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

## TL;DR

This paper calculates the Isgur-Wise function for $B_c$ decays to charmonium states using the Bethe-Salpeter method, highlighting the importance of relativistic corrections for accurate form factor descriptions.

## Contribution

The paper introduces a method to extract the Isgur-Wise function for $B_c$ decays to various charmonium states using the Bethe-Salpeter approach, emphasizing relativistic effects.

## Key findings

- Relativistic corrections significantly impact the Isgur-Wise function.
- The method provides numerical results for multiple charmonium final states.
- Overlapping integrals of wave functions are used to derive the Isgur-Wise function.

## Abstract

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.