Twisted Spectral Triple for the Standard Model and Spontaneous Breaking of the Grand Symmetry

Abstract

Grand symmetry models in noncommutative geometry, characterized by a non-trivial action of functions on spinors, have been introduced to generate minimally (i.e. without adding new fermions) and in agreement with the first order condition an extra scalar field beyond the standard model, which both stabilizes the electroweak vacuum and makes the computation of the mass of the Higgs compatible with its experimental value. In this paper, we use a twist in the sense of Connes-Moscovici to cure a technical problem due to the non-trivial action on spinors, that is the appearance together with the extra scalar field of unbounded vectorial terms. The twist makes these terms bounded and -thanks to a twisted version of the first-order condition that we introduce here -also permits to understand the breaking to the standard model as a dynamical process induced by the spectral action, as conjectured in [24]. This is a spontaneous breaking from a pre-geometric Pati-Salam model to the almost-commutative geometry of the standard model, with two Higgs-like fields: scalar and vector.

Lorentz signature and twisted spectral triples

“…One then shows [25] that (A, H, D Aρ ; ρ) is a real twisted spectral triple, with the same real structure and KO-dimension as (A, H, D; ρ).…”

“…It relies on an enlargement of the Standard Model algebra, and ultimately allows one to obtain the field σ in agreement with the NCG principles, namely as an internal part of a connection. As shown in [25] however, in order to make the grand symmetry extension work, one is required to twist the noncommutative geometry of the Standard Model, in the sense of Connes-Moscovici [26]. Besides solving some technical difficulties, the twist also permits one to understand the breaking of the grand symmetry down to the Standard Model symmetries in a dynamical process induced by the spectral action.…”

“…Another condition that plays an important role is the order one condition [29], which for twisted spectral triples one writes as [25,30] […”

A new algebraic structure in the standard model of particle physics

“…(ii) It is interesting to consider how our formalism interacts with other recent interesting proposals for how to go beyond the traditional NCG formulation of the standard model (see e.g. [38][39][40][41][42][43][44][45][46][47]). (iii) We believe that there are interesting issues to be sorted out involving the relation between euclidean and lorentzian signature -see the comment in subsection 5.2.…”

Spectral Triples in Particle Physics

“…Nevertheless, the idea of mixing spinorial and gauge indices is important one for study possible extensions of SM. The problem of symmetry breaking from Grand Symmetry into the SM can be solved by using twisted version of Dirac operator as was proposed in [12]. Nowadays, we have more general notion of twisted spectral triples, which can be use as an alternative approach to study particle physic models by the consideration of so-called Twisted Standard Model [16].…”