Manifestations of the $a^0_0(980)-f_0(980)$ mixing in $D^0\to K^0_S\pi^+\pi^-$ and $D^0\to K^0_S\eta\pi^0$ decays

TL;DR

This paper investigates how the mixing of the $a^0_0(980)$ and $f_0(980)$ resonances affects decay spectra in $D^0$ meson decays, revealing significant distortions in the $ ho^+ ho^-$ mass spectrum due to isospin-breaking effects.

Contribution

It provides a detailed analysis of the $a^0_0(980)$-$f_0(980)$ mixing manifestations in specific $D^0$ decay channels, highlighting the impact on the $ ho^+ ho^-$ spectrum and the dependence on phase differences.

Findings

The $a^0_0(980)$-$f_0(980)$ mixing significantly influences the $ ho^+ ho^-$ mass spectrum.

The $f_0(980)$ peak can be distorted due to mixing effects.

The magnitude of distortions depends on the phase between decay amplitudes.

Abstract

Possible manifestations of the isospin-breaking mixing of the $a^0_0(980)$ and $f_0(980)$ resonances are analyzed in the $D^0\to K^0_S \pi^+\pi^-$ and $D^0\to K^0_S\eta\pi^0$ decays. It is shown that the $a^0_0(980)-f_0(980)$ mixing has the most influence on the $\pi^+\pi^-$ mass spectrum in the decay $D^0\to K^0_S\pi^+\pi^-$. Owing to the $a^0_0(980)-f_0(980)$ mixing, the $f_0(980)$ peak in $D^0 \to K^0_Sf_0(980)\to K^0_S \pi^+\pi^-$ can experience distortions comparable with its size. The effect essentially depends on the relative phase between the amplitudes of the $D^0\to K^0_Sf_0(980)$ and $D^0\to K^0_Sa^0_0(980)$ transitions.

Equations

$$\frac{d{N}_{{\pi }^{+}{\pi }^{-}}}{dm}=2m{P}_{K_S^0}(m){\rho }_{{\pi }^{+}{\pi }^{-}}(m)\frac{g_{f_0{\pi }^{+}{\pi }^{-}}^2}{16\pi }{\left|\frac{C_1}{D_{f_0}(m)}+\frac{e^{i\phi }{C}_2{\Pi }_{a_0^0{f}_0}(m)}{D_{a_0^0}(m)D_{f_0}(m)-{\Pi }_{a_0^0{f}_0}^2}\right|}^2,$$

$$\frac{d{N}_{\eta {\pi }^0}}{dm}=2m{P}_{K_S^0}(m){\rho }_{\eta {\pi }^0}(m)\frac{g_{a_0^0\eta {\pi }^0}^2}{16\pi }{\left|\frac{e^{i\phi }{C}_2}{D_{a_0^0}(m)}+\frac{C_1{\Pi }_{a_0^0{f}_0}(m)}{D_{a_0^0}(m)D_{f_0}(m)-{\Pi }_{a_0^0{f}_0}^2}\right|}^2,$$

$$BR(D^0\to K_S^0{f}_0(980)\to K_S^0{\pi }^{+}{\pi }^{-})=\left(1.23_{-0.24}^{+0.40}\right)\cdot 10^{-3}$$

$$=\underset{2m_{\pi }^{+}}{\overset{m_{D^0}-m_{K^0}}{\int }}2m{P}_{K_S^0}(m){\rho }_{{\pi }^{+}{\pi }^{-}}(m)\frac{g_{f_0{\pi }^{+}{\pi }^{-}}^2}{16\pi }{\left|\frac{C_1}{D_{f_0}(m)}\right|}^{2}dm,$$

$$BR(D^0\to K_S^0{a}_0^0(980)\to K_S^0\eta {\pi }^0)=\left(6.6\pm 2.0\right)\cdot 10^{-3}$$

$$=\underset{m_{\eta }+m_{{\pi }^0}}{\overset{m_{D^0}-m_{K^0}}{\int }}2m{P}_{K_S^0}(m){\rho }_{\eta {\pi }^0}(m)\frac{g_{a_0^0\eta {\pi }^0}^2}{16\pi }{\left|\frac{C_2}{D_{a_0^0}(m)}\right|}^{2}dm.$$

$$\Delta BR({\pi }^{+}{\pi }^{-})=\frac{1}{1.23\cdot 10^{-3}}\underset{2m_{\pi }^{+}}{\overset{m_{D^0}-m_{K^0}}{\int }}\frac{d{N}_{{\pi }^{+}{\pi }^{-}}}{dm}dm-1$$

$$\Delta BR(\eta {\pi }^0)=\frac{1}{6.6\cdot 10^{-3}}\underset{m_{\eta }+m_{{\pi }^0}}{\overset{m_{D^0}-m_{K^0}}{\int }}\frac{d{N}_{\eta {\pi }^0}}{dm}dm-1.$$

