ar X iv : h ep - t h / 95 06 09 2 v 1 1 4 Ju n 19 95 KANAZAWA - 95 - 05 May 1995 Gauge - Yukawa Unification and The Top Quark Mass †

The principles of finiteness and reduction of couplings can be applied to achieve GaugeYukawa Unification. It is found that the observed top-bottom hierarchy and the top quark mass naturally follow if there exists Gauge-Yukawa Unification which is a simple functional relation among the gauge coupling and the Yukawa couplings of the third generation in various susy unified gauge models. We briefly outline the basic idea of these principles and present the main results of the Gauge-Yukawa Unified models that have recently been studied in detail. † Presented by G. Zoupanos at 30th Recontres de Moriond on Electroweak Interactions and Unified Theories, Les Arcs, France, March 11-18, 1995, to appear in the proceedings. Partially supported by the C.E.U. projects (SC1-CT91-0729; CHRX-CT93-0319). Why is the top quark so heavy, and who orders the top-bottom hierarchy? These are the questions to which we have recently addressed ourselves . Obviously, these questions cannot be answered within the framework of the traditional GUT idea and new proposals are required going beyond GUTs. In a series of our resent studies , we have found that in a class of susy unified gauge models the top-bottom hierarchy as well as the top mass, consistent with the present experimental knowledge , can be predicted, if the Yukawa couplings of the third generation are related in a certain way to the gauge couplings of the standard models at the unification scale–very similarly to the way the hierarchy of the gauge couplings follows in GUTs . This observation might indicate that Gauge-Yukawa-Unification (GYU) has a realistic meaning, as far as the Yukawa couplings of the third generation are concerned. In these studies , we have considered the GYU which is based on the principles of reduction of couplings 6),7),2),3) and also finiteness . These principles, which are formulated in perturbation theory, are not explicit symmetry principles, although they might imply symmetries. The former principle is based on the existence of renormalization group invariant (RGI) relations among couplings which preserve perturbative renormalizability. Similarly, the latter one is based on the fact that it is possible to find RGI relations among couplings that keep finiteness in perturbation theory, even to all orders . Applying these principles, one can relate the gauge and Yukawa couplings without introducing necessarily a symmetry, thereby improving the predictive power of a model. In what follows, we briefly outline the basic tool of this GYU scheme and its application to various models. A RGI relation among couplings can be expressed in an implicit form Φ(g1, · · · , gN) = 0 , (1) which has to satisfy the partial differential equation (PDE) μ dΦ/dμ = ∑N i=1 βi ∂Φ/∂gi = 0, where βi is the β-function of gi. There exist (N − 1) independent Φ’s, and finding the complete set of these solutions is equivalent to solve the so-called reduction equations , βg dgi dg = βi , i = 1, · · · , N , (2)