The Width of f0(980) in Isospin-Violating Decays

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The Width of f₀(980) in Isospin-Violating Decays

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14 May 2023

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15 May 2023

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Abstract

The scalar meson f0(980) has been a long-standing puzzle in light hadron physics. The mass and width of the f0(980) in normal decay processes are estimated to be M=990±20 MeV/c2 and Γ=40−100 MeV, respectively. Theoretically, the internal structure of the f0(980) is proposed to be a conventional quark-antiquark meson, tetraquarks state, KK¯ molecule, or quark-antiquark gluon hybrid. So far, explanations about the nature of f0(980) have been controversial for a long time. Recently, anomalously narrow width f0(980) were observed by the BESIII experiment in five independent isospin-violating decay channels. Based on the experimental data, we performed a simultaneous fit to ππ invariant mass distributions, and the mass and width in isospin-violating decays are determined to be M=990.0±0.4 MeV/c2 and Γ=11.4±1.1 MeV, respectively. In addition, we use the parameterized Flatté formula to simultaneously fit the same ππ invariant mass distributions, the two coupling constants of gfππ and gfKK¯ are measured to be gfππ=0.46±0.03 and gfKK¯=1.24±0.32, respectively. According to the joint confidence regions of gfππ and gfKK¯, we conclude that the experimental data tend to support the KK¯ molecule model and the quark-antiquark (qq¯) model, but tend not to support the tetraquarks (q2q¯2) model and the quark-antiquark gluon (qq¯g) hybrid model.

Keywords:

Subject: Physical Sciences - Particle and Field Physics

1. Introduction

The scalar meson

f_{0} (980) (I^{G} J^{P C} = 0^{+} 0^{+ +})

was established experimentally 40 years ago; however, so far, its nature has been a long-standing puzzle. This scalar meson dominantly decays to

π π

final states, but also decays to

K \bar{K}

final states with a small branching fraction [1]. Resonant parameters of

f_{0} (980)

were obtained via measuring decay channels, such as

ϕ \to π^{0} π^{0} γ

by SND [2] and KLOE [3], and

J / ψ \to ϕ π^{+} π^{-}

by BESII [4], or via analyzing scattering processes, such as

e^{+} e^{-} \to K^{+} K^{-} π^{+} π^{-} / K^{+} K^{-} π^{0} π^{0}

by BABAR [5],

π π

scattering data and

K_{l 4}

decay data [6,7]. Based on those experimental measurements and data analysis, the mass and width of

f_{0} (980)

estimated by the Particle Data Group (PDG) were

M = 990 \pm 20

MeV/

c^{2}

and

Γ = 40

to 100 MeV until 2016 [8]. It should be noted that not all uncertainties come from experimental data. The shape of the

f_{0} (980)

varies for different decay processes, and that is why the mass and width are quoted from their process-independent pole position at an unphysical sheet of the complex energy plane, defined as

\sqrt{s_{p o l e}} = M - i Γ / 2

In 2012, the BESIII Collaboration first observed anomalously narrow widths of about 10 MeV of

f_{0} (980)

via isospin-violating decays of

J / ψ \to γ η (1405), η (1405) \to π^{0} f_{0} (980) \to π^{0} π^{+} π^{-} / π^{0} π^{0} π^{0}

[9]. The isospin-violating ratio between

η (1405) \to π^{0} f_{0} (980) \to π^{0} π^{+} π^{-}

and

η (1405) \to π^{0} a_{0}^{0} (980) \to π^{0} η π^{0}

is up to

(17.9 \pm 0.42) %

. Three years later, the BESIII Collaboration once again observed the similarly narrow widths of

f_{0} (980)

via isospin-violating decays of

J / ψ \to ϕ f_{1} (1285), f_{1} (1285) \to π^{0} f_{0} (980) \to π^{0} π^{+} π^{-} / π^{0} π^{0} π^{0}

, in which the isospin-violating ratio is measured to be

(3.6 \pm 1.4) %

[10]. Since then the width of

f_{0} (980)

in PDG has been updated to be

Γ =

10 to 100 MeV [1]. In 2018, the anomalously narrow width of

f_{0} (980)

was further confirmed by the BESIII Collaboration via an isospin-violating decay of

χ_{c 1} \to π^{0} f_{0} (980) \to π^{0} π^{+} π^{-}

[11]. It is worth noting that the narrow-width

f_{0} (980)

mesons are only produced in isospin-violating decays, which apparently disagrees with the normal width of

f_{0} (980)

in isospin-conserving decays.

