We discuss a minimal flavon-free model with twin modular symmetries, leading to trimaximal TM 1 lepton mixing in which the first column of the tri-bimaximal lepton mixing matrix is preserved. The model involves two modular S 4 groups, one acting in the neutrino sector, associated with a modulus field value τ SU with residual Z SU 2 symmetry, and one acting in the charged lepton sector, associated with a modulus field value τ T with residual Z T 3 symmetry. Apart from the predictions of TM 1 mixing, the model leads to a new neutrino mass sum rule which implies lower bounds on neutrino masses close to current limits from neutrinoless double beta decay experiments and cosmology.

Grand Unified Theories (GUT) predict proton decay as well as the formation of cosmic strings which can generate gravitational waves. We determine which non-supersymmetric SO(10) breaking chains provide gauge unification in addition to a gravitational signal from cosmic strings. We calculate the GUT and intermediate scales for these SO(10) breaking chains by solving the renormalisation group equations at the two-loop level. This analysis predicts the GUT scale, hence the proton lifetime, in addition to the scale of cosmic string generation and thus the associated gravitational wave signal. We determine which SO(10) breaking chains survive in the event of the null results of the next generation of gravitational waves and proton decay searches and determine the correlations between proton decay and gravitational waves scales if these observables are measured.

Proton decay is a smoking gun signature of Grand Unified Theories (GUTs). Searches by Super-Kamiokande have resulted in stringent limits on the GUT symmetry breaking scale. The largescale multipurpose neutrino experiments DUNE, Hyper-Kamiokande and JUNO will either discover proton decay or further push the symmetry breaking scale above 10 16 GeV. Another possible observational consequence of GUTs is the formation of a cosmic string network produced during the breaking of the GUT to the Standard Model gauge group. The evolution of such a string network in the expanding Universe produces a stochastic background of gravitational waves which will be tested by a number of gravitational wave detectors over a wide frequency range. We demonstrate the non-trivial complementary between the observation of proton decay and gravitational waves produced from cosmic strings in determining SO(10) GUT breaking chains. We show that such observations could exclude SO(10) breaking via flipped SU (5) × U (1) or standard SU (5), while breaking via a Pati-Salam intermediate symmetry, or standard SU (5) × U (1), may be favoured if a large separation of energy scales associated with proton decay and cosmic strings is indicated.

We develop a general formalism for multiple moduli and their associated modular symmetries. We apply this formalism to an example based on three moduli with finite modular symmetries S A 4 , S B 4 and S C 4 , associated with two right-handed neutrinos and the charged lepton sector, respectively. The symmetry is broken by two bi-triplet scalars to the diagonal S 4 subgroup. The low energy effective theory involves the three independent moduli fields τ A , τ B and τ C , which preserve the residual modular subgroups Z A 3 , Z B 2 and Z C 3 , in their respective sectors, leading to trimaximal TM 1 lepton mixing, consistent with current data, without flavons.

The radiative decay of neutral fermions has been studied for decades but CP violation induced within such a paradigm has evaded attention. CP violation in these processes can produce an asymmetry between circularly polarised directions of the radiated photons and produces an important source of net circular polarisation in particle and astroparticle physics observables. The results presented in this work outlines the general connection between CP violation and circular polarisation for both Dirac and Majorana fermions and can be used for any class of models that produce such radiative decays. The total CP violation is calculated based on a widely studied Yukawa interaction considered in both active and sterile neutrino radiative decay scenarios as well as searches for dark matter via direct detection and collider signatures. Finally, the phenomenological implications of the formalism on keV sterile neutrino decay, leptogenesis-induced right-handed neutrino radiative decay and IceCube-driven heavy dark matter decay are discussed. PACS number(s):

We consider for the first time level 7 modular invariant flavour models where the lepton mixing originates from the breaking of modular symmetry and couplings responsible for lepton masses are modular forms. The latter are decomposed into irreducible multiplets of the finite modular group Γ 7 , which is isomorphic to PSL(2, Z 7), the projective special linear group of two dimensional matrices over the finite Galois field of seven elements, containing 168 elements, sometimes written as PSL 2 (7) or Σ(168). At weight 2, there are 26 linearly independent modular forms, organised into a triplet, a septet and two octets of Γ 7. A full list of modular forms up to weight 8 are provided. Assuming the absence of flavons, the simplest modular-invariant models based on Γ 7 are constructed, in which neutrinos gain masses via either the Weinberg operator or the type-I seesaw mechanism, and their predictions compared to experiment.

In models with discrete flavour symmetries, flavons are critical to realise specific flavour structures. Leptonic flavour mixing originates from the misalignment of flavon vacuum expectation values which respect different residual symmetries in the charged lepton and neutrino sectors. Flavon cross couplings are usually forbidden, in order to protect these symmetries. Contrary to this approach, we show that cross couplings can play a key role and give raise to necessary corrections to flavour-mixing patterns, including a non-zero value for the reactor angle and CP violation. For definiteness, we present two models based on A 4 . In the first model, all flavons are assumed to be real or pseudo-real, with 7 real degrees of freedom in the flavon sector in total. A sizable reactor angle associated with nearly maximal CP violation is achieved, and, as both originate from the same cross coupling, a sum rule results with a precise prediction for the value of the Dirac CP-violating phase. In the second model, the flavons are taken to be complex scalars, which can be connected with supersymmetric models and multi-Higgs models. The complexity properties of flavons provide new sources for generating the reactor angle. Models in this new approach introduce very few degrees of freedom beyond the Standard Model and can be more economical than those in the framework of extra dimension or supersymmetry.

We propose a new mechanism to generate a lepton asymmetry based on the vacuum CP-violating phase transition (CPPT). This approach differs from classical thermal leptogenesis as a specific seesaw model, and its UV completion, need not be specified. The lepton asymmetry is generated via the dynamically realised coupling of the Weinberg operator during the phase transition. This mechanism provides a connection with low-energy neutrino observables.