On the spectral functions of scalar mesons

Francesco Giacosa; Giuseppe Pagliara

REVIEW

On the spectral functions of scalar mesons

Francesco Giacosa, Giuseppe Pagliara

High Energy Physics - Phenomenology · Submitted 24 Jul 2007 · 30 references

Pith's verdict: A careful one-loop study whose normalization point is sound but whose decay-rate integral is an ansatz, and whose a0/f0 branching ratios carry an unstated error budget.

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arXiv ID0707.3594 (version 2)

Pith ID7RP37D7V, Pith's permanent record of this paper and its review

CitationFrancesco Giacosa, Giuseppe Pagliara. "On the spectral functions of scalar mesons." arXiv:0707.3594 (2007). https://pith.science/paper/7RP37D7V

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Bottom line: this is a careful, honest one-loop study of light scalar line shapes, and the methodological point — if your propagator satisfies Källén-Lehmann, the spectral function comes out normalized for free, so you don't hand-normalize a Breit-Wigner — is worth taking seriously. But the decay-rate formula they build on top of it is an ansatz, not a derivation, and the a0/f0 branching ratios inherit an error budget the paper never states. What's genuinely new is modest and clearly delimited: the explicit numerical spectral functions with f_Λ = (1+q²/Λ²)^{-1}, the weighted decay integrals, and the a0/f0 application with the subthreshold KK tail. The observation that Flatté emerges as the weak-coupling limit is credited to Achasov & Kiselev [22], and the authors say so. The one-channel σ/k results showing deviations from Breit-Wigner are a useful illustration. Soft spots, in order of how much they matter. First, Eq. (22)/(35) defines the width as ∫dx d_S(x) Γ^{t-l}(x). That's an impulse approximation — a classical mixture of on-shell masses times the on-shell two-body width. Appendix B justifies it heuristically via B→AS, S→φφ; it is not derived from the full three-body amplitude. The Conclusions call it a correct definition, which is stronger than what's shown. For a broad σ or near the KK threshold this matters. Second, the cutoff insensitivity is demonstrated only for σ→ππ (Fig. 5). The headline ~50% probability M > 2m_K and the KK branching ratios are integrals over exactly the subthreshold tail where the 3d cutoff is least controlled, and no normalization value or cutoff scan is reported for those parameters. Third, g_f0KK is tuned to reproduce the PDG total width before the branching ratios are quoted, so the agreement in Fig. 7 is partly by construction. The a0 KK/πη ratio coming out at 0.3-0.4 against PDG's 0.183 is noted but not treated as a tension. None of this kills the normalization claim, which is the paper's real content. But the branching-ratio numbers should be read as indicative, not predictive. Who should read it: anyone parametrizing scalar line shapes for experimental fits, or working on a0/f0 spectroscopy. It deserves a referee — the questions above are answerable and the paper is worth the round.

Referee report

Significance

If the construction holds, it is a useful methodological constraint: it says that line shapes used to extract scalar-meson couplings and widths should come from propagators satisfying the Källén-Lehmann representation, rather than from hand-normalized Breit-Wigner or Flatté forms. The observation that the Flatté distribution is the weak-coupling limit of a loop-derived spectral function is a clean and quotable statement, and the demonstration that a subthreshold channel can produce a non-zero physical decay rate through the high-mass tail is physically well motivated and relevant to the ongoing a₀(980)/f₀(980) spectroscopy debate. The paper is honest about the limits of its one-loop treatment in places (normalization fails for large g; covariance is lost with the 3d cutoff), and the σ/k results are presented as illustrative rather than as precision determinations. The value of the paper therefore rests on the methodological point and on the a₀/f₀ application, and it is on the latter that the error budget is thinnest.

Major objections (4)

