Determination of the \(K^{+}\bar{K}^{0}\) scattering length and effective range and its relation to the \(a_{0}^{+}(980)\) from the \(\chi _{c1}\rightarrow \pi ^{+}\pi ^{-}\eta \) reaction

We analyze the clean cusp, seen in the \(\eta \pi \) mass distribution with high precision of the \(\chi _{c1} \rightarrow \eta \pi ^+ \pi ^-\) reaction in the BESIII experiment, with the aim of making a precise determination of the scattering length a and effective range \(r_0\) of \(K^+ \bar{K}^{0}\). For that, we follow a previous theoretical work that gave a good reproduction of these data using the chiral unitary approach for the meson-meson interaction, and allow some flexibility in the input to carry a better fit to the data. The important task of determining the uncertainties in the scattering parameters is done using the resampling method and an accuracy in a and \(r_0\) is obtained better than \(20\%\). The effective range is determined for the first time with this analysis.