Modular invariance approach to the flavour problem (from bottom up)

We review aspects of the modular invariance approach to the flavour problem. Harald Fritzsch was among the first to realise the existence and the fundamental nature of the quark and lepton flavour problems in particle physics, that symmetries may be the key to the solution(s) of these problems and to propose in 1978 and 1979 a solution to the quark flavour problem in the form of the Fritzsch quark mass matrices with texture zeros. After introducing the general ingredients of the modular invariance approach, we describe the formalism that allows to construct models in which fermion (charged-lepton and quark) mass hierarchies follow solely from the properties of the modular forms, avoiding the fine-tuning of the constant parameters present in the fermion mass matrices and the need to introduce extra fields. Focusing on the lepton sector, we show how the indicated formalism can be used in lepton flavour models to obtain the charged lepton mass hierarchies without parameter fine-tuning. Only a very limited number of models satisfy the non-fine-tuning requirement in explaining the observed fermion mass hierarchies. We review next the general conditions under which also the peculiar pattern of Pontecorvo, Maki, Nakagawa and Sakata (PMNS) lepton mixing consisting of two large and one small angles can be reproduced without fine-tuning. We then give an example of a phenomenologically viable “minimal” modular lepton flavour model in which both the lepton mixing and charged lepton mass hierarchies are generated without fine-tuning. We conclude by listing some of the open questions and problems in the modular invariant approach to the flavour problem.