[§II.A, Eq. (22); §III.A, Eq. (35); Appendix B] The identification of the physical width with ∫dx d_S(x) Γ^{t-l}(x) is presented as a 'generalization' in §II.A and as 'a correct definition of decay amplitudes' in the Conclusions, but it is an impulse/factorization ansatz rather than a derived result. Appendix B justifies it through a two-step B → A S, S → φφ decomposition with a small coupling c, which is a heuristic argument at the level of rates; it is not obtained from the full B → Aφφ amplitude, where off-shell and interference contributions would appear. The ansatz is exact only in the narrow-width limit, and the paper's own examples (the σ near threshold, the a₀/f₀ with M within a few MeV of 2m_K) are precisely where the narrow-width limit is least reliable. The authors should either derive Eq. (22) at the amplitude level in a controlled limit and estimate the leading correction, or soften the language in §II.A and the Conclusions from 'correct definition' to 'a consistent prescription that reduces to the tree-level result as g → 0'. As written, the strongest wording outruns what Appendix B establishes.
[§III.B, Fig. 6 and Fig. 7] The headline numbers of the two-channel application — the statement that there is a 'large probability, ∼50%' that M > 2m_K, and the KK branching ratios plotted in Fig. 7 — are integrals over exactly the subthreshold tail where the one-loop, non-covariant 3d-cutoff treatment is least controlled. The claim that cutoff dependence is mild is demonstrated only for σ → ππ in Fig. 5 (Λ = 1 and 2 GeV); no analogous variation is shown for the a₀/f₀ case, and no explicit value of ∫d_S(x)dx is reported for the quoted parameter set (g_f₀KK = 3 GeV, Λ = 1.5 GeV). Since the paper itself uses the failure of normalization as its validity criterion, the reader needs (i) the numerical normalization for the a₀/f₀ parameters, (ii) the same curves of Fig. 6 and Fig. 7 for at least Λ = 1 and 2 GeV, and (iii) an uncertainty band on the '∼50%' figure. Without these, the abstract's main claim [abstract:s3] carries an unstated error budget.
[§III.B, Eq. (36) and the determination of g_f₀KK] The coupling ratios in Eq. (36) are taken from the fit of Ref. [4], and g_f₀KK is then fixed by requiring the total theoretical width to match the PDG value ('g_f₀KK has to lie between 3-4 GeV'). The subsequent comparison of the individual branching ratios with PDG is therefore partly a consistency check of an already-tuned parameter rather than an independent prediction, and the paper should say so explicitly. Relatedly, the obtained Γ_{a₀KK}/Γ_{a₀πη} ≈ 0.3-0.4 versus the PDG average 0.183 ± 0.024 is a genuine tension that is mentioned in passing but not discussed; it deserves a sentence on whether it reflects the one-loop truncation, the choice of cutoff, or the input ratios of Eq. (36).
[§II.A, regularization and Eq. (15)] The use of f_Λ(q) = f_Λ(q⃗²) explicitly breaks covariance, as the authors acknowledge, and the paper states that different forms of f_Λ were tried without changing the picture. However, all forms tried appear to be functions of q⃗², so the sensitivity to the loss of covariance itself is not tested. Because the real part R(x) enters the mass definition of Eq. (15) and hence the position of the distribution, a single comparison with a covariant regulator (for example f_Λ(q²) or a Pauli-Villars subtraction) for one representative case (σ at M = 0.6 GeV, or f₀ with g_f₀KK = 3 GeV) would substantiate the claim that the regulator choice is immaterial. This is a bounded check and would materially strengthen the paper's central methodological message.

Minor objections (9)

[Abstract and throughout] The name is spelled 'Källén-Lehmann' (double n); the manuscript writes 'Källén-Lehman' in the abstract and in several places in the body. 'Flatte'' should be 'Flatté' throughout, and 'Breit-Wigner' is occasionally hyphenated inconsistently.
[§II.A, paragraph after Eq. (21)] 'a picked distribution d_S(x)' should read 'a peaked distribution'; similarly 'deviations form the usually employed tree-level formula' in the Introduction should read 'from'.
[Fig. 6 caption] The right panel (a₀) caption reads 'g_f₀KK = 3 GeV'; this should presumably be g_{a₀KK}. Please also state explicitly in the caption which mass value (M_{a₀} = 984.7 MeV, M_{f₀} = 980 MeV) and which m_π, m_K, m_η inputs were used.
[§II.B, Fig. 4 and Fig. 5] The statement that increasing g_σππ 'leads outside the range of validity of the normalization' would be more useful with a number: for which g does ∫d_S dx fall below, say, 0.99? A short table of normalization versus g and Λ would let the reader judge where the one-loop treatment can be trusted.
[Appendix B] 'tree-body decay' should read 'three-body decay' (two occurrences). The argument would also be clearer if the narrow-width expansion of Eq. (B3) were written explicitly to first order in the width.
[Appendix A, Eq. (A1)] It would help to state that Eq. (A1) reduces to the standard closed-form one-loop self-energy in the Λ → ∞ limit, or at least to note the correspondence with the usual Passarino-Veltman scalar integral, so that the reader can check the normalization and overall sign conventions.
[§III.B, final paragraph] The claim that 'the f₀ meson turns out to have typically a larger width than a₀' is supported only visually by Fig. 7; please quote the numerical values at the preferred coupling for both states.
[Table 1] The caption states 'we use m = m_π' but does not give the numerical value; please state m_π and Λ explicitly in the caption so the table is self-contained.
[References] Several references are given in shortened or inconsistent form (e.g. [3], [12], [26]); please check that all arXiv identifiers and journal references are complete and consistent with the journal style.

