64. Scalar Mesons below 1 GeV

164. Scalar Mesons below 1 GeV
Revised August 2023 by T. Gutsche (Tübingen U.), C. Hanhart (FZ Jülich), R.E. Mitchell (IndianaU.) and S. Spanier (Tennessee U.).

64.1 Introduction
The identification of the light scalar mesons is a long-standing puzzle. Scalar resonances are
difficult to resolve experimentally as they can have large decay widths, which cause a strong overlap
between neighboring resonances and with background. In addition, in some cases, several decay
channels open up within a short mass interval (e.g. at the K K̄ and ηη thresholds), producing cusps
in the line shapes of the nearby resonances. Furthermore, one expects non-q q̄ scalar objects, suchas hadronic molecules and multiquark states, in the mass range of interest (for reviews see, e.g.,
Refs. [1–6]).
Light scalars are produced, for example, in πN scattering on polarized/unpolarized targets, pp̄
annihilation, central hadronic production, J/ψ(1S), B-, D- and K-meson decays, γγ formation, andφ radiative decays. Especially for the lightest scalar mesons simple parameterizations like BreitWigner functions and variants thereof fail — this is demonstrated explicitly on the example of thef0 (500) or σ, e.g., in Ref. [7]. Accordingly, more advanced theory tools are necessary to extract theresonance parameters from data. In the analyses available in the literature, fundamental propertiesof the amplitudes such as unitarity, analyticity, Lorentz invariance, and chiral and flavor symmetry
are implemented at different levels of rigor. Especially, chiral symmetry implies the appearanceof zeros, the so-called Adler zeros, close to the threshold in elastic S-wave scattering amplitudes
involving soft (pseudo) Goldstone bosons [8, 9], which may be shifted or removed in associatedproduction processes [10]. Moreover, especially for the lightest non-strange and strange scalar
resonance precision extractions of pole parameters get complicated by the presence of both left-handcuts as well as circular cuts (for a recent review on the subject see Ref. [5]). The methods employed
are the K-matrix formalism, the N/D-method, the Dalitz-Tuan ansatz, the inverse amplitude
method, unitarized quark models with coupled channels, effective chiral field theories and the linear
sigma model, etc. Dynamics near the lowest two-body thresholds in some analyses are describedby crossed channel (t, u) meson exchange or with an effective range parameterization instead of,or in addition to, resonant features in the s-channel. Dispersion theoretical approaches are applied
to pin down the location of resonance poles for the low mass states [11–16] unambigously — inparticular the existence of a nonet of light scalar is not questioned anymore.
In parallel to the developments sketched above also lattice QCD entered the field of precision
spectroscopy, since it was acknowledged that a study of resonances calls for a proper treatment of
the scattering process employing e.g. the Lüscher method [17], later extended to coupled channels
in Refs. [18–23]. This allowed for determinations of the poles of the I = 1/2 [24], the I = 1 [25]
as well as the I = 0 [26, 27] light scalar mesons. Besides and additional firm confirmation of theexistence of a nonet of light scalar mesons, via the quark mass dependence of the pole locations(here the most sophisticated analysis is reported in Ref. [28]) these studies allow for an alternative
look on their nature [29–31].
The mass and width of a resonance are found from the position of the nearest pole in the process
amplitude (S- or T -matrix) at an unphysical sheet of the complex energy plane, traditionally labeledasqsPole = M − i Γ/2 .
(64.1)
It is important to note that in general the pole of a Breit-Wigner parameterization does not agreewith the S- or T -matrix pole. For a detailed discussion of this issue we refer to the review on
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:19am

putting lab
2

64. Scalar Mesons below 1 GeVResonances in this Review of Particle Physics (RPP).
In this review we present proposed values for the mass parameters of the scalar resonances below
1 GeV. Note that those are labeled as ‘our estimate’ — it is not an average over the quoted analyses,
but is chosen to include the bulk of the analyses. An averaging procedure is not justified, since
the analyses use overlapping or sometimes even identical data sets so that they are not statisticallyindependent.
In this note, we discuss the light scalars below 1 GeV organized in the Listings under the entries
K0∗ (700) (or κ) with isospin I = 1/2, a0 (980) with I = 1, as well as f0 (500) (or σ) and f0 (980)
both with I = 0. The I = 2 ππ and I = 3/2 Kπ partial waves do not exhibit resonant behaviour.

Abstract

64.2 The K0∗ (700), also known as κ (I = 1/2)

The K0∗ (700) shows up as a pole in the low energy πK scattering, although its presence andproperties are difficult to establish, since it appears to have a very large width (Γ ≈ 500 MeV)
and resides close to the Kπ threshold. Hadronic D- and B-meson decays provide additional data
points in the vicinity of the Kπ threshold and are discussed in detail in the Review on Multibody
Charm Analyses in this RPP. With a few exceptions discussed there, the three- or more-hadron
final states are usually treated as non-interacting two-body systems. Precise information from

Mπ +MK

Figure 64.1: Location of the K0∗ (700) (or κ) poles in the complex energy plane. Red circles
denote the results of the most sophisticated analyses based on dispersion relations (see text for
details), poles extracted from Breit-Wigner fits are shown as blue squares, while all other analyses
quoted in the Listings are denoted by black triangles. The corresponding references are given in
the listing. The arrow shows the location of the πK thresholds. The grey box indicates the rangeof pole locations classified as ’our estimate’ .
31st May, 2024

64. Scalar Mesons below 1 GeV

3semileptonic D decays, where the strongly interacting two-particle final states could be treated
without approximation, is not available. BES II [32] (re-analyzed in [33]) finds a K0∗ (700)-likestructure in J/ψ(1S) decays to K̄ ∗0 (892)K + π − where K0∗ (700) recoils against the K ∗ (892). The
decay τ − → KS0 π − ντ can be considered clean with respect to final-state interactions and is studied
by Belle [34], with K0∗ (700) parameters fixed to those of Ref. [32].
Some authors find a K0∗ (700) pole in their phenomenological analysis (see, e.g., [35–44]), whileothers do not need to include it in their fits (see, e.g., [45–48]; note that in Ref. [49] only a K0∗ meson
above 825 MeV was discarded using scattering data fits and analyticity, which is not in conflict withthe properties quoted below). All works including constraints from chiral symmetry at low energies
naturally find a light K0∗ (700) below 800 MeV, see, e.g., [50–56]. The analyses of Ref. [15, 16] arebased on the Roy-Steiner equations, which include analyticity and crossing symmetry constraints.
Ref. [57] uses the Padé method to extract pole parameters after refitting scattering data constrained
to satisfy forward dispersion relations. All three arrive at compatible pole positions for the K0∗ (700)that are also consistent with the pole parameters deduced from other theoretical methods. Dueto their large uncertainties, the pole locations deduced from the Breit-Wigner fits appear to bejust about consistent with the other determinations, but the real parts of all those analyses lie
systematically higher. Moreover, phase shifts extracted from the Breit-Wigner functions for theK0∗ (700) are very different from the known scalar πK phase shifts. The various poles are shown inin Fig. 64.1. The compilation in this figure motivates the pole parameters of the K0∗ (700), which
we quote as ‘our estimate’, namely,qsκPole = (630 − 730) − i(260 − 340) MeV .

