The symmetry approach to quark and lepton masses and mixing

Abstract: The Standard Model lacks an organizing principle to describe quark and lepton flavours''. Neutrino oscillation experiments show that leptons mix very differently from quarks, adding a major challenge to the flavour puzzle. We briefly sketch the seesaw and the dark-matter-mediatedscotogenic'' neutrino mass generation approaches. We discuss the limitations of popular neutrino mixing patterns and examine the possibility that they arise from symmetry, giving a bottom-up approach to residual flavour and CP symmetries. We show how such family and/or CP symmetries can yield novel, viable and predictive mixing patterns. Model-independent ways to predict lepton mixing and neutrino mass sum rules are reviewed. We also discuss UV-complete flavour theories in four and more space-time dimensions. As benchmark examples we present an $A_4$ scotogenic construction with trimaximal mixing pattern TM2 and another with $S_4$ flavour symmetry and generalized CP symmetry. Higher-dimensional flavour completions are also briefly discussed, such as 5-D warped flavordynamics with a $T^\prime$ symmetry yielding a TM1 mixing pattern, detectable neutrinoless double beta decay rates and a very good global fit of flavour observables. We also mention 6-D orbifolds as a way to fix the structure of the 4-D family symmetry. We give a scotogenic benchmark orbifold model predicting the ``golden'' quark-lepton mass relation, stringent neutrino oscillation parameter regions, and an excellent global flavour fit, including quark observables. Finally, we discuss promising recent progress in tackling the flavor issue through the use of modular symmetries.