  In the wake of measurements on $B_c^+ \to J/\psi K^+$, $B_c^+ \to J/\psi
\pi^+\pi^-\pi^+$, and $B_c^+ \to J/\psi K^+ K^-\pi^+$ at Large Hadron Collider
experiments, we propose to study the decays $B_c^+ \to J/\psi M^+$
comprehensively, with $M$ being the light charged pseudoscalar ($P$), vector
($V$), scalar ($S$), axial-vector ($A$), and tensor ($T$) mesons, within the
improved perturbative QCD (iPQCD) formalism at leading order in the Standard
Model. The theoretical predictions for experimental observables such as
branching fractions, relative ratios, and longitudinal polarization fractions
in the iPQCD formalism await near future examinations relying on the upgraded
Large Hadron Collider, even the forthcoming Circular Electron-Positron
Collider. We emphasize that the investigations on the
factorizable-emission-suppressed or -forbidden decays like $B_c^+ \to J/\psi
S^+$, $B_c^+ \to J/\psi A^+_{1^1\!P_1}$, and $B_c^+ \to J/\psi T^+$, should go
definitely beyond naive factorization to explore the rich dynamics, which
could, in turn, further help understand the QCD nature of $B_c$ meson, as well
as that of related hadrons. The future confirmations on those predictions about
the relative ratios between the branching fractions of $B_c^+ \to J/\psi
b_1(1235)^+ (a_0(980)^+, a_0(1450)^+, a_2(1320)^+)$ and $B_c^+ \to J/\psi
\pi^+$ could further examine the reliability of this iPQCD formalism. Because
of containing only tree-level $\bar b \to \bar c$ transitions, the {\it CP}
asymmetries in the $B_c^+ \to J/\psi M^+$ decays exhibit naturally zero.

High Energy Physics - Phenomenology;High Energy Physics - Experiment