Observing true tauonium via two-photon fusion at $e^+e^-$ and hadron colliders

The feasibility of observing true tauonium, the bound state of two tau leptons, $\mathcal{T}_0\equiv(\tau^+\tau^-)_0$, via photon-photon collisions at $e^+e^-$ colliders and at the LHC, is studied. The production cross sections of the process $\gamma\gamma\to\mathcal{T}_0\to\gamma\gamma$ -- as well as those of all relevant backgrounds: spin-0 and 2 charmonium resonances decaying to diphotons, and light-by-light scattering -- are computed in the equivalent photon approximation for $e^+e^-$ collisions at BES III ($\sqrt{s} = 3.8$ GeV), Belle II ($\sqrt{s} = 10.6$ GeV), and FCC-ee ($\sqrt{s} = 91.2$ GeV), as well as for ultraperipheral p-p, p-Pb, and Pb-Pb collisions at the LHC. Despite small $\mathcal{T}_0$ production cross sections and a final state swamped by decays from overlapping pseudoscalar and tensor charmonium states -- the $\mathrm{\chi_{c2}}$, $\mathrm{\eta_{c}(2S)}$, and $\mathrm{\chi_{c0}}$ states have masses only 2.5, 84, and 139 MeV away, respectively, from the $\mathcal{T}_0$ peak -- evidence and observation of the ground state of the heaviest leptonium appears feasible at Belle II and FCC-ee, respectively, with in-situ high-precision measurements of the irreducible backgrounds.