Theoretically, the internal structure of the

f_{0} (980)

is not only considered as the conventional quark-antiquark, but also proposed to be tetraquarks [12],

K \bar{K}

molecule [13], or quark-antiquark gluon hybrid [14]. However, explanations about the nature of

f_{0} (980)

have been controversial to date. The most famous theoretical study about

f_{0} (980)

is the

a_{0}^{0} (980)

f_{0} (980)

mixing mechanism, which was first proposed in the late 1970s [15]. Because

a_{0}^{0} (980)

and

f_{0} (980)

both can decay into

K \bar{K}

, the charged and neutral kaon mass thresholds are different by about 8 MeV due to isospin-violating effects. The mixing amplitude between

a_{0}^{0} (980)

and

f_{0} (980)

is dominated by the unitary cuts of the intermediate two-kaon system and proportional to the phase-space difference between charged and neutral kaon systems. As a consequence, a narrow peak of about 8 MeV in width is predicted between the charged and neutral

K \bar{K}

mass thresholds. The Feynman diagram of

a_{0}^{0} (980) \to f_{0} (980)

mixing in the decay of

X \to π^{0} a_{0}^{0} (980) \to π^{0} f_{0} (980) \to 3 π

is shown in Figure 1(a), where X can be

η (1405)

f_{1} (1285)

χ_{c 1}

. The

a_{0}^{0} (980)

f_{0} (980)

mixing mechanism has been investigated extensively for a long time, and many decay processes have been discussed [16,17,18,19,20,21,22,23,24,25,26,27]. There was no experimental results until the BESIII Collaboration reported

a_{0}^{0} (980)

f_{0} (980)

mixing via the decays of

J / ψ \to ϕ f_{0} (980) \to ϕ a_{0}^{0} (980) \to ϕ η π^{0}

and

χ_{c 1} \to π^{0} a_{0}^{0} (980) \to π^{0} f_{0} (980) \to π^{0} π^{+} π^{-}

[11,28]. The mixing intensity of

a_{0}^{0} (980) \to f_{0} (980)

, i.e. the isospin-violating ratio, is measure to be

0.40 \pm 0.07 \pm 0.14 \pm 0.07

, which is less than

1.0 %

. Obviously, the

a_{0}^{0} (980)

f_{0} (980)

mixing mechanism can not completely describe the large isospin-violating ratio in the decays of

η (1405) \to π^{0} f_{0} (980) \to π^{0} π^{+} π^{-} / π^{0} π^{0} π^{0}

and

f_{1} (1285) \to π^{0} f_{0} (980) \to π^{0} π^{+} π^{-} / π^{0} π^{0} π^{0}

Since the anomalously narrow widths of about 10 MeV of

f_{0} (980)

was reported by the BESIII experiment [9], a novel scenario called triangle singularity mechanism was proposed to explain the internal behavior in isospin-violating decay processes [29]. Taking the decay

η (1405) \to π^{0} f_{0} (980) \to 3 π

as an example, the intermediate

K {\bar{K}}^{*} +

c.c. pair can exchange an on-shell kaon, the three interaction vertices satisfy the energy-momentum conservation, the physical amplitude has a logarithmic triangle singularity, and the kinematic effects result in a narrow peak in the

π π

invariant mass distributions. The Feynman diagram of the triangle singularity mechanism is shown in Figure 1(b). The triangle singularity mechanism can well explain the narrow width of the

f_{0} (980)

and the large isospin-violating ratio in the decays of

η (1405) \to π^{0} f_{0} (980) \to 3 π

. The triangle singularity mechanism was continuously discussed more extensively and deeply [30,31,32], and applied to other decay processes [33,34,35,36,37].

