Are neutrino masses modular forms? (1706.08749v2)

Abstract: We explore a new class of supersymmetric models for lepton masses and mixing angles where the role of flavour symmetry is played by modular invariance. The building blocks are modular forms of level N and matter supermultiplets, both transforming in representations of a finite discrete group Gamma_N. In the simplest version of these models, Yukawa couplings are just modular forms and the only source of flavour symmetry breaking is the vacuum expectation value of a single complex field, the modulus. In the special case where modular forms are constant functions the whole construction collapses to a supersymmetric flavour model invariant under Gamma_N, the case treated so far in the literature. The framework has a number of appealing features. Flavon fields other than the modulus might not be needed. Neutrino masses and mixing angles are simultaneously constrained by the modular symmetry. As long as supersymmetry is exact, modular invariance determines all higher-dimensional operators in the superpotential. We discuss the general framework and we provide complete examples of the new construction. The most economical model predicts neutrino mass ratios, lepton mixing angles, Dirac and Majorana phases uniquely in terms of the modulus vacuum expectation value, with all the parameters except one within the experimentally allowed range. As a byproduct of the general formalism we extend the notion of non-linearly realised symmetries to the discrete case.

Overview of "Are neutrino masses modular forms?"

This paper, authored by Ferruccio Feruglio, explores a novel approach to the modeling of lepton masses and mixing angles within a supersymmetric framework, wherein modular invariance supplant traditional flavor symmetries. The paper introduces supersymmetric models where modular forms of a given level $N$ and matter supermultiplets, with their transformations governed by corresponding finite discrete groups $\Gamma_N$, provide the mechanism for flavor symmetry.

Framework and Fundamental Features

The paper posits that in its simplest embodiment, the flavor-breaking effects are sourced solely from the vacuum expectation value (VEV) of a single complex field, the modulus $\tau$. The Yukawa couplings themselves are expressed as modular forms, and hence, the modular symmetry directly constraints the neutrino masses and mixing angles. During unbroken supersymmetry, this symmetry extends to determine all higher-dimensional operators in the superpotential, presenting a theoretically elegant setup that potentially minimizes the model’s dependence on arbitrary parameters.

In instances where the modular forms are functionally constant, the setup reduces to models with a traditional supersymmetric flavor symmetry under the group $\Gamma_N$, recognizable from existing literature. Fascinatingly, the introduction of non-trivial $\tau$-dependent modular forms invites exploration of an entirely new class of models.

Numerical Results and Theoretical Implications

The paper illustrates the framework through concrete models, highlighting its potential to predict neutrino mass ratios, lepton mixing angles, and expectations for Dirac and Majorana phases strictly as functions of the modulus VEV. Notably, in some minimalist models, all but one parameter falls within the experimentally observed range. For example, one such model predicts $\sin^2\theta_{13} \approx 0.045$, which, while slightly outside current experimental bounds, underscores the approach's potential predictive power that extends beyond existing models founded on discrete symmetries.

This rendition of lepton mass modeling, by constraining higher-dimensional operator contributions and mitigating dependence on explicit symmetry-breaking flavons, offers a particularly stringent and predictive theoretical framework. It uniquely integrates the roles of modular invariance and supersymmetry in governing neutrino properties.

Novel Insights into Discrete Symmetries

An auxiliary outcome of the formalism discussed is an extended interpretation of non-linear symmetry realizations applicable to discrete groups. This investigation explores novel constraints on levels of modular forms, prompting potential reinterpretations and applications within relativistic quantum field theories.

The paper methodically introduces this framework, balancing technical rigor with exploratory forays into its implications and constraints. It systematically advances the symmetry-based dialogue in neutrino physics, hinting at untapped areas for symmetry-based model exploration like the intriguing prospects of leveraging discrete modular symmetries in a more generalized context.

Prospective studies will likely focus on refining these predictions and expanding the conceptual apparatus to address potential extensions beyond the scope of lepton and neutrino sectors. In summary, this paper draws an elegant bridge between modular forms and neutrino physics, marking a step forward in the systematic pursuit of symmetries underpinning fundamental particle properties.