A higher order QED calculation of the ultraperipheral heavy ion cross section for mu+ mu- pair production at RHIC and LHC is carried out. The so-called Coulomb corrections lead to an even greater percentage decrease of mu+ mu- production from perturbation theory than the corresponding decrease for e+ e- pair production. Unlike the e+ e- case, the finite charge distribution of the ions (form factor) and the necessary subtraction of impact parameters with matter overlap are significant effects in calculation an observable ultraperipheral mu+ mu- total cross section.
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