General normal form procedure to correct tune-shift and non-linear chromaticity for large accelerators like the LHC

In future hadron colliders such as the LHC very high fields are needed to reach the design energy. Only superconducting magnets can produce such high fields and only at the cost of strong multipolar errors up to high order. This leads to a large non-linear shift of the tunes (detuning) both in amplitude and momentum, which may forbid a safe operation of the accelerator. The best solution to decrease these effects is to introduce a quasi-local correction via placing a set of non-linear elements in each cell near the source of the errors. A sequence of programs were used to perform this kind of correction. SIXTRACK was used to produce the high order transfer map in five variables using the DA-package of Berz (1989). The tune-shift functions are derived with the Lielib package of Forest (1989). Then, based on the approach proposed by Bazzani and Turchetti (1988) and first applied by Todesco (1990), we developed a correction procedure to minimize these detuning functions up to fifth order (decapole contribution) considering four-dimensional tune-shift with the momentum deviation as a parameter. For different machine versions we computed correction schemes and compared the results with tracking simulations. In all cases a considerable improvement of the detuning was established