Isgur-Wise function in $B_c$ decays to charmonium with the Bethe-Salpeter method

The Isgur-Wise function vastly reduces the weak-decay form factors of hadrons containing one heavy quark. In this paper, we extract the Isgur-Wise functions from the instantaneous Bethe-Salpeter method, and give the numerical results for the $B_c$ decays to charmonium where the final states include $1S$, $1P$, $2S$ and $2P$. The overlapping integral of the wave functions for the initial and final states is the Isgur-Wise function, as the heavy quark effective theory does. In the case of accurate calculation, describing form factors need to introduce more relativistic corrections which are the overlapping integrals with the relative momentum between the quark and antiquark to Isgur-Wise function. The relativistic corrections to Isgur-Wise function provide greater contributions especially involving the excited state, and therefore are necessary to be adopted.