Full text

Recently, we have examined the interference phenomena in the $\eta\pi^{0}$ mass spectrum in the $D^{+}_{s}\to\eta\pi^{0}\pi^{+}$ decay AS17 caused by the isospin-breaking $a^{0}_{0}(980)-f_{0}(980)$ mixing ADS79 ; AS16 . Their detection would help clarify the production and decay mechanisms of the light scalars $f_{0}(980)$ and $a^{0}_{0}(980)$ in weak three-body hadronic $D^{+}_{s}$ decays. A detailed list of references concerning the $a^{0}_{0}(980)-f_{0}(980)$ mixing can be found, for example, in Refs. AS16 ; AKS16 .

In Ref. AS17 , it has been noted that the investigations of the $a^{0}_{0}(980)-f_{0}(980)$ mixing in three-body decays of the $D^{0}$ meson, such as $D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$, $D^{0}\to K^{0}_{S}\eta\pi^{0}$, $D^{0}\to\bar{K}^{0}K^{-}K^{+}$, $D^{0}\to K^{-}K^{+}\pi^{0}$, and $D^{0}\to\pi^{+}\pi^{-}\pi^{0}$, are also promising and interesting. In the present work, we analyze a possible manifestation of the $a^{0}_{0}(980)-f_{0}(980)$ mixing in the decays $D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$ and $D^{0}\to K^{0}_{S}\eta\pi^{0}$. We show that the $a^{0}_{0}(980)-f_{0}(980)$ mixing has the most impact on the $\pi^{+}\pi^{-}$ mass spectrum in $D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$. Owing to $a^{0}_{0}(980)-f_{0}(980)$ mixing, the $f_{0}(980)$ peak in $D^{0}\to K^{0}_{S}f_{0}(980)\to K^{0}_{S}\pi^{+}\pi^{-}$ can experience distortions comparable with its size. The effect essentially depends on the relative phase between the amplitudes of the $D^{0}\to K^{0}_{S}f_{0}(980)$ and $D^{0}\to K^{0}_{S}a^{0}_{0}(980)$ transitions.

The $\pi^{+}\pi^{-}$ and $\eta\pi^{0}$ mass spectra in the $D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$ and $D^{0}\to K^{0}_{S}\eta\pi^{0}$ decays caused by the $f_{0}(980)$ and $a^{0}_{0}(980)$ resonance contributions, respectively, and the $a^{0}_{0}(980)-f_{0}(980)$ mixing have the form

[TABLE]

[TABLE]

where $m$ is the invariant mass of the $\pi^{+}\pi^{-}$ or $\eta\pi^{0}$ system, and $\varphi$ is the relative phase between the amplitudes of the $D^{0}\to K^{0}_{S}f_{0}(980)$ and $D^{0}\to K^{0}_{S}a^{0}_{0}(980)$ transitions. The kinematic factors $P_{K^{0}_{S}}(m)=[m^{4}_{D^{0}}-2m^{2}_{D^{0}}(m^{2}_{K^{0}}+m^{2})+(m^{2}_{K^{0}}-m^{2})^{2}]^{1/2}/(2m_{D^{0}})$, $\rho_{\pi^{+}\pi^{-}}(m)=(1-4m^{2}_{\pi}/m^{2})^{1/2}$, and $\rho_{\eta\pi^{0}}(m)=[1-2(m^{2}_{\eta}+m^{2}_{\pi})/m^{2}+(m^{2}_{\eta}-m^{2}_{\pi})^{2}/m^{4}]^{1/2}$; $m_{\pi}=m_{\pi}^{+}=m_{\pi}^{0}=0.135$ GeV. Formulas for the inverse propagators $D_{f_{0}}(m)$ and $D_{a^{0}_{0}}(m)$ of the $f_{0}(980)$ and $a^{0}_{0}(980)$ resonances and the amplitude $\Pi_{a^{0}_{0}f_{0}}(m)$ describing the $a^{0}_{0}(980)\to(K^{+}K^{-}+K^{0}\bar{K}^{0})\to f_{0}(980)$ transition, together with the values of coupling constants $g_{f_{0}ab}$, $ab=(\pi^{+}\pi^{-},\,\pi^{0}\pi^{0},\,K^{+}K^{-},\,K^{0}\bar{K}^{0},\,\eta\eta)$, and $g_{a^{0}_{0}ab}$, $ab=(\eta\pi^{0},\,K^{+}K^{-},\,K^{0}\bar{K}^{0},\,\eta^{\prime}\pi^{0})$, which we use here, have been written in detail in Refs. AKS16 ; AS17 . The values of $C_{1}=0.047$ GeV*-1/2* and $C_{2}=0.095$ GeV*-1/2* are fixed taking into account the CLEO Collaboration results CLEO02 ; CLEO04 , together with the Particle Data Group information PDG16 , and the relations

