Effect of space charge on the transverse mode-coupling instability threshold
In plain words
The beam's own repulsion shifts its oscillation frequencies, and theories disagree on whether this raises or lowers the intensity at which bunches start to shake apart.
Precise statement
For a single proton bunch with transverse impedance $Z$ and incoherent space-charge tune shift $\Delta Q_{\mathrm{sc}}$ comparable to or larger than the synchrotron tune $Q_s$, compute the transverse mode-coupling instability threshold intensity as a function of $\Delta Q_{\mathrm{sc}}/Q_s$, wake shape (broadband, resonator, resistive wall) and longitudinal profile. Reconcile with CERN SPS measurements in which TMCI is suppressed but other instabilities appear at high space charge.
What would settle it
A self-consistent theory or converged simulation that reproduces the measured SPS thresholds versus space-charge strength without free parameters.
Status in the literature
Models published 2016-2018 give thresholds rising or falling with space charge depending on wake sign and phase advance; SPS experiments reported in 2022 saw TMCI replaced by other instabilities at high space charge.