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

1. Introduction

The scalar meson $f_0\left(980\right)(I^G{J}^{PC}=0^{+}{0}^{++})$ was established experimentally 40 years ago; however, so far, its nature has been a long-standing puzzle. This scalar meson dominantly decays to $\pi \pi$ final states, but also decays to $K\overline{K}$ final states with a small branching fraction [1]. Resonant parameters of $f_0\left(980\right)$ were obtained via measuring decay channels, such as $\varphi \to {\pi }^0{\pi }^0\gamma$ by SND [2] and KLOE [3], and $J/\psi \to \varphi {\pi }^{+}{\pi }^{-}$ by BESII [4], or via analyzing scattering processes, such as $e^{+}{e}^{-}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}/K^{+}{K}^{-}{\pi }^0{\pi }^0$ by BABAR [5], $\pi \pi$ scattering data and $K_{l4}$ decay data [6,7]. Based on those experimental measurements and data analysis, the mass and width of $f_0\left(980\right)$ estimated by the Particle Data Group (PDG) were $M=990\pm 20$ MeV/$c^2$ and $\mathsf{\Gamma }=40$ to 100 MeV until 2016 [8]. It should be noted that not all uncertainties come from experimental data. The shape of the $f_0\left(980\right)$ 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 $\sqrt{s_{pole}}=M-i\mathsf{\Gamma }/2$.

In 2012, the BESIII Collaboration first observed anomalously narrow widths of about 10 MeV of $f_0\left(980\right)$ via isospin-violating decays of $J/\psi \to \gamma \eta \left(1405\right),\eta \left(1405\right)\to {\pi }^0{f}_0\left(980\right)\to {\pi }^0{\pi }^{+}{\pi }^{-}/{\pi }^0{\pi }^0{\pi }^0$ [9]. The isospin-violating ratio between $\eta \left(1405\right)\to {\pi }^0{f}_0\left(980\right)\to {\pi }^0{\pi }^{+}{\pi }^{-}$ and $\eta \left(1405\right)\to {\pi }^0{a}_0^0\left(980\right)\to {\pi }^0\eta {\pi }^0$ is up to $(17.9\pm 0.42)\%$. Three years later, the BESIII Collaboration once again observed the similarly narrow widths of $f_0\left(980\right)$ via isospin-violating decays of $J/\psi \to \varphi f_1\left(1285\right),f_1\left(1285\right)\to {\pi }^0{f}_0\left(980\right)\to {\pi }^0{\pi }^{+}{\pi }^{-}/{\pi }^0{\pi }^0{\pi }^0$, in which the isospin-violating ratio is measured to be $(3.6\pm 1.4)\%$ [10]. Since then the width of $f_0\left(980\right)$ in PDG has been updated to be $\mathsf{\Gamma }=$10 to 100 MeV [1]. In 2018, the anomalously narrow width of $f_0\left(980\right)$ was further confirmed by the BESIII Collaboration via an isospin-violating decay of ${\chi }_{c1}\to {\pi }^0{f}_0\left(980\right)\to {\pi }^0{\pi }^{+}{\pi }^{-}$ [11]. It is worth noting that the narrow-width $f_0\left(980\right)$ mesons are only produced in isospin-violating decays, which apparently disagrees with the normal width of $f_0\left(980\right)$ in isospin-conserving decays.

Theoretically, the internal structure of the $f_0\left(980\right)$ is not only considered as the conventional quark-antiquark, but also proposed to be tetraquarks [12], $K\overline{K}$ molecule [13], or quark-antiquark gluon hybrid [14]. However, explanations about the nature of $f_0\left(980\right)$ have been controversial to date. The most famous theoretical study about $f_0\left(980\right)$ is the $a_0^0\left(980\right)$-$f_0\left(980\right)$ mixing mechanism, which was first proposed in the late 1970s [15]. Because $a_0^0\left(980\right)$ and $f_0\left(980\right)$ both can decay into $K\overline{K}$, the charged and neutral kaon mass thresholds are different by about 8 MeV due to isospin-violating effects. The mixing amplitude between $a_0^0\left(980\right)$ and $f_0\left(980\right)$ 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 $K\overline{K}$ mass thresholds. The Feynman diagram of $a_0^0\left(980\right)\to f_0\left(980\right)$ mixing in the decay of $X\to {\pi }^0{a}_0^0\left(980\right)\to {\pi }^0{f}_0\left(980\right)\to 3\pi$ is shown in Figure 1(a), where X can be $\eta \left(1405\right)$, $f_1\left(1285\right)$ or ${\chi }_{c1}$. The $a_0^0\left(980\right)$-$f_0\left(980\right)$ 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 $a_0^0\left(980\right)$-$f_0\left(980\right)$ mixing via the decays of $J/\psi \to \varphi f_0\left(980\right)\to \varphi a_0^0\left(980\right)\to \varphi \eta {\pi }^0$ and ${\chi }_{c1}\to {\pi }^0{a}_0^0\left(980\right)\to {\pi }^0{f}_0\left(980\right)\to {\pi }^0{\pi }^{+}{\pi }^{-}$ [11,28]. The mixing intensity of $a_0^0\left(980\right)\to f_0\left(980\right)$, 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 $a_0^0\left(980\right)$-$f_0\left(980\right)$ mixing mechanism can not completely describe the large isospin-violating ratio in the decays of $\eta \left(1405\right)\to {\pi }^0{f}_0\left(980\right)\to {\pi }^0{\pi }^{+}{\pi }^{-}/{\pi }^0{\pi }^0{\pi }^0$ and $f_1\left(1285\right)\to {\pi }^0{f}_0\left(980\right)\to {\pi }^0{\pi }^{+}{\pi }^{-}/{\pi }^0{\pi }^0{\pi }^0$.

