Semileptonic $B_c$ decays to $P-$wave charmonium and the nature of $\chi_{c1}(3872)$

The hadronic form factors of $B_c$ semileptonic decays to the $P-$wave charmonium 4-plet can be expressed near the zero-recoil point in terms of universal functions, performing a systematic expansion in QCD in the relative velocity of the heavy quarks and in $1/m_Q$. Such functions are independent of the member of the multiplet involved in the transitions. We present the results of an NLO calculation up to $O(1/m_Q^2)$ classifying the universal functions at this order. We work out a set of relations among the form factors of the same mode and of different modes, which should be reproduced by explicit calculations, reducing the hadronic uncertainty affecting such channels. The approach is also helpful to investigate the debated nature of $\chi_{c1}(3872)$, studying the production in $B_c$ semileptonic decays and comparing it to the modes involving the other $2P$ charmonia.