Next-to-leading-logarithm $k_T$ resummation for $B_c\to J/\psi$ decays

We derive the $k_T$ resummation for a transverse-momentum-dependent charmonium wave function, which involves the bottom quark mass $m_b$, the charm quark mass $m_c$, and the charm quark transverse momentum $k_T$, up to the next-to-leading-logarithm (NLL) accuracy under the hierarchy $m_b\gg m_c \gg k_T$. The resultant Sudakov factor is employed in the perutrbative QCD (PQCD) approach to the $B_c\to J/\psi$ transition form factor $A_0^{B_c \to J/\psi}(0)$ and the $B_c^+ \to J/\psi \pi^+$ decay. We compare the NLL resummation effect on these processes with the leading-logarithm one in the literature, and find a $(5-10)\%$ enhancement to the form factor $A_0^{B_c \to J/\psi}(0)$ and a $(10-20)\%$ enhancement to the decay rate ${\rm BR}(B_c^+ \to J/\psi \pi^+)$. The improved $k_T$ resummation formalism is applicable to the PQCD analysis of heavy meson decays to other charmonia.