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The paper

The paper claims

Dressing a scalar meson's propagator with meson loops yields a spectral function that is normalized for free, and decay rates become averages of the tree-level width over that distribution.

The reading

The paper asks how to write the line shape — the mass distribution — of a light scalar meson such as the $f_0(600)$ (sigma), the $k$, the $a_0(980)$ or the $f_0(980)$, given that these states are broad enough for their own decay products to feed back into the propagator. Its answer is to dress the propagator with one-loop bubbles of pseudoscalar mesons, regularize the loop with a momentum cutoff that can be read as a nonlocal interaction vertex, and then check numerically that the resulting propagator satisfies the Källén-Lehmann representation. When it does, the spectral function $d_S(x) = (2x/π)|Im Δ_S(x)|$ integrates to one automatically, so it can be treated as an honest probability distribution over masses, and a decay rate can be written as an average of the tree-level width over that distribution. The payoff is a consistent way to fold finite-width effects into decay rates, plus a criterion — loss of normalization — for when the one-loop treatment has been pushed too far. Applied to $a_0(980)$ and $f_0(980)$, the same machinery gives a nonzero $K̄K$ rate even though both masses sit below the $K̄K$ threshold.

What carries the argument

The load-bearing object is the resummed scalar propagator $Δ_S(x) = 1/(x^2 - M_0^2 + R(x) + iI(x) + iε)$, where $R$ and $I$ are the real and imaginary parts of the mesonic self-energy with the coupling factored out. Two ingredients make it work: a cutoff function $f_Λ(q) = (1 + q⃗^2/Λ^2)^{-1}$ that regularizes the logarithmically divergent loop and is shown to correspond to a nonlocal vertex in the Lagrangian, and the optical theorem, which ties $I(x)$ to $x$ times the tree-level width so that the imaginary part knows about the physical channel opening at threshold. The Källén-Lehmann representation is then the consistency check: if it holds, the spectral function $d_S(x) = (2x/π)|Im Δ_S(x)|$ is a normalized mass distribution, the residue $Z$ below threshold measures the admixture of virtual two-meson cloud, and the weighted-integral decay formula follows.

Extended reading notes

Core claim

The central claim is that a one-loop dressed scalar propagator, when it respects the Källén-Lehmann representation, supplies a spectral function that is normalized to unity without any hand adjustment, and that this normalized distribution should replace the hand-normalized Breit-Wigner and Flatté forms usually fitted to light scalar resonances. On this basis the authors define the decay rate as $Γ_{Sφφ} = ∫ dx d_S(x) Γ^{t-l}_{Sφφ}(x)$, an integral of the tree-level two-body width weighted by the mass distribution, which reduces to the tree-level result as the coupling goes to zero and differs from it increasingly as the coupling grows. They show that for the $\sigma$ and the $k$ the resulting shape can depart substantially from a Breit-Wigner form, especially when the mass sits near threshold, and that the mass parameter defined by the zero of the real part of the inverse propagator is not the peak of the distribution. In the two-channel case with one subthreshold channel, the loop-derived distribution turns out to be well approximated by a Flatté form, which the authors therefore present as the weak-coupling limit of their construction; for $a_0(980)$ and $f_0(980)$ they find roughly a 50 percent probability that the mass lies above $2m_K$, which is what makes the tree-level forbidden $K̄K$ decay go through. They also note that when the coupling is large and the mass near threshold the normalization integral falls below one, and they use that failure as the boundary of validity of the one-loop study.

Load-bearing premise

The load-bearing premise is that the physical decay rate equals the spectral function times the on-shell tree-level width, integrated over mass — a factorization that treats the resonance as a classical mixture of masses rather than deriving the rate from the full three-body amplitude.