Since the anomalously narrow widths of about 10 MeV of $f_0\left(980\right)$ 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 $\eta \left(1405\right)\to {\pi }^0{f}_0\left(980\right)\to 3\pi$ as an example, the intermediate $K{\overline{K}}^{*}+$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 $\pi \pi$ 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 $f_0\left(980\right)$ and the large isospin-violating ratio in the decays of $\eta \left(1405\right)\to {\pi }^0{f}_0\left(980\right)\to 3\pi$. 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 $\pi \pi$ invariant mass distributions reported by the BESIII Collaboration, thereby obtaining the accurate mass and width of the $f_0\left(980\right)$ for isospin-violating decays in Section 2. Secondly, the energy-dependent Flatté formula as the shape of the $f_0\left(980\right)$ is used to simultaneously fit the $\pi \pi$ invariant mass distributions to determine the coupling constants of $f_0\left(980\right)\to K\overline{K}$ and $f_0\left(980\right)\to \pi \pi$, i.e., $g_{fK\overline{K}}$ and $g_{f\pi \pi }$ in Section 3. Thirdly, we obtain the joint confidence region of the two coupling constants between $g_{fK\overline{K}}$ and $g_{f\pi \pi }$ as a way to provide quantitative constraints on different theoretical models of the $f_0\left(980\right)$ meson in Section 4. Finally, we summarize the whole article in Section 5.

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

So far, the BESIII Collaboration has reported anomalously narrow width of the $f_0\left(980\right)$ meson in five independent isospin-violating decay processes, they are $J/\psi \to \gamma \eta \left(1405\right),\eta \left(1405\right)\to {\pi }^0{f}_0\left(980\right)\to {\pi }^0{\pi }^{+}{\pi }^{-}/{\pi }^0{\pi }^0{\pi }^0$ [9], $J/\psi \to \varphi f_1\left(1285\right),f_1\left(1285\right)\to {\pi }^0{f}_0\left(980\right)\to {\pi }^0{\pi }^{+}{\pi }^{-}/{\pi }^0{\pi }^0{\pi }^0$ [10], and $\psi \left(2S\right)\to \gamma {\chi }_{c1},{\chi }_{c1}\to {\pi }^0{f}_0\left(980\right)\to {\pi }^0{\pi }^{+}{\pi }^{-}$ [11], respectively. In order to accurately determine the mass and width of the $f_0\left(980\right)$ in isospin-violating decays, we perform a simultaneous fit to the $\pi \pi$ invariant mass spectra of the above five decay channels. In the simultaneous fit, the signal shape of the narrow $f_0\left(980\right)$ 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 $f_0\left(980\right)$ 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 $\pi \pi$ invariant mass distributions in the decays of $\eta \left(1405\right)\to {\pi }^0{f}_0\left(980\right)$, $f_0\left(980\right)\to {\pi }^0{\pi }^{+}{\pi }^{-}$ and ${\pi }^0{\pi }^0{\pi }^0$; Figure 2 (c) and (d) are the fit results of the $\pi \pi$ invariant mass spectra in the decays of $f_1\left(1285\right)\to {\pi }^0{f}_0\left(980\right)$, $f_0\left(980\right)\to {\pi }^0{\pi }^{+}{\pi }^{-}$ and ${\pi }^0{\pi }^0{\pi }^0$; Figure 2 (e) is the fit result of the $\pi \pi$ invariant mass distribution in the decay of ${\chi }_{c1}\to {\pi }^0{f}_0\left(980\right)$, ${\chi }_{c1}\to {\pi }^0{\pi }^{+}{\pi }^{-}$. The mass and width of the $f_0\left(980\right)$ 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 $\mathsf{\Gamma }=11.4\pm 1.1$ 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 $g_{f\pi \pi }$ and $g_{fK\overline{K}}$

