Electroweak Symmetry Breaking as a Consequence of Compact Dimensions

It has been shown recently that the Higgs doublet may be composite, with the left-handed top-bottom doublet and a new right-handed anti-quark as constituents bound by some four-quark operators with non-perturbative coefficients. I show that these operators are naturally induced if there are extra spatial dimensions with a compactification scale in the multi-TeV range. The Higgs compositness is due mainly to the Kaluza-Klein modes of the gluons, while flavor symmetry breaking may be provided by various fields propagating in the compact dimensions. I comment briefly on the embedding of this scenario in string theory. ∗e-mail address: [email protected] Recently it has been shown that a bound state formed of the left-handed top-bottom quark doublet and a new right-handed anti-quark can play viably the role of the Standard Model (SM) Higgs doublet [1, 2]. The only ingredients necessary in this framework are some four-quark interactions, suppressed by a multi-TeV scale M1. One is then led to ask what is the origin of these non-renormalizable interactions. The traditional answer is that they are produced by the exchange of some heavy gauge bosons associated with the breaking of a larger gauge group down to the QCD group. The simplest choice of this sort is the topcolor [3] scheme: SU(Nc)1 × SU(Nc)2 → SU(Nc)C, where Nc = 3, the third generation quarks are charged under SU(Nc)1, and the first two generations are charged under SU(Nc)2. This choice has several nice features. For example, the asymptotic freedom of the topcolor gauge group allows the heavy gauge bosons to be strongly coupled at the scale M1 without problems with a Landau pole. Also, it is technically convenient because it allows the use of the large-Nc expansion. The drawback is that one has to include additional structures in order to spontaneously break topcolor. Moreover, this embedding of the SU(Nc)C color group may be seen as artificial. The embedding can be improved by allowing all quarks to transform only under SU(Nc)1 [4] which is then embedded in a gauged flavor or family symmetry [5, 6]. However, such an extension requires a more complicated topcolor breaking sector. It is therefore legitimate to ask whether one can avoid the extension of the gauge group while inducing the desired four-quark operators and retaining the nice features of topcolor ? In this letter I point out that the required four-quark operators are naturally induced if the SM gauge fields propagate not only in the usual four-dimensional Minkowski space-time but also in a compact space. The first step in showing this is to recall that the existence of the extra dimensions is manifested within the 4-dimensional space-time through a tower of Kaluza-Klein (KK) modes associated with each of the fields propagating in the compact manifold (see e.g., [7, 8]). The KK modes of a massless gauge boson have masses between M1 = R −1 max , where Rmax is the largest compactification radius, and the fundamental scale associated with quantum gravity, M∗. Assuming for simplicity that the fermions propagate at the fixed points of an orbifold, so that they have only zero-modes, it follows that the couplings of the KK modes of the gauge bosons to the quarks and leptons are identical (up to an overall normalization) with those of the SM gauge bosons. Thus, the KK excitations of