Gauge flavour unification: from the flavour puzzle to stable protons

Abstract: The idea of unification attempts to explain the structure of the Standard Model (SM) in terms of fewer fundamental forces and/or matter fields. However, traditional grand unified theories based on $SU(5)$ and $\mathrm{Spin}(10)$ shed no light on the existence of three generations of fermions, nor the distinctive pattern of their Yukawa couplings to the Higgs. We discuss two routes for unifying the SM gauge symmetry with its flavour symmetries: firstly, unifying flavour with electroweak symmetries via the group $SU(4) \times Sp(6)_L \times Sp(6)_R$; secondly, unifying flavour and colour via $SU(12) \times SU(2)_L \times SU(2)_R$. In either case, all three generations of SM fermions are unified into just two fundamental fields. In the larger part of this proceeding, we describe how the former model of `electroweak flavour unification' offers a new explanation of hierarchical fermion masses and CKM angles. As a postscript, we show that gauge flavour unification can have unexpected spin-offs not obviously related to flavour. In particular, the $SU(12) \times SU(2)_L \times SU(2)_R$ symmetry, when broken, can leave behind remnant discrete gauge symmetries that exactly stabilize protons to all orders.