In this article, firstly, we use the non-relativistic Breit-Wigner formula to simultaneously fit five

π π

invariant mass distributions reported by the BESIII Collaboration, thereby obtaining the accurate mass and width of the

f_{0} (980)

for isospin-violating decays in Section 2. Secondly, the energy-dependent Flatté formula as the shape of the

f_{0} (980)

is used to simultaneously fit the

π π

invariant mass distributions to determine the coupling constants of

f_{0} (980) \to K \bar{K}

and

f_{0} (980) \to π π

, i.e.,

g_{f K \bar{K}}

and

g_{f π π}

in Section 3. Thirdly, we obtain the joint confidence region of the two coupling constants between

g_{f K \bar{K}}

and

g_{f π π}

as a way to provide quantitative constraints on different theoretical models of the

f_{0} (980)

meson in Section 4. Finally, we summarize the whole article in Section 5.

2. Simultaneous fit to $π π$ invariant mass distributions in isospin-violating decays

So far, the BESIII Collaboration has reported anomalously narrow width of the

f_{0} (980)

meson in five independent isospin-violating decay processes, they are

J / ψ \to γ η (1405), η (1405) \to π^{0} f_{0} (980) \to π^{0} π^{+} π^{-} / π^{0} π^{0} π^{0}

[9],

J / ψ \to ϕ f_{1} (1285), f_{1} (1285) \to π^{0} f_{0} (980) \to π^{0} π^{+} π^{-} / π^{0} π^{0} π^{0}

[10], and

ψ (2 S) \to γ χ_{c 1}, χ_{c 1} \to π^{0} f_{0} (980) \to π^{0} π^{+} π^{-}

[11], respectively. In order to accurately determine the mass and width of the

f_{0} (980)

in isospin-violating decays, we perform a simultaneous fit to the

π π

invariant mass spectra of the above five decay channels. In the simultaneous fit, the signal shape of the narrow

f_{0} (980)

meson is described by the same non-relativistic Breit-Wigner function convolved with a Gaussian mass resolution of each decay channel. The mass resolution has an important influence on determining the width of the

f_{0} (980)

meson, therefore the mass resolution is obtained in advance using Monte Carlo simulations, listed in Table 1. The backgrounds are represented by first-order or second-order Chebyshev polynomials, where the polynomial order is kept consistent with the fit in original papers published by BESIII.

The results of the simultaneous fit are illustrated in Figure 2. Figure 2 (a) and (b) are the fit results of the

π π

invariant mass distributions in the decays of

η (1405) \to π^{0} f_{0} (980)

f_{0} (980) \to π^{0} π^{+} π^{-}

and

π^{0} π^{0} π^{0}

; Figure 2 (c) and (d) are the fit results of the

π π

invariant mass spectra in the decays of

f_{1} (1285) \to π^{0} f_{0} (980)

f_{0} (980) \to π^{0} π^{+} π^{-}

and

π^{0} π^{0} π^{0}

; Figure 2 (e) is the fit result of the

π π

invariant mass distribution in the decay of

χ_{c 1} \to π^{0} f_{0} (980)

χ_{c 1} \to π^{0} π^{+} π^{-}

. The mass and width of the

f_{0} (980)

meson obtained from the simultaneous fit, as well as the results reported by the BESIIII Collaboration, are listed in Table 1. The mass and width from the simultaneous fit are

M = 990.0 \pm 0.4

MeV/

c^{2}

and

Γ = 11.4 \pm 1.1

MeV, respectively. The fit results are consistent with the mass and width reported by the BESIIII Collaboration for each channel, but the errors of the mass and width are improved remarkably. We can also obtain the joint confidence regions of the mass and width from the simultaneous fit, as shown in Figure 2 (e), in which the circles from inside to outside represent the confidence level of the two parameters from 1 to 6 standard deviations in order.

