Model with Strong $\gamma_4$ $T$-violation

We extend the $T$ violating model of the paper on "Hidden symmetry of the CKM and neutrino-mapping matrices" by assuming its $T$-violating phases $\chi_\uparrow$ and $\chi_\downarrow$ to be large and the same, with $\chi=\chi_\uparrow=\chi_\downarrow$. In this case, the model has 9 real parameters: $\alpha_\uparrow, \beta_\uparrow, \xi_\uparrow, \eta_\uparrow$ for the $\uparrow$-quark sector, $\alpha_\downarrow, \beta_\downarrow, \xi_\downarrow, \eta_\downarrow$ for the $\downarrow$ sector and a common $\chi$. We examine whether these nine parameters are compatible with ten observables: the six quark masses and the four real parameters that characterize the CKM matrix (i.e., the Jarlskog invariant ${\cal J}$ and three Eulerian angles). We find that this is possible only if the $T$violating phase $\chi$ is large, between $-120^0$ to $-135^0$. In this strong $T$ violating model, the smallness of the Jarlskog invariant ${\cal J}\cong 3\times 10^{-5}$ is mainly accounted for by the large heavy quark masses, with $\frac{m_c}{m_t} <\frac{m_s}{m_b} \approx .02$, as well as the near complete overlap of $t$ and $b$ quark, with $(c|b)=-.04$.