Restricting $q^2 l^2$ operators from $\pi^0 \rightarrow \mu e$

In this paper we consider semileptonic lepton flavor violating operators of the type $q^2l^2$ in low energy effective theory (LEFT). At the chiral scale, we match these operators to chiral perturbation theory ($\chi$PT) and place constraints from the process $\pi^0\rightarrow \mu^+ \,\, e^-$. These bounds are shown to depend on the chiral nature of the operators. The scalar operators are significantly more constrained compared to the vector operators. We then compare the limits from $\mu \to e$ conversion in Nuclei and show that the limits on scalar operators are within an order of magnitude of the corresponding limits from $\mu \to e$ conversion in Ti. On the other hand, the limits on vector operators are however much weaker. Towards the end, we evolve the LEFT operators to W-boson mass scale using RGE, and match them to the Standard Model effective field theory (SMEFT) operators. We, then, derive the constraints on the parameter space of Leptoquark models that could generate these SMEFT operators at tree level.