3. Determination of coupling constants of $g_{f π π}$ and $g_{f K \bar{K}}$

In general, the

f_{0} (980)

meson can be descibed by the Flatté form of the propagator [29]:

G_{f} = \frac{1}{s - m_{f}^{2} + i \sqrt{s} [Γ_{f π π} (s) + Γ_{f K \bar{K}} (s)]},

(1)

where, s is the square of

π π

invariant mass,

m_{f}^{2}

is the square of the nominal mass of

f_{0} (980)

in PDG [1],

Γ_{f π π} (s)

and

Γ_{f K \bar{K}} (s)

are energy-dependent partial widths of

f_{0} (980) \to π π

and

f_{0} (980) \to K \bar{K}

, respectively. They are defined as:

\begin{matrix} Γ_{f π π} (s) & = & \frac{g_{f π π}^{2}}{16 π \sqrt{s}} [ρ (π^{0}, π^{0}) + 2 ρ (π^{+}, π^{-})], \end{matrix}

(2)

\begin{matrix} Γ_{f K \bar{K}} (s) & = & \frac{g_{f K \bar{K}}^{2}}{16 π \sqrt{s}} [ρ (K^{0}, {\bar{K}}^{0}) + ρ (K^{+}, K^{-})] . \end{matrix}

(3)

where,

g_{f π π}

and

g_{f K \bar{K}}

are coupling constants of

f_{0} (980) \to K \bar{K}

and

f_{0} (980) \to π π

, and

ρ (A, B) = \sqrt{(s - {(m_{A} + m_{B})}^{2}) (s - {(m_{A} - m_{B})}^{2})} / 2 s

is the momentum of the particle A or B in the center-of-mass frame of the two-body decay.

In order to obtain the coupling constants of

g_{f π π}

and

g_{f K \bar{K}}

, a similar simultaneous fit to the

π π

invariant mass spectra of the five decay channels is performed. In the new fit, only the shape of the

f_{0} (980)

meson is replaced by the Flatté formula from the original Breit-Wigner function, and the remaining parts keep the same as the previous fit. The two coupling constants of

g_{f π π}

and

g_{f K \bar{K}}

from the simultaneous fit are determined to be

g_{f π π} = 0.46 \pm 0.03

and

g_{f K \bar{K}} = 1.24 \pm 0.32

, respectively, which are listed in Table 2.

4. Joint confidence regions of the coupling constants $g_{f π π}$ and $g_{f K \bar{K}}$

Theoretically, the

f_{0} (980)

is not only considered as the conventional quark-antiquark (

q \bar{q}

) meson, but also the internal structures are proposed to be tetraquarks (

q^{2} {\bar{q}}^{2}

) state [12],

K \bar{K}

molecule [13], or quark-antiquark gluon (

q \bar{q} g

) hybrid [14]. For those different theoretical models, the correspending coupling constants,

g_{f π π}

and

g_{f K \bar{K}}

, in the decays of

f_{0} (980) \to K \bar{K}

and

f_{0} (980) \to π π

have been predicted [27], which are also listed in Table 2 for comparison.

In order to distinguish the different theoretical models using the experimental data, we also draw the joint confidence regions of these two coupling constants from the simultaneous fit, shown in Figure 3. The dashed lines from inside to outside represent the confidence level of the two coupling constants from 1 to 5 standard deviations in order. The predicted positions from different theoretical models are also marked on the same plot, we find that only the prediction of the

K \bar{K}

molecule model is located within the region of 5 standard deviations, while the predictions from other theoretical models are all outside the region of the five standard deviations, and the position of the quark-antiquark (

q \bar{q}

) model is very close to the boundary of the five standard deviations. The predicted positions of the tetraquarks (

q^{2} {\bar{q}}^{2}

) and the quark-antiquark gluon (

q \bar{q} g

) hybrid are far from the region of the five standard deviations. Therefore, the experimental data tend to support the

K \bar{K}

molecule model and the quark-antiquark (

q \bar{q}

) model, but tend not to support the tetraquarks (

q^{2} {\bar{q}}^{2}

) model and the quark-antiquark gluon (

q \bar{q} g

) hybrid model.

