Optimizing integrated luminosity of future hadron colliders

The integrated luminosity, a key figure of merit for any particle-physics collider, is closely linked to the peak luminosity and to the beam lifetime. The instantaneous peak luminosity of a collider is constrained by a number of boundary conditions, such as the available beam current, the maximum beam-beam tune shift with acceptable beam stability and reasonable luminosity lifetime (i.e., the empirical ``beam-beam limit''), or the event pileup in the physics detectors. The beam lifetime at high-luminosity hadron colliders is largely determined by particle burn off in the collisions. In future highest-energy circular colliders synchrotron radiation provides a natural damping mechanism, which can be exploited for maximizing the integrated luminosity. In this article, we derive analytical expressions describing the optimized integrated luminosity, the corresponding optimum store length, and the time evolution of relevant beam parameters, without or with radiation damping, while respecting a fixed maximum value for the total beam-beam tune shift or for the event pileup in the detector. Our results are illustrated by examples for the proton-proton luminosity of the existing Large Hadron Collider (LHC) at its design parameters, of the High-Luminosity Large Hadron Collider (HL-LHC), and of the Future Circular Collider (FCC-hh).