$a_0(980) - f_0(980)$ mixing in $\chi_{c1} \to \pi^0 f_0(980) \to \pi^0 \pi^+ \pi^-$ and $\chi_{c1} \to \pi^0 a_0(980) \to \pi^0 \pi^0 \eta$

TL;DR

This paper investigates the isospin breaking effects in $ ext{chi}_{c1}$ decays related to $a_0(980)$ and $f_0(980)$ mixing, using a chiral unitary approach to match experimental observations.

Contribution

The study applies a previously developed theoretical model to new decay channels, demonstrating the importance of isospin violation inside the $T$ matrix for accurate predictions.

Findings

Isospin breaking inside the $T$ matrix enhances the mixing amplitude.

Constructive interference of $ ext{pi}^0 ext{eta} o ext{pi}^+ ext{pi}^-$ and $Kar{K} o ext{pi}^+ ext{pi}^-$ is crucial.

Model successfully reproduces experimental measurements of $a_0(980)-f_0(980)$ mixing.

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.