Low energy theorems and the unitarity bounds in the extra $U(1)$ superstring inspired $E_6$ 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 $\eta$ model, emanating from $E_6$ group of superstring theory. We have made use of an alternative procedure given by Durand & 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 $M_G$. These are found to be consistent with finiteness over the entire range $M_Z\le \sqrt{s}\le M_G$ i.e. from grand unification scale to weak scale. For completeness, the similar approach has been made use of in other models i.e., $\chi$, $\psi$, and $\nu$ models emanating from $E_6$ and it has been noticed that at weak scale, the unitarity bounds on Yukawa couplings do not differ among $E_6$ extra $U(1)$ models significantly except for the case of χ model in $\mathbf{16}$ representations. For the case of the $E_6$-$\eta$ model $({\beta }_E\cong 9.64)$, the analysis using the unitarity constraints leads to the following bounds on various parameters: ${\lambda }_{t(\max .)}(M_Z)=1.294$, ${\lambda }_{b(\max .)}(M_Z)=1.278$, ${\lambda }_{H(\max .)}(M_Z)=0.955$, ${\lambda }_{D(\max .)}(M_Z)=1.312$. The analytical analysis of RGE at the one-loop level provides the following critical bounds on superpotential couplings: ${\lambda }_{t,c}(M_Z)\cong 1.295$, ${\lambda }_{b,c}(M_Z)\cong 1.279$, ${\lambda }_{H,c}(M_Z)\cong 0.968$, ${\lambda }_{D,c}(M_Z)\cong 1.315$. 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 $m_t\le 225\mathrm{GeV}$, (ii) the upper bound on the lightest neutral Higgs boson mass at the tree level is $m_{H_2^0}^{\mathrm{tree}}\le 169\mathrm{GeV}$, and after the inclusion of the one-loop radiative correction it is $m_{H_2^0}\le 229\mathrm{GeV}$ when ${\lambda }_t\ne {\lambda }_b$ at the grand unified theory scale. On the other hand, these are $m_{H_2^0}^{\mathrm{tree}}\le 159\mathrm{GeV}$, $m_{H_2^0}\le 222\mathrm{GeV}$, respectively, when ${\lambda }_t={\lambda }_b$ at the grand unified theory scale. A plausible range on $D$-quark mass as a function of mass scale $M_{Z_2}$ is $m_D\approx O(3\mathrm{TeV})$ for $M_{Z_2}\approx O(1\mathrm{TeV})$ for the favored values of $\tan \beta \le 1$. The bounds on aforesaid physical parameters in the case of $\chi$, $\psi$, and $\nu$ models in the $\mathbf{27}$ representation are almost identical with those of $\eta$ model and are consistent with the present day experimental precision measurements.

Figure 1

(a,b,c): Tree-level Feynman diagrams contributing to the $M_{\mathrm{gauge}}$ part of the amplitude for the process $W_L^{+}{W}_L^{-}\to Z_L{Z}_L$.

Figure 2

$s$-channel tree-level Feynman diagram contributing to the $M_{\mathrm{SB}}$ part of the amplitude for the process $W_L^{+}{W}_L^{-}\to Z_L{Z}_L$.

Figure 3

$t$-channel and $u$-channel Feynman diagrams contributing to the $M_{\mathrm{SB}}$ part of the amplitude for the process $W_L^{+}{W}_L^{-}\to Z_{iL}{Z}_{jL}$ ($i,j=1,2$) in the extra $U(1)$ superstring inspired model.

Figure 4

The unitarity surface in the subspace of the scalar superpotential couplings ${\lambda }_t,{\lambda }_H,{\lambda }_D$ for ${\lambda }_b=0$ for the energy range $M_Z\ll \sqrt{s}\le M_G$

Figure 5

The unitarity surface of Yukawa couplings at weak scale ($M_Z$) obtained by evolving the surface at ($M_G$) shown in Fig. 4 from GUT scale to weak scale for the case of $\eta$ model for $({\beta }_E\cong 9.64)$.

Figure 6

(a) The projection of unitarity boundaries are shown in the plane ${\lambda }_t,{\lambda }_H$ plane at GUT scale. This leads to unitarity constrained region of Yukawa couplings at weak scale and are shown in (b) $\chi (16)$ $({\beta }_E\cong 6.023)$, (c) $\chi (27)$ $({\beta }_E\cong 9.023)$, (d) $\psi$ $({\beta }_E\cong 9.028)$, (e) $\eta$ $({\beta }_E\cong 9.64)$, and (f) $\nu$ $({\beta }_E\cong 9.9)$ $E_6$ models. The effect of evolution for ${\lambda }_t\ne {\lambda }_b$ (${\lambda }_b=0$) and ${\lambda }_t={\lambda }_b$ are represented by curve (i) and curve (ii), respectively.

Figure 7

(a) The possible values of $x/\upsilon$ for the favored values of $\tan \beta$ of $E_6$ extra $U(1)$ models for various $Z_2$ masses. (b) A plausible range for $D$-quark mass against various $M_{Z_2}$.