Improved covariant analysis of $B_c^+ \to χ_{c1}(nP)$ decays and implications for the nature of $χ_{c1}(3872)$
We study the weak decays $B_c^+\toχ_{c1}(nP)\ell^+ ν_{\ell}$ and $B_c^+\toχ_{c1}(nP)X$ (n=1,2,3) within the Bethe-Salpeter formalism, treating $χ_{c1}(3872)$ as the conventional $χ_{c1}(2P)$ charmonium. We upgrade the covariant hadronic transition amplitude to consistently evaluate the final-state wave function in its rest frame, enabling a reliable description of large-recoil processes and sizable relativistic corrections pertinent to highly excited charmonium. Our results provide a novel platform to probe the internal structure of $χ_{c1}(3872)$ via $B_c$ decays. While the LHCb search for $B_c^+\toχ_{c1}(3872)π^+$ yielded only an upper limit, we demonstrate that this is primarily due to insufficient luminosity, approximately 20 times the current $B_c$ data sample would be required for a definitive observation. In contrast, we identify the semileptonic mode $B_c^+\toχ_{c1}(3872)μ^+ν_μ$ as a far more promising channel, which could become accessible with merely twice the existing data set. Our predictions offer concrete guidance for upcoming LHCb analyses and complement ongoing efforts to resolve the nature of $χ_{c1}(3872)$.