5. Summary

In summary, we perform a simultaneous fit to five

π π

invariant mass distributions reported by the BESIII Collaboration, and determine the mass and width of the

f_{0} (980)

meson in isospin-violating decay processes to be

M = 990.0 \pm 0.4

MeV/

c^{2}

and

Γ = 11.4 \pm 1.1

MeV, respectively. The results are consistent with the mass and width reported by the BESIII experiment, but the errors of the mass and width are improved remarkably. We also use the parameterized Flatté formula to simultaneously fit the same

π π

invariant mass distributions, and obtain the model-independent coupling constants of

g_{f π π}

and

g_{f K \bar{K}}

in the decays of

f_{0} (980) \to K \bar{K}

and

f_{0} (980) \to π π

. Finally, we obtain the joint confidence regions of these two coupling constants from the simultaneous fit. We draw a conclusion that the experimental data tend to support the

K \bar{K}

molecule model and the quark-antiquark (

q \bar{q}

) model, but tend not to support the tetraquarks (

q^{2} {\bar{q}}^{2}

) model and the quark-antiquark gluon (

q \bar{q} g

) hybrid model.

Author Contributions

Conceptualization, Y.L. and W.Y.; methodology, Y.L. and W.Y.; formal analysis, Y.L. and W.Y.; investigation, Y.L. and W.Y.; resources, Y.L. and W.Y.; data curation, Y.L. and W.Y.; writing—original draft preparation, W.Y.; writing—review and editing, W.Y. and D.L.; visualization, W.Y.; supervision, D.L.; project administration, W.Y. and D.L.; funding acquisition, W.Y. All authors have read and agreed to the published version of the manuscript.

Funding

This work is supported in part by National Natural Science Foundation of China (NSFC) under Contracts Nos. 11705006.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

We thank the BESIII Collaboration for publishing the data.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:

