  The rare process of $W$ radiative decay into the $B_c$ meson can serve as an
interesting play ground for testing two influential perturbative QCD
approaches: non-relativistic quantum chromodynamics (NRQCD) and ligh-cone (LC)
factorization. Since the $B_c$ meson is a genuine heavy quarkonium which is
composed of two different species of heavy quarks, it is natural to employ
NRQCD factorization to tackle this exclusive quarkonium production process; on
the other hand, since the mass of the $W$ boson is much greater than that of
the $B_c$, the collinear factorization is also a viable approach. To fully
disentangle the contributions from several distinct scales $M_W$, $m_Q$, $m_Q
v$ ($v$ represents the typical velocity of the $c/\bar{b}$ quarks inside
$B_c$), and $\Lambda_{\rm QCD}$, we carry out a joint analysis that combines
both light-cone and NRQCD factorization approaches, and compute the
order-$\alpha_s$ correction to $W \to B_c+\gamma$. With the aid of the
celebrated Efremov-Radyushkin-Brodsky-Lepage (ERBL) evolution equation, we
resum the leading collinear logarithms $\alpha_s \ln\! {M_W\over m_Q}$ to all
orders in $\alpha_s$. We also explore some phenomenological implications of our
predictions.
