Grand unification, small vs large representations, hadron colliders and all that

We review minimal realistic grand unified models based on $SU(5)$ and $SO(10)$ gauge groups. The models with small Higgs representations and higher dimensional operators - under the assumption of no cancellations in proton decay amplitudes - predict nearby oases with new light particles. Two of them stand out: real fermion and scalar weak triplets. The former generates dynamically neutrino Majorana mass through the so-called type III seesaw mechanism, while the latter naturally gets a small vacuum expectation value and thus generically modifies $W$-boson mass. On the contrary, the renormalisable versions of these theories fail to determine the particle spectra. In particular, in the renormalisable version of the $SO(10)$ theory the mass of the right-handed gauge boson - normally assumed (and allowed) to lie far in the desert - could be even accessible at the LHC. Last but not least, we show that in the minimal supersymmetric $SU(5)$ theory all the superpartners can lie orders of magnitude above the weak scale, unless one requires strict naturalness. Thus, the so-called split supersymmetry does not imply any light states.