ar X iv : h ep - p h / 95 09 43 4 v 1 2 O ct 1 99 5 KANAZAWA - 95 - 15 August 1995 Gauge - Yukawa Unification †

Gauge-Yukawa Unification (GYU) yields a functional relation among the gauge and Yukawa couplings, and it may follow from the usual Grand Unification if it is supplemented with some additional principles. Postulating the principles of finiteness and reduction of couplings, we have achieved Gauge-Yukawa Unification in various supersymmetric unified models leading, among other things, to interesting predictions on the top quark mass. † Presented by G. Zoupanos at SUSY 95, Ecole Polytechnique, Palaiseau, France, 11-18 May, 1995, to appear in the proceedings. Partially supported by the C.E.U. projects (SC1-CT91-0729; CHRX-CT93-0319). The traditional way of reducing the number of the free parameters of the standard model (SM) is to require that the theory is more symmetric at higher energies. This approach has been applied, e.g., in GUTs with a certain success in the gauge and independently in the Yukawa sectors of the theory. However, this attractive principle has its limitation as it is well known that increasing the gauge symmetry of a GUT (e.g., SO(10), E6, E7, E8 ) one does not necessarily increase the predictive power of the theory. This is because, to construct realistic models, one has to understand the breaking of these symmetries, which requires introducing additional free parameters in general. Alternatively, we suggest [1]-[3] that a natural gradual extension of the GUT idea, in prospect of increasing the predictability of the low energy parameters of the theory, is to attempt to relate the couplings of the gauge and Yukawa sectors, i.e., to achieve Gauge-Yukawa Unification (GYU). Searching for a symmetry that would provide GYU, one is led to consider N = 2 supersymmetric theories [4], which however proved to have more serious phenomenological problems than the SM. The last comment holds also for superstring theories and composite models which could in principle lead to relations among the gauge and Yukawa couplings. In our recent studies [1]-[3], we have considered the GYU which is based on the principles of reduction of couplings [7, 8, 2, 3] and also finiteness [9, 12, 1]. 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 [12]. 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, 2 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 [7], βg dgi dg = βi , i = 1, · · · , N , (2) where g and βg are the primary coupling and its β-function, and i does not include it. Using all the (N − 1)Φ’s to impose RGI relations, one can in principle express all the couplings in terms of a single coupling g. The complete reduction, which formally preserve perturbative renormalizability, can be achieved by demanding power series solution