The impedance of an accelerator induces coherent tune shifts of the
bunch oscillation modes. For a single bunch, below the threshold
current for the transverse mode-coupling instability, these tune
shifts are real and may lead to a loss of Landau damping when they
exceed the incoherent tune spread. This is a potential problem in the
LHC, where the broad-band impedance is small (and capacitive) at
injection energy and becomes larger (and inductive) at collision
energy [1], owing to the 
dependence of the
space charge impedance. Since the incoherent tune spread due to
direct space charge also decreases as , transverse
head-tail oscillation modes may become unstable during the ramp.
Thus, the situation is more critical at collision energy than at lower
energies.
Feedback cannot stabilize the higher-order head-tail modes. Thus, it is necessary to have octupoles (which induce tune spread) to damp these modes. However, for simplicity, we compute the stability threshold due to Landau damping for rigid dipole modes because they generally have larger tune shifts than the higher-order head-tail modes. The computation of the stability threshold for the rigid dipole modes therefore provides an upper bound for the required octupole strengths.
In the next section we sketch an elementary derivation of the beam
transfer function and of the corresponding dispersion integral for
rigid dipole oscillations
. This derivation
cannot be extended to higher order head-tail modes because we neglect
the longitudinal degree of freedom. Then, in Section 3, we assume a
Gaussian beam distribution in the two transverse planes and derive an
analytic formula for the stability limit when the detunings are linear
combinations of the two betatron action variables.
A Gaussian distribution gives rise to an infinite tune spread, whereas in reality, the tune spread in the beam will be finite. Also, the tails of the distribution will not extend out to infinity as for a Gaussian. So, one is tempted to consider a Gaussian distribution which is truncated beyond a certain amplitude. However, the stability limit exhibits complicated pathologies for a truncated Gaussian distribution and, to obtain quantitative results concerning the loss of Landau damping for a given real coherent tune shift, in Section 4 we discuss the case of a ``quasi-parabolic'' distribution in the two transverse planes. The integrated octupole strength required to stabilize transverse collective oscillations in the LHC at collision energy is discussed in a companion paper [2].
For a general introduction to the physics and mathematics of Landau damping, as well as for a good collection of references, we refer the reader to [3].