  We calculate the order-v^2 relativistic correction of J/\psi, including
electromagnetic correction, to \chi_{bJ}-> J/\psi J/\psi, in the framework of
NRQCD factorization formula. The relativistic order-v^2 effect is found to
increase the lower-order prediction for the decay width by about 10%, while the
electromagnetism contribution is very small about 0.2% for \chi_{b0} and
\chi_{b2}. The total branching ratios are predicted to be of order 10^{-5} for
\chi_{b0,b2}-> J/\psi J/\psi, but 10^{-11} for \chi_{b1}-> J/\psi J/\psi, since
there is only electromagnetism contribution in this channel. We predict it is
possible to observe these reactions in LHC. Finally, we estimate the decay
width and branch ratio of \chi_{cJ}-> \omega\omega,\phi\phi) in the constituent
quark model by our formula at the leading-order of relativistic correction and
electromagnetic correction. The obtained branch ratio of \chi_{c0,2}-> \phi\phi
is in agreement with the experimental measurement in the order of magnitude.