(64.2)For an extensive discussion about the πK system in general and the κ meson in particular, see
Ref. [58].

64.3 The a0 (980) (I = 1)
The a0 (980) couples strongly to the channels πη and K K̄. Independent of any model, the K K̄component must be large in the a0 (980) wave function, since the mass of the a0 (980) lies very
close to the opening of the K K̄ channel, to which it strongly couples [59, 60]. This generates apronounced cusp-like behavior in the resonant amplitude and to reveal its true coupling constants
the presence of the K K̄ channel cannot be ignored. All listed a0 (980) measurements agree on amass position value near 980 MeV, but the width deduced from the imaginary part of the polelocation has values between 50 and 100 MeV, mostly due to the different models. For example,
the analysis of the pp̄ annihilation data [59] using a unitary K-matrix description finds a width
determined from the T -matrix pole of 92 ± 8 MeV. Note that the width of the a0 (980) line shapeis typically smaller than what could be expected from the pole location.
The relative coupling K K̄/πη is determined indirectly from f1 (1285) [61–63] or η(1410) decays[64–66], from the line shape observed in the πη decay mode [67–70], or from the coupled-channel
analysis of the ππη and K K̄π final states of pp̄ annihilation at rest [59].
For the extraction of the a0 (980) pole locations, Refs. [52, 53, 55, 71] use unitarized chiral perturbation theory. Ref. [72] uses a similar formalism to extract the pole of the a0 (980), employing
the amplitude fixed in Ref. [73]. A dispersion theoretical approach to the isovector scalar πη − K K̄system is presented in Ref. [74] that may be refined further from studies of heavy meson decays [75].
Those efforts lead to a rather precise determination of the a0 (980) pole location [76]. A value consistent for the mass parameter, but with a larger width, is found in the analysis of p̄p annihilationin flight data employing a K-matrix [77]. The poles presented in Refs. [55, 59, 70, 76, 77] are shownin Fig. 64.2 together with the range of pole parameters estimated by us from this compilation,
31st May, 2024

64. Scalar Mesons below 1 GeV

4namely,qa (980)0sPole= (970 − 1020) − i(30 − 70) MeV(64.3)

indicated by the box.

M

+M −
K+
K






M

K

− +M

M

+M 0
K0
K̄




K0

Figure 64.2: Location of the a0 (980) poles from different extractions in the complex energy plane.
The corresponding references are given in the Listings. Also shown are the thresholds for the K + K −and K 0 K̄ 0 channels, relevant for a0 (980)0 , and for the K − K 0 channel, relevant for the a0 (980)− .
The grey box indicates the range of pole locations classified as ’our estimate’. Note that by default
we take the pole located closest to the physical axis; thus the poles shown are not all on the same
sheet.

64.4 The f0 (500), also known as σ-meson (I = 0)
For discussions of the ππ S wave below the K K̄ threshold and on the long history of thef0 (500), which was suggested in linear sigma models more than 50 years ago, see the review [5].
Information on the ππ S-wave phase shift δJI = δ00 was already extracted many years ago from
πN scattering [78–80], and near the ππ threshold from Ke4 decays [81]. The kaon decays werelater revisited leading to consistent data with very much improved statistics [82, 83]. The reported
ππ → K K̄ cross sections [84–87] have analyzed in large uncertainties. The πN data have beencombination with high-statistics
data
(see
entrieslabeledasRVUE
forre-analysesofM + +M −M 0 +M 0 the data). The
K
K̄
K
K
0
2π invariant mass spectra, extracted from pp̄ annihilation at rest into 3π 0 [88,89] and into 5π 0 [90]and from central pp collision [91] do not show a distinct resonance structure below 900 MeV, butthese data sets are consistently described with the standard solution for the ππ scalar isoscalar
S-wave extracted from high energy πN → ππN data [79, 92], which allows for the existence of
31st May, 2024

64. Scalar Mesons below 1 GeV

5

the broad f0 (500). An enhancement is observed in the π + π − invariant mass near threshold in the
decays D+ → π + π − π + [93–95] and J/ψ(1S) → ωπ + π − [96, 97], and in ψ(2S) → J/ψ(1S)π + π −with very limited phase space [98, 99].
The precise f0 (500) (or σ) pole is difficult to establish because of its large width. The ππscattering amplitude shows an unusual energy dependence due to the presence of the Adler zeroin the unphysical regime close to the threshold [8, 9], required by chiral symmetry. However, mostof the analyses listed under f0 (500) agree on a pole position near (500 − i 250 MeV). In particular,
analyses of ππ data that include unitarity, are consistent with the near threshold ππ data fromKe4 decays, and the chiral symmetry constraints from Adler zeroes and/or scattering lengths finda light f0 (500), see, e.g., [100, 101].
Precise pole positions with an uncertainty of less than 20 MeV (see our table for the T -matrix
pole in the Listings) are extracted using the Roy equations, which are twice subtracted dispersion
relations derived from crossing symmetry and analyticity. In Ref. [12] the subtraction constants are
fixed to the S-wave scattering lengths a00 and a20 derived from matching the Roy equations and twoloop chiral perturbation theory [11]. The only additional relevant input to fix the f0 (500) pole isthe ππ-wave phase shift at some higher energy point, chosen as 800 MeV. The analysis is improved
further in Ref. [14]. Alternatively, in Ref. [13] Roy equations are used as constraints for a fit to data.
In that reference also once-subtracted Roy–like equations, called GKPY equations, are used, since
the extrapolation into the complex plane based on the twice-subtracted equations leads to larger
uncertainties, mainly due to the limited experimental information on the isospin 2 ππ scattering
length. Ref. [102] uses Padé approximants for the analytic continuation. All these extractions findconsistent results. Using only analyticity and unitarity to describe data from K2π and Ke4 decays,
Ref. [103] finds consistent values for the pole position and the scattering length a00 . The importanceof the ππ scattering data for fixing the f0 (500) pole is nicely illustrated by comparing analyses ofp̄p → 3π 0 omitting [88, 104] or including [89, 105] information on ππ scattering: while the former
analyses find an extremely broad structure above 1 GeV, the latter find f0 (500) masses of the orderof 400 MeV.
From Fig. 64.3 we estimate the range of pole positions for the f0 (500), namely,qsσPole = (400 − 550) − i(200 − 350) MeV .

