Strong isospin breaking at production of light scalars

TL;DR

This paper explores how breaking isotopic symmetry can be used to understand the production mechanisms and nature of light scalar mesons, providing insights into their underlying physics.

Contribution

It introduces a novel approach using isospin breaking to investigate the properties and origins of light scalar mesons.

Findings

Isospin breaking effects are significant in scalar meson production.

The study offers new perspectives on the internal structure of light scalars.

Results suggest specific mechanisms for scalar meson formation.

Abstract

It is discussed breaking the isotopic symmetry as the tool of studying the mechanism production and nature of light scalar mesons.

Equations

$$\displaystyle\Pi_{a^{0}_{0}f_{0}}(m)=\frac{g_{a^{0}_{0}K^{+}K^{-}}g_{f_{0}K^{+}K^{-}}}{16\pi}\Biggl{[}\,i\,\Bigl{(}\rho_{K^{+}K^{-}}(m)$$

$$\displaystyle-\rho_{K^{0}\bar{K}^{0}}(m)\Bigr{)}-\frac{\rho_{K^{+}K^{-}}(m)}{\pi}\ln\frac{1+\rho_{K^{+}K^{-}}(m)}{1-\rho_{K^{+}K^{-}}(m)}$$

$$\displaystyle+\frac{\rho_{K^{0}\bar{K}^{0}}(m)}{\pi}\ln\frac{1+\rho_{K^{0}\bar{K}^{0}}(m)}{1-\rho_{K^{0}\bar{K}^{0}}(m)}\,\,\Biggl{]}$$

$$\displaystyle\approx\frac{g_{a^{0}_{0}K^{+}K^{-}}g_{f_{0}K^{+}K^{-}}}{16\pi}\,i\,\Bigl{(}\rho_{K^{+}K^{-}}(m)-\rho_{K^{0}\bar{K}^{0}}(m)\Bigr{)},$$

$$|{\Pi }_{a_0^0{f}_0}(m)|\approx \frac{|g_{a_0^0{K}^{+}{K}^{-}}{g}_{f_0{K}^{+}{K}^{-}}|}{16\pi }\sqrt{\frac{2(m_{K^0}-m_{K^{+}})}{m_{K^0}}}$$

$$\approx 0.127\frac{|g_{a_0{K}^{+}{K}^{-}}{g}_{f_0{K}^{+}{K}^{-}}|}{16\pi }\simeq 0.03\text{mbox}{GeV}^2$$

$$\approx m_K\sqrt{m_{K^0}^2-m_{K^{+}}^2}\approx m_K^{3/2}\sqrt{m_d-m_u}.$$

$$BR(f_0(980)\to K\bar{K}\to a_0^0(980)\to \eta {\pi }^0)$$

$$=\int {\left|\frac{{\Pi }_{a_0^0{f}_0}(m)}{D_{a_0^0}(m)D_{f_0}(m)-{\Pi }_{a_0^0{f}_0}^2(m)}\right|}^2$$

$$\displaystyle\times\frac{2m^{2}\Gamma_{a^{0}_{0}\to\eta\pi^{0}}(m)}{\pi}dm\approx 0.3\%\,,\ \ \$$

$$BR(a_0^0(980)\to K\bar{K}\to f_0(980)\to \pi \pi )$$

$$=\int {\left|\frac{{\Pi }_{a_0^0{f}_0}(m)}{D_{a_0^0}(m)D_{f_0}(m)-{\Pi }_{a_0^0{f}_0}^2(m)}\right|}^2$$

$$\displaystyle\times\frac{2m^{2}\Gamma_{f_{0}\to\pi\pi}(m)}{\pi}dm\approx 0.2\%\,,\ \ \$$

$$\frac{d^3\sigma }{dtdmd\psi }=\frac{1}{2\pi }[|M_{++}{|}^2+|M_{+-}{|}^2$$

$$+2\Im (M_{++}{M}_{+-}^{\ast })P\cos \psi ],$$

$$\frac{BR(J/\psi \to \varphi f_0(980)\to \varphi a_0^0(980)\to \varphi \eta {\pi }^0)}{BR(J/\psi \to \varphi f_0(980)\to \varphi \pi \pi )}$$

