Chiral dynamics and S-wave contributions in $${\bar{B}}^0/D^0 \rightarrow \pi ^{\pm }\eta \ l^{\mp } \nu $$ decays

In this work, we analyze the semi-leptonic decays $$\bar{B}^0/D^0 \rightarrow (a_0(980)^{\pm }\rightarrow \pi ^{\pm }\eta ) l^{\mp } \nu $$ within light-cone sum rules. The two and three-body light-cone distribution amplitudes (LCDAs) of the B meson and the only available two-body LCDA of the D meson are used. To include the finite-width effect of the $$a_0(980)$$ , we use a scalar form factor to describe the final-state interaction between the $$\pi \eta $$ mesons, which was previously calculated within unitarized Chiral Perturbation Theory. The result for the decay branching fraction of the $$D^0$$ decay is in good agreement with that measured by the BESIII Collaboration, while the branching fraction of the $${\bar{B}}^0$$ decay can be tested in future experiments.