Maximum tolerable space-charge tune shift in synchrotrons
In plain words
Rings tolerate only a limited spread of particle oscillation frequencies caused by space charge before losses grow; this limit is known only empirically.
Precise statement
High-intensity synchrotrons operate with incoherent space-charge tune shift $\left|\Delta Q_{\mathrm{sc}}\right|$ up to approximately 0.2-0.5 over storage times of 1e3 to 1e5 turns. Derive, from the resonance structure of a given lattice and the periodic resonance crossing caused by synchrotron motion in a bunched beam, the maximal $\left|\Delta Q_{\mathrm{sc}}\right|$ compatible with emittance growth below 10 percent and loss below 1e-3 over $N$ turns, and compare with CERN PS and SPS, J-PARC and GSI SIS18 data.
What would settle it
A scaling law or simulation that matches measured limits in at least three rings with differing lattices and storage times.