[TABLE]

[TABLE]

The current experimental situation with the decays under consideration will be discussed below. Here we only note that the essential numerical input in Eqs. (Manifestations of the $a^{0}_{0}(980)-f_{0}(980)$ mixing in $D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$ and $D^{0}\to K^{0}_{S}\eta\pi^{0}$ decays) and (Manifestations of the $a^{0}_{0}(980)-f_{0}(980)$ mixing in $D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$ and $D^{0}\to K^{0}_{S}\eta\pi^{0}$ decays) is based now on a very limited experimental statistics using the isobar model of the decay amplitudes, and more precise information would be very desirable. Unfortunately, in more recent analyses of the $BABAR$ BaBar08 and Belle Belle14 Collaborations (see also Ref. BaBar10 ), the $f_{0}(980)$ contribution is not separately treated in the fit to the data on the $D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$ decay. Nevertheless, the $f_{0}(980)$ peak is clearly visible in the $\pi^{+}\pi^{-}$ mass spectrum in this decay BaBar08 ; Belle14 .

The fractions of the $D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$ and $D^{0}\to K^{0}_{S}\eta\pi^{0}$ branching ratios caused by the $a^{0}_{0}(980)-f_{0}(980)$ mixing are given by, respectively,

[TABLE]

and

[TABLE]

$\Delta BR(\pi^{+}\pi^{-})$ and $\Delta BR(\eta\pi^{0})$ as functions of the relative phase $\varphi$ between the amplitudes of the $D^{0}\to K^{0}_{S}a^{0}_{0}(980)$ and $D^{0}\to K^{0}_{S}f_{0}(980)$ decays are shown in Fig. 1 by the solid and dashed curves, respectively. The solid and dashed horizontal lines in Fig. 1 show the incoherent contributions to $\Delta BR(\pi^{+}\pi^{-})$ and $\Delta BR(\eta\pi^{0})$ from the $a^{0}_{0}(980)-f_{0}(980)$ mixing [i.e., the modules squared of the second terms in the sums in Eqs. (1) and (2)] that are equal to $\approx 1.7\%$ and $\approx 0.17\%$, respectively.

As is seen from Fig. 1, the maximal constructive (destructive) interference between the contribution of the $f_{0}(980)$ resonance and that of the $a^{0}_{0}(980)-f_{0}(980)$ mixing in the $\pi^{+}\pi^{-}$ channel corresponds to the phase $\varphi\approx 245^{\circ}$ ($70^{\circ}$), while the maximal constructive (destructive) interference between the contributions from the $a^{0}_{0}(980)$ and the $a^{0}_{0}(980)-f_{0}(980)$ mixing in the $\eta\pi^{0}$ channel corresponds to the phase $\varphi\approx 110^{\circ}$ ($290^{\circ}$). Figure 2 shows the mass spectra $dN_{\pi^{+}\pi^{-}}/dm$ and $dN_{\eta\pi^{0}}/dm$ for these ultimate interference patterns.

The fact that $BR(D^{0}\to K^{0}_{S}a^{0}_{0}(980)\to K^{0}_{S}\eta\pi^{0})$ is about 5 times greater than $BR(D^{0}\to K^{0}_{S}f_{0}(980)\to K^{0}_{S}\pi^{+}\pi^{-})$ [see Eqs. (Manifestations of the $a^{0}_{0}(980)-f_{0}(980)$ mixing in $D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$ and $D^{0}\to K^{0}_{S}\eta\pi^{0}$ decays) and (Manifestations of the $a^{0}_{0}(980)-f_{0}(980)$ mixing in $D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$ and $D^{0}\to K^{0}_{S}\eta\pi^{0}$ decays)] leads to an increase in the influence of the $a^{0}_{0}(980)-f_{0}(980)$ mixing on the $\pi^{+}\pi^{-}$ mass spectrum in comparison with its influence in the $\eta\pi^{0}$ channel. Owing to interference, the integral effect from the $a^{0}_{0}(980)-f_{0}(980)$ mixing contribution can reach about $11\%$ (see Fig. 1). This is very large for the isotopic symmetry-breaking effect. As is seen from Fig. 2, the $a^{0}_{0}(980)-f_{0}(980)$ mixing can result in the essential distortions of the $f_{0}(980)$ line shape in the $\pi^{+}\pi^{-}$ channel. For example, it can lead to a narrowing of the $f_{0}(980)$ peak by about 1.5 times and to an increase of its height up to 60% or even to the formation of two peaks. The effect essentially depends on the relative phase between the $D^{0}\to K^{0}_{S}f_{0}(980)$ and $D^{0}\to K^{0}_{S}a^{0}_{0}(980)$ decay amplitudes. Certainly, a good mass resolution and high statistics are required to detect phenomena of such a kind.

