The Quest for $mu to e gamma$ and its Experimental Limiting Factors at Future High Intensity Muon Beams

The search for the Lepton Flavor Violating decay mu into e gamma will reach an unprecedented level of sensitivity within the next five years thanks to the MEG-II experiment. This experiment will take data at the Paul Scherrer Institut where continuous muon beams are delivered at a rate of about 10^8 muons per second. On the same time scale, accelerator upgrades are expected in various facilities, making it feasible to have continuous beams with an intensity of 10^9 or even 10^10 muons per second. We investigate the experimental limiting factors that will define the ultimate performances, and hence the sensitivity, in the search for mu into e gamma with a continuous beam at these extremely high rates. We then consider some conceptual detector designs and evaluate the corresponding sensitivity as a function of the beam intensity.