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The mode coupling instability for coasting beams has been discussed in aprevious paper using macroparticle tracking simulations from the pyHeadTailcode and a simple analytical formula which was proposed as an extension of theansatz used for the single-particle formalism. In this paper, we propose aself-consistent derivation of this formula based on the linearized Vlasovequation. The proposed mode coupling instability for coasting beams was neverpredicted or discussed in the past and we believe that the reason is twofold.First, to derive it analytically from the linearized Vlasov equation, oneshould not make the usual approximation $\sin(\phi)\simeq (e^{j\phi})/(2j)$,where $\phi$ is the transverse betatron phase, but really consider the twoterms of $\sin(\phi)=(e^{j\phi}-e^{-j\phi})/(2j)$ as the second term is the oneresponsible for the mode coupling in coasting beams. It should be stressed herethat mode coupling is found already with driving impedance only. Note that theprevious approximation is also usually made for bunched beams and this caseshould therefore also be carefully reviewed in the future. Second, by includingthe detuning impedance, the coupling is much stronger and this is what we foundalso in pyHeadTail simulations.

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