$$=(0.60\pm 0.20(stat.)\pm 0.12(sys.)\pm 0.26(para.))\%$$

$$\approx \frac{BR(f_0(980)\to K\bar{K}\to a_0^0(980)\to \eta {\pi }^0)}{BR(f_0(980)\to \pi \pi )},$$

$$\frac{BR({\chi }_{c1}\to a_0^0(980){\pi }^0\to f_0(980){\pi }^0\to {\pi }^{+}{\pi }^{-}{\pi }^0)}{BR({\chi }_{c1}\to a_0^0(980){\pi }^0\to \eta {\pi }^0{\pi }^0)}$$

$$=(0.31\pm 0.16(stat.)\pm 0.14(sys.)\pm 0.03(para.))\%$$

$$\approx \frac{BR(a_0^0(980)\to K\bar{K}\to f_0(980)\to {\pi }^{+}{\pi }^{-})}{BR(a_0^0(980)\to \eta {\pi }^0)}.$$

$$\frac{BR(f_1(1285)\to a_0^0(980){\pi }^0\to f_0(980){\pi }^0\to {\pi }^{+}{\pi }^{-}{\pi }^0)}{BR(f_1(1285)\to a_0^0(980){\pi }^0\to \eta {\pi }^0{\pi }^0)}$$

$$=(2.5\pm 0.9)\%$$

$$\approx \frac{BR(a_0^0(980)\to K\bar{K}\to f_0(980)\to {\pi }^{+}{\pi }^{-})}{BR(a_0^0(980)\to \eta {\pi }^0)},$$

$$\frac{g_{f_0{\pi }^{+}{\pi }^{-}}^2}{4\pi }=1.2\text{mbox}{GeV}^2,\frac{g_{f_0{K}^{+}{K}^{-}}^2}{4\pi }=5.7\text{mbox}{GeV}^2,$$

$$\frac{g_{a_0^0\eta {\pi }^0}^2}{4\pi }=1.9\text{mbox}{GeV}^2,\frac{g_{a_0^0{K}^{+}{K}^{-}}^2}{4\pi }=9.9\text{mbox}{GeV}^2.$$

$$\frac{BR(f_1(1285)\to f_0(980){\pi }^0\to {\pi }^{+}{\pi }^{-}{\pi }^0)}{BR(f_1(1285)\to K\bar{K}\pi )}=0.0022$$

$$\frac{BR(f_1(1285)\to f_0(980){\pi }^0\to {\pi }^{+}{\pi }^{-}{\pi }^0)}{BR(f_1(1285)\to K\bar{K}\pi )}$$

$$=0.033\pm 0.010.$$

$$BR(f_1(1285)\to f_0(980){\pi }^0\to {\pi }^{+}{\pi }^{-}{\pi }^0)\approx 0.0255\%$$

$$BR(f_1(1285)\to f_0(980){\pi }^0\to {\pi }^{+}{\pi }^{-}{\pi }^0)$$

$$=(0.30\pm 0.09)\%.$$

$${\mathcal{M}}_{f_1(1285)\to f_0(980){\pi }^0}(m)=g_{f_0{K}^{+}{K}^{-}}\{A(m)$$

$$\displaystyle\times i[\rho_{K^{+}K^{-}}(m)-\rho_{K^{0}\bar{K}^{0}}(m)]+B(m)[\rho^{2}_{K^{+}K^{-}}(m)\$$

$$-{\rho }_{K^0{\bar{K}}^0}^2(m)]+O[{\rho }_{K^{+}{K}^{-}}^3(m)-{\rho }_{K^0{\bar{K}}^0}^3(m)]+\cdot \cdot \cdot \}.$$