Editorial extensions

If this is right

Decay widths computed from loop-dressed propagators come out systematically below the tree-level values, with the gap growing as the coupling grows — about 60 MeV at $g_{σππ} = 4$ GeV in the sigma case.
Branching ratios for $a_0(980)$ and $f_0(980)$ can be predicted once the $K̄K$ coupling is fixed, and the authors find $g_{f_0K̄K}$ between 3 and 4 GeV reproduces the measured total widths.
The Flatté distribution is demoted from a model to a small-coupling approximation of the loop-derived spectral function, with visible differences in the peak position for the strongly kaon-coupled $f_0$.
A kaonic-cloud fraction of about 38 percent for the $f_0(980)$ follows from the residue factor, supporting a picture in which a compact core is dressed by kaon pairs.
Failure of the normalization integral gives a practical diagnostic for when a one-loop spectral function can no longer be trusted and higher orders are required.

Reading between the lines

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

If the weighted-integral prescription is right, the same logic applies to any broad resonance whose width is comparable to its distance from threshold, so the prescription is a general constraint on line-shape parametrizations rather than a statement about scalars only.
The paper's normalization criterion could be turned into a fitting rule: a parametrization that fails to integrate to one is not merely inconvenient but inconsistent, which gives experimentalists a cheap numerical check on any extracted line shape.
The stated breakdown of normalization at strong coupling suggests the interesting physics — a genuinely broad sigma — sits just outside the one-loop window, so a two-loop or dispersive treatment is the natural next step rather than a refinement.
Because the cutoff enters both the loop and the tree-level vertex, the mild cutoff dependence the authors report is itself a testable prediction: line-shape fits should be insensitive to the choice of cutoff form over the 1–2 GeV range.

Original abstract

In this work we study the spectral functions of scalar mesons in one- and two-channel cases. When the propagators satisfy the K\"allen-Lehman representation a normalized spectral function is obtained, allowing to take into account finite-width effects in the evaluation of decay rates. In the one-channel case, suitable to the light sigma and k mesons, the spectral function can deviate consistently from a Breit-Wigner shape. In the two-channel case with one subthreshold channel the evaluated spectral function is well approximated by a Flatte' distribution; when applying the study to the a_0(980) and f_0(980) mesons the tree-level forbidden KK decay is analysed.

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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.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.26 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.41 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

In this work we study the spectral functions of scalar mesons in one- and two-channel cases. When the propagators satisfy the K\"allen-Lehman representation a normalized spectral function is obtained, allowing to take into account finite-width effects in the evaluation of decay rates. In the one-channel case, suitable to the light sigma and k mesons, the spectral function can deviate consistently from a Breit-Wigner shape. In the two-channel case with one subthreshold channel the evaluated spectral function is well approximated by a Flatte' distribution; when applying the study to the a_0(980) and f_0(980) mesons the tree-level forbidden KK decay is analysed.

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	In this work we study the spectral functions of scalar mesons in one- and two-channel cases.	contribution 1.00	0.90	0.05	asserted 0.79
s1	When the propagators satisfy the K\"allen-Lehman representation a normalized spectral function is obtained, allowing to take into account finite-width effects in the evaluation of decay rates.	method 0.64	0.38	0.16	asserted 0.99
s2	In the one-channel case, suitable to the light sigma and k mesons, the spectral function can deviate consistently from a Breit-Wigner shape.	result 0.99	0.75	0.18	hedged possibility 0.72
s3	In the two-channel case with one subthreshold channel the evaluated spectral function is well approximated by a Flatte' distribution; when applying the study to the a_0(980) and f_0(980) mesons the tree-level forbidden KK decay is analysed.	result 0.88	0.90	0.23	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 0.87.

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full text
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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.** FIG. 2: Real and imaginary part of the mesonic loop for

**Figure 3.** FIG. 3: Spectral functions in the cases

**Figure 4.** FIG. 4: Spectral functions of

**Figure 5.** FIG. 5: The full (solid) and the tree-level (dashed) decay ra

**Figure 6.** FIG. 6: Left panel: spectral function of

**Figure 7.** FIG. 7: Left panel: decay rates of

Figures reproduced from arXiv:0707.3594 by the authors.

