Proposal of an Inverse Logarithm Scaling Law for the Luminosity Evolution

A scaling law for the time-dependence of the dynamic aperture, i.e., the region of phase space where stable motion occurs, was proposed in previous papers, about ten years ago. It was showed that dynamic aperture has a logarithmic dependence on time, which would be suggested by some fundamental theorems of the theory of dynamical systems. Such a scaling law was recently extended also to the intensity evolution in a storage ring. In this paper, inspired by these results, and inverse logarithm scaling law for the luminosity in a circular collider is proposed. Such a time-dependence is then tested against the data from the LHC physics runs and also with some examples from other machines. The results are presented and discussed in detail.