$${\mathcal{M}}_{f_1(1285)\to f_0(980){\pi }^0}(m)=g_{f_0{K}^{+}{K}^{-}}A(m)$$

$$\times i[{\rho }_{K^{+}{K}^{-}}(m)-{\rho }_{K^0{\bar{K}}^0}(m)].$$

$$\frac{d{\Gamma }_{f_1(1285)\to f_0(980){\pi }^0\to {\pi }^{+}{\pi }^{-}{\pi }^0}(m)}{dm}$$

$$=\frac{1}{16\pi }|{\mathcal{M}}_{f_1(1285)\to f_0(980){\pi }^0}(m){|}^2$$

$$\times p^3(m)\frac{2m^2{\Gamma }_{f_0\to {\pi }^{+}{\pi }^{-}}(m)}{\pi |D_{f_0}(m){|}^2},$$

$$\frac{d{\Gamma }_{f_1(1285)\to K^{+}{K}^{-}{\pi }^0}}{dm}$$

$$\displaystyle\qquad=\frac{2\,m}{\pi}\,\rho_{K^{+}K^{-}}(m)\,p^{3}(m)\,|A(m)|^{2}\,.\ \$$

$$\displaystyle\Gamma_{f_{1}(1285)\to f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}}\ \ \ \$$

Full text

I Introduction

The thirty seven years ago we discovered theoretically a threshold phenomenon known as the mixing of $a^{0}_{0}(980)$ and $f_{0}(980)$ resonances that breaks the isotopic invariance considerably, since the effect $\sim\sqrt{2(M_{K^{0}}-M_{K^{+}})/M_{K^{0}}}\approx 0,13$ in the module of the amplitude ADS79 ; see also Ref. ADS81 . This effect appears as the narrow, $2(M_{K^{0}}-M_{K^{+}})\approx 8$ MeV, resonant structure between the $K^{+}K^{-}$ and $K^{0}\bar{K}^{0}$ thresholds, $a_{0}^{0}(980)\to K\bar{K}\to f_{0}(980)$ and vice versa. Since that time many new proposals were appeared, concerning both the searching it and estimating the effects related with this phenomenon [3-29].

Nowadays, this phenomenon has been discovered experimentally and studied with the help of detectors VES in Protvino Do08 ; Do11 and BESIII in Beijing Ab1 ; Ab2 ; Ab3 in the processes

(a)   $\pi^{-}N\to\pi^{-}f_{1}(1285)N\to\pi^{-}f_{0}(980)\pi^{0}N\to$

$\ \ \ \ \to$ $\pi^{-}\pi^{+}\pi^{-}\pi^{0}N$  Do08 ; Do11 ,

(b)   $J/\psi\to\phi f_{0}(980)\to\phi a_{0}(980)\to\phi\eta\pi^{0}$  Ab1 ,

(c)    $\chi_{c1}$ $\to$ $a_{0}(980)\pi^{0}$ $\to$ $f_{0}(980)\pi^{0}$ $\to$ $\pi^{+}\pi^{-}\pi^{0}$  Ab1 ,

(d)   $J/\psi\to\gamma\eta(1405)\to\gamma f_{0}(980)\pi^{0}\to\gamma\,3\pi$  Ab2 ,

(e)   $J/\psi\to\phi f_{0}(980)\pi^{0}\to\phi\,3\pi$  Ab3 ,

(f)   $J/\psi\to\phi f_{1}(1285)\to\phi f_{0}(980)\pi^{0}\to\phi\,3\pi$  Ab3

It has become clear AKS15 ; AKS16 that the similar isospin breaking effect can appear not only due to the $a^{0}_{0}(980)-f_{0}(980)$ mixing, but also for any mechanism of the production of the $K\bar{K}$ pairs in the $S$ wave, $X\to K\bar{K}\to f_{0}(980)/a^{0}_{0}(980)$. 222Each such mechanism reproduces both the narrow resonant peak and the sharp jump of the phase of the amplitude between the $K^{+}K^{-}$ and $K^{0}\bar{K}^{0}$ thresholds. Thus a new tool to study the production mechanism and nature of light scalars is emerged.

