We develop a bottom-up approach to flavour models which combine modular symmetry with orbifold constructions. We first consider a 6d orbifold 𝕋2 /ℤ N , with a single torus defined by one complex coordinate z and a single modulus field τ, 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 τ 1 , τ 2 , τ 3 transforming under three finite modular groups. Assuming supersymmetry, consistent with the holomorphicity requirement, we consider all 10d orbifolds of the form (𝕋2)3 /(ℤ N  × 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   <math> <msubsup> <mi>S</mi> <mn>4</mn> <mn>3</mn> </msubsup> </math>  $$ {S}_4^3 $$  modular symmetry based on (𝕋2)3 /(ℤ4 × ℤ2) where τ 1 = i, τ 2 = i + 2 are constrained by the orbifold, while τ 3 = ω 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 −   <math> <msqrt> <mn>6</mn> </msqrt> </math>  $$ \sqrt{6} $$ .

Mexico

Francisco Anda

United Kingdom

Stephen King

Modular flavour symmetry and orbifolds

We propose a theory of quark and lepton mass and mixing with non-universal Z′ couplings based on a 5d Standard Model with quarks and leptons transforming as triplets under a new gauged SO(3) isospin. In the 4d effective theory, the SO(3) isospin is broken to U(1)′, through a S 1/(ℤ2 ×   <math> <msubsup> <mi>ℤ</mi> <mn>2</mn> <mi>t</mi> </msubsup> </math>  $$ {\mathrm{\mathbb{Z}}}_2^t $$ ) orbifold, then subsequently dynamically broken, resulting in a massive Z′. Quarks and leptons in the 5d bulk appear as massless modes, with zero Yukawa couplings to the Higgs on the brane, and zero couplings to Z′, at leading order, due to the U(1)′ symmetry. However, after the U(1)′ breaking, both Yukawa couplings and non-universal Z′ couplings are generated by heavy Kaluza-Klein exchanges. Hierarchical quark and lepton masses result from a hierarchy of 5d Dirac fermion masses. Neutrino mass and mixing arises from a novel type Ib seesaw mechanism, mediated by Kaluza-Klein Dirac neutrinos. The non-universal Z′ couplings may contribute to semi-leptonic B decay ratios which violate μ − e universality. In this model such couplings are related to the corresponding quark and lepton effective Yukawa couplings.

Quark and lepton mass and mixing with non-universal Z′ from a 5d Standard Model with gauged SO(3)

We propose a flavour theory in which the family symmetry results naturally from a six-dimensional orbifold compactification. “Diracness” of neutrinos is a consequence of the spacetime dimensionality, and the fact that right-handed neutrinos live in the bulk. Dark matter is incorporated in a scotogenic way, as a result of an auxiliary ℤ3 symmetry, and its stability is associated to the conservation of a “dark parity” symmetry. The model leads naturally to a “golden” quark-lepton mass relation.

Switzerland

France

Ignatios Antoniadis

Spain

José Valle

Carlos Vaquera-Araujo

Scotogenic dark matter in an orbifold theory of flavor

In this paper we propose the idea that flavons can emerge from extra dimensional gauge fields, referred to as gauge-flavon unification (GFU) analogous to gauge-Higgs unification (GHU). We assume that there is a gauged family symmetry in extra dimensions and that the flavons are the extra dimensional components of the gauge field. This provides a simple mechanism to align the VEVs of the flavons through a combination of Wilson lines and orbifold symmetry breaking. We present some simple 5d examples of GFU based on SO(3) and SU(4) gauged family symmetry, the latter case yielding SU(3) × U(1) gauged family symmetry in 4d, broken by triplet and antitriplet flavons, with effective couplings to fermions. We also present a general formalism for Wilson lines and orbifolds, in any number of dimensions, including non-commutative aspects Wilson lines, which may be useful for aligning additional flavons as required for realistic models.

Alfredo Aranda

Gauge-flavon unification

We systematically develop the formalism necessary for ensuring that boundary conditions of flavon fields in extra dimensions are consistent with heterotic string theory. Having developed a set of consistency conditions on the boundary conditions, we explore a series of examples of orbifolds in various dimensions to see which ones can satisfy them. In addition we impose the further phenomenological requirements of having non-trivial flavon vacuum alignments and also of having quarks and leptons located appropriately in extra dimensions. The minimal successful case seems to be a 10d theory with a SU(3)fl gauged flavour symmetry, where the six-dimensional torus is compactified on a 𝕋6 /∆(54) orbifold. We construct a realistic SU(5) grand unified theory along these lines, leading to tribimaximal-reactor lepton mixing, which we show to be consistent with current neutrino data.

