Softly Broken Finite Supersymmetric Grand Unified Theory

Abstract

In the context of the standard SUSY GUT scenario, we present a detailed analysis of the softly broken finite supersymmetric grand unified theory. The model, albeit non-minimal, remains very rigid due to the requirement of finiteness. It is based on the $SU(5)$ gauge group and is UV finite to all orders of perturbation theory. It contains three generations of the matter fields together with four pairs of Higgses. The requirement of UV finiteness fixes all the Yukawa couplings at the GUT scale. Imposing the condition of universality on the soft couplings at the Planck scale and then extending the condition of finiteness to them, one gets a completely finite unified theory above $M_{GUT}.$ This makes the fine-tuning procedure more meaningful and leads to the usual Minimal Supersymmetric Standard Model below $M_{GUT}$. All the masses of the ordinary particles including Higgses are obtained due to the Higgs mechanism at the electro-weak scale. The hierarchy of quark and lepton masses is related to that of v.e.v.'s of the Higgs fields and is governed by the Higgs mixing matrix in the generation space. Superpartners develop their masses according to the RG equations starting from the soft terms at the Planck scale. The suggestion of complete finiteness and maximal simplicity of the unified theory leads to the connection between the initial values of soft SUSY breaking parameters, namely $m_0^2=1/3m_{1/2}^2,\ A_t=A_b=A_τ=-m_{1/2}, \ B=-m_{1/2}$, so that the number of free parameters is less than that of the MSSM.