References

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hep-ph 2020-09 conditional novelty 5.0 of 10

In a unitarized scalar phi^4 model, bound-state number density is the standard thermal integral, but the bound-state pressure is only partially realized (zeta ~ 0.5-0.6) because the two-particle interaction cancels pa...
Vacuum Properties of Mesons in a Linear Sigma Model with Vector Mesons and Global Chiral Invariance
hep-ph 2010-03 conditional novelty 5.0 of 10

In a globally chirally invariant linear sigma model with vector mesons, the f0(600) two-pion width comes out at most about 145 MeV against 600-1000 MeV measured, disfavoring a quark-antiquark interpretation of the lig...

Show 1 more citing papers

From controversy to precision on the sigma meson: a review on the status of the non-ordinary $f_0(500)$ resonance
hep-ph 2015-10 accept novelty 3.0 of 10

The sigma meson, renamed f0(500), is established as a broad non-quark-antiquark resonance with pole position about 449 - i275 MeV, from a synthesis of dispersive analyses of pion-pion scattering.

How later papers used it

57 later papers cite it

Evidence: some. 10 of the citing papers share no author with it, and 4 of those carry a confident label from Pith's citation classifier.

Pith read 25 of the 117 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

2 later papers by other authors are listed below: 1 used its method or tool and 1 used its data or values. 3 more papers by the same authors are listed: 1 built on it and 2 used its method or tool.

Builds on it 1 paper · 1 by the same authors

By the same authors

Generalized this paper's study by considering a nonlocal interaction Lagrangian of derivative type.

Radiative $\phi$ decays with derivative interactions (2008) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

Uses its method or tool 3 of 10 papers with this label · 2 by the same authors

Adopted the form of g(beta) given in this paper.

\eta\ transitions between charmonia with meson loop contributions (2012) · 0.96 · paraphrased: the licence does not allow quoting

By the same authors

Included finite width effects by integrating the decay width weighted with the spectral functions of the involved (axial-)vector and pseudovector particles over the particle masses, following this paper.

Phenomenology of axial-vector and pseudovector mesons: decays and mixing in the kaonic sector (2013) (same authors) · 0.97 · paraphrased: the licence does not allow quoting
Gave the partial width of the full process following this paper.

Radiative production and decays of the exotic $\eta_1^\prime(1855)$ and its siblings (2023) (same authors) · 0.95 · paraphrased: the licence does not allow quoting

Uses its data or values 1 paper

Fit its coupling using the value of Lambda_1 from this paper.

Investigating $\eta^\prime_{1}(1855)$ exotic states in $J/\psi\to\eta^\prime_{1}(1855)\eta^{(\prime)}$ decays (2022) · 0.81 · paraphrased: the licence does not allow quoting

How the literature uses it

55 arXiv papers cite this in 113 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 · 77 sentences without a confident label

Builds on it 1 paper

We aim to do it by properly taking into account loops and finite-width effects using the formalism developed in Ref. [ref] extended to the case of derivative couplings.

Radiative $\phi$ decays with derivative interactions · Introduction · 0.82

Uses its method or tool 9 papers

Being this not the case for some of the decay channels studied in this work, we include the finite width effects by integrating the decay width weighted with the spectral functions of the involved (axial-)vector and pseudovector particles over the particle masses [ref] .

Phenomenology of axial-vector and pseudovector mesons: decays and mixing in the kaonic sector · The model: Lagrangian and decay widths · 0.97
The form of $g(β)$ adopted in this work is $g(β)=(1+q⃗^2/β^2)^-1$, where $q⃗$ is the momentum of $J/ψ$ in the rest frame of $ψ(4040)$ and parameter $β$ is of the order of 1 GeV [ref] .

\eta\ transitions between charmonia with meson loop contributions · Finite width effects on $ψ(4040)/ψ(4160)$ decays · 0.96
This result can be achieved by using a nonlocal Lagrangian and its corresponding Feynman rules, see Appendix A.2.1 and Refs. [ref] .

Spectral function of a scalar boson coupled to fermions · The fermionic loop · 0.89

Compares with it 1 paper

Similar comments can be made concerning the resonances below $1$ GeV by regarding at the works done in Refs. [ref] .

The role of the next-to-leading order triangle-shaped diagram in two-body hadronic decays · Specific examples · 0.90

Cites it as support 5 papers

Namely, even if there are low-energy threshold(s) and high-energy distortions, as long as the ratio $Γ_S→ϕϕ^tree/m_S$ is a small number, the role of hadronic loop contributions is small [ref] .

