  We investigate the physics potential of the annihilation decays
$\bar{B}^0_{s,d} \to J/\psi$ $\gamma$ in the Standard Model and beyond. In
naive factorization approach, the branching ratios are estimated to be
$\mathcal{B}(\bar{B}^0_s \to J/\psi\gamma)=1.40\times 10^{-6}$ and
$\mathcal{B}(\bar{B}^0_d \to J/\psi\gamma)=5.29\times 10^{-8}$. In the
framework of QCD factorization, we compute the non-factorizable corrections and
get $\mathcal{B}(\bar{B}^0_s \to J/\psi\gamma) = 2.11\times10^{-7}$,
$\mathcal{B}(\bar{B}^0_d \to J/\psi\gamma) =7.65\times10^{-9}$. Future
measurements of these decays would be useful for testing the factorization
frameworks. The smallness of these decays in the SM make them sensitive probes
of New Physics. As an example, we will consider the possible admixture of (V+A)
charge current to the standard (V-A) current. This admixture will give
significant contributions to the decays.
