Self-consistent derivation of the transverse mode coupling instability for coasting beams using the linearized Vlasov equation

The mode coupling instability for coasting beams has been discussed in a previous paper using macroparticle tracking simulations from the pyHeadTail code and a simple analytical formula which was proposed as an extension of the ansatz used for the single-particle formalism. In this paper, we propose a self-consistent derivation of this formula based on the linearized Vlasov equation. The proposed mode coupling instability for coasting beams was never predicted or discussed in the past and we believe that the reason is twofold. First, to derive it analytically from the linearized Vlasov equation, one should not make the usual approximation $\sin(\phi)\simeq (e^{j\phi})/(2j)$, where $\phi$ is the transverse betatron phase, but really consider the two terms of $\sin(\phi)=(e^{j\phi}-e^{-j\phi})/(2j)$ as the second term is the one responsible for the mode coupling in coasting beams. It should be stressed here that mode coupling is found already with driving impedance only. Note that the previous approximation is also usually made for bunched beams and this case should therefore also be carefully reviewed in the future. Second, by including the detuning impedance, the coupling is much stronger and this is what we found also in pyHeadTail simulations.