Normal form approaches and resonance analysis of LHC models

In order to understand the dynamic aperture limitations in LHC optics versions 4 and 5 at injection energy (450 GeV), a thorough resonance analysis is performed through Lie perturbation methods and Normal Form construction. In this respect, a simple numerical tool has been developed, the Graphical Representation of Resonances (GRR), allowing the evaluation and graphical representation of the resonance strengths and detuning, up to a desired order. The resonance analysis performed by means of GRR enabled us to understand the effect of the large errors in some special quadrupoles of LHC optics version 5. We were also able to identify and minimize the resonance which was correlated with the drop of the dynamic aperture, following the introduction of a large skew octupole bias in the LHC optics version 5, using the ''target'' error table. As shown by subsequent tracking studies, the proposed correction procedures led to a considerable improvement of the dynamic aperture of the studied LHC models