Practical criterion of transverse coupled-bunch head-tail stability

Analytical outcome of the paper is a few formulae to simplify practical threshold calculations of transverse coupled-bunch head-tail instability caused by narrow-band impedances in a proton synchrotron, which provide a useful quantitative view on how to keep the instability under control with chromaticity and cubic-nonlinearity correctors of the magnetic field. The formulae include: (i) the envelopes of head-tail mode form-factors expressed via a pair of averages over a bunch longitudinal distribution, and (ii) expressions of the effective betatron tune spread introduced by partial spreads in 2-D function ω_y(𝒥_x, 𝒥_z) of transverse action variables in x, z-directions, y=x, z. The tolerable values of transverse coupling impedances at parasitic higher-order E_1np-modes of the UNK accelerating cavities are estimated as an example of application