We study models of lepton masses and mixing based on broken modular invariance. We consider invariance under the finite modular group Γ 4 S 4 and focus on the minimal scenario where the expectation value of the modulus is the only source of symmetry breaking, such that no flavons need to be introduced. After constructing a basis for the lowest weight modular forms, we build two minimal models, one of which successfully accommodates charged lepton masses and neutrino oscillation data, while predicting the values of the Dirac and Majorana CPV phases.

The formalism of combined finite modular and generalised CP (gCP) symmetries for theories of flavour is developed. The corresponding consistency conditions for the two symmetry transformations acting on the modulus τ and on the matter fields are derived. The implications of gCP symmetry in theories of flavour based on modular invariance described by finite modular groups are illustrated with the example of a modular S 4 model of lepton flavour. Due to the addition of the gCP symmetry, viable modular models turn out to be more constrained, with the modulus τ being the only source of CP violation.

We investigate models of charged lepton and neutrino masses and lepton mixing based on broken modular symmetry. The matter fields in these models are assumed to transform in irreducible representations of the finite modular group Γ 4 S 4 . We analyse the minimal scenario in which the only source of symmetry breaking is the vacuum expectation value of the modulus field. In this scenario there is no need to introduce flavon fields. Using the basis for the lowest weight modular forms found earlier, we build minimal phenomenologically viable models in which the neutrino masses are generated via the type I seesaw mechanism. While successfully accommodating charged lepton masses, neutrino mixing angles and mass-squared differences, these models predict the values of the lightest neutrino mass (i.e., the absolute neutrino mass scale), of the Dirac and Majorana CP violation (CPV) phases, as well as specific correlations between the values of the atmospheric neutrino mixing parameter sin 2 θ 23 and i) the Dirac CPV phase δ, ii) the sum of the neutrino masses, and iii) the effective Majorana mass in neutrinoless double beta decay. We consider also the case of residual symmetries Z ST 3 and Z S 2 respectively in the charged lepton and neutrino sectors, corresponding to specific vacuum expectation values of the modulus.

In the framework of the modular symmetry approach to lepton flavour, we consider a class of theories where matter superfields transform in representations of the finite modular group Γ 5 A 5 . We explicitly construct a basis for the 11 modular forms of weight 2 and level 5. We show how these forms arrange themselves into two triplets and a quintet of A 5 . We also present multiplets of modular forms of higher weight. Finally, we provide an example of application of our results, constructing two models of neutrino masses and mixing based on the supersymmetric Weinberg operator.

We develop the formalism of the finite modular group Γ 4 ≡ S 4 , a double cover of the modular permutation group Γ 4 S 4 , for theories of flavour. The integer weight k > 0 of the level 4 modular forms indispensable for the formalism can be even or odd. We explicitly construct the lowest-weight (k = 1) modular forms in terms of two Jacobi theta constants, denoted as ε(τ ) and θ(τ ), τ being the modulus. We show that these forms furnish a 3D representation of S 4 not present for S 4 . Having derived the S 4 multiplication rules and Clebsch-Gordan coefficients, we construct multiplets of modular forms of weights up to k = 10. These are expressed as polynomials in ε and θ, bypassing the need to search for non-linear constraints. We further show that within S 4 there are two options to define the (generalised) CP transformation and we discuss the possible residual symmetries in theories based on modular and CP invariance. Finally, we provide two examples of application of our results, constructing phenomenologically viable lepton flavour models.

In modular-invariant models of flavour, hierarchical fermion mass matrices may arise solely due to the proximity of the modulus τ to a point of residual symmetry. This mechanism does not require flavon fields, and modular weights are not analogous to Froggatt-Nielsen charges. Instead, we show that hierarchies depend on the decomposition of field representations under the residual symmetry group. We systematically go through the possible fermion field representation choices which may yield hierarchical structures in the vicinity of symmetric points, for the four smallest finite modular groups, isomorphic to S3, A4, S4, and A5, as well as for their double covers. We find a restricted set of pairs of representations for which the discussed mechanism may produce viable fermion (charged-lepton and quark) mass hierarchies. We present two lepton flavour models in which the charged-lepton mass hierarchies are naturally obtained, while lepton mixing is somewhat fine-tuned. After formulating the conditions for obtaining a viable lepton mixing matrix in the symmetric limit, we construct a model in which both the charged-lepton and neutrino sectors are free from fine-tuning.

The observation of neutrinoless double beta decay would allow to establish lepton number violation and the Majorana nature of neutrinos. The rate of this process in the case of 3-neutrino mixing is controlled by the neutrinoless double beta decay effective Majorana mass | m |. For a neutrino mass spectrum with normal ordering, which is favoured over the spectrum with inverted ordering by recent global fits, | m | can be significantly suppressed. Taking into account updated data on the neutrino oscillation parameters, we investigate the conditions under which | m | in the case of spectrum with normal ordering exceeds 10 −3 (5 × 10 −3 ) eV: | m |NO > 10 −3 (5 × 10 −3 ) eV. We analyse first the generic case with unconstrained leptonic CP violation Majorana phases. We show, in particular, that if the sum of neutrino masses is found to satisfy Σ > 0.10 eV, then | m |NO > 5 × 10 −3 eV for any values of the Majorana phases. We consider also cases where the values for these phases are either CP conserving or are in line with predictive schemes combining flavour and generalised CP symmetries.

We point out that hints of deviations from unitarity in the first row of the CKM matrix may be explained by the presence of a single vector-like top. We study how the stringent experimental constraints arising from CP Violation in the kaon sector and from meson mixing such as $$ {D}^0\hbox{-} {\overline{D}}^0,{K}^0\hbox{-} {\overline{K}}^0 $$ D 0 ‐ D ¯ 0 , K 0 ‐ K ¯ 0 and $$ {B}_{d,s}^0\hbox{-} {\overline{B}}_{d,s}^0 $$ B d , s 0 ‐ B ¯ d , s 0 can be satisfied in the proposed framework. In order for the deviations from unitarity to be of the required size while keeping the theory perturbative, the new top quark should have a mass mT ≲ 7 TeV which could be probed in upcoming experiments at the energy frontier.