(64.4)The plot contains the poles from Refs. [40, 51, 53, 55, 56, 69, 71, 78, 89, 95, 98–101, 103, 105–124] aswell as the advanced dispersion analyses [11–14, 102]. The extracted f0 (500) pole position is very
sensitive to the high accuracy low energy ππ scattering data [82,83]. In fact, all analyses consistent
with this data find poles within the accepted range indicated in the figure. As in case of theK0∗ (700), poles extracted from Breit-Wigner analyses are shown as blue squares. Again we seethat those poles have the tendency to appear at higher masses, although here the effect is notas pronounced as in case of the K0∗ (700). One should not take this as an indication that using
Breit-Wigners is justified in case of the f0 (500), since ππ phase shifts extracted from Breit-Wigners
are in strong discrepancy with the scattering phase shifts.
If one uses just the most advanced dispersive analyses of Refs. [11–14], shown as solid dots in
Fig. 64.3 to determine the pole location of the f0 (500), the range narrows down to [5]qsσPole = (449+22−16 ) − i(275 ± 12) MeV ,(64.5)

which is labeled as ’conservative dispersive estimate’ in this reference.
Besides ππ, the only other decay channel of the f0 (500) is two photons. Due to the large full
width of the f0 (500) an extraction of its two-photon width directly from data is not possible. Thus,
31st May, 2024

6

64. Scalar Mesons below 1 GeV

Figure 64.3: Location of the f0 (500) (or σ) poles in the complex energy plane. Circles denote
the recent analyses based on Roy(-like) dispersion relations, poles extracted from Breit-Wigner fits
are shown as blue squares, while all other analyses are denoted by triangles. The correspondingreferences are given in the Listings. The grey box indicates the range of pole locations classified as’our estimate’ .
the values for Γ (γγ) quoted in the literature as well as the in Listings are based on the expressionpin the narrow width approximation [125] Γ (γγ) ' α2 |gγ |2 /(4Re( sσPole )), where gγ is derived fromthe residue at the f0 (500) pole to two photons and α denotes the electromagnetic fine-structure
constant (see also the review on Resonances in this issue of the RPP). The explicit form of theexpression may vary between different authors due to different definitions of the coupling constant,however, the expression given for Γ (γγ) is free of ambiguities. According to Refs. [126–129], ππand γγ scattering data sets are consistently described including f0 (500) via the two step processof γγ → π + π − with pion exchange in the t- and u-channel, followed by a final state interaction
π + π − → π 0 π 0 . The same conclusion is drawn in Ref. [130], where the f0 (500) → γγ decay
width is dominated by re-scattering. Therefore, it might be difficult to learn anything new about
the nature of the f0 (500) from its γγ coupling. For the most recent work on γγ → ππ, see
Refs. [56, 71, 103–105, 112–125, 128–133]. There are strong indications (e.g., [134–164]) that thePreprint submitted to Elsevier

31st May, 2024

64. Scalar Mesons below 1 GeV7

f0 (500) pole cannot be classified as a q q̄ state.

64.5 The f0 (980) (I = 0)
The f0 (980) couples predominantly to the ππ and K K̄ channels and its signal overlaps strongly
with the background represented mainly by the f0 (500) and the f0 (1370). This can lead to a dip
in the ππ spectrum at the K K̄ threshold. It changes from a dip into a peak structure in theπ 0 π 0 invariant mass spectrum of the reaction π − p → π 0 π 0 n [138], with increasing four momentum
transfer to the π 0 π 0 system, which means increasing the a1 (1260) exchange contribution in the¯ source is switched to a ss̄
amplitude, while the π exchange decreases. Also when a (uū + dd)
source, as it appears when moving from Bd → J/ψ(1S)ππ to Bs → J/ψ(1S)ππ, the f0 signal
switches from a dip to a peak [152]. The f0 (500) and the f0 (980) are also observed in the data forradiative φ decays (φ → f0 γ) from SND [139, 140], CMD2 [141], and KLOE [142, 143].

M

K

+ +M




M

K−

K

0 +M


K̄ 0

Figure 64.4: Location of the f0 (980) poles from different extractions in the complex energy plane.
The corresponding references are given in the Listings. Also shown are the thresholds for the K + K −and K 0 K̄ 0 channels. The grey box indicates the range of pole locations classified as ’our estimate’.
Unitarized chiral perturbation theory is employed to extract the pole of the f0 (980) in Refs. [52,53, 55, 71, 72]. Two different dispersive analyses are used in Ref. [13] to simultaneously pin down
the pole parameters of both the f0 (500) and the f0 (980). The poles extracted in Refs. [13, 14, 55,
56, 77, 108, 165, 166] are shown in Fig. 64.4, together with the range of pole parameters estimatedby us from this compilation, namely,qf (980)0sPole= (980 − 1010) − i(20 − 35) MeV(64.6)

31st May, 2024

October 2, 2023

8

64. Scalar Mesons below 1 GeV

indicated by the box. A disclaimer is important here: Both the poles of a0 (980) and of f0 (980) arelocated very close to the kaon thresholds, with the charged and neutral thresholds being 8 MeV
apart — to illustrate this point the pertinent thresholds are shown explicitly in Figs. 64.2 and 64.4.
This observation leads to the prediction of an enhanced a0 −f0 mixing [167–170]. On the other
hand, all analyses employed in the pole determinations quoted above are done assuming isospin
symmetry. Future studies need to show the impact of isospin violation on the extraction of thea0 (980)/f0 (980) pole parameters.
Analyses of γγ → ππ data [144–146] underline the importance of the K K̄ coupling of thef0 (980), while the resulting two-photon width of the f0 (980) cannot be determined precisely [147].
The prominent appearance of the f0 (980) in the semileptonic Ds decays and decays of B and Bs
mesons implies a dominant (s̄s) component: those decays occur via weak transitions that alternatively result in φ(1020) production. Ratios of decay rates of B and/or Bs mesons into J/ψ(1S)plus f0 (980) or f0 (500) are proposed to extract the flavor mixing angle and to probe the tetraquark
nature of those mesons within a certain model [148,149]. The resulting phenomenological fits of the
LHCb collaboration [150, 151] lead the authors to conclude that their data are incompatible witha model where f0 (500) and f0 (980) are formed from two quarks and two antiquarks (tetraquarks).
However, a dispersive analysis of the same data that allows for a model independent inclusion of thehadronic final state interactions in Ref. [152] puts into question the conclusions of Ref. [150]. Moreover, the assumption underlying Ref. [150] that the production dynamics of f0 (500) and f0 (980)
are equal such that they cancel in ratios appears to be not justified [5].

