  Radiative decay of the sub-threshold $1_1^-$ and $2_1^+$ states of $^{16}$O
is studied in cluster effective field theory. The wave function normalization
factors for initial and final states of the radiative decay amplitudes are
deduced by using the phase shift data of elastic $\alpha$-$^{12}$C scattering
for $l=0,1,2$, which are related to the asymptotic normalization coefficients
of $0_1^+$, $1_1^-$, $2_1^+$ states of $^{16}$O for the two-body
$\alpha$-$^{12}$C channel; then only one unfixed parameter remains in each of
the radiative decay amplitudes. We fit the parameters to the experimental
radiative decay rates and apply the fitted parameters to the study of radiative
$\alpha$ capture on $^{12}$C, $^{12}$C($\alpha$,$\gamma$)$^{16}$O. The order of
magnitude of astrophysical $S$ factors of $E1$ and $E2$ transitions of
$^{12}$C($\alpha$,$\gamma$)$^{16}$O is reproduced compared to the experimental
data, and we discuss an improvement in the calculation of $S$ factors of
$^{12}$C($\alpha$,$\gamma$)$^{16}$O.
