Numerical Methods for the Optimization of the LHC Luminosity Computation

One of the most significant merit figures in the Large Hadron Collider, LHC, is the luminosity. Larger values of luminosity allow for more interesting physical observations by the experiments, and it is a metric to quantify the instantaneous rate of the particles interacting in the collider. The luminosity is a scalar observable that indicates how much “Physics” is created during the run in the LHC, hence it is directly connected to LHC’s capability for discovery. The intensity of the two beams, the number of colliding bunches, the bunch profiles, the frequency of revolution, etc. . . are all physical quantities that contribute to the luminosity. Several facets of the accelerator and the beam properties are integrated into this figure. The objective of this thesis is to employ various numerical and mathematical techniques in order to enhance the efficiency of luminosity computation. At the beginning, we will present a mathematical model for the calculation of colliding bunches, followed by a com- prehensive examination of an optimized Python implementation. Subsequently, our objec- tive will be to optimize the number of colliding bunches through the strategic arrangement of these bunches, more focusing on a number theory approach. The primary difficulty is in the vast number of potential combinations and the numerous limits that must be adhered to. Consequently, achieving our fundamental goal becomes almost unreachable, as we shall soon discover. Subsequently, we will analyse the bunch profiles, attempting to calculate them based on the observed luminosity. The latter refers to a scalar quantity that is derived through a convolution in three-dimensional space and time of these bunch profiles. Our objective is to obtain distinct values from a single scalar using an analytical approach. This presents a problem that is inherently ill-conditioned. However, as we will demonstrate later, we are able to obtain intriguing results. By pursuing this approach, we will endeavour to solve the same ill-conditioned problem but in the presence of noise.