64.6 Interpretation of the scalars below 1 GeV
In the literature, many structures are discussed for the light scalar mesons, such as conventionalq q̄ mesons, compact (qq)(q̄ q̄) structures (tetraquarks), or meson-meson bound states (hadronicmolecules). In reality, there can be superpositions of these components, and one often depends
on models to determine the dominant one. Although we have seen progress in recent years, thisquestion remains open. Here, we mention some of the present conclusions.
The f0 (980) and a0 (980) are often interpreted as compact tetraquark states [161–164,171] or K K̄bound states [172]. The insight into their internal structure using two-photon widths [140,173–179]
is not conclusive. The f0 (980) appears as a peak structure in Ds decays without f0 (500) background.
Based on that observation it is suggested that f0 (980) has a large ss̄ component, which according
to Ref. [180] is surrounded by a virtual K K̄ cloud (see also Ref. [181]). The inclusive production
property of the f0 (980) in Z 0 decays is consistent with mesons of other nonets [182]. However,as stated by the authors, the results should be taken as model-dependent measurements while
presently no reliable predictions for production rates of non-q q̄ states in Z 0 decay are available.
Data sets on radiative decays (φ → f0 γ and φ → a0 γ) from SND, CMD2, and KLOE (see above)
are consistent with a prominent role of kaon loops. This observation is interpreted as evidence for
a compact four-quark structure of the light scalars in Ref. [183], while it is claimed to point at amolecular nature in Ref. [184,185]. Details of this controversy are given in the comments [186,187];
see also Ref. [188]. There is now a rather broad consensus that the states f0 (980) and a0 (980),
together with the f0 (500) and the K0∗ (700), form a nonet of predominantly four-quark states, whereat larger distances the quarks recombine into a pair of pseudoscalar mesons creating a meson cloud(see, e.g., Ref. [189]). Different QCD sum rule studies [190–195] do not agree on a tetraquark
configuration for the same particle group.
Models that start directly from chiral Lagrangians, either in non-linear [52, 55, 100, 184] or in
linear [196–202] realization, predict the existence of the f0 (500) meson near 500 MeV. Here thef0 (500), a0 (980), f0 (980), and K0∗ (700) (in some models the K0∗ (1430)) would form a nonet (notnecessarily q q̄). In the linear sigma models the lightest pseudoscalars appear as their chiral partners.
31st May, 2024

964. Scalar Mesons below 1 GeV

In the non-linear approaches of Refs. [52] and [100] the above resonances together with the
low mass vector states are generated starting from chiral perturbation theory predictions near the
first open channel, and then by shifting the states to the resonance locations, using unitarity andanalyticity.
Ref. [196] uses a framework with explicit resonances that are unitarized and coupled to the light
pseudoscalars in a chirally invariant way. Evidence for a dominant non-q̄q nature of the lightest
scalar resonances is derived from their mixing scheme. In Ref. [197] the scheme is extended andapplied to the decay η 0 → ηππ, which leads to the same conclusions. In Ref. [203] the large Nc
behavior of the poles is studied to identify the nature of the resonances generated from scattering
equations. This leads to the observation that, while the light vector states behave consistent withwhat is predicted for q̄q states, the light scalars behave very differently. This finding provides strong
support for a dominant non-q̄q nature of the light scalar resonances. Note, the more refined studyof Ref. [134] which finds, in the case of the f0 (500), indications for a subdominant q̄q component
located around 1 GeV in addition to a dominant non-q̄q nature. Additional support for the dominant
non-q q̄ nature of the f0 (500) is given in Ref. [204], where the connection between the pole ofresonances and their Regge trajectories is analyzed. All works including constraints from chiralsymmetry at low energies naturally find a light K0∗ (700) below 800 MeV and a f0 (500) below 600
MeV, see, e.g., [50–55]. In these works the K0∗ (700), f0 (500), f0 (980), and a0 (980) appear to form anonet [51,54]. Additional evidence for this assignment is presented in Ref. [14], where the couplingsof the nine states to q̄q sources are compared. Similar conclusions are reached in Ref. [205] wherein the analyzed data the f0 (980) for example appears as a peak structure in J/ψ(1S) → φπ + π −while in J/ψ(1S) → ωπ + π − the signal is suppressed. The same low mass scalar nonet is also foundearlier in the unitarized quark model of Ref. [111]. A recent phenomenological study based on a q q̄flavor scheme for the light scalars concludes that their glueball content is ruled out because of low
production rates in radiative J/ψ(1S) decays [206].
There are, however, alternative interpretations of the light scalars. For example Ref. [207] (fora more recent, condensed discussion of the idea see Ref. [208]), also builds on chiral symmetry, but
expands around an infrared fixed point such that the f0 (500) appears as a QCD dilaton with a mass
driven by the QCD scale anomaly. The phenomenology studied in that work appears consistent
with this proposal. In Ref. [108,130,209,210] data sets on ππ − K̄K scattering, as well as γγ → ππ,
are analysed and the authors conclude that especially the f0 (500) should have a significant gluon
content. An extensive analysis on mass, width and decay couplings of the light scalars in a QCD
spectral sum rule study [211] leads to ambiguous results for their internal structure. The f0 (500)can equally well have tetraquark, molecular or gluonic content, while the states f0 (980) and a0 (980)
can be of molecular or tetraquark nature. A scenario where the f0 (500) and f0 (980) result fromquarkonia-glueball mixing is also possible. Note, however, that the large similarities of the featuresof the κ and the σ meson put into question both a dilaton and a glueball contribution to the latter,
since neither of them can be present in the former.
A model independent method to identify hadronic molecules goes back to a proposal by Weinberg [212] (an extension of the formalism to virtual states is provided in Ref. [31]), which is shown
to be equivalent to the pole counting arguments of [167–170, 213–218] in Ref. [216]. The formalism
allows one to extract the amount of molecular component in the wave function from the effective coupling constant of a physical state to a nearby continuum channel. It can be applied tonear threshold states only and provides strong evidence that the f0 (980) is predominantly a K̄K
molecule, while the situation turns out to be less clear for the a0 (980) (see also Refs. [177, 179]).
This is in line with the findings of Ref. [72], which reports an important role of the πη channel tothe formation of the a0 (980) in addition to the K̄K channel, while the f0 (980) also in this work
appears to be predominantly a K̄K molecule. The relevance of both the K̄K and the πη channels
31st May, 2024