In general, the $f_0\left(980\right)$ meson can be descibed by the Flatté form of the propagator [29]:

$$G_f=\frac{1}{s-m_f^2+i\sqrt{s}[{\mathsf{\Gamma }}_{f\pi \pi }\left(s\right)+{\mathsf{\Gamma }}_{fK\overline{K}}\left(s\right)]},$$

(1)

where, s is the square of $\pi \pi$ invariant mass, $m_f^2$ is the square of the nominal mass of $f_0\left(980\right)$ in PDG [1], ${\mathsf{\Gamma }}_{f\pi \pi }\left(s\right)$ and ${\mathsf{\Gamma }}_{fK\overline{K}}\left(s\right)$ are energy-dependent partial widths of $f_0\left(980\right)\to \pi \pi$ and $f_0\left(980\right)\to K\overline{K}$, respectively. They are defined as:

$$\begin{array}{ccc}\hfill {\mathsf{\Gamma }}_{f\pi \pi }\left(s\right)& =& \frac{g_{f\pi \pi }^2}{16\pi \sqrt{s}}[\rho ({\pi }^0,{\pi }^0)+2\rho ({\pi }^{+},{\pi }^{-})],\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill {\mathsf{\Gamma }}_{fK\overline{K}}\left(s\right)& =& \frac{g_{fK\overline{K}}^2}{16\pi \sqrt{s}}[\rho (K^0,{\overline{K}}^0)+\rho (K^{+},K^{-})].\hfill \end{array}$$

(3)

where, $g_{f\pi \pi }$ and $g_{fK\overline{K}}$ are coupling constants of $f_0\left(980\right)\to K\overline{K}$ and $f_0\left(980\right)\to \pi \pi$, and $\rho (A,B)=\sqrt{(s-{(m_A+m_B)}^2)(s-{(m_A-m_B)}^2)}/2s$ 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 $g_{f\pi \pi }$ and $g_{fK\overline{K}}$, a similar simultaneous fit to the $\pi \pi$ invariant mass spectra of the five decay channels is performed. In the new fit, only the shape of the $f_0\left(980\right)$ 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 $g_{f\pi \pi }$ and $g_{fK\overline{K}}$ from the simultaneous fit are determined to be $g_{f\pi \pi }=0.46\pm 0.03$ and $g_{fK\overline{K}}=1.24\pm 0.32$, respectively, which are listed in Table 2.

4. Joint confidence regions of the coupling constants $g_{f\pi \pi }$ and $g_{fK\overline{K}}$

Theoretically, the $f_0\left(980\right)$ is not only considered as the conventional quark-antiquark ($q\overline{q}$) meson, but also the internal structures are proposed to be tetraquarks ($q^2{\overline{q}}^2$) state [12], $K\overline{K}$ molecule [13], or quark-antiquark gluon ($q\overline{q}g$) hybrid [14]. For those different theoretical models, the correspending coupling constants, $g_{f\pi \pi }$ and $g_{fK\overline{K}}$, in the decays of $f_0\left(980\right)\to K\overline{K}$ and $f_0\left(980\right)\to \pi \pi$ 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 $K\overline{K}$ 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 ($q\overline{q}$) model is very close to the boundary of the five standard deviations. The predicted positions of the tetraquarks ($q^2{\overline{q}}^2$) and the quark-antiquark gluon ($q\overline{q}g$) hybrid are far from the region of the five standard deviations. Therefore, the experimental data tend to support the $K\overline{K}$ molecule model and the quark-antiquark ($q\overline{q}$) model, but tend not to support the tetraquarks ($q^2{\overline{q}}^2$) model and the quark-antiquark gluon ($q\overline{q}g$) hybrid model.

5. Summary

In summary, we perform a simultaneous fit to five $\pi \pi$ invariant mass distributions reported by the BESIII Collaboration, and determine the mass and width of the $f_0\left(980\right)$ meson in isospin-violating decay processes to be $M=990.0\pm 0.4$ MeV/$c^2$ and $\mathsf{\Gamma }=11.4\pm 1.1$ 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 $\pi \pi$ invariant mass distributions, and obtain the model-independent coupling constants of $g_{f\pi \pi }$ and $g_{fK\overline{K}}$ in the decays of $f_0\left(980\right)\to K\overline{K}$ and $f_0\left(980\right)\to \pi \pi$. 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 $K\overline{K}$ molecule model and the quark-antiquark ($q\overline{q}$) model, but tend not to support the tetraquarks ($q^2{\overline{q}}^2$) model and the quark-antiquark gluon ($q\overline{q}g$) hybrid model.