Comment on"Fast-slow mode coupling instability for coasting beams in the presence of detuning impedance"

In this comment we show untenability of key points of the recent article of N. Biancacci, E. Metral and M. Migliorati [Phys. Rev. Accel. Beams 23, 124402 (2020)], hereafter the Article and the Authors. Specifically, the main Eqs. (23), suggested to describe mode coupling, are shown to be unacceptable even as an approximation. The Article claims the solution of this pair of equations to be in"excellent agreement"with the pyHEADTAIL simulations for CERN PS, which is purportedly demonstrated by Fig. 6. Were it really so, it would be a signal of a mistake in the code. However, the key part of the simulation results is not actually shown, and the demonstrated agreement has all the features of an illusion.

Due to the driving impedance properties, only the slow modes can be unstable. An illustrative sketch of the spectrum is presented in Fig. 1, assuming smooth approximation and focusing detuning impedance, when the modes can cross but not couple.
For the Article's PS example with lattice tunes Q β = ω β /ω 0 = 6.4, a slow mode with frequency ω β − 7ω 0 = −0.6ω 0 has the nearest mode −ω β + 6ω 0 = −0.4ω 0 , the backward one. The Article, including the title, calls the latter mode "fast," which is a terminological mistake: the value of the phase velocity of that allegedly "fast" mode is actually smallest among all the modes.
In linear systems with time-independent coefficients, modes can couple only when their frequencies coincide. Clearly, positive-based modes can couple only with the negative-based ones, and vice versa; one of the modes must be stable (fast, zero, or backward), and another unstable (slow). For the PS example, the positive-based slow mode with n = n 1 = −7 might couple with the negative-based backward mode with n = n 2 = +6. The coupling can happen if the lattice tune difference between the two modes, 0.6 − 0.4 = 0.2, is compensated by the detuning impedance, presumably able to shift the betatron tune up by 0.1. Note that the difference between the harmonic numbers of the coupled modes, n 2 − n 1 = 13 = ceil(2Q β ), is just above the doubled betatron tune; the same is true for any pair of coupled modes.
We beg pardon for this pedantic textbook explanation, but we feel obliged to make it since in the Article the terms are confused and the harmonic numbers are given without signs, making a false impression that the modes of the neighbor harmonics, 6 and 7, can sometimes be coupled.
Let us now come back to Eqs. (23). They are derived from Eq. (22) by an ansatz that the collective oscillation y(s, t) is a linear combination of two harmonics, n 1 and n 2 . In where Z driv (Ω, s)β(s) = const, these cross terms are equal to zero. Thus, instead of Eqs. (23) of the Article, the mode coupling problem should be described by the following equations; Here ∆Ω tot ∝ i ds[Z det (0, s) + Z driv (Ω, s)]β(s) is the conventional uncoupled coherent tune shift at the sought-for frequency Ω ≈ ±ω β + nω 0 , and the cross-coefficients can be expressed as ∆Ω driv with ∆Ω driv  As to the bottom plot of Fig. 6, we see there that the mode tunes are locked in the half-integer resonance. For the simulations, something like that has to be expected just on the ground of the sufficiently strong detuning quadrupole for a lattice with nonzero harmonic 13. For a perfectly smooth lattice, however, the half-integer tune 6.5 would be as good as any other tune; thus, the result of the simulations must be sensitive to the lattice smoothness. Since Eqs. (23) are fully insensitive to the phase advance per cell or other smoothness parameters, the agreement between the pyHEADTAIL simulations and theory in the bottom plot of Fig. 6 can be only accidental.
We'd like also to note that, although the half integer resonance is not presented in the smooth approximation, it plays a significant role in any real machine. Approaching this resonance results in a big variation of beta-functions and, consequently, fast increase of effective impedance Z driv (Ω, s)β(s) and its coupling-related harmonic, mentioned above.
We hope that our disagreement with key issues of the Article is clearly expressed, and we would appreciate a response of the Authors.