Narrow Results

In the presence of anomaly induced sphaleron process, only a B - L asymmetry can be partially converted to the baryon asymmetry while any B + L asymmetry would be completely erased. Thus in any successful baryogenesis theories, B - L is usually violated above the electroweak scale to explain the observed matter–antimatter asymmetry of the universe. However, if any lepton asymmetry is not affected by the sphaleron processes, a B - L conserved theory can still realize the baryogenesis. We present here an SU(5) GUT realization of this scenario, which naturally accommodates small masses of Dirac neutrinos.

It is investigated what constraints are imposed on the so-called yukawaon model when we regard quarks and leptons as of SU(5) (i = 1, 2, 3: index of U(3) family symmetry). In past yukawaon models, the effective Yukawa coupling constants are given by vacuum expectation values of fields Y_f ("yukawaons") as . In order to build a model without a cutoff scale Λ, vector-like fields are introduced in addition to the conventional quarks and leptons. The U(3) family symmetry is broken at 〈Y_f〉~10¹³GeV.

We associate the lepton–quark families with the vertices of the 4D polytopes 5-cell ${(0001)}_{A4}$ and the rectified 5-cell ${(0100)}_{A4}$ derived from the $SU(5)$ Coxeter–Dynkin diagram. The off-diagonal gauge bosons are associated with the root polytope ${(1001)}_{A4}$ whose facets are tetrahedra and the triangular prisms. The edge-vertex relations are interpreted as the $SU(5)$ charge conservation. The Dynkin diagram symmetry of the $SU(5)$ diagram can be interpreted as a kind of particle-antiparticle symmetry. The Voronoi cell of the root lattice consists of the union of the polytopes ${{{{(1000)}_{A4}+(0100)}_{A4}+(0010)}_{A4}+(0001)}_{A4}$ whose facets are 20 rhombohedra. We construct the Delone (Delaunay) cells of the root lattice as the alternating 5-cell and the rectified 5-cell, a kind of dual to the Voronoi cell. The vertices of the Delone cells closest to the origin consist of the root vectors representing the gauge bosons. The faces of the rhombohedra project onto the Coxeter plane as thick and thin rhombs leading to Penrose-like tiling of the plane which can be used for the description of the 5-fold symmetric quasicrystallography. The model can be extended to $SO(10)$ and even to $SO(11)$ by noting the Coxeter–Dynkin diagram embedding $A_4\subset D_5\subset B_5$. Another embedding can be made through the relation $A_4\subset D_5\subset E_6$ for more popular $GU{T}^{\prime }s$. Appendix A includes the quaternionic representations of the Coxeter–Weyl groups $W(A_4)\subset {W(H}_4)$ which can be obtained directly from ${W(E}_8)$ by projection. This leads to relations of the $SU(5)$ polytopes with the quasicrystallography in 4D and $E_8$ polytopes. Appendix B discusses the branching of the polytopes in terms of the irreducible representations of the Coxeter–Weyl group $W(A_4)\approx S_5$.