Normal Mode Laslett Coefficients for One (co-Axial) and Two (Offset) Liners in a Circular Magnetic Yoke

A general framework for computing the normal mode coherent and incoherent Laslett coefficients for one (co-axial) and two (offset, parallel) perfectly conducting liners surrounded by a circular bore magnetic yoke is presented, including electric and magnetic image fields. The formalism is then extended to a (simplified) twin-liner LHC geometry. It is seen that the interaction is dominated by the static part of the magnetic beam-beam force. The offset of the liners with respect to the yoke axis produces a nontrivial dependence of the tune shift on the azimuthal coordinate in the pipe, while leaving its order of magnitude almost unchanged from the single beam value.