  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.
