Accidental permutation symmetries as a test for Grand Unification: the supersymmetric $SU(5)$ case

Unification of matter fields implies the existence of accidental permutation symmetries, which potentially remain immune to large quantum corrections up to the TeV scale. We investigate the case of a supersymmetric $SU(5)$ grand unified theory, where such a permutation symmetry is present in the up-type squark sector. We present a variety of tests allowing to challenge the $SU(5)$ hypothesis based on the observation of squarks at the LHC. These tests appear as relations among observables involving flavour-violating or chirality-flipping decays of squarks. Moreover, they rely on top-polarimetry and charm-tagging. As an example, we discuss the application to the scenario of Natural Supersymmetry, while more examples can be found in the related journal publications.