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We calculate the order-v^2 relativistic correction of J/\psi, includingelectromagnetic correction, to \chi_{bJ}-> J/\psi J/\psi, in the framework ofNRQCD factorization formula. The relativistic order-v^2 effect is found toincrease the lower-order prediction for the decay width by about 10%, while theelectromagnetism 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, sincethere is only electromagnetism contribution in this channel. We predict it ispossible to observe these reactions in LHC. Finally, we estimate the decaywidth and branch ratio of \chi_{cJ}-> \omega\omega,\phi\phi) in the constituentquark model by our formula at the leading-order of relativistic correction andelectromagnetic correction. The obtained branch ratio of \chi_{c0,2}-> \phi\phiis in agreement with the experimental measurement in the order of magnitude.

Exclusive decay of P-wave Bottomonium into double J/\psi