We present the first example of a grand unified theory (GUT) with a modular symmetry interpreted as a family symmetry. The theory is based on supersymmetric <math><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mn>5</mn><mo>)</mo></mrow></math> in 6d, where the two extra dimensions are compactified on a <math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math> orbifold. We have shown that, if there is a finite modular symmetry, then it can only be <math><msub><mi>A</mi><mn>4</mn></msub></math> with an (infinite) discrete choice of moduli, where we focus on <math><mrow><mi>τ</mi><mo>=</mo><mi>ω</mi><mo>=</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mn>2</mn><mi>π</mi><mo>/</mo><mn>3</mn></mrow></msup></mrow></math>, the unique solution with <math><mrow><mo>|</mo><mi>τ</mi><mo>|</mo><mo>=</mo><mn>1</mn></mrow></math>. The fields on the branes respect a generalized <math><mrow><mi>C</mi><mi>P</mi></mrow></math> and flavor symmetry <math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>⋉</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math> which is isomorphic to <math><msub><mi>S</mi><mn>4</mn></msub></math> which leads to an effective <math><mi>μ</mi><mo>−</mo><mi>τ</mi></math> reflection symmetry at low energies, implying maximal atmospheric mixing and maximal leptonic <math><mrow><mi>C</mi><mi>P</mi></mrow></math> violation. We construct an explicit model along these lines with two triplet flavons in the bulk, whose vacuum alignments are determined by orbifold boundary conditions, analogous to those used for <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>5</mn><mo>)</mo></math> breaking with doublet-triplet splitting. There are two right-handed neutrinos on the branes whose Yukawa couplings are determined by modular weights. The charged lepton and down-type quarks have diagonal and hierarchical Yukawa matrices, with quark mixing due to a hierarchical up-quark Yukawa matrix.

Mexico

Francisco J. de Anda

United Kingdom

Stephen F. King

Elena Perdomo

<math><mi>S</mi><mi>U</mi><mo>(</mo><mn>5</mn><mo>)</mo></math> grand unified theory with <math><msub><mi>A</mi><mn>4</mn></msub></math> modular symmetry

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.

Francisco Anda

Stephen King

Germany

Patrick Vaudrevange

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

We study a version of Starobinsky-like inflation in no-scale supergravity (SUGRA) where a Polonyi term in the hidden sector breaks supersymmetry (SUSY) after inflation, providing a link between the gravitino mass and inflation. We extend the theory to the visible sector and calculate the soft-SUSY breaking parameters depending on the modular weights in the superpotential and choice of Kähler potential. We are led to either no-scale SUGRA or pure gravity mediated SUSY breaking patterns, but with inflationary constraints on the Polonyi term setting a strict upper bound on the gravitino mass m 3/2 &lt; 103 TeV. Since gaugino masses are significantly lighter than m 3/2, this suggests that SUSY may be discovered at the LHC or FCC.

Starobinsky-like inflation and soft-SUSY breaking

Here we correct some typesetting errors in affiliations.

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

We discuss a flavourful Z′ portal model with a coupling to fourth-family singlet Dirac neutrino dark matter. In the absence of mixing, the Z′ is fermiophobic, having no couplings to the three chiral families, but does couple to a fourth vector-like family. Due to mixing effects, the Z′ gets induced couplings to second family left-handed lepton doublets and third family left-handed quark doublets. This model can simultaneously account for the measured B-decay ratios R  K  and R  K*  and for the observed relic abundance of dark matter. We identify the parameter space where this explanation is consistent with existing experimental constraints from dark matter direct and indirect detection, LHC searches, and precision measurements of flavour mixing and neutrino processes.

France

Adam Falkowski

Mathias Pierre

Flavourful Z′ portal for vector-like neutrino dark matter and  <math><msub><mi>R</mi><msup><mi>K</mi><mfenced><mo>*</mo></mfenced></msup></msub></math> $$ {R}_{K^{\left(*\right)}} $$

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.