$a_0(980)-f_0(980)$ mixing in ${\chi }_{c1}\to {\pi }^0{f}_0(980)\to {\pi }^0{\pi }^{+}{\pi }^{-}$ and ${\chi }_{c1}\to {\pi }^0{a}_0(980)\to {\pi }^0{\pi }^0\eta$

Abstract

We study the isospin breaking in the reactions ${\chi }_{c1}\to {\pi }^0{\pi }^{+}{\pi }^{-}$ and ${\chi }_{c1}\to {\pi }^0{\pi }^0\eta$ and its relation to the $a_0(980)-f_0(980)$ mixing, which was measured by the BESIII Collaboration. We show that the same theoretical model previously developed to study the ${\chi }_{c1}\to \eta {\pi }^{+}{\pi }^{-}$ reaction (also measured by BESIII), and further explored in the predictions to the ${\eta }_c\to \eta {\pi }^{+}{\pi }^{-}$, can be successfully employed in the present study. We assume that the ${\chi }_{c1}$ behaves as an $SU(3)$ singlet to find the weight in which trios of pseudoscalars are created, followed by the final state interaction of pairs of mesons to describe how the $a_0(980)$ and $f_0(980)$ are dynamically generated, using the chiral unitary approach in coupled channels. The isospin violation is introduced through the use of different masses for the charged and neutral kaons, either in the propagators of the pairs of mesons created in the ${\chi }_{c1}$ decay, or in the propagators inside the $T$ matrix, constructed through the unitarization of the scattering and transition amplitudes of pairs of pseudoscalar mesons. We find that violating isospin inside the $T$ matrix makes the ${\pi }^0\eta \to {\pi }^{+}{\pi }^{-}$ amplitude nonzero, which gives an important contribution and also enhances the effect of the $K\bar{K}$ term. Also, we find that in the total amplitude the most important effect is the isospin breaking inside the $T$ matrix, due to the constructive sum of ${\pi }^0\eta \to {\pi }^{+}{\pi }^{-}$ and $K\bar{K}\to {\pi }^{+}{\pi }^{-}$, which is essential to get a good agreement with the experimental measurement of the mixing.

arXiv commentrevised manuscript, new figures, new references; 8 pages, 6 figures

References in corpus (39)

