Photoproduction of the double $J/psi$ ($Upsilon$) at the LHC with forward proton tagging

We calculate the photoproduction of double $J/psi$ ($Upsilon$) to leading order based on the nonrelativistic quantum chromodynamics factorization framework at the Large Hadron Collider with forward proton tagging. The numerical results of double $J/psi$ photoproduction pp $rightarrow$ p$gamma$p $rightarrow$ $J/psi$ + $J/psi$ with different forward detector acceptances ($xi$) are presented. The total cross section of double $J/psi$ photoproduction is less than 200 fb with 0.1 $<$ $xi$ $<$ 0.5, but can reach about 1.37(1.27) pb with 0.0015 $<$ $xi$ $<$ 0.5 ( 0.0015 $<$ $xi$ $<$ 0.15 ). The double $J/psi$ photoproduction may have the potential to be detected and provide an interesting signature, thus is useful for studying the mechanism of heavy quarkonium production. We also predict the double $Upsilon$ photoproduction and find they are, unfortunately, small (with less than 10 fb).