  We study breakup of the deuteron induced by neutrinos in the neutral $\nu
d\to \nu np$, $\bar{\nu} d\to \bar{\nu} np$ and the charged $\bar{\nu} d\to e^+
n n$, $\nu d\to e^- pp$ processes. Pionless effective field theory with
dibaryon fields is used to calculate the total cross sections for neutrino
energies $E_\nu$ from threshold to 20 MeV. Amplitudes are expanded up to
next-to-leading order, and the partial wave is truncated at $P$-waves. The
Coulomb interaction between two protons is included nonperturbatively in the
reaction amplitudes, and an analytic expression of the amplitudes is obtained.
The contribution of the next-to-leading order to the total cross section is in
the range of 5.2$-$9.9\% in magnitude, and that of the $P$-wave is 2.4$-$2.8\%
at $E_\nu = 20$ MeV. Uncertainty arising from an axial isovector low-energy
constant is estimated to be on the order of 1\%.
