Correlation between $R_{D^{(*)}}$ and top quark FCNC decays in leptoquark models

Some interpretations of $R_{D^{(*)}}$ anomaly in $B$ meson decay using leptoquark (LQ) models can also generate top quark decays through Flavor Changing Neutral Current (FCNC). In this work we focus on two LQs, i.e. scalar $S_1$ and vector $U_1$ which are both singlet under the $SU(2)_L$ gauge group in the Standard Model (SM). We investigate their implications on the 3-body top FCNC decays $t\to c \ell_i \ell_j$ at tree level and the 2-body $t\to c V$ at one-loop level, with $\ell$ being the SM leptons and $V=\gamma, Z, g$ being the SM gauge bosons. We utilize the $2\sigma$ parameter fitting ranges of the LQ models and find that $Br(t\to c \ell_i \ell_j)$ at tree level can reach $\mathcal{O}(10^{-6})$ and $Br(t\to c V)$ at one-loop level can reach $\mathcal{O}(10^{-10})$. Some quick collider search prospects are also analyzed.