PDG	Particle Data Group
BESIII	Beijing Spectrometer III

References

R. L. Workman et al. (Particle Data Group). Review of Particle Physics. Prog. Theor. Exp. Phys. 2022, 2022, 083C01.
M.N. Achasov et al. (SND Collaboration). The ϕ(1020)→π⁰π⁰γ decay. Phys. Lett. B 2000, 485, 349–356.
A. Aloisio et al. (KLOE Collaboration). Study of the decay ϕ→π⁰π⁰γ with the KLOE detector. Phys. Lett. B 2002, 537, 21–27. [CrossRef]
M. Ablikim et al. (BESIII Collaboration). Resonances in J/ψ→ϕπ⁺π⁻ and ϕK⁺K⁻. Phys. Lett. B 2005, 607, 243–253.
B. Aubert et al. (BABAR Collaboration). e⁺e⁻→K⁺K⁻π+π⁻, K⁺K⁻π⁰π⁰ and K⁺K⁻K⁺K⁻ cross sections measured with initial-state radiation. Phys. Rev. D 2007, 76, 012008. [CrossRef]
R. García-Martín, R. Kamiński, J. R. Peláez, J. Ruiz de Elvira and F. J. Ynduráin. Pion-pion scattering amplitude. IV. Improved analysis with once subtracted Roy-like equations up to 1100 MeV. Phys. Rev. D 2011, 83, 074004.
R. García-Martín, R. Kamiński, J. R. Peláez, and J. Ruiz de Elvira. Precise Determination of the f₀(600) and f₀(980) Pole Parameters from a Dispersive Data Analysis. Phys. Rev. Lett. 2011, 107, 072001. [CrossRef]
K.A. Olive et al. (Particle Data Group). Review of Particle Physics. Chin. Phys. C 2014, 38, 090001.
M.Ablikim et al. (BESIII Collaboration). First Observation of η(1405) Decays into f₀(980)π⁰. Phys. Rev. Lett. 2012, 108, 182001. [CrossRef]
M. Ablikim et al. (BESIII Collaboration). Observation of the isospin-violating decay J/ψ→ϕπ⁰f₀(980). Phys. Rev. D 2015, 92, 012007. [CrossRef]
M. Ablikim et al. (BESIII Collaboration). Observation of $a_{0}^{0}$ (980)−f₀(980) Mixing. Phys. Rev. Lett. 2018, 121, 022001. [CrossRef]
J. Weinstein and N. Isgur qq qq system in a potential model. Phys. Rev. D 2015, 27, 588.
J. Weinstein and N. Isgur KK molecules. Phys. Rev. D 1990, 41, 2236. [CrossRef]
S. Ishida et al., In Proceedings of the 6th International Conference on Hadron Spectroscopy, Manchester, United Kingdom, 10th-14th July 1995 (World Scientific, Singapore, 1995), p.454.
N. N. Achasov, S. A. Devyanin, and G. N. Shestakov S*-δ⁰ mixing as a threshold phenomenon. Phys. Lett. B 1979, 88, 367–371. [CrossRef]
B. Kerbikov and F. Tabakin. Mixing of the f₀ and a₀ scalar mesons in threshold photoproduction. Phys. Rev. C 2000, 62, 064601. [CrossRef]
N. N. Achasov and G. N. Shestakov. Proposed Search for $a_{0}^{0}$ (980)−f₀(980) Mixing in Polarization Phenomena. Phys. Rev. Lett. 2004, 92, 182001. [CrossRef]
N.N. Achasov and G. N. Shestakov. Manifestation of the $a_{0}^{0}$ (980)-f₀(980) mixing in the reaction π^-p→ηπ⁰n on a polarized target. Phys. Rev. D 2004, 70, 074015. [CrossRef]
A. E. Kudryavtsev and V. E. Tarasov. On the possibility of observation of a₀-f₀ mixing in the pn→da₀ reaction. JETP Lett. 2000, 72, 410. [CrossRef]
A.E. Kudryavtsev, V. E. Tarasov, J. Haidenbauer, C. Hanhart, and J. Speth. Angular asymmetries in the reactions $\vec{p}$ p→dπ⁺η and $\vec{p}$ n→dπ⁰η and a₀-f₀ mixing. Phys. At. Nucl. 2003, 66, 1946.
A. E. Kudryavtsev, V. E. Tarasov, J. Haidenbauer, C. Hanhart, and J. Speth. Aspects of a₀-f₀ mixing in the reaction $\vec{p} \vec{n}$ da₀. Phys. Rev. C 2002, 66, 015207. [CrossRef]
V.Y. Grishina, L. A. Kondratyuk, M. Büscher, W. Cassing, and H. Ströher. a₀(980)-f₀(980) mixing and isospin violation in the reactions pN→da₀, pd→³He/³Ha₀ and dd→⁴Hea₀. Phys. Lett. B 2001, 521, 217–224. [CrossRef]
F.E. Close and A. Kirk. Isospin breaking exposed in f₀(980)-a₀(980) mixing. Phys. Lett. B 2000, 489, 24–28. [CrossRef]
F.E. Close and A. Kirk. Large isospin mixing in ϕ radiative decay and the spatial size of the f₀(980)-a₀(980) mesons. Phys. Lett. B 2001, 515, 13–16. [CrossRef]
C.Hanhart, B. Kubis, and J. R. Pelaez. Investigation of a₀–f₀ mixing. Phys. Rev. D 2007, 76, 074028. [CrossRef]
J.J. Wu, Q. Zhao and B. S. Zou. Possibility of measuring $a_{0}^{0}$ (980)−f₀(980) mixing from J/ψ→ϕ $a_{0}^{0}$ (980). Phys. Rev. D 2007, 75, 114012. [CrossRef]
J.J. Wu and B. S. Zou. Study of $a_{0}^{0}$ (980)−f₀(980) mixing from $a_{0}^{0}$ (980)→f₀(980) transition. Phys. Rev. D 2008, 78, 074017. [CrossRef]
M.Ablikim et al. (BESIII Collaboration). Study of $a_{0}^{0}$ (980)-f₀(980) mixing. Phys. Rev. D 2011, 83, 032003.
J.-J. Wu, X.-H. Liu, Q. Zhao and B.-S. Zou. Puzzle of Anomalously Large Isospin Violations in η(1405/1475)→3π. Phys. Rev. Lett. 2012, 108, 081803. [CrossRef]
F. Aceti, W. H. Liang, E. Oset, J. J. Wu and B. S. Zou. Isospin breaking and f₀(980)-a₀(980) mixing in the η(1405)→π⁰f₀(980) reaction. Phys. Rev. D 2012, 86, 114007. [CrossRef]
Meng-Chuan Du and Qiang Zhao. Internal particle width effects on the triangle singularity mechanism in the study of the η(1405) and η(1475) puzzle. Phys. Rev. D 2019, 100, 036005. [CrossRef]
Meng-Chuan Du, Yin Cheng, and Qiang Zhao. Vertex corrections due to the triangle singularity mechanism in the light axial vector meson couplings to K^*K+c.c. Phys. Rev. D 2022, 106, 054019. [Google Scholar] [CrossRef]
En Wang, Ju-Jun Xie, Wei-Hong Liang, Feng-Kun Guo, and Eulogio Oset. Role of a triangle singularity in the γp→K⁺Λ(1405) reaction. Phys. Rev. C 2017, 95, 015205. [Google Scholar] [CrossRef]
Hao-Jie Jing, Shuntaro Sakai, Feng-Kun Guo and Bing-Song Zou. Triangle singularities in J/ψ→ηπ⁰ϕ and π⁰π⁰ϕ. Phys. Rev. D 2019, 100, 114010. [Google Scholar] [CrossRef]
Shuntaro Sakai, Eulogio Oset, and Feng-Kun Guo. Triangle singularity in the B^-→K^-π⁰X(3872) reaction and sensitivity to the X(3872) mass. Phys. Rev. D 2020, 101, 054030. [Google Scholar] [CrossRef]
Natsumi Ikeno, Raquel Molina, and Eulogio Oset. Triangle singularity mechanism for the pp→π⁺d fusion reaction. Phys. Rev. C 2021, 104, 014614. [Google Scholar] [CrossRef]
A. Feijoo, R. Molina, L. R. Dai, Eulogio Oset Λ(1405) mediated triangle singularity in the K⁻d→pΣ⁻ reaction. Eur. Phys. J. C 2022, 82, 1028.