II The $a^{0}_{0}(980)-f_{0}(980)$ mixing

The main contribution to the $a^{0}_{0}(980)-f_{0}(980)$ mixing amplitude, caused by the diagrams shown in Fig. 1, has the form

[TABLE]

where $m$ (invariant virtual mass of scalar resonances) $\geq 2m_{K^{0}}$ and $\rho_{K\bar{K}}(m)=\sqrt{1-4m_{K}^{2}/m^{2}}$; in the region $0\leq m\leq 2m_{K}$, $\rho_{K\bar{K}}(m)$ should be replaced by $i|\rho_{K\bar{K}}(m)|$. The modulus and the phase of $\Pi_{a^{0}_{0}f_{0}}(m)$ are shown in Fig. 2.

In the region between the $K^{+}K^{-}$ and $K^{0}\bar{K}^{0}$ thresholds, which is the 8 MeV wide,

[TABLE]

Note that $|\Pi_{\rho^{0}\omega}|\approx|\Pi_{\pi^{0}\eta}|\approx 0.003\mbox{ GeV}^{2}\approx(m_{d}-m_{u})\times 1\mbox{ GeV}.$

The branching ratios of the isospin-breaking decays $f_{0}(980)\to\eta\pi^{0}$ and $a^{0}_{0}(980)\to\pi^{+}\pi^{-}$, caused by the $a^{0}_{0}(980)-f_{0}(980)$ mixing, are AKS16

[TABLE]

[TABLE]

where $D_{a^{0}_{0}}(m)$ and $D_{f_{0}}(m)$ are the propagators of the $a^{0}_{0}(980)$ and $f_{0}(980)$ resonances, respectively.

Figure 3 shows the mass spectra correspond to the integrands in the above equations. 333Here we use the values of the coupling constants of the $f_{0}(980)$ and $a^{0}_{0}(980)$ resonances with the $\pi\pi$, $K\bar{K}$, and $\eta\pi$ channels obtained in Ref. AKS16 from the BESIII data on the intensities of the $f_{0}(980)\to a^{0}_{0}(980)$ and $a^{0}_{0}(980)\to f_{0}(980)$ transitions measured in the reactions (b) and (c) Ab1 .

III Polarization phenomena

The phase jump (see Fig. 2(b)) suggests the idea to study the $a_{0}^{0}(980)-f_{0}(980)$ mixing in polarization phenomena AS04a ; AS04b . If the process amplitude with the spin configuration is dominated by the $a_{0}^{0}(980)-f_{0}(980)$ mixing then the spin asymmetry of the cross section jumps near the $K\bar{K}$ thresholds. An example is the reaction $\pi^{-}p_{\uparrow}\to\left(a_{0}^{0}(980)+f_{0}(980)\right)n\to a_{0}^{0}(980)n\to\eta\pi^{0}\,n$ on a polarized proton target. The corresponding differential cross section has the form

[TABLE]

and the dimensionless normalized spin asymmetry is $A(t,m)=2\,\Im(M_{++}M^{*}_{+-})/(\,|M_{++}|^{2}+|M_{+-}|^{2}\,)$, $\,-1\leq A(t,m)\leq 1$. 444Here $M_{+-}$ and $M_{++}$ are the $s$-channel helicity amplitude with ana without nucleon helicity flip, $\psi$ is the angle between the normal to the reaction plain formed by the momenta of the $\pi^{-}$ and $\eta\pi^{0}$ system, and the transverse (to the $\pi^{-}$ beam axis) polarization of the the proton target, and $P$ is a degree of this polarization. Figure 4 illustrates the strong asymmetry jump which is straightforward manifestation of the $a_{0}(980)-f_{0}(980)$ mixing amplitude interfering with the isospin preserving one in the $\rho_{2}$ and $\pi$ Regge exchange model. Details and various variants may be found in Refs. AS04a ; AS04b .

These effects are still in waiting for their studies.

