Low energy theorems and the unitarity bounds in the extra U(1) superstring inspired E6 models

The conventional method using low energy theorems derived by Chanowitz et al. [Phys. Rev. Lett. 57, 2344 (1986);] does not seem to lead to an explicit unitarity limit in the scattering processes of longitudinally polarized gauge bosons for the high energy case in the extra U(1) superstring inspired models, commonly known as η model, emanating from E6 group of superstring theory. We have made use of an alternative procedure given by Durand and Lopez [Phys. Lett. B 217, 463 (1989);], which is applicable to supersymmetric grand unified theories. Explicit unitarity bounds on the superpotential couplings (identified as Yukawa couplings) are obtained from both using unitarity constraints as well as using renormalization group equations (RGE) analysis at one-loop level utilizing critical couplings concepts implying divergence of scalar coupling at MG. These are found to be consistent with finiteness over the entire range MZ≤√(s)≤MG i.e. from grand unification scale to weak scale. For completeness, the similar approach has been made use of in other models i.e., χ, ψ, and ν models emanating from E6 and it has been noticed that at weak scale, the unitarity bounds on Yukawa couplings do not differ among E6 extra U(1) models significantly except for the case of χ model in 16 representations. For the case of the E6-η model (βE congruent with 9.64), the analysis using the unitarity constraints leads to the following bounds on various parameters: λt(max.)(MZ)=1.294, λb(max.)(MZ)=1.278, λH(max.)(MZ)=0.955, λD(max.)(MZ)=1.312. The analytical analysis of RGE at the one-loop level provides the following critical bounds on superpotential couplings: λt,c(MZ) congruent with 1.295, λb,c(MZ) congruent with 1.279, λH,c(MZ) congruent with 0.968, λD,c(MZ) congruent with 1.315. Thus superpotential coupling values obtained by both the approaches are in good agreement. Theoretically we have obtained bounds on physical mass parameters using the unitarity constrained superpotential couplings. The bounds are as follows: (i) Absolute upper bound on top quark mass mt≤225 GeV (ii) the upper bound on the lightest neutral Higgs boson mass at the tree level is mH20tree≤169 GeV, and after the inclusion of the one-loop radiative correction it is mH20≤229 GeV when λt≠λb at the grand unified theory scale. On the other hand, these are mH20tree≤159 GeV, mH20≤222 GeV, respectively, when λt=λb at the grand unified theory scale. A plausible range on D-quark mass as a function of mass scale MZ2 is mD≅O(3 TeV) for MZ2≅O(1 TeV) for the favored values of tanβ≤1. The bounds on aforesaid physical parameters in the case of χ, ψ, and ν models in the 27 representation are almost identical with those of η model and are consistent with the present day experimental precision measurements