Constraints on the Higgs and Top Quark Masses From Effective Potential and Non-Commutative Geometry

We consider the standard model in the formulationo of non-commutative geometry, for a Euclidean space-time consisting of two copies. The electroweak scale is set by the vacuum expectation value of a scalar field and is undetermined at the classical level. By adding the Coleman-Weinberg effective potential, that scale turns out to be fixed. Provided that the renormalized form of the Lagrangian maintains the vanishing of the cosmological constant, we show that the only solutions for the minimization equations of the total potential occur in the narrow band 146.2 ≤ mt ≤ 147.4 Gev for the top quark mass, with the corresponding Higgs mass 117.3 ≤ mH ≤ 142.6 Gev. * Supported in part by the Swiss National Foundation (SNF) † Permanent address: Theoretische Physik, ETH, CH 8093 Zürich Switzerland Connes’ [1] framework of non-commutative geometry provides a geometrical interpretation of the Higgs field necessary to break the gauge symmetry spontaneously. The structure of space-time is taken to be a product of a continuous four-dimensional Riemannian manifold times a discrete set of two points. For such a structure of spacetime the usual methods of differential geometry fail and must be replaced with the more general framework of non-commutative geometry. Using the non-commutative setting, Connes and Lott [2] recovered the standard model with all its parameters, at the classical level. At present, it is not known how to quantize the non-commutative action directly. We are left only with the possibility of quantizing the resulting theory in the usual way. The quantum corrections are then given by familiar expressions. There is, however, one important difference between this approach and the standard analysis connected with the gravitational effects of the discrete geometry. This have, under certain conditions, very surprising consequences. Explaining and exploiting these effects is the main concern of this note. Geometrically, the distance between the two copies of the four-dimensional Minkowski space is the inverse of the electoweak scale. At smaller scales the flat manifold is curved and the distance between the two copies becomes a dynamical scalar field [3]. The leptonic Dirac operator associated with Connes-Lott space takes the form Dl = ( γea(∂μ + . . .)⊗ 12 ⊗ 13 γ5e ⊗M12 ⊗ k γ5e −κσ ⊗M∗ 12 ⊗ k γea(∂μ + . . .)⊗ 13 )