The role of the next-to-leading order triangle-shaped diagram in two-body hadronic decays · The model and the triangle diagram · 0.98
However, the results of Ref. [ref] suggest that unitarity corrections are small for resonances whose ratio of decay width to mass is small as well.

Excited Scalar and Pseudoscalar Mesons in the Extended Linear Sigma Model · Introduction · 0.99
For the decays that we will examine, the ratio `width'/`mass' ($Γ/M$) is safely below 1, ensuring that loop corrections do not change much the tree-level results [ref] .

Strong and radiative decays of excited vector mesons and predictions for a new $\phi(1930)$ resonance · Decay widths: theoretical expressions · 0.98

Cites it as background 12 papers

(For details and phenomenology of the spectral function, see Refs. [ref] ).

The light scalar $K_{0}^{\ast}(700)$ in the vacuum and at nonzero temperature · The light $κ$ in the vacuum · 0.96
Our starting point is a toy model with two scalar fields, $S$ and $ϕ$, representing the $f_0(500)$ state and (neutral) pions, and containing the one-channel decay process $S→ϕϕ$: [equation] For general studies of this model see Refs. [ref] , while detailed works concerning its pole structure can be found in Refs. [ref] .

Dynamical generation of hadronic resonances · Simple toy model for $f_0(500)$ · 0.94
We now turn to the concrete examples of the well-known scalar resonances $f_0(500)$, $f_0(980)$, $f_0(1370)$ and $f_0(1500)$ and calculate their decays into pions [ref] (for a discussion of the internal structure of these states in terms of quarks and gluons see also e.g. Refs. [ref] ).

The role of the next-to-leading order triangle-shaped diagram in two-body hadronic decays · Specific examples · 0.85

What it cites · 33 sentences without a confident label

Uses data or values from 1 paper

For definiteness we use the following ratios obtained in the experimental analysis of [ref]

Decays of sigma, kappa, a0(980) and fo(980) · Application to $a_0(980)$ and $f_0(980)$ · 1.00

Compares with 2 papers

In Section III we turn to the two-channel case, with particular attention to the resonances $a_0(980)$ and $f_0(980)$, their decay rates and spectral functions in comparison with the Flattè distribution [ref] .

Flatte-like distributions and the a_0(980)/f_0(980) mesons · Introduction · 0.99
In Ref. [ref] a width $150$ MeV larger is obtained.

Mass and width of the lowest resonance in QCD · Application to the light scalar mesons $σ$ and $k$ · 0.92

Cites as support 2 papers

When the Kä llen-Lehmann representation is satisfied, as verified at one-loop level in the case of light scalar mesons for large ranges of parameters [ref] , the spectral function (proportional to the imaginary part of the propagator) is correctly normalized and is interpreted as a `mass distribution' for the scalar state.

Propagators of light scalar mesons · Introduction · 0.98
This number increases for increasing coupling strength $g_f_0KK.$ This discussion confirms the interpretation put forward in Ref. [ref] , where the light scalar mesons posses a tetraquark core but are dressed by kaonic clouds.

Scalar mesons above and below 1 GeV · Application to $a_0(980)$ and $f_0(980)$ · 0.99

Cites as background 5 papers

The scalar mesons below $2$ GeV are in center of debate since many years [ref] .

Scalar mesons above and below 1 GeV · Introduction · 1.00
The scalar mesons below $2$ GeV are in center of debate since many years [ref] .

Exotica · Introduction · 1.00
The scalar mesons below $2$ GeV are in center of debate since many years [ref] .

Decays of sigma, kappa, a0(980) and fo(980) · Introduction · 1.00

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.

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a similar equation to (10) (where s = − →q 2 and fΛ = G(s) in the notation of [25]) represents the starting point of the analysis. The function G(s) in the above cited works is taken to be a Gaussian, Λ is of the order o f 1 GeV. The present approach shows the link between such form factor fΛ = G(s) and a nonlocal Lagrangian. However, we will not concentr...

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The resonance k(800) is now listed in the compilation of the Particle Data Gr oup [5] but it still needs conﬁrmation and is omitted from the summary table. The resonance is also found i n many recent theoretical and experimental works ([4, 19], E. Van Beveren, T. A. Rijken, K. Metzger, C. Dullemond, G. Rup p and J. E. Ribeiro, Z. Phys. C 30 (1986) 615. S....

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