Description of the luminosity evolution for the CERN LHC including dynamic aperture effects. Part I: the model

In recent years, modelling the evolution of beam losses in circular proton machines starting from the evolution of the dynamic aperture has been the focus of intense research. Results from single-particle, non-linear beam dynamics have been used to build simple models that proved to be in good agreement with beam measurements. These results have been generalised, thus opening the possibility to describe also the luminosity evolution in a circular hadron collider. In this paper, the focus is on the derivation of scaling laws for luminosity, which include both burn off and additional pseudo-diffusive effects. It is worthwhile stressing that time-dependence of some beam parameters can be taken into account in the proposed framework. The proposed models are applied to the analysis of a subset of the data collected during the CERN Large Hadron Collider (LHC) Run~1 in a companion paper (Part II).