Extended Higgs sector on noncommutative geometry and nonassociative lepton chiral symmetry

Abstract

ABSTRACT: Connes’ noncommutative geometry (NCG) provides a generalization of Riemannian geometry by turning our attention away from manifolds to instead focus on the algebra of functions defined on them. In this setting, the main object is the so-called ‘spectral triple’ A, H, D, which consists of an algebra A and a Dirac operator D represented on a Hilbert space H. Furthermore, this framework enjoys a great physical interest since it offers a geometric reinterpretation for the standard model (SM) of particle physics coupled with gravity. In particular, in this picture, there is an underlying finite space attached to each space-time point where the Higgs boson appears as the ‘connection’ associated with this new ‘dimensionless’ space. However, after the discovery of the Higgs boson in 2012, the spectral approach to the SM revealed an inconsistent value for the Higgs boson mass. In this thesis, we build particle physics models that might meaningfully contribute to the Higgs sector in the noncommutative geometry standard model (NCG SM). In chapter 3, we show that there are as many Higgs doublets as Yukawa couplings are in the fluctuated Dirac operator. Then, we construct the two Higgs doublet model (2HDM) in the NCG context. We deduce the boundary conditions for the renormalization group equations (RGEs) of the scalar couplings to calculate the mass spectrum for each one of these models. In particular, we study the parameter space of the required couplings to have a phenomenologically viable noncommutative geometry two Higgs doublet model (NCG 2HDM) type II. In chapter 4, we focus on the non-associative ‘Bison algebra’ B2, which has the automorphisms group SU(2) U(1). We identify a natural representation for the SM leptons together with an exotic fermion degree of freedom. Then, we define a Hermitian Dirac operator containing the minimal Yukawa interaction. We end up with a ‘twisted’ spectral triple describing the electroweak theory for the SM lepton sector.