A Theory of the Beam Transfer Function (BTF) with Chromaticity Induced Head-Tail Phase Shift

The feasibility of chromaticity determination by measuring head-tail phase shifts has been demonstrated both in the CERN-SPS, at HERA (DESY) and at RHIC (BNL). This method, however, requires to apply sizable transverse kick to the beam which might turn to be detrimental to the LHC performance in view of the tight emittance growth budget allowed for this machine. The aim of this paper is to extend the theory to the case of a coherent beam excitation of small amplitude, in particular at a frequency equal to the beam betatron tune (Phase Locked Loop mode), with, as a result, the possibility of envisaging a chromaticity feed-back system based on this technique for the LHC.