Theory of the tune shift due to linear coupling

The presence of skew quadrupole fields will linearly couple the x and y motions. The x and y motions can then be written as the sum of two normal modes >. This paper presents analytical perturbation theory results for the tunes of the normal modes. The results for the normal mode tunes are first found correct to lowest order in the skew quadrupole fields. The results are then carried one step further to include the next higher order terms in the skew quadrupole fields. These analytical results show that for the higher order shift in the tune, the important harmonics of the skew quadrupole field are the harmonics near the sum of the tunes. However the harmonics closest to the sum of the tunes do not contribute to the higher order tune splitting, the seperation of the tunes, as they shift the two tunes about equally.This results in a lack of a dominant harmonic for the higher order part of the tune splitting, which complicates the understanding and correction of the higher order part of the tune splitting.