$J\mathrm{/}\psi$ Pair Production at the Tevatron with $\sqrt{s}=1.96~$\mathrm{T}\mathrm{e}\mathrm{V}$

We revisit the $J\mathrm{/}\psi$ pair production issue at the Fermilab Tevatron Run II with the center-of-mass energy $\sqrt{s}=1.96\mathrm{T}\mathrm{e}\mathrm{V}$. Both color-singlet and -octet production mechanisms are considered. Our result shows that the transverse momentum($p_T$) scaling behaviors of double $J\mathrm{/}\psi$ differential cross sections in color-singlet and -octet deviate distinctively from each other while $p_T$ is larger than $8\mathrm{G}\mathrm{e}\mathrm{V}$, and with the luminosity of $5\mathrm{f}{\mathrm{b}}^{-1}$ the $J\mathrm{/}\psi$ pair events from color-singlet scheme are substantially measurable in Tevatron experiments, even with certain lower transverse momentum cut. Hence the Tevatron is still possibly a platform to check the heavy quarkonium production mechanism.