Study on the $P$-wave form factors contributing to $B_s$ to $D_s$ inclusive semileptonic decays from lattice simulations

We present a pilot study on extracting the form factors of the semileptonic decay of a $B_s$ meson to the $P$-wave $D_s^{\ast \ast }$ states from $B_s$ four-point correlators. With their inclusive nature, four-point correlators include contributions from all possible final states. From the extracted $P$-wave form factors, we obtain numerical results for the corresponding Isgur-Wise form factors. The results suggest significant contributions from radial excitations to the Uraltsev sum rule at zero-recoil. In this pilot study, a coarse lattice of ${24}^3\times 64$ with lattice spacing of $0.11\mathrm{f}\mathrm{m}$ is used for the analysis.