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The $γγ\to J/ψJ/ψ$ reaction and the $J/ψJ/ψ$ pair production in exclusive ultraperipheral ultrarelativistic heavy ion collisions

We calculate the cross section for the $γγ\to J/ψJ/ψ$ process. Two mechanisms are considered: box (two-loop) diagrams of the order of $O(α_{em}^2 α_s^2)$ and two-gluon exchange of the order of $O(α_{em}^2 α_s^4)$. The first mechanism is calculated in the heavy-quark non-relativistic approximation while the second case we also include the effects of quantum motion of quarks in the bound state. The box contribution dominates at energies close to the threshold ($W <$ 15 GeV) while the two-gluon mechanism takes over at $W >$ 15 GeV. Including the bound-state wave function effects for the two-gluon exchange mechanism gives a cross section 0.1 - 0.4 pb, substantially smaller than that in the non-relativistic limit (0.4 - 1.6 pb). We also find a strong infrared sensitivity which manifests itself in a rather strong dependence on the mass for the $t$-channel gluons. The elementary cross section is then used in the Equivalent Photon Approximation (EPA) in the impact parameter space to calculate the cross section for $^{208}Pb+^{208}Pb \to ^{208}Pb + J/ψJ/ψ+ ^{208}Pb$ reaction. Distributions in rapidity of the $J/ψJ/ψ$ pair and invariant mass of the pair are shown.