Figure 1. The Feynman diagram of the hadronic level: (a) The diagram of

a_{0}^{0} (980) \to f_{0} (980)

mixing in the decay of

X \to π^{0} a_{0}^{0} (980) \to π^{0} f_{0} (980) \to 3 π

. (b) The diagram of triangle singularity mechanism in the decay of

X \to π^{0} f_{0} (980) \to 3 π

. Here, X in the two diagrams can be

η (1405)

f_{1} (1285)

χ_{c 1}

Figure 1. The Feynman diagram of the hadronic level: (a) The diagram of

a_{0}^{0} (980) \to f_{0} (980)

mixing in the decay of

X \to π^{0} a_{0}^{0} (980) \to π^{0} f_{0} (980) \to 3 π

. (b) The diagram of triangle singularity mechanism in the decay of

X \to π^{0} f_{0} (980) \to 3 π

. Here, X in the two diagrams can be

η (1405)

f_{1} (1285)

χ_{c 1}

Figure 2. The simultaneous fit to the

π π

invariant mass spectra of the decay channels of (a)

η (1405) \to π^{0} f_{0} (980) \to π^{0} π^{+} π^{-}

, (b)

η (1405) \to π^{0} f_{0} (980) \to π^{0} π^{0} π^{0}

; (c)

f_{1} (1285) \to π^{0} f_{0} (980) \to π^{0} π^{+} π^{-}

, (d)

f_{1} (1285) \to π^{0} f_{0} (980) \to π^{0} π^{0} π^{0}

; (e)

χ_{c 1} \to π^{0} f_{0} (980) \to π^{0} π^{+} π^{-}

. The dots with error bars are the data reported by BESIII, the solid curves represent the total fit, the dashed curves represent the Breit-Wigner function of the

f_{0} (980)

meson, and the long-dashed curves represent the background polynomials. (f) The joint confidence regions of the mass and width of the

f_{0} (980)

meson from the simultaneous fit.

