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Domain Walls in $A_4$ Flavour Models

Bowen Fu, Stephen F. King, Luca Marsili, Jessica Turner, Ye-Ling Zhou

TL;DR

Domain walls arising from spontaneous breaking of a non-Abelian flavour symmetry $A_4$ can form in the early Universe and annihilate to produce a stochastic gravitational-wave background. The paper analyzes three flavon realizations—real, complex, and SUSY—and classifies the resulting non-Abelian walls (including $S$-type, $T$-type, and their complex variants $S^c$, $T^c$, $T'^c$), their tensions, stabilities, and decay channels. It embeds these structures into explicit leptonic flavour models and discusses distinctive GW signatures, including multi-peak spectra from multi-scale wall collapse and string-wall networks. The cosmological implications are explored both in non-SUSY and SUSY contexts, with biases from explicit breaking or soft terms providing annihilation triggers and connecting GW signals to the underlying flavour sector.

Abstract

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.

Domain Walls in $A_4$ Flavour Models

TL;DR

Domain walls arising from spontaneous breaking of a non-Abelian flavour symmetry can form in the early Universe and annihilate to produce a stochastic gravitational-wave background. The paper analyzes three flavon realizations—real, complex, and SUSY—and classifies the resulting non-Abelian walls (including -type, -type, and their complex variants , , ), their tensions, stabilities, and decay channels. It embeds these structures into explicit leptonic flavour models and discusses distinctive GW signatures, including multi-peak spectra from multi-scale wall collapse and string-wall networks. The cosmological implications are explored both in non-SUSY and SUSY contexts, with biases from explicit breaking or soft terms providing annihilation triggers and connecting GW signals to the underlying flavour sector.

Abstract

The spontaneous breaking of an 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 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 group symmetries. We investigate the properties of supersymmetric domain walls, and highlight the BPS walls. Finally we show how these results may be achieved in leptonic flavour models, with and without supersymmetry, and discuss their distinctive gravitational wave signatures.

Paper Structure

This paper contains 27 sections, 139 equations, 16 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Domain Walls from $A_4$ Symmetry
  3. Domain Walls from Real Scalar Fields
  4. Domain Walls from Complex Scalar Fields
  5. $\mathrm{S}^c$-type domain walls
  6. $\mathrm{T}^c$-type domain walls
  7. $\mathrm{T}'^c$-type domain walls
  8. Supersymmetric $A_4$ Domain Walls
  9. Simplest SUSY Domain Walls
  10. Domain Walls with R-symmetry
  11. Domain Walls from $A_4$ Lepton Flavour Models
  12. Real Flavon $A_4$ Models
  13. Complex flavon $A_4$ models
  14. SUSY flavon $A_4$ models
  15. Cosmological Solutions to the non-Abelian Domain Wall Problem
  16. ...and 12 more sections

Figures (16)

  • Figure 1: $\mathrm{S}$-type vacua (left panel) and $T_+$-type ($T_-$-type) vacua (right panel) in teal (purple) colour. The fields are rescaled as $\tilde{\phi}_i = \phi_i \sqrt{3g_1+2g_2}/\mu$ and $a=0.1$ is chosen as a benchmark case.
  • Figure 2: $Z_3$-preserving vacua of $A_4$ and three topologically different types of DWs. TI, TII and TIII DWs are given in the left, middle and right panels.
  • Figure 3: The left panel shows the domain wall solutions $i.e.$ solutions to EoM Eq. \ref{['eq:EoM']} between the vacua $u_1$ and $u_8$. The right panel shows the normalised energy density as a function of the normalised $z$ -coordinate for the different domain wall solutions. The red line marks $\Delta V$.
  • Figure 4: The ratio $R_\sigma=\sigma_\mathrm{1p}/\sigma_\mathrm{2p}$ for different $\beta$ and $a$. The benchmark points in Fig. \ref{['fig:path_and_en']} are marked out by the star and the triangle.
  • Figure 5: Relation between the vacua under $A_4$ group transformations and CP conjugation.
  • ...and 11 more figures