New phenomenological tests for supersymmetric theories with SU(5)-like unification

In this thesis, we explore the consequences at low energies of supersymmetric grand unified theories (SUSY-GUT) based on the Lie group SU(5). These models constitute the prototype of minimal SUSY GUT and as such, have been intensively studied in the last decades. In particular, the quark-lepton unification in the Yukawa sector has received a lot of attention. The originality of our approach is to stay within the up-squark sector, where we have shown that as of a run noticed SU(5) induced symmetry relation emerges, way less model dependent than the one aforementioned. Indeed, we demonstrate that a supersymmetric model, equipped with a SU(5) gauge symmetry generically implies that the up-squark trilinear term A_U is symmetric at the GUT scale, namely that A_U = A^T_U. This relation is only mildly spoiled by the renormalization group running, and remains well preserved at the TeV scale, leaving thus a SU(5) footprint in the low scale SUSY spectrum. This allows to build a series of phenomenological SU(5) tests on low scale SUSY spectra, potentially accessible at the LHC. Most of these tests generically consist in determining a certain relation among low scale observables. We exhibit such relations for three different sort of SUSY spectra, namely 'Heavy SUSY' spectra which feature a SUSY breaking scale well above the TeV scale, 'Natural SUSY' spectra which feature a light third scalar generation and 'Top Charm SUSY' spectra in which only the first scalar generation is kept heavy. We have developed a generic method, based on a p-value frequentist statistical test to quantify to which significance one can assess a departure from these low scale constraints relations, in the three above-mentioned types of SUSY spectra. Typically, for each of the tests, Ο(10-100) events are necessary in the processes considered, to assess a departure of 50% at 3σ level of significance. It should be mentioned that experimental techniques such as charm tagging or top - polarimetry play a crucial role to apply these tests at current colliders, such as the LHC. The second part of this thesis is devoted to a more global numerical analysis, based on Bayesian model comparison methods, and applicable to any kind of spectrum whatever its hierarchy. In this context, we have developed a numerical method based on a Markov Chain Monte Carlo algorithm allowing to test, given an observed low energy spectrum, with which significance it points or not toward a high scale SU(5) dynamic. Again, we have considered several cases depending on the amount of information that will be collected on the squark spectrum at the LHC. We find that typically, the constraining power of these would be collected information is too low to draw any conclusion on a possible SU(5)-like high scale dynamic, except in certain cases, when the uncertainties attached to the observables considered are reduced to Ο(1%) and/or when spectra far from being SU(5)-symmetric are considered. In this last case, this analysis seems to disfavor the presence of a GUT scale SU(5) symmetry with weak evidence. (author)