Yukawa unification as a window into the soft supersymmetry breaking Lagrangian

We study Yukawa unification, including the effects of a physical neutrino mass consistent with the SuperKamiokande observations, in a string or D-brane inspired $\mathrm{SU}\left(4\right)\mathrm{}\otimes {\mathrm{SU}\left(2\right)\mathrm{}}_L\otimes {\mathrm{SU}\left(2\right)\mathrm{}}_R$ model which allows the most general non-universal scalar and gaugino masses, including the usual D-term contributions which arise in $\mathrm{SO}\left(10\right).$ We investigate how the tight constraints from rare decays such as $\overrightarrow{b}s\gamma$ and $\overrightarrow{\tau }\mu \gamma$ can provide information about the family dependent supersymmetry breaking soft Lagrangian, for example the trilinears associated with the second and third family. Many of our results also apply to $\mathrm{SO}\left(10\right)\mathrm{}$ to which the model approximately reduces in a limiting case. In both models we find that Yukawa unification is perfectly viable providing the non-universal soft masses have particular patterns. In this sense Yukawa unification acts as a window into the soft supersymmetry breaking Lagrangian.