10

64. Scalar Mesons below 1 GeV

in a dynamically generated a0 (980) is also pointed out in the description of the χc1 → ηπ + π −and Ds+ → π + π 0 η reactions [219, 220]. A recent analysis on the compositeness of the ππ − K K̄and πη − K K̄ channels for the f0 (980) and a0 (980) resonances based on pole values and decaybranchings is performed in Ref. [221]. While the K K̄ configuration dominates in the f0 (980), themeson-meson components are subdominant in a0 (980). Further insights into a0 (980) and f0 (980)
are expected from their mixing [167–170]. A corresponding signal predicted in Refs. [168, 169] isreported by BES III [222]. The importance of the molecular structure of a0 (980), f0 (980) andof their mixing in the reactions B̄s0 → J/ψ(1S)π + π − and B̄s0 → J/ψ(1S)π 0 η is pointed out in
Ref. [223].
In the unitarized quark model with coupled q q̄ and meson-meson channels, the light scalars
are interpreted as additional manifestations of bare q q̄ confinement states, strongly mass shiftedfrom the 1.3 - 1.5 GeV region and very distorted due to the strong 3 P0 coupling to S-wave twomeson decay channels [218, 224]. Thus, in these models the light scalar nonet comprising thef0 (500), f0 (980), K0∗ (700), and a0 (980), as well as the nonet consisting of the f0 (1370), f0 (1500)(or f0 (1710)), K0∗ (1430), and a0 (1450), respectively, are seen as two manifestations of the same bare
input states (see also Ref. [225]). It should not remain unmentioned, however, that the heavier nonet
lies rather close to the input nonet and that the light scalar one emerges only once the coupling
to the two-meson channels is switched on, again highlighting that the meson-meson interaction isindispensable for the light scalars.
Acknowledgement
The authors would like to thank Eberhard Klempt, and Jose Ramon Pelaez for helpful input
and discussions in the preparation of the review.
References[1] F. E. Close and N. A. Tornqvist, J. Phys. G28, R249 (2002), [hep-ph/0204205].
[2] C. Amsler and N. A. Tornqvist, Phys. Rept. 389, 61 (2004).
[3] D. V. Bugg, Phys. Rept. 397, 257 (2004), [hep-ex/0412045].
[4] E. Klempt and A. Zaitsev, Phys. Rept. 454, 1 (2007), [arXiv:0708.4016].
[5] J. R. Pelaez, Phys. Rept. 658, 1 (2016), [arXiv:1510.00653].
[6] J. R. Peláez, A. Rodas and J. R. de Elvira, Eur. Phys. J. ST 230, 6, 1539 (2021),[arXiv:2101.06506].
[7] S. Gardner and U.-G. Meissner, Phys. Rev. D 65, 094004 (2002), [hep-ph/0112281].
[8] S. L. Adler, Phys. Rev. 137, B1022 (1965), [,140(1964)].
[9] S. L. Adler, Phys. Rev. 139, B1638 (1965), [,152(1965)].
[10] J. A. Oller, Phys. Rev. D71, 054030 (2005), [hep-ph/0411105].
[11] G. Colangelo, J. Gasser and H. Leutwyler, Nucl. Phys. B603, 125 (2001), [hep-ph/0103088].
[12] I. Caprini, G. Colangelo, and H. Leutwyler, Phys. Rev. Lett. 96, 132001 (2006).
[13] R. Garcia-Martin et al., Phys. Rev. Lett. 107, 072001 (2011), [arXiv:1107.1635].
[14] B. Moussallam, Eur. Phys. J. C71, 1814 (2011), [arXiv:1110.6074].
[15] S. Descotes-Genon and B. Moussallam, Eur. Phys. J. C 48, 553 (2006), [hep-ph/0607133].
[16] J. R. Peláez and A. Rodas, Phys. Rev. Lett. 124, 17, 172001 (2020), [arXiv:2001.08153].
[17] M. Luscher, Nucl. Phys. B 354, 531 (1991).
[18] S. He, X. Feng and C. Liu, JHEP 07, 011 (2005), [hep-lat/0504019].
[19] M. Doring et al., Eur. Phys. J. A 47, 139 (2011), [arXiv:1107.3988].
31st May, 2024

11

64. Scalar Mesons below 1 GeV[20] M. Doring et al., Eur. Phys. J. A 47, 163 (2011), [arXiv:1108.0676].
[21] M. T. Hansen and S. R. Sharpe, Phys. Rev. D 86, 016007 (2012), [arXiv:1204.0826].
[22] P. Guo et al., Phys. Rev. D 88, 1, 014501 (2013), [arXiv:1211.0929].
[23] R. A. Briceno and Z. Davoudi, Phys. Rev. D 88, 9, 094507 (2013), [arXiv:1204.1110].
[24] J. J. Dudek et al. (Hadron Spectrum), Phys. Rev. Lett. 113, 18, 182001 (2014),[arXiv:1406.4158].
[25] J. J. Dudek, R. G. Edwards and D. J. Wilson (Hadron Spectrum), Phys. Rev. D 93, 9, 094506(2016), [arXiv:1602.05122].
[26] R. A. Briceno et al., Phys. Rev. Lett. 118, 2, 022002 (2017), [arXiv:1607.05900].
[27] R. A. Briceno et al., Phys. Rev. D 97, 5, 054513 (2018), [arXiv:1708.06667].
[28] A. Rodas, J. J. Dudek and R. G. Edwards (2023), [arXiv:2304.03762].
[29] C. Hanhart, J. R. Pelaez and G. Rios, Phys. Rev. Lett. 100, 152001 (2008), [arXiv:0801.2871].
[30] C. Hanhart, J. R. Pelaez and G. Rios, Phys. Lett. B 739, 375 (2014), [arXiv:1407.7452].
[31] I. Matuschek et al., Eur. Phys. J. A 57, 3, 101 (2021), [arXiv:2007.05329].
[32] M. Ablikim et al. (BES), Phys. Lett. B633, 681 (2006), [hep-ex/0506055].
[33] F.-K. Guo et al., Nucl. Phys. A773, 78 (2006), [hep-ph/0509050].
[34] D. Epifanov et al. (Belle), Phys. Lett. B654, 65 (2007), [arXiv:0706.2231].
[35] C. Cawlfield et al. (CLEO), Phys. Rev. D74, 031108 (2006), [hep-ex/0606045].
[36] A. V. Anisovich and A. V. Sarantsev, Phys. Lett. B413, 137 (1997), [hep-ph/9705401].
[37] R. Delbourgo and M. D. Scadron, Int. J. Mod. Phys. A13, 657 (1998), [hep-ph/9807504].
[38] C. M. Shakin and H. Wang, Phys. Rev. D63, 014019 (2001).
[39] M. D. Scadron et al., Nucl. Phys. A724, 391 (2003), [hep-ph/0211275].
[40] D. V. Bugg, Phys. Lett. B572, 1 (2003), [Erratum: Phys. Lett.B595,556(2004)].
[41] M. Ishida, Prog. Theor. Phys. Suppl. 149, 190 (2003), [hep-ph/0212383].
[42] H. Q. Zheng et al., Nucl. Phys. A733, 235 (2004), [hep-ph/0310293].
[43] Z. Y. Zhou and H. Q. Zheng, Nucl. Phys. A775, 212 (2006), [hep-ph/0603062].
[44] J. M. Link et al. (FOCUS), Phys. Lett. B653, 1 (2007), [arXiv:0705.2248].
[45] B. Aubert et al. (BaBar), Phys. Rev. D76, 011102 (2007), [arXiv:0704.3593].
[46] S. Kopp et al. (CLEO), Phys. Rev. D63, 092001 (2001), [hep-ex/0011065].
[47] J. M. Link et al. (FOCUS), Phys. Lett. B535, 43 (2002), [hep-ex/0203031].
[48] J. M. Link et al. (FOCUS), Phys. Lett. B621, 72 (2005), [hep-ex/0503043].
[49] S. N. Cherry and M. R. Pennington, Nucl. Phys. A688, 823 (2001), [hep-ph/0005208].
[50] M. Jamin, J. A. Oller and A. Pich, Nucl. Phys. B587, 331 (2000), [hep-ph/0006045].
[51] D. Black et al., Phys. Rev. D64, 014031 (2001), [hep-ph/0012278].
[52] J. A. Oller, E. Oset and J. R. Pelaez, Phys. Rev. D59, 074001 (1999), [Erratum: Phys.
Rev.D75,099903(2007)], [hep-ph/9804209].
[53] J. A. Oller and E. Oset, Phys. Rev. D60, 074023 (1999), [hep-ph/9809337].
[54] J. A. Oller, Nucl. Phys. A727, 353 (2003), [hep-ph/0306031].
[55] J. R. Pelaez, Mod. Phys. Lett. A19, 2879 (2004), [hep-ph/0411107].
31st May, 2024

