Planck-scale unification and dynamical symmetry breaking.

We explore the possibility of unification of gauge couplings near the Planck scale in models of extended technicolor. We observe that models of the form G×SU(3)c×SU(2)L×U(1)Y cannot be realized, due to the presence of massless neutral Goldstone bosons (axions) and light charged pseudo-Goldstone bosons; thus, unification of the known forces near the Planck scale cannot be achieved. The next simplest possibility, G × SU(4)PS × SU(2)L × U(1)T3R , cannot lead to unification of the Pati-Salam and weak gauge groups near the Planck scale. However, superstring theory provides relations between couplings at the Planck scale without the need for an underlying grand-unified gauge group, which allows unification of the SU(4)PS and SU(2)L couplings. ∗ Present address: Department of Physics, University of Illinois, 1110 West Green St., Urbana, IL 61801 The standard model of the strong and electroweak interactions is based on the gauge group SU(3)c×SU(2)L×U(1)Y , with SU(2)L×U(1)Y spontaneously broken to U(1)EM at the weak scale, ( √ 2GF ) −1/2 = 246 GeV. Although the coupling strengths of the three gauge forces are apparently unrelated at ordinary energies, it is attractive to hypothesize that, as a result of their evolution, they are related at some higher energy [1]. One realization of this conjecture is grand unification, in which the standard gauge group is embedded in a larger gauge group, which is spontaneously broken at one or more scales above the weak scale [2]. The simplest example is minimal SU(5) [2], which nearly succeeds in unifying the known gauge forces at a scale of around 10 GeV [1], far above the weak scale. A well-known difficulty with attempts at grand unification is the enormous disparity between the weak scale and the grand-unified scale. It is not natural for such a hierarchy of scales to occur if the gauge symmetries are broken by the vacuumexpectation values of fundamental scalar fields [1, 3]. Furthermore, a hierarchy based on fundamental scalar fields is unstable due to quadratic divergences in the renormalization of the parameters of the scalar-field potential [3]. A generic means to stabilize this hierarchy is to invoke low-energy supersymmetry (SUSY) [4]. Supersymmetry itself must be softly broken, but at a scale not far above the weak scale if it is to protect the hierarchy. The introduction of supersymmetry requires the existence of the superpartners of the standard particles, with masses of order the SUSY breaking scale, as well as an additional Higgs doublet and its superpartner. These additional particles influence the evolution of the three gauge couplings [5]. As is well known, minimal SUSY SU(5) succeeds in unifying the three known gauge forces, at a scale of about 10 GeV [6]. This is often considered to be indirect evidence of the fundamental correctness of both SU(5) grand unification and supersymmetry. The other known force, gravity, is not a gauge interaction. At ordinary energies, gravity is described by a classical field theory. The scale at which quantum gravity becomes relevant is (8πGN) −1/2 ≈ 2.4 × 10 GeV, which we will refer to as the Planck scale. It is compelling to hypothesize that this is a fundamental scale of physics, and that unification of the four known forces should occur there. The fact that the minimal SU(5) grand-unified scale is close to the Planck scale also suggests that gravity and unification are related [1]. Despite the success of the minimal SUSY SU(5) grand-unified scenario, we wish to explore models of Planck-scale unification based on dynamical symmetry breaking [3, 7, 8]. There are several motivations for doing so. First, dynamical symmetry breaking is the only other known generic mechanism besides supersymmetry to maintain the hierarchy between the Planck scale (or grand-unified scale) and the weak scale [1, 3]. Thus it is the only realistic alternative to the SUSY grand-unified sceThe energy G −1/2 N = 1.22 × 10 GeV is usually called the Planck scale. The factor 8π comes from the Einstein field equation, G = 8πGNT μν .