Modular flavour symmetry and orbifolds

Modular flavour symmetry and orbifolds

We develop a bottom-up approach to flavour models which combine modular symmetry with orbifold constructions. We first consider a 6d orbifold $\mathbb{T}^2/\mathbb{Z}_N$, with a single torus defined by one complex coordinate $z$ and a single modulus field $\tau$, playing the role of a flavon transforming under a finite modular symmetry. We then consider 10d orbifolds with three factorizable tori, each defined by one complex coordinate $z_i$ and involving the three moduli fields $\tau_1, \tau_2, \tau_3$ transforming under three finite modular groups. Assuming supersymmetry, consistent with the holomorphicity requirement, we consider all 10d orbifolds of the form $(\mathbb{T}^2)^3/(\mathbb{Z}_N\times\mathbb{Z}_M)$, and list those which have fixed values of the moduli fields (up to an integer). The key advantage of such 10d orbifold models over 4d models is that the values of the moduli are not completely free but are constrained by geometry and symmetry. To illustrate the approach we discuss a 10d modular seesaw model with $S_4^3$ modular symmetry based on $(\mathbb{T}^2)^3/(\mathbb{Z}_4\times\mathbb{Z}_2)$ where $\tau_1=i,\ \tau_2=i+2$ are constrained by the orbifold, while $\tau_3=\omega$ is determined by imposing a further remnant $S_4$ flavour symmetry, leading to a highly predictive example in the class CSD$(n)$ with $n=1-\sqrt{6}$.

hep-ph

Anda, Francisco J. de

dbb27610-c99c-4727-9158-5a2c155f61de

King, Stephen F.

f8c616b7-0336-4046-a943-700af83a1538

Anda, Francisco J. de

dbb27610-c99c-4727-9158-5a2c155f61de

King, Stephen F.

f8c616b7-0336-4046-a943-700af83a1538

[Unknown type: UNSPECIFIED]

## Abstract

We develop a bottom-up approach to flavour models which combine modular symmetry with orbifold constructions. We first consider a 6d orbifold $\mathbb{T}^2/\mathbb{Z}_N$, with a single torus defined by one complex coordinate $z$ and a single modulus field $\tau$, playing the role of a flavon transforming under a finite modular symmetry. We then consider 10d orbifolds with three factorizable tori, each defined by one complex coordinate $z_i$ and involving the three moduli fields $\tau_1, \tau_2, \tau_3$ transforming under three finite modular groups. Assuming supersymmetry, consistent with the holomorphicity requirement, we consider all 10d orbifolds of the form $(\mathbb{T}^2)^3/(\mathbb{Z}_N\times\mathbb{Z}_M)$, and list those which have fixed values of the moduli fields (up to an integer). The key advantage of such 10d orbifold models over 4d models is that the values of the moduli are not completely free but are constrained by geometry and symmetry. To illustrate the approach we discuss a 10d modular seesaw model with $S_4^3$ modular symmetry based on $(\mathbb{T}^2)^3/(\mathbb{Z}_4\times\mathbb{Z}_2)$ where $\tau_1=i,\ \tau_2=i+2$ are constrained by the orbifold, while $\tau_3=\omega$ is determined by imposing a further remnant $S_4$ flavour symmetry, leading to a highly predictive example in the class CSD$(n)$ with $n=1-\sqrt{6}$.

Text

** 2304.05958v2
- Author's Original**
## More information

Submitted date: 12 April 2023

Additional Information:
20 pages, 3 figures. v2: corrected typos

Keywords:
hep-ph

## Identifiers

Local EPrints ID: 478120

URI: http://eprints.soton.ac.uk/id/eprint/478120

PURE UUID: 0701298e-df7f-406a-a88d-4469e412c448

## Catalogue record

Date deposited: 22 Jun 2023 16:31

Last modified: 17 Mar 2024 01:58

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## Contributors

Author:
Francisco J. de Anda

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