• Chiral Symmetry Amplitudes in the S-Wave Isoscalar and Isovector Channels and the σ, f_0(980), a_0(980) Scalar Mesons
• Chiral symmetry amplitudes in the s-wave isoscalar and isovector channels and the $\sigma ,f_0(980),a_0(980)$ scalar mesons
• Resonances in $J/\psi \to \varphi {\pi }^{+}{\pi }^{-}$ and $\varphi K^{+}{K}^{-}$
• The puzzle of anomalously large isospin violations in $\eta (1405/1475)\to 3\pi$
• The nature of $\sigma$, $\kappa$, $a_0(980)$ and $f_0(980)$
• Search for the scalar $a_0$ and $f_0$ mesons in the reactions $e^{+}{e}^{-}\to \gamma {\pi }^0{\pi }^0(\eta )$
• First observation of $\eta (1405)$ decays into $f_0(980){\pi }^0$
• Isospin breaking and $f_0(980)$-$a_0(980)$ mixing in the $\eta (1405)\to {\pi }^0{f}_0(980)$ reaction
• Understanding the property of $\eta (1405/1475)$ in the $J/\psi$ radiative decay
• $f_1(1285)$ decays into $a_0(980){\pi }^0$, $f_0(980){\pi }^0$ and isospin breaking
• Isospin breaking decay $\eta (1405)\to f_0(980){\pi }^0\to 3\pi$
• Possibility of measuring a0(980)-f0(980) mixing from J/psi -> phi a0
• Investigation of a0-f0 mixing
• Meson-meson scattering: $K\overline{K}$-thresholds and $f_0(980)-a_0(980)$ mixing
• Isospin breaking exposed in f0(980)- a0(980) mixing
• Amplitude analysis of the ${\chi }_{c1}\to \eta {\pi }^{+}{\pi }^{-}$ decays
• Extended chiral Khuri-Treiman formalism for $\eta \to 3\pi$ and the role of the $a_0(980)$, $f_0(980)$ resonances
• Study a0(980)-f0(980) mixing from a0(980) -> f0(980) transition
• Study of a0(980)--f0(980) mixing
• a0(980)-f0(980) mixing and isospin violation in the reactions pN -> d a0, pd -> 3He/3H a0 and dd -> 4He a0
• Mixing of the $f_0$ and $a_0$ scalar mesons in threshold photoproduction
• Once more on mixing of the a_0(980) and f_0(980) mesons
• $f_0(500)$, $f_0(980)$ and $a_0(980)$ production in the ${\chi }_{c1}\to \eta {\pi }^{+}{\pi }^{-}$ reaction
• Constraint on K Kbar compositeness of the a0(980) and f0(980) resonances from their mixing intensity
• Proposed Search for $a_0^0(980)-f_0(980)$ Mixing in Polarization Phenomena
• Predictions for ${\eta }_c\to \eta {\pi }^{+}{\pi }^{-}$ producing $f_0(500)$, $f_0(980)$ and $a_0(980)$
• Large Isospin mixing in phi radiative decay and the spatial size of the f0(980)- a0(980) meson
• Tetraquarks, hadronic molecules, meson-meson scattering and disconnected contributions in lattice QCD
• Isospin violation in J/Psi to phi pi0 eta decay and the f0-a0 mixing
• Search for the $a_0(980)-f_0(980)$ mixing in weak decays of $D_s/B_s$ mesons
• On the possibility of observation of $a_0$-$f_0$ mixing in the $pn\to d{a}_0$ reaction
• Mechanisms of the isospin-breaking decay $f_1(1285)\to f_0(980){\pi }^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$
• Interference phenomena in the decay $D_s^{+}\to \eta {\pi }^0{\pi }^{+}$ induced by the $a_0^0(980)-f_0(980)$ mixing
• chi_{cJ} Decays to h^+h^-h^0
• chi_cJ Decays to h^+h^-h^0
• Manifestations of the $a_0^0(980)-f_0(980)$ mixing in $D^0\to K_S^0{\pi }^{+}{\pi }^{-}$ and $D^0\to K_S^0\eta {\pi }^0$ decays
• Amplitude Analysis of the Decays ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}{\pi }^0$ and ${\eta }^{\prime }\to {\pi }^0{\pi }^0{\pi }^0$
• Scalar mesons in eta' -> eta pi0 pi0 decay
• Scalar mesons in ${\eta }^{\prime }\to 3{\pi }^0,{\pi }^0{\pi }^{+}{\pi }^{-}$ decays

Cited by in corpus (7)

• Observation of $a_0^0(980)$-$f_0(980)$ Mixing
• The $D_s^{+}\to a_0(980)e^{+}{\nu }_e$ reaction and the $a_0(980)-f_0(980)$ mixing
• Strong isospin symmetry breaking in light scalar meson production
• Investigation of $J/\psi \to \gamma {\pi }^0\eta ({\pi }^{+}{\pi }^{-},{\pi }^0{\pi }^0)$ radiative decays including final-state interactions
• Study of the isospin breaking decay $\mathbf{Y}(2175)\to \mathbf{\varphi }{\mathbf{f}}_0(980)\to \mathbf{\varphi \eta }{\mathbf{\pi }}^0$ at BESIII
• Study of $a_0^0(980)$-$f_0(980)$ mixing from $a_0(1450)\to a_0^0(980)f_0(500)\to {\pi }^{+}{\pi }^{-}{f}_0(500)$
• The ${\bar{B}}_s^0\to J/\psi {\pi }^0\eta$ decay and the $a_0(980)-f_0(980)$ mixing