IV The decay $f_{1}(1285)\to f_{0}(980)\pi^{0}\to 3\pi$

Estimated are the contributions of the following mechanisms responsible for the decay $f_{1}(1285)\to f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}$ AKS16 :

(1) the contribution of the $a^{0}_{0}(980)-f_{0}(980)$ mixing, $f_{1}(1285)\to a_{0}(980)\pi^{0}\to(K^{+}K^{-}+K^{0}\bar{K}^{0})\pi^{0}\to f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}$,

(2) the contribution of the transition $f_{1}(1285)\to(K^{+}K^{-}+K^{0}\bar{K}^{0})\pi^{0}\to f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}$, arising due to the pointlike decay $f_{1}(1285)\to K\bar{K}\pi^{0}$,

(3) the contribution of the transition $f_{1}(1285)\to(K^{*}\bar{K}+\bar{K}^{*}K)\to(K^{+}K^{-}+K^{0}\bar{K}^{0})\pi^{0}\to f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}$, where $K^{*}=K^{*}(892)$, and

(4) the contribution of the transition $f_{1}(1285)\to(K^{*}_{0}\bar{K}+\bar{K}^{*}_{0}K)\to(K^{+}K^{-}+K^{0}\bar{K}^{0})\pi^{0}\to f_{0}(980)\to\pi^{+}\pi^{-}\pi^{0}$, where $K^{*}_{0}=K^{*}_{0}(800)$ (or $\kappa$) and $K^{*}_{0}(1430)$.

These mechanisms break the conservation of the isospin due to the nonzero mass difference of the $K^{+}$ and $K^{0}$ mesons. They result in the appearance of the narrow resonance structure in the $\pi^{+}\pi^{-}$ mass spectrum in the region of the $K\bar{K}$ thresholds, with the width $\approx 2m_{K^{0}}-2m_{K^{+}}\approx 8$ MeV. The observation of such a structure in experiment is the direct indication on the $K\bar{K}$ loop mechanism of the breaking of the isotopic invariance.

We point out that existing data should be more precise, and it is difficult to explain them using the single specific mechanism from those listed above. Taking the decay $f_{1}(1285)\to f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}$ as the example, we discuss the general approach to the description of the $K\bar{K}$ loop mechanism of the breaking of isotopic invariance.

(1) The matter is that the $J/\psi$ $\to$ $\phi f_{0}(980)$ $\to$ $\phi a_{0}(980)$ $\to$ $\phi\eta\pi^{0}$ Ab1 and $\chi_{c1}$ $\to$ $a_{0}(980)\pi^{0}$ $\to$ $f_{0}(980)\pi^{0}$ $\to$ $\pi^{+}\pi^{-}\pi^{0}$ Ab1 decays are described by the $a^{0}_{0}(980)-f_{0}(980)$ mixing well enough:

[TABLE]

[TABLE]

As for the $f_{1}(1285)\to f_{0}(980)\pi^{0}\to 3\pi$ decay Do11 , its description requires the “terrible” $a^{0}_{0}(980)-f_{0}(980)$ mixing:

[TABLE]

and, as a result, the inconvenient coupling constants of the scalar mesons with the pseudo-scalar mesons in the many cases

[TABLE]

For example, due to the very strong coupling of $a^{0}_{0}(980)$ with the $K\bar{K}$ channel, the width of the $a^{0}_{0}(980)$ resonance in the $\eta\pi^{0}$ mass spectrum turns out to be near 15 MeV.

(2) The the pointlike decay $f_{1}(1285)\to K\bar{K}\pi^{0}$ gives

[TABLE]

instead of the experimental value

[TABLE]

The $\pi^{+}\pi^{-}$ mass spectrum in the decay $f_{1}(1285)\to(K^{+}K^{-}+K^{0}\bar{K}^{0})\pi^{0}\to f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}$ looks similar to the curves in Fig. 3 for the $a_{0}(980)$-$f_{0}(980)$ mixing case. However, it is clear that the pointlike mechanism of the decay $f_{1}(1285)\to K\bar{K}\pi$ cannot by itself provide the considerable probability of the $f_{1}(1285)\to f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}$ transition.

