Models of Electroweak Interactions in Non-Commutative Geometry: A Comparison

Alain Connes' construction of the standard model is based on a generalized Dirac-Yukawa operator and the K-cycle (\HD ,D), with \HD a fermionic Hilbert space. If this construction is reformulated at the level of the differential algebra then a direct comparison with the alternative approach by the Marseille-Mainz group becomes possible. We do this for the case of the toy model based on the structure group $U(1)\times U(1)$ and for the $SU(2)\times U(1)$ of electroweak interactions. Connes' results are recovered without the somewhat disturbing $\gamma_{5}$-factors in the fermion mass terms and Yukawa couplings. We discuss both constructions in the same framework and, in particular, pinpoint the origin of the difference in the Higgs potential obtained by them.Comment: 9p, MZ-TH/93-2

Supersymmetry and Noncommutative Geometry

The purpose of this article is to apply the concept of the spectral triple, the starting point for the analysis of noncommutative spaces in the sense of A.~Connes, to the case where the algebra \cA contains both bosonic and fermionic degrees of freedom. The operator \cD of the spectral triple under consideration is the square root of the Dirac operator und thus the forms of the generalized differential algebra constructed out of the spectral triple are in a representation of the Lorentz group with integer spin if the form degree is even and they are in a representation with half-integer spin if the form degree is odd. However, we find that the 2-forms, obtained by squaring the connection, contains exactly the components of the vector multiplet representation of the supersymmetry algebra. This allows to construct an action for supersymmetric Yang-Mills theory in the framework of noncommutative geometry.Comment: 26pp., LaTe

Gauge parameter dependence in the background field gauge and the construction of an invariant charge

By using the enlarged BRS transformations we control the gauge parameter dependence of Green functions in the background field gauge. We show that it is unavoidable -- also if we consider the local Ward identity -- to introduce the normalization gauge parameter $\xi_o$, which enters the Green functions of higher orders similarly to the normalization point $\kappa$. The dependence of Green functions on $\xi_o$ is governed by a further partial differential equation. By modifying the Ward identity we are able to construct in 1-loop order a gauge parameter independent combination of 2-point vector and background vector functions. By explicit construction of the next orders we show that this combination can be used to construct a gauge parameter independent RG-invariant charge. However, it is seen that this RG-invariant charge does not satisfy the differential equation of the normalization gauge parameter $\xi_o$, and, hence, is not $\xi_o$-independent as required.Comment: 29 pages, LaTe

Connes' Model Building Kit

Alain Connes' applications of non-commutative geometry to interaction physics are described for the purpose of model building.Comment: 35 pages, LATeX, CPT-93/P.296

The Standard Model as a noncommutative geometry: the low energy regime

We render a thorough, physicist's account of the formulation of the Standard Model (SM) of particle physics within the framework of noncommutative differential geometry (NCG). We work in Minkowski spacetime rather than in Euclidean space. We lay the stress on the physical ideas both underlying and coming out of the noncommutative derivation of the SM, while we provide the necessary mathematical tools. Postdiction of most of the main characteristics of the SM is shown within the NCG framework. This framework, plus standard renormalization technique at the one-loop level, suggest that the Higgs and top masses should verify 1.3 m_top \lesssim m_H \lesssim 1.73 m_top.Comment: 44 pages, Plain TeX with AMS fonts, mass formulae readjusted, some references added, to appear in Physics Report

The Callan-Symanzik equation of the electroweak Standard Model and its 1-loop functions

We derive the Callan-Symanzik equation of the electroweak Standard Model in the QED-like on-shell parameterization. The various coefficient functions, the $\beta$-functions and anomalous dimensions, are determined in one-loop order in the most general linear gauge compatible with rigid symmetry. In this way the basic elements for a systematic investigation of higher-order leading logarithmic contributions in the Standard Model are provided. The one-loop $\beta$-function of the electromagnetic coupling turns out to be independent of mass ratios and it is QED-like in this sense. Besides the QED-contributions of fermions it contains non-abelian contributions from vectors and ghosts with negative sign, which overcompensate the contributions of the fermions if one restricts the latter to one fermion generation. We also compare our results with the symmetric theory and give relations between the $\beta$-functions of the spontaneously broken and the symmetric theory valid in one-loop order.Comment: 39 pages, LaTe