12

64. Scalar Mesons below 1 GeV[56] I. Danilkin, O. Deineka and M. Vanderhaeghen, Phys. Rev. D 103, 11, 114023 (2021),[arXiv:2012.11636].
[57] J. R. Peláez, A. Rodas and J. Ruiz de Elvira, Eur. Phys. J. C77, 2, 91 (2017),[arXiv:1612.07966].
[58] J. R. Peláez and A. Rodas, Phys. Rept. 969, 1 (2022), [arXiv:2010.11222].
[59] A. Abele et al., Phys. Rev. D57, 3860 (1998).
[60] M. Bargiotti et al. (OBELIX), Eur. Phys. J. C26, 371 (2003).
[61] D. Barberis et al. (WA102), Phys. Lett. B440, 225 (1998), [hep-ex/9810003].
[62] M. J. Corden et al., Nucl. Phys. B144, 253 (1978).
[63] C. Defoix et al., Nucl. Phys. B44, 125 (1972).
[64] Z. Bai et al. (MARK-III), Phys. Rev. Lett. 65, 2507 (1990).
[65] T. Bolton et al., Phys. Rev. Lett. 69, 1328 (1992).
[66] C. Amsler et al. (Crystal Barrel), Phys. Lett. B353, 571 (1995).
[67] S. M. Flatte, Phys. Lett. 63B, 224 (1976).
[68] C. Amsler et al. (Crystal Barrel), Phys. Lett. B333, 277 (1994).
[69] G. Janssen et al., Phys. Rev. D52, 2690 (1995), [arXiv:nucl-th/9411021].
[70] D. V. Bugg, Phys. Rev. D78, 074023 (2008), [arXiv:0808.2706].
[71] J. A. Oller and E. Oset, Nucl. Phys. A620,
Phys.A652,407(1999)], [hep-ph/9702314].

438 (1997),[Erratum:Nucl.

[72] H. A. Ahmed and C. W. Xiao, Phys. Rev. D 101, 9, 094034 (2020), [arXiv:2001.08141].
[73] C. W. Xiao, U. G. Meißner and J. A. Oller, Eur. Phys. J. A 56, 1, 23 (2020),[arXiv:1907.09072].
[74] M. Albaladejo and B. Moussallam, Eur. Phys. J. C 75, 10, 488 (2015), [arXiv:1507.04526].
[75] M. Albaladejo et al., JHEP 04, 010 (2017), [arXiv:1611.03502].
[76] J. Lu and B. Moussallam, Eur. Phys. J. C 80, 5, 436 (2020), [arXiv:2002.04441].
[77] M. Albrecht et al. (Crystal Barrel), Eur. Phys. J. C 80, 5, 453 (2020), [arXiv:1909.07091].
[78] S. D. Protopopescu et al., Phys. Rev. D7, 1279 (1973).
[79] G. Grayer et al., Nucl. Phys. B75, 189 (1974).
[80] H. Becker et al. (CERN-Cracow-Munich), Nucl. Phys. B151, 46 (1979).
[81] L. Rosselet et al., Phys. Rev. D15, 574 (1977).
[82] S. Pislak et al. (BNL-E865), Phys. Rev. Lett. 87, 221801 (2001), [Erratum: Phys. Rev.
Lett.105,019901(2010)], [hep-ex/0106071].
[83] J. R. Batley et al. (NA48-2), Eur. Phys. J. C70, 635 (2010).
[84] W. Wetzel et al., Nucl. Phys. B115, 208 (1976).
[85] V. A. Polychronakos et al., Phys. Rev. D19, 1317 (1979).
[86] D. H. Cohen et al., Phys. Rev. D22, 2595 (1980).
[87] A. Etkin et al., Phys. Rev. D25, 1786 (1982).
[88] C. Amsler et al., Phys. Lett. B342, 433 (1995).
[89] C. Amsler et al. (Crystal Barrel), Phys. Lett. B355, 425 (1995).
[90] A. Abele et al. (Crystal Barrel), Phys. Lett. B380, 453 (1996).
31st May, 2024

13

64. Scalar Mesons below 1 GeV[91] D.M. Alde et al., Phys. Lett. B397, 350 (1997).
[92] R. Kaminski, L. Lesniak and K. Rybicki, Z. Phys. C74, 79 (1997), [hep-ph/9606362].
[93] E. M. Aitala et al. (E791), Phys. Rev. Lett. 86, 770 (2001), [hep-ex/0007028].
[94] J. M. Link et al. (FOCUS), Phys. Lett. B585, 200 (2004), [hep-ex/0312040].
[95] G. Bonvicini et al. (CLEO), Phys. Rev. D76, 012001 (2007), [arXiv:0704.3954].
[96] J. E. Augustin et al. (DM2), Nucl. Phys. B320, 1 (1989).
[97] M. Ablikim et al. (BES), Phys. Lett. B598, 149 (2004), [hep-ex/0406038].
[98] A. Gallegos, J. L. Lucio M. and J. Pestieau, Phys. Rev. D69, 074033 (2004), [hepph/0311133].
[99] M. Ablikim et al. (BES), Phys. Lett. B645, 19 (2007), [hep-ex/0610023].
[100] A. Dobado and J. R. Pelaez, Phys. Rev. D56, 3057 (1997), [hep-ph/9604416].
[101] I. Caprini, Phys. Rev. D77, 114019 (2008), [arXiv:0804.3504].
[102] P. Masjuan, J. Ruiz de Elvira, J.J. Sanz-Cillero, Phys. Rev. D90, 097901 (2014).
[103] R. Garcia-Martin, J. R. Pelaez and F. J. Yndurain, Phys. Rev. D76, 074034 (2007), [hepph/0701025].
[104] V.V. Anisovich et al., Sov. Phys. Usp. 41, 419 (1998).
[105] V. V. Anisovich, Int. J. Mod. Phys. A21, 3615 (2006), [hep-ph/0510409].
[106] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 118, 1, 012001 (2017), [arXiv:1606.03847].
[107] M. Albaladejo and J. A. Oller, Phys. Rev. D 86, 034003 (2012), [arXiv:1205.6606].
[108] G. Mennessier, S. Narison and X. G. Wang, Phys. Lett. B 688, 59 (2010), [arXiv:1002.1402].
[109] M. Ablikim et al. (BES), Phys. Lett. B 598, 149 (2004), [hep-ex/0406038].
[110] P. Estabrooks, Phys. Rev. D 19, 2678 (1979).
[111] E. van Beveren et al., Z. Phys. C30, 615 (1986), [arXiv:0710.4067].
[112] B. S. Zou and D. V. Bugg, Phys. Rev. D48, R3948 (1993).
[113] N. A. Tornqvist and M. Roos, Phys. Rev. Lett. 76, 1575 (1996), [hep-ph/9511210].
[114] R. Kaminski, L. Lesniak and J. P. Maillet, Phys. Rev. D50, 3145 (1994), [hep-ph/9403264].
[115] N. N. Achasov and G. N. Shestakov, Phys. Rev. D49, 5779 (1994).
[116] M. P. Locher, V. E. Markushin and H. Q. Zheng, Eur. Phys. J. C4, 317 (1998), [hepph/9705230].
[117] T. Hannah, Phys. Rev. D60, 017502 (1999), [hep-ph/9905236].
[118] R. Kaminski, L. Lesniak and B. Loiseau, Phys. Lett. B413, 130 (1997), [hep-ph/9707377].
[119] R. Kaminski, L. Lesniak and B. Loiseau, Eur. Phys. J. C9, 141 (1999), [hep-ph/9810386].
[120] M. Ishida et al., Prog. Theor. Phys. 104, 203 (2000), [hep-ph/0005251].
[121] Y.S. Surovtsev et al., Phys. Rev. D61, 054024 (2001).
[122] M. Ishida et al., Phys. Lett. B518, 47 (2001).
[123] Z. Y. Zhou et al., JHEP 02, 043 (2005), [hep-ph/0406271].
[124] D. V. Bugg, J. Phys. G34, 151 (2007), [hep-ph/0608081].
[125] D. Morgan and M. R. Pennington, Z. Phys. C48, 623 (1990).
[126] J. F. Donoghue, B. R. Holstein and Y. C. Lin, Phys. Rev. D 37, 2423 (1988).
[127] A. Dobado and J. R. Pelaez, Z. Phys. C 57, 501 (1993).
31st May, 2024

