Domain Walls in $A_4$ Flavour Models

The spontaneous breaking of an $A_4$ flavour symmetry, often used to predict leptonic mixing, can lead to the formation of domain walls which can annihilate and generate a stochastic gravitational wave background. We study this phenomenon in three scenarios where the nature of the scalar field responsible for breaking the $A_4$ symmetry spontaneously differs: real, complex, and supersymmetric. For the real scalar, a biased potential produces metastable walls that decay into oscillating two-wall systems with important consequences for gravitational wave signals. In the complex scalar case, we discuss the interplay between domain walls and global strings and classify the types of domain walls that form in terms of the $A_4$ group symmetries. We investigate the properties of supersymmetric $A_4$ domain walls, and highlight the BPS walls. Finally we show how these results may be achieved in leptonic $A_4$ flavour models, with and without supersymmetry, and discuss their distinctive gravitational wave signatures.

💡 Research Summary

This paper presents a comprehensive study of the cosmological consequences arising from the spontaneous breaking of the A4 discrete flavor symmetry, a popular framework for explaining lepton mixing patterns. The central focus is on the formation and evolution of topological defects, specifically domain walls, and their potential to generate an observable stochastic gravitational wave (GW) background.

The analysis is structured around three primary scenarios based on the nature of the scalar “flavon” field responsible for symmetry breaking: real scalar triplet, complex scalar triplet, and supersymmetric (SUSY) models. For the real scalar case with the most general renormalizable potential, the vacuum manifold contains two classes: six Z2-preserving “S-type” vacua and eight Z3-preserving “T-type” vacua. The cubic A4-invariant term splits the eight T-type vacua into two energy levels, creating metastable domain walls between vacua of different depths (T+ and T-). These metastable walls decay under the influence of a biased potential, and the study details their classification into topological types (TI, TII, TIII) and computes their tensions. Unstable wall trajectories are also identified.

Extending to a complex scalar triplet within an A4×U(1) symmetry, the breaking of the global U(1) factor leads to the formation of global cosmic strings alongside the A4 domain walls. The authors provide a group-theoretic classification of the possible domain walls into S-type and T-type categories, further subdivided based on relative phases between vacuum expectation values. This results in the possibility of composite string-wall networks, whose dynamics and GW emission are more complex.

In the SUSY scenario, the potential is assumed to be F-term dominated. This allows the domain wall equations to satisfy first-order (BPS-like) conditions, enabling the derivation of exact wall profiles and tensions. The inclusion of soft SUSY-breaking terms introduces a bias that lifts the degeneracy between different vacua, ensuring the eventual decay of the domain wall network and linking the SUSY-breaking scale directly to the peak frequency of the generated GW signal.

The theoretical findings are subsequently embedded into concrete lepton flavor models based on A4. The study demonstrates how S-type and T-type vacua, and consequently their associated domain walls, emerge in minimal real flavon models, complex flavon models with an additional U(1) symmetry (leading to strings), and SUSY A4 models with driving fields and R-symmetries.

The cosmological implications are addressed in two key aspects. First, solutions to the notorious domain wall problem are discussed, emphasizing how explicit breaking terms (bias) in both non-SUSY and SUSY contexts can pressure the wall network to annihilate. Second, the GW signatures from the collapse of these multi-scale domain wall networks are qualitatively analyzed. The paper highlights that models with multiple flavons (e.g., separate flavons for charged leptons and neutrinos) can lead to sequential wall collapses at different cosmic times, potentially producing a multi-peak GW spectrum. This connects the parameters of particle physics flavor models (symmetry breaking scales, coupling constants) to observable features in the GW background, opening a new avenue for indirect testing of flavor symmetry paradigms with future GW observatories like LISA, DECIGO, and Einstein Telescope.