Elena Perdomo

Germany

Patrick Vaudrevange

Flavon alignments from orbifolding: SU(5) × SU(3) model with 𝕋6 /∆(54)

Here we correct some typesetting errors in affiliations.

Erratum to: SO(10) × S 4 grand unified theory of flavour and leptogenesis

We discuss a simple and elegant SU(3) × SO(10) family unified gauge theory in 6d compactified on a torus with the orbifold T 2/Z  2 3  and supplemented by a Z 6 × Z 3 discrete symmetry. The orbifold boundary conditions generate all the desired SU(3) breaking vacuum alignments, including the (0, 1, −1) and (1, 3, −1) alignments of the Littlest Seesaw model for atmospheric and solar neutrino mixing, as well as the usual SO(10) breaking with doublet-triplet splitting. The absence of driving and messenger fields considerably simplifies the field content of the model. It naturally explains why there are three families of quarks and leptons, and accounts for all their masses, mixing angles and CP phases via rather elegant looking Yukawa and Majorana matrices in the theory basis. The resulting model controls proton decay and allows successful Leptogenesis.

SU(3) × SO(10) in 6d

We propose a 6d model with a SUSY SU(5) gauge symmetry. After compactification, it explains the origin of the S 4 Family Symmetry with CSD3 vacuum alignment, as well as SU(5) breaking with doublet-triplet splitting. The model naturally accounts for all quark and lepton (including neutrino) masses and mixings, incorporating the highly predictive Littlest Seesaw structure. It spontaneously breaks CP symmetry, resulting in successful CP violation in the quark and lepton sectors, while solving the Strong CP problem. It also explains the Baryon Asymmetry of the Universe (BAU) through leptogenesis, with the leptogenesis phase directly linked to the Dirac and Majorana phases.

An S 4 × SU(5) SUSY GUT of flavour in 6d

We propose a Grand Unified Theory of Flavour, based on SO(10) together with a non-Abelian discrete group S 4, under which the unified three quark and lepton 16-plets are unified into a single triplet 3′. The model involves a further discrete group ℤ  4  R   × ℤ  4 3  which controls the Higgs and flavon symmetry breaking sectors. The CSD2 flavon vacuum alignment is discussed, along with the GUT breaking potential and the doublet-triplet splitting, and proton decay is shown to be under control. The Yukawa matrices are derived in detail, from renormalisable diagrams, and neutrino masses emerge from the type I seesaw mechanism. A full numerical fit is performed with 15 input parameters generating 19 presently constrained observables, taking into account supersymmetry threshold corrections. The model predicts a normal neutrino mass ordering with a CP oscillation phase of 260°, an atmospheric angle in the first octant and neutrinoless double beta decay with m  ββ  = 11 meV. We discuss N 2 leptogenesis, which fixes the second right-handed neutrino mass to be M 2 ≃ 2 × 1011 GeV, in the natural range predicted by the model.

SO(10) × S 4 grand unified theory of flavour and leptogenesis

We propose a natural S 4 × SO(10) supersymmetric grand unified theory of flavour with an auxiliary ℤ 4 2 × ℤ 4 R $$ {\mathbb{Z}}_4^2\times {\mathbb{Z}}_4^R $$ symmetry, based on small Higgs representations (nothing larger than an adjoint) and hence a type-I seesaw mechanism. The Yukawa structure of all fermions is determined by the hierarchical vacuum expectation values of three S 4 triplet flavons, with CSD3 vacuum alignments, where up-type quarks and neutrinos couple to one Higgs 10 , and the down-type quarks and charged leptons couple to a second Higgs 10 . The Yukawa matrices are obtained from sums of low-rank matrices, where each matrix in the sum naturally accounts for the mass of a particular family, as in sequential dominance in the neutrino sector, which predicts a normal neutrino mass hierarchy. The model accurately fits all available quark and lepton data, with predictions for the leptonic CP phase in 95% credible intervals given by 281° &lt; δ ℓ &lt; 308° and 225° &lt; δ ℓ &lt; 253°. The model reduces to the MSSM, with the two Higgs doublets emerging from the two Higgs 10 s without mixing, and we demonstrate how a μ term of O T e V $$ \mathcal{O}\left(\mathrm{T}\mathrm{e}\mathrm{V}\right) $$ can be realised, as well as doublet-triplet splitting, with Planck scale operators controlled by symmetry, leading to acceptable proton decay.