14

64. Scalar Mesons below 1 GeV[128] M. R. Pennington, Phys. Rev. Lett. 97, 011601 (2006).
[129] M. R. Pennington, Mod. Phys. Lett. A22, 1439 (2007), [arXiv:0705.3314].
[130] G. Mennessier, S. Narison and W. Ochs, Phys. Lett. B665, 205 (2008), [arXiv:0804.4452].
[131] R. Garcia-Martin and B. Moussallam, Eur. Phys. J. C70, 155 (2010), [arXiv:1006.5373].
[132] M. Hoferichter, D. R. Phillips and C. Schat, Eur. Phys. J. C71, 1743 (2011),[arXiv:1106.4147].
[133] L.-Y. Dai and M. R. Pennington, Phys. Rev. D90, 3, 036004 (2014), [arXiv:1404.7524].
[134] J. R. Pelaez and G. Rios, Phys. Rev. Lett. 97, 242002 (2006), [hep-ph/0610397].
[135] H.-X. Chen, A. Hosaka and S.-L. Zhu, Phys. Lett. B650, 369 (2007), [hep-ph/0609163].
[136] F. Giacosa, Phys. Rev. D75, 054007 (2007), [hep-ph/0611388].
[137] L. Maiani et al., Eur. Phys. J. C50, 609 (2007), [hep-ph/0604018].
[138] N. N. Achasov and G. N. Shestakov, Phys. Rev. D58, 054011 (1998), [hep-ph/9802286].
[139] M. N. Achasov et al., Phys. Lett. B479, 53 (2000), [hep-ex/0003031].
[140] M. N. Achasov et al., Phys. Lett. B485, 349 (2000), [hep-ex/0005017].
[141] R. R. Akhmetshin et al. (CMD-2), Phys. Lett. B462, 371 (1999), [hep-ex/9907005].
[142] A. Aloisio et al. (KLOE), Phys. Lett. B536, 209 (2002), [hep-ex/0204012].
[143] F. Ambrosino et al. (KLOE), Eur. Phys. J. C49, 473 (2007), [hep-ex/0609009].
[144] M. Boglione and M. R. Pennington, Eur. Phys. J. C9, 11 (1999), [hep-ph/9812258].
[145] T. Mori et al. (Belle), Phys. Rev. D75, 051101 (2007), [hep-ex/0610038].
[146] N. N. Achasov and G. N. Shestakov, Phys. Rev. D77, 074020 (2008), [arXiv:0712.0885].
[147] M. R. Pennington et al., Eur. Phys. J. C56, 1 (2008), [arXiv:0803.3389].
[148] R. Fleischer, R. Knegjens and G. Ricciardi, Eur. Phys. J. C71, 1832 (2011), [arXiv:1109.1112].
[149] S. Stone and L. Zhang, Phys. Rev. Lett. 111, 6, 062001 (2013), [arXiv:1305.6554].
[150] R. Aaij et al. (LHCb), Phys. Rev. D90, 1, 012003 (2014), [arXiv:1404.5673].
[151] R. Aaij et al. (LHCb), Phys. Rev. D89, 9, 092006 (2014), [arXiv:1402.6248].
[152] J. T. Daub, C. Hanhart and B. Kubis, JHEP 02, 009 (2016), [arXiv:1508.06841].
[153] D. Barberis et al. (WA102), Phys. Lett. B462, 462 (1999), [hep-ex/9907055].
[154] D. Barberis et al. (WA102), Phys. Lett. B479, 59 (2000), [hep-ex/0003033].
[155] M. Gaspero, Nucl. Phys. A562, 407 (1993).
[156] A. Adamo et al., Nucl. Phys. A558, 13C (1993).
[157] C. Amsler et al. (Crystal Barrel), Phys. Lett. B322, 431 (1994).
[158] A. Abele et al. (Crystal Barrel), Eur. Phys. J. C19, 667 (2001).
[159] A. Abele et al. (CRYSTAL BARREL), Eur. Phys. J. C21, 261 (2001).
[160] D. Barberis et al. (WA102), Phys. Lett. B471, 440 (2000), [hep-ex/9912005].
[161] R. L. Jaffe, Phys. Rev. D15, 267 (1977).
[162] M. G. Alford and R. L. Jaffe, Nucl. Phys. B578, 367 (2000), [hep-lat/0001023].
[163] L. Maiani et al., Phys. Rev. Lett. 93, 212002 (2004), [hep-ph/0407017].
[164] L. Maiani, A. D. Polosa and V. Riquer, Phys. Lett. B651, 129 (2007), [hep-ph/0703272].
[165] V. V. Anisovich and A. V. Sarantsev, Int. J. Mod. Phys. A 24, 2481 (2009).
31st May, 2024

