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The $a_0(980)$ and $Λ(1670)$ in the $Λ^+_c \to π^+ ηΛ$ decay

We propose to study the $a_0(980)$ and the $Λ(1670)$ resonances in the $Λ^+_c \to π^+ ηΛ$ decay via the final state interactions of the $π^+ η$ and $ηΛ$ pairs. The weak interaction part proceeds through the $c$ quark decay process: $c(ud) \to (s + u + \bar d)(ud)$, while the hadronization part takes place in two different mechanisms. In the first mechanism, the $sud$ cluster picks up a $q\bar{q}$ pair from the vacuum to form the $ηΛ$ meson-baryon pair while the $u\bar{d}$ pair from the weak decay hadronizes into a $π^+$. In the second, the $sud$ cluster turns into a $Λ$, while the $u\bar{d}$ pair from the $c$ decay picks up a $q\bar{q}$ pair and hadronizes into a meson-meson pair ($πη$ or $K\bar{K}$). Because the final $π^+ η$ and $ηΛ$ states are in pure isospin $I = 1$ and $I=0$ combinations, the $Λ^+_c \to π^+ ηΛ$ decay can be an ideal process to study the $a_0(980)$ and $Λ(1670)$ resonances. Describing the final state interaction in the chiral unitary approach, we find that the $π^+ η$ and $ηΛ$ invariant mass distributions, up to an arbitrary normalization, show clear cusp and peak structures, which can be associated with the $a_0(980)$ and $Λ(1670)$ resonances, respectively. The proposed mechanism can provide valuable information on the nature of these resonances and can in principle be test by facilities such as BEPCII.