Spontaneous symmetry breaking and masses numerical results in Doplicher-Fredenhagen-Roberts noncommutative space-time

Abstract

With the elements of the Doplicher, Fredenhagen and Roberts (DFR) noncommutative formalism, we have constructed the standard electroweak model. We have introduced the spontaneous symmetry breaking and the hypercharge in DFR framework. The electroweak symmetry breaking was analyzed and the masses of the new bosons were computed.PACS numbers: 11.15.-q; 11.10.Ef; 11.10.Nx Keywords: DFR noncommutativity, electroweak standard modelThe fact that we have infinities that destroy the final results of several calculations in QFT have motivated theoretical physicists to ask if a continuum space-time would be really necessary. One of the possible solutions would be to create a discrete space-time with a noncommutative (NC) algebra, where the position coordinates would be promoted to operatorsX µ (µ = 0, 1, 2, 3) and they must satisfy commutation relationswhere ℓ is a length parameter, θ µν is an antisymmetric constant matrix and1 1 is the identity operator. In this way, we would have a kind of fuzzy space-time where, from this commutator, we have an uncertainty in the position coordinate. In order to put these ideas together, Snyder [1] have published the first work that considers the space-time as being NC. However, the frustrated result have doomed Snyder's NC theory to years of ostracism [2]. After the relevant result that the algebra obtained from string theory embedded in a magnetic background is NC, a new flame concerning noncommutativity (NCY) was rekindle [3]. One of the paths (the most famous at least) of introducing NCY is through the MoyalWeyl product where the NC parameter, i.e. θ µν , is an antisymmetric constant matrix, namely,However, at superior orders of calculations, the MoyalWeyl product turns out to be highly nonlocal. This fact forced us to work with low orders in θ µν . Although it keeps the translational invariance, the Lorentz symmetry is not preserved [4]. For instance, concerning the case of the hydrogen atom, it breaks the rotational symmetry of the model, which removes the degeneracy of the energy levels [5]. Other subjects where the objective is * Electronic address: mariojr@ufrrj.br † Electronic address: evertonabreu@ufrrj.br to introduce NC effects in gravity [6], in anyon models [7] and symmetries [8]. NCY through path integrals and coherent states was devised in [9]. For more generalized NC issues and reviews, the interested reader can look at [10] and the references therein.One way to work with the NCY was introduced by Doplicher, Fredenhagen and Roberts (DFR). They have considered the parameter θ µν as an ordinary coordinate of the system [11,12]. This extended and new NC spacetime has ten dimensions: four relative to the Minkowski space-time and six relative to θ-space. Recently, in [13][14][15] it was demonstrated that the DFR formalism have in fact a canonical momentum associated with θ µν [14,16,17] (for a review, [18]). The DFR framework is characterized by a field theory constructed in a spacetime with extra-dimensions (4 + 6), and which does not need necessarily the presence of a le...

Faddeev–Jackiw approach of noncommutative space–time electromagnetic theories

“…where these commutators indicate operators in Snyder's formulation and the θ-parameter is constant, which is known as the canonical NCY. There are other approaches where the θ-parameter is not constant [10,11], something like θ = θ(x) or θ is a coordinate of the space-time [10,12]. Notice that we have talked only about NC formalism concerning the θ-parameter.…”

Strong interaction model in DFR noncommutative space–time

“…To get it we use the mass of G µ boson so obtained in the NC electroweak model [20] m G µ = 770 GeV ,…”

“…The immediate application is the version of Glashow-Salam-Weinberg model by the symmetry U ⋆ (2)×U ⋆ (1)×U ⋆ (2)×U ⋆ (1) whose the extended Higgs sector gives masses for the new bosons. We estimated these masses using some experimental constraints at the Z 0 mass, for more details, see [20]. In this paper, we discuss with more details the extended Yang-Mills symmetry U ⋆ (N) × U ⋆ (N), and as application, we propose the unification of Strong Interaction with the electromagnetism…”

“…It has revealed that at the one loop order, the ultraviolet/infrared divergence mixing phenomena does not emerge in the DFR scalar model ⋆φ 4 , for more details, see [19]. Posteriorly, the Field Theory formulation has inspired to study the gauge symmetry in the DFR context when the Dirac/scalar actions are putted under local symmetry [20]. Following these steps, the composed symmetry Higgs sector gives masses for the new bosons.…”

The standard electroweak model in the noncommutative DFR space–time

“…In the last few years, we have been concentrated in the construction of a NC QFT concerning the DFR formalism [22][23][24][25]. In this work we, following this DFR QFT quest, have constructed the DFR basis for a standard model after the construction of a DFR metric tensor.…”