The required kernels, i.e., the potentials which describe the interaction of the different pairs of the system are calculated using chiral Lagrangians [ref] .

Searching for exotic states in the $N \pi K $ system · Formalism and Results · 0.81

Uses its data or values 4 papers

However, by using the two-pion decay width $Γ(f_0→ππ)$=19.5 MeV derived in [ref] within the framework of a chiral unitarity approach, we can estimate the effective coupling $g_KK̅π⃗π⃗$ around 10 GeV.

f0(980) meson as a K bar K molecule in a phenomenological Lagrangian approach · The strong decay · 1.00
[ref] that is slightly better for this quantum number (dashed vs. solid line in figure [ref]).

Finite volume effects in pion-kaon scattering and reconstruction of the kappa(800) resonance · P-wave states: Extraction of the K*(892) and the rho(770) · 0.93
The values of the $O(p^4)$ coupling constants are given in Table II of Ref. [ref] and the cutoff momentum $q_max=1.02$.

Form factors of the $\rho$ meson from effective field theory and the lattice · Parameterization of the $ππ$ amplitude in the vicinity of the pole · 0.99

Compares with it 4 papers

Furthermore, small differences between two parameterizations can be parameterized by introducing the quantity ϵ_i(k^2)=2(G_i(k^2)-G_i'(k^2))/|G_i(k^2)|+|G_i'(k^2)| , i=1,2,3 , where $G_i(k^2)$ and $G_i'(k^2)$ are calculated with the use of the parameterizations from Ref. [ref] and Ref. [ref] , respectively.

Form factors of the $\rho$ meson from effective field theory and the lattice · Numerical implementation: contact contributions to the form factors in NREFT · 0.87
[ref] (dashed lines in fig.

Finite volume effects in pion-kaon scattering and reconstruction of the kappa(800) resonance · The pion-kaon amplitude in the infinite volume limit with next-to-leading order terms · 0.99
The results are shown in Figure [ref] and are in agreement with previous works on the subject [ref] : $(i)$ We observe a rapidly falling inelasticity when the $KK̅$ threshold is reached.

Dynamical generation of hadronic resonances in effective models with derivative interactions · Phase shifts and inelasticity · 0.99

Cites it as support 2 papers

This has been shown to be sufficient to reproduce the available scattering data on the scalar channel and generates the observed resonances [ref] .

Investigation of a0-f0 mixing · Form factors and unitarization · 0.97
This procedure has been shown to accurately describe low-lying resonances in multiple channels with a relatively small set of parameters (see, for example, Refs. [ref] ).

From QCD to Physical Resonances · METHOD AND RESULTS · 0.99

Cites it as background 22 papers

The application of the BSE methodology to the light (up and down) quark sector have been plentiful both in the meson [ref] as well as baryon sector [ref] , which also includes extensions to coupled channels.

Towards a theory of hadron resonances · Light quark sector: Meson-meson scattering · 0.90
The $ππ$-$KK̅$-$πη$ coupled-channel system around $1$ was studied in this framework in Ref. [ref] with a particular interest in the $f_0$ and $a_0$ resonances followed by the studies with similar approaches [ref] [The scalar mesons around 1 GeV have been studied for a long time, and many studies were devoted for it from various viewpoints, e.g., as done in Refs. [ref] (see also Ref. [ref] for a review article).] and the applications to many reactions.

Role of the triangle mechanism in the $\Lambda_b\rightarrow \Lambda_c\pi^-f_0(980)$ reaction · $KK̅→π^+π^-$ scattering amplitude · 0.80
In the literature, three schemes were proposed to unitarize coupled-channel amplitudes, i.e., the inverse amplitude method [ref] , the N/D method [ref] , and the Bethe-Salpeter method [ref] .

Radiative decays of the $\Lambda(1520)$ as a dynamically generated resonance · Introduction · 0.89

Discussion

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