Let us briefly discuss the experimental situation PDG16 ; CLEO04 ; BaBar05 ; CLEO02 ; BaBar08 ; Belle14 ; BaBar10 . There is only one experiment on the $D^{0}\to K^{0}_{S}\eta\pi^{0}$ decay performed by the CLEO Collaboration CLEO04 . The Dalitz analysis of 155 selected $D^{0}\to K^{0}_{S}\eta\pi^{0}$ candidates showed that the $D^{0}\to K^{0}_{S}\eta\pi^{0}$ decay proceeds mainly via $K^{0}_{S}a^{0}_{0}(980)$ and $K^{*0}(892)\eta$ intermediate states, the first of which is dominant CLEO04 . Note that the $a^{0}_{0}(980)$ production has been observed not only in the $D^{0}\to K^{0}_{S}a^{0}_{0}(980)\to K^{0}_{S}\eta\pi^{0}$ decay [for which $BR(D^{0}\to K^{0}_{S}a^{0}_{0}(980)\to K^{0}_{S}\eta\pi^{0})=(6.6\pm 2.0)\cdot 10^{-3}$ CLEO04 ; PDG16 ] but also in the channel $D^{0}\to K^{0}_{S}a^{0}_{0}(980)\to K^{0}_{S}K^{+}K^{-}$ BaBar05 ; BaBar10 . According to Ref. BaBar05 , $BR(D^{0}\to K^{0}_{S}a^{0}_{0}(980)\to K^{0}_{S}K^{+}K^{-})=(3.0\pm 0.4)\cdot 10^{-3}$, while from Supplemental Material to Ref. BaBar10 (see Ref. [18] in BaBar10 ) it follows that the central value for $BR(D^{0}\to K^{0}_{S}a^{0}_{0}(980)\to K^{0}_{S}K^{+}K^{-})\approx 2.3\cdot 10^{-3}$. The relation of the above branching ratios is typical for the $a^{0}_{0}(980)$ resonance in the $q^{2}\bar{q}^{2}$ model ADS81 ; AI89 ; A98 , in which the $a_{0}(980)$ is strongly coupled with the $\eta\pi$ and $K\bar{K}$ channels.

In the CLEO CLEO02 , $BABAR$ BaBar08 , and Belle Belle14 experiments 5299, 487 000, and 1 231 731 $D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$ candidates were selected, respectively. The $D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$ Dalitz distributions have a rich structure. Among the possible intermediate states are such as $K^{*-}(982)\pi^{+}$, $K^{*-}(1430)\pi^{+}$, $K^{0}_{S}\rho^{0}$, $K^{0}_{S}f_{0}(980)$, $K^{0}_{S}f_{2}(1270)$, and $K^{0}_{S}f_{0}(1370)$. According to the CLEO analysis CLEO02 , the $K^{0}_{S}f_{0}(980)\to K^{0}_{S}\pi^{+}\pi^{-}$ fraction constitutes of $(4.3^{+1.4}_{-0.8})\%$. The $K^{0}_{S}$ production together with the $\pi^{+}\pi^{-}$ system in the $S$ wave, $K^{0}_{S}(\pi^{+}\pi^{-})_{S}$, constitutes about 12% of the total branching ratio of the $D^{0}$ decay into $K^{0}_{S}\pi^{+}\pi^{-}$ BaBar08 ; Belle14 . The $\pi^{+}\pi^{-}$ mass spectrum in the 1 GeV region was scanned in the $BABAR$ BaBar08 and Belle Belle14 experiments with an $\approx\,$5-MeV-wide step. It is interesting to note that the whole visible $f_{0}(980)$ peak contains only 6–7 points; i.e., its width is less than 25 MeV. Such a narrowness of the $f_{0}(980)$ peak can be related to the effect of the $a^{0}_{0}(980)-f_{0}(980)$ mixing. Of course, it is impossible to completely eliminate the effect of interference of the $f_{0}(980)$ contribution with the background from other intermediate states. There is hope that further investigations will clarify this issue. The most clear information about the $D^{0}\to K^{0}_{S}f_{0}(980)\to K^{0}_{S}\pi^{+}\pi^{-}$ decay channel, as well as about the possible role of background contributions, is given by the distribution of $D^{0}\to K^{0}_{S}\pi^{+}\pi^{-}$ events on the ($m^{2}_{\pi^{+}\pi^{-}},\,m^{2}_{K^{0}_{S}\pi^{\pm}}$)-Dalitz plot.

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.