(3) The isospin-breaking decay $f_{1}(1285)\to(K^{*}\bar{K}+\bar{K}^{*}K)\to(K^{+}K^{-}+K^{0}\bar{K}^{0})\pi^{0}$ $\to$ $f_{0}(980)\pi^{0}$ $\to$ $\pi^{+}\pi^{-}\pi^{0}$ is induced by the diagram shown in Fig. 5,

because the contributions from the $K^{+}K^{-}$ and $K^{0}\bar{K}^{0}$ pair production are not compensated entirely. The transition $f_{1}(1285)\to(K^{*}\bar{K}+\bar{K}^{*}K)\to(K^{+}K^{-}+K^{0}\bar{K}^{0})\pi^{0}\to f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}$ gives the shape of the $\pi^{+}\pi^{-}$ spectrum practically coincides with the corresponding spectrum caused by the $a^{0}_{0}(980)-f_{0}(980)$ mixing, but its

[TABLE]

is much less then the experimental value

[TABLE]

So, the $f_{1}(1285)\to(K^{*}\bar{K}+\bar{K}^{*}K)\to(K^{+}K^{-}+K^{0}\bar{K}^{0})\pi^{0}\to f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}$ transition mechanism alone is also insufficient to understand the experimental data.

(4) The variant $f_{1}(1285)\to(K^{*}_{0}(800)\bar{K}+\bar{K}^{*}_{0}(800)K)\to(K^{+}K^{-}+K^{0}\bar{K}^{0})\pi^{0}\to f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}$ is rejected by the shapes of the $K\pi$ and $K\bar{K}$ mass spectra in the decay $f_{1}(1285)\to K\bar{K}\pi$. As for $f_{1}(1285)\to(K^{*}_{0}(1430)\bar{K}+\bar{K}^{*}_{0}(1430)K)\to(K^{+}K^{-}+K^{0}\bar{K}^{0})\pi^{0}\to f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}$, it provides the results similar to $f_{1}(1285)\to(K^{*}\bar{K}+\bar{K}^{*}K)\to(K^{+}K^{-}+K^{0}\bar{K}^{0})\pi^{0}\to f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}$ and consequently cannot describe the data alone.

V The consistency condition

The isospin breaking amplitude $\mathcal{M}_{f_{1}(1285)\to f_{0}(980)\pi^{0}}(m)$ can be expanded near the $K\bar{K}$ threshold into the series in $\rho_{K\bar{K}}(m)=\sqrt{1-4m^{2}_{K}/m^{2}}$ :

[TABLE]

With a good accuracy

[TABLE]

The amplitude $A(m)$ contains the information about all possible mechanisms of production of the $K\bar{K}$ system with isospin $I=1$ in $S$ wave in the process $f_{1}(1285)\to K\bar{K}\pi$.

From the data on the decay $f_{1}(1285)\to f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}$  one can extract the information about $|A(m)|^{2}$ in the region of the $K^{+}K^{-}$ and $K^{0}\bar{K}^{0}$ thresholds,

[TABLE]

where $p(m)$ = $[m^{4}_{f_{1}}-2m^{2}_{f_{1}}(m^{2}+m^{2}_{\pi})+(m^{2}-m^{2}_{\pi})^{2}]^{1/2}\\ /(2m_{f_{1}})$. Moreover, the information about $|A(m)|^{2}$ at $m>2m_{K}$ can be obtained from the data on the $K\bar{K}$ mass spectra measured in the decays $f_{1}(1285)\to K\bar{K}\pi$. For instance, the $K^{+}K^{-}$ spectrum in the decay $f_{1}(1285)\to K^{+}K^{-}\pi^{0}$ can be represented in the form

[TABLE]

Fitting the data on $d\Gamma_{f_{1}\to K^{+}K^{-}\pi^{0}}/dm$, one can find the value $|A(2m_{K^{+}})|^{2}$ and obtain the following approximate estimate AKS16

[TABLE]

Thus its comparison with the data on the decay $f_{1}(1285)\to\pi^{+}\pi^{-}\pi^{0}$ permits one to verify their consistence with the data on the decay $f_{1}(1285)\to K\bar{K}\pi$ and with the idea of the breaking of isotopic invariance caused by the mass difference of $K^{+}$ and $K^{0}$ mesons.

