  Electron spin relaxation due to the D'yakonov-Perel' mechanism is
investigated in bilayer graphene with only the lowest conduction band being
relevant. The spin-orbit coupling is constructed from the symmetry group
analysis with the parameters obtained by fitting to the numerical calculation
according to the latest report by Konschuh {\it et al.} [Phys. Rev. B {\bf 85},
115423 (2012)] from first principles. In contrast to single-layer graphene, the
leading term of the out-of-plane component of the spin-orbit coupling in
bilayer graphene shows a Zeeman-like term with opposite effective magnetic
fields in the two valleys. This Zeeman-like term opens a spin relaxation
channel in the presence of intervalley scattering. It is shown that the
intervalley electron-phonon scattering, which has not been reported in the
previous literature, strongly suppresses the in-plane spin relaxation time at
high temperature whereas the intervalley short-range scattering plays an
important role in the in-plane spin relaxation especially at low temperature. A
marked nonmonotonic dependence of the in-plane spin relaxation time on
temperature with a minimum of several hundred picoseconds is predicted in the
absence of the short-range scatterers. This minimum is comparable to the
experimental data. Moreover, a peak in the electron density dependence of the
in-plane spin relaxation time at low temperature, which is very different from
the one in semiconductors, is predicted. We also find a rapid decrease in the
in-plane spin relaxation time with increasing initial spin polarization at low
temperature, which is opposite to the situation in both semiconductors and
single-layer graphene. ......(The remaining is cut due to the limit of space)
