Effects of SO(10) D-Term on Yukawa Unification and Unstable Minima of the Supersymmetric Scalar Potential

We study the effects of SO(10) D-terms on the allowed parameter space (APS) in models with $t - b - \tau$ and $b - \tau$ Yukawa unifiction. The former is allowed only for moderate values of the D-term, if very precise ($\le$ 5%) unification is required. Next we constrain the parameter space by looking for different dangerous directions where the scalar potential may be unbounded from below (UFB1 and UFB3). The common trilinear coupling $A_0$ plays a significant role in constraing the APS. For very precise $t - b - \tau$ Yukawa unification, $-m_{16} < or \approx A_0 < or \approx m_{16}$ can be probed at the LHC, where $m_{16}$ is the common soft breaking mass for the sfermions. Moreover, an interesting mass hierarchy with very heavy sfermions but light gauginos, which is strongly disfavoured in models without D-terms, becomes fairly common in the presence of the D-terms. The APS exhibits interesting characteristics if $m_{16}$ is not the same as the soft breaking mass $m_{10}$ for the Higgs sector. In $b - \tau$ unification models with D-terms, the APS consistent with Yukawa unification and radiative electroweak symmetry breaking, increases as the UFB1 constraint becomes weaker. However for $A_0 \leq 0$, a stronger UFB3 condition still puts, for a given $m_{16}$, a stringent upper bound on the common gaugino mass ($m_{1/2}$) and a lower bound on $m_{16}$ for a given $m_{1/2}$. The effects of sign of $\mu$ on Yukawa unification and UFB constraints are also discussed.