15

64. Scalar Mesons below 1 GeV[166] A. V. Sarantsev et al., Phys. Lett. B 816, 136227 (2021), [arXiv:2103.09680].
[167] N. N. Achasov, S. A. Devyanin and G. N. Shestakov, Phys. Lett. 88B, 367 (1979).
[168] J.-J. Wu, Q. Zhao and B. S. Zou, Phys. Rev. D75, 114012 (2007), [arXiv:0704.3652].
[169] C. Hanhart, B. Kubis and J. R. Pelaez, Phys. Rev. D76, 074028 (2007), [arXiv:0707.0262].
[170] L. Roca, Phys. Rev. D88, 014045 (2013), [arXiv:1210.4742].
[171] G. ’t Hooft et al., Phys. Lett. B662, 424 (2008), [arXiv:0801.2288].
[172] J. D. Weinstein and N. Isgur, Phys. Rev. D41, 2236 (1990).
[173] T. Barnes, Phys. Lett. 165B, 434 (1985).
[174] Z. P. Li, F. E. Close and T. Barnes, Phys. Rev. D43, 2161 (1991).
[175] R. Delbourgo, D.-s. Liu and M. D. Scadron, Phys. Lett. B446, 332 (1999), [hep-ph/9811474].
[176] J. L. Lucio Martinez and M. Napsuciale, Phys. Lett. B454, 365 (1999), [hep-ph/9903234].
[177] C. Hanhart et al., Phys. Rev. D75, 074015 (2007), [hep-ph/0701214].
[178] R. H. Lemmer, Phys. Lett. B650, 152 (2007), [hep-ph/0701027].
[179] T. Branz, T. Gutsche and V. E. Lyubovitskij, Eur. Phys. J. A37, 303 (2008),[arXiv:0712.0354].
[180] A. Deandrea et al., Phys. Lett. B502, 79 (2001), [hep-ph/0012120].
[181] K. M. Ecklund et al. (CLEO), Phys. Rev. D80, 052009 (2009), [arXiv:0907.3201].
[182] K. Ackerstaff et al. (OPAL), Eur. Phys. J. C 4, 19 (1998), [hep-ex/9802013].
[183] N. N. Achasov and V. N. Ivanchenko, Nucl. Phys. B315, 465 (1989).
[184] J. A. Oller, Nucl. Phys. A714, 161 (2003), [hep-ph/0205121].
[185] Yu. S. Kalashnikova et al., Eur. Phys. J. A24, 437 (2005), [hep-ph/0412340].
[186] Yu. S. Kalashnikova et al., Phys. Rev. D78, 058501 (2008), [arXiv:0711.2902].
[187] N. N. Achasov and A. V. Kiselev, Phys. Rev. D78, 058502 (2008), [arXiv:0806.2993].
[188] M. Boglione and M. R. Pennington, Eur. Phys. J. C30, 503 (2003), [hep-ph/0303200].
[189] F. Giacosa and G. Pagliara, Phys. Rev. C76, 065204 (2007), [arXiv:0707.3594].
[190] S. Narison, Nucl. Phys. B Proc. Suppl. 96, 244 (2001).
[191] H.-J. Lee, Eur. Phys. J. A30, 423 (2006), [hep-ph/0512212].
[192] H.-X. Chen, A. Hosaka and S.-L. Zhu, Phys. Rev. D76, 094025 (2007), [arXiv:0707.4586].
[193] J. Sugiyama et al., Phys. Rev. D76, 114010 (2007), [arXiv:0707.2533].
[194] T. Kojo and D. Jido, Phys. Rev. D78, 114005 (2008), [arXiv:0802.2372].
[195] H.-J. Lee, K. S. Kim and H. Kim, Phys. Rev. D 100, 3, 034021 (2019), [arXiv:1904.12311].
[196] D. Black et al., Phys. Rev. D59, 074026 (1999), [hep-ph/9808415].
[197] A. H. Fariborz et al., Phys. Rev. D90, 3, 033009 (2014), [arXiv:1407.3870].
[198] M. D. Scadron, Eur. Phys. J. C6, 141 (1999), [hep-ph/9710317].
[199] M. Ishida, Prog. Theor. Phys. 101, 661 (1999), [hep-ph/9902260].
[200] N. A. Tornqvist, Eur. Phys. J. C11, 359 (1999), [hep-ph/9905282].
[201] M. Napsuciale and S. Rodriguez, Phys. Lett. B603, 195 (2004), [hep-ph/0403072].
[202] M. Napsuciale and S. Rodriguez, Phys. Rev. D70, 094043 (2004), [hep-ph/0407037].
[203] J. R. Pelaez, Phys. Rev. Lett. 92, 102001 (2004), [hep-ph/0309292].
31st May, 2024

16

64. Scalar Mesons below 1 GeV[204] J. T. Londergan et al., Phys. Lett. B729, 9 (2014), [arXiv:1311.7552].
[205] L. Roca et al., Nucl. Phys. A 744, 127 (2004), [hep-ph/0405228].
[206] E. Klempt, Phys. Lett. B 820, 136512 (2021), [arXiv:2104.09922].
[207] R. J. Crewther and L. C. Tunstall, Phys. Rev. D 91, 3, 034016 (2015), [arXiv:1312.3319].
[208] R. J. Crewther, Universe 6, 7, 96 (2020), [arXiv:2003.11259].
[209] R. Kaminski, G. Mennessier and S. Narison, Phys. Lett. B 680, 148 (2009), [arXiv:0904.2555].
[210] G. Mennessier, S. Narison and X. G. Wang, Phys. Lett. B 696, 40 (2011), [arXiv:1009.2773].
[211] R. Albuquerque, S. Narison and D. Rabetiarivony, Nucl. Phys. A 1039, 122743 (2023),[arXiv:2305.02421].
[212] S. Weinberg, Phys. Rev. 130, 776 (1963).
[213] D. Morgan and M. R. Pennington, Phys. Lett. B258, 444 (1991), [Erratum:
Lett.B269,477(1991)].

Phys.

[214] D. Morgan, Nucl. Phys. A543, 632 (1992).
[215] N. A. Tornqvist, Phys. Rev. D51, 5312 (1995), [hep-ph/9403234].
[216] V. Baru et al., Phys. Lett. B586, 53 (2004), [hep-ph/0308129].
[217] M. Ablikim et al. (BESIII), Phys. Rev. D83, 032003 (2011), [arXiv:1012.5131].
[218] N. A. Tornqvist, Z. Phys. C68, 647 (1995), [hep-ph/9504372].
[219] W.-H. Liang, J.-J. Xie and E. Oset, Eur. Phys. J. C 76, 12, 700 (2016), [arXiv:1609.03864].
[220] R. Molina et al., Phys. Lett. B 803, 135279 (2020), [arXiv:1908.11557].
[221] Z.-Q. Wang et al., Phys. Rev. D 105, 7, 074016 (2022), [arXiv:2201.00492].
[222] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 121, 2, 022001 (2018), [arXiv:1802.00583].
[223] J.-T. Li et al., Chin. Phys. C 46, 8, 083108 (2022), [arXiv:2203.13786].
[224] E. van Beveren and G. Rupp, Eur. Phys. J. C22, 493 (2001), [hep-ex/0106077].
[225] M. Boglione and M. R. Pennington, Phys. Rev. D65, 114010 (2002), [hep-ph/0203149].

31st May, 2024