Figure 2. The simultaneous fit to the

π π

invariant mass spectra of the decay channels of (a)

η (1405) \to π^{0} f_{0} (980) \to π^{0} π^{+} π^{-}

, (b)

η (1405) \to π^{0} f_{0} (980) \to π^{0} π^{0} π^{0}

; (c)

f_{1} (1285) \to π^{0} f_{0} (980) \to π^{0} π^{+} π^{-}

, (d)

f_{1} (1285) \to π^{0} f_{0} (980) \to π^{0} π^{0} π^{0}

; (e)

χ_{c 1} \to π^{0} f_{0} (980) \to π^{0} π^{+} π^{-}

. The dots with error bars are the data reported by BESIII, the solid curves represent the total fit, the dashed curves represent the Breit-Wigner function of the

f_{0} (980)

meson, and the long-dashed curves represent the background polynomials. (f) The joint confidence regions of the mass and width of the

f_{0} (980)

meson from the simultaneous fit.

Figure 3. The joint confidence regions of the two coupling constants,

g_{f π π}

and

g_{f K \bar{K}}

, from the simultaneous fit. The dashed lines from inside to outside represent the confidence level of the two coupling constants from one to five standard deviations in order. The predicted coupling constants from different theoretical models are also marked on the plot. The triangle represents the position of the traditional quark-antiquark model, the reversed triangle represents the position of the

K \bar{K}

molecule model, and the round dot indicates the position of the quark-antiquark gluon hybrid, and the pentagon represents the position of the tetraquark model.

Figure 3. The joint confidence regions of the two coupling constants,

g_{f π π}

and

g_{f K \bar{K}}

K \bar{K}

molecule model, and the round dot indicates the position of the quark-antiquark gluon hybrid, and the pentagon represents the position of the tetraquark model.

Table 1. The mass resolutions, the mass and width of

f_{0} (980)

reported by the BESIIII Collaboration. The bottom line is the results from the simultaneous fit.

Table 1. The mass resolutions, the mass and width of

f_{0} (980)

reported by the BESIIII Collaboration. The bottom line is the results from the simultaneous fit.

Decay Channels	Resolution (MeV)	M (MeV/ $c^{2}$ )	$Γ$ (MeV)
$η (1405) \to π^{+} π^{-} π^{0}$	3.3	$989.9 \pm 0.4$	$9.5 \pm 1.1$
$η (1405) \to π^{0} π^{0} π^{0}$	10.1	$987.0 \pm 1.4$	$4.6 \pm 5.1$
$f_{1} (1285) \to π^{+} π^{-} π^{0}$	3.5	$989.0 \pm 1.4$	$15.4 \pm 4.9$
$f_{1} (1285) \to π^{0} π^{0} π^{0}$	9.1	$995.2 \pm 4.9$	$15.5 \pm 14.6$
$χ_{c 1} \to π^{+} π^{-} π^{0}$	3.5	$989.8 \pm 1.4$	$10.0 \pm 4.0$
Simultaneous fit	—	$990.0 \pm 0.4$	$11.4 \pm 1.1$

Table 2. The central mass and coupling constants,

g_{f π π}

and

g_{f K \bar{K}}

, from the different theoretical predictions and the simultaneous fit.

Table 2. The central mass and coupling constants,

g_{f π π}

and

g_{f K \bar{K}}

, from the different theoretical predictions and the simultaneous fit.

Models	M (MeV/ $c^{2}$ )	$g_{f π π}$ (GeV)	$g_{f K \bar{K}}$ (GeV)
$q \bar{q}$	975	0.64	1.80
$q^{2} {\bar{q}}^{2}$	975	1.90	5.37
$K \bar{K}$	980	0.65	2.74
$q \bar{q} g$	975	1.54	1.70
Simultaneous fit	$990.0 \pm 0.4$	$0.46 \pm 0.03$	$1.24 \pm 0.32$

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The Width of f0(980) in Isospin-Violating Decays

Abstract

1. Introduction

2. Simultaneous fit to π π invariant mass distributions in isospin-violating decays

3. Determination of coupling constants of g f π π and g f K K ¯

4. Joint confidence regions of the coupling constants g f π π and g f K K ¯

5. Summary

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

Abbreviations

References

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The Width of f₀(980) in Isospin-Violating Decays

2. Simultaneous fit to $π π$ invariant mass distributions in isospin-violating decays

3. Determination of coupling constants of $g_{f π π}$ and $g_{f K \bar{K}}$

4. Joint confidence regions of the coupling constants $g_{f π π}$ and $g_{f K \bar{K}}$