VI The decay $J/\psi$ $\to$ $\gamma\eta(1405)$ $\to$ $\gamma f_{0}(980)\pi^{0}\to\gamma\pi^{+}\pi^{-}\pi^{0}$

According to BESIII Ab2 , the mass and the width of the $\eta(1405)$ peak in the $\pi^{+}\pi^{-}\pi^{0}$ channel are $1409.0\pm 1.7$ MeV and $48.3\pm 5.2$ MeV, respectively, while the branching ratio is

[TABLE]

In addition, the BESIII gives the ratio

[TABLE]

that rules out practically the explanation of the discovered effect by means of the $a_{0}(980)-f_{0}(980)$ mixing.

We discuss the possibility of the theoretical explanation of the large breaking of isotopic invariance in the decay $\eta(1405)$ $\to$ $f_{0}(980)\pi^{0}$ $\to$ $\pi^{+}\pi^{-}\pi^{0}$ by means of the anomalous Landau thresholds (the logarithmic triangle singularities), which are in the transition $\eta(1405)\to(K^{*}\bar{K}+\bar{K}^{*}K)\to(K^{+}K^{-}+K^{0}\bar{K}^{0})\pi^{0}\to f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}$ (see Fig. 6), and show that the account of the finite width of the $K^{*}(892)$ ($\Gamma_{K^{*}\to K\pi}\approx 50$ MeV) smoothes the logarithmic singularities in the amplitude and results in the suppression of the calculated width of the decay $\eta(1405)\to f_{0}(980)\pi^{0}\to 3\pi$ by the factor of $6-8$ in comparison with the case of $\Gamma_{K^{*}\to K\pi}=0$ AKS15 .

The accounting of the finite width of the $K^{*}$ resonance, i.e., the averaging of the amplitude over the resonance Breit–Wigner distribution in accord with the spectral Källén–Lehmann representation for the propagator of the unstable $K^{*}$ meson, smoothes the logarithmic singularities of the amplitude and hence makes the compensation of the contributions of the $K^{*+}K^{-}+K^{*-}K^{+}$ and $K^{*0}\bar{K}^{0}+\bar{K}^{*0}K^{0}$ intermediate states more strong. This results in both the diminishing of the calculated width of the decay $\eta(1405)\to\pi^{+}\pi^{-}\pi^{0}$ by a number of times in comparison with the case of $\Gamma_{K^{*}\to K\pi}=0$, and in the concentration of the main effect of the isospin breaking in the domain of the $\pi^{+}\pi^{-}$ invariant mass between the $K\bar{K}$ thresholds.

Assuming the dominance of the $\eta(1405)\to(K^{*}\bar{K}+\bar{K}^{*}K)\to K\bar{K}\pi^{0}$ decay, one obtains

[TABLE]

that reasonably agrees with experiment.

[TABLE]

We also analyze the difficulties related with the assumption of the dominance of the $\eta(1405)\to(K^{*}\bar{K}+\bar{K}^{*}K)\to K\bar{K}\pi$ decay mechanism and discuss the possible dynamics of the decay $\eta(1405)\to\eta\pi\pi$ AKS15 . The decisive improvement of the experimental data on the $K\bar{K}$, $K\pi$, $\eta\pi$, and $\pi\pi$ mass spectra in the decay of the resonance structure $\eta(1405/1475)$ to $K\bar{K}\pi$ and $\eta\pi\pi$, and on the shape of the resonance peaks themselves in the $K\bar{K}\pi$ and $\eta\pi\pi$ decay channels is necessary for the further establishing the $\eta(1405)\to 3\pi$ decay mechanism.

ACKNOWLEDGMENTS

The present work is partially supported by the Russian Foundation for Basic Research Grant No. 16-02-00065 and the Presidium of the Russian Academy of Sciences project No. 0314-2015-0011.