A proposal on complementary determination of the electro-weak mixing angles ${\rm sin^2\theta_W}$ at a super Z-factory

A proposal on determination of the effective electro-weak mixing angle ${\rm sin^2\theta_{eff}^f\;(f=c,b)}$ at a super Z-factory (an $e^+e^-$ collider designed with the possible highest luminosity and running around the center mass energy $\sqrt s=m_Z$) is proposed. It is to determine how the effective mixing angles depend on the flavors, especially the flavors $c$, $b$ (with the determined effective ones, the electro-weak mixing angles ${\rm sin^2\theta_W^f\;(f=c,b)}$ of Standard Model can be derived out precisely). The key point of the proposal is that at such a super Z-factory, via measuring the forward-backward ($A_{FB}$), left-right ($A_{LR}$) and combined left-right forward-backward ($A^{FB}_{LR}$) asymmetries of the produced doubly heavy-flavored hadrons (such as $B_c$, $B_c^*$ and the baryons $H_{QQ'q}$: $\Xi_{cc}$, $\Xi_{bc}$ and $\Xi_{bb}$ etc) in the relevant production processes $e^+e^-\to B_c+\cdots$ and $e^+e^-\to H_{QQ'q}+\cdots$ one may complement the final determination of the mixing angle ${\rm sin^2\theta_W^f\;(f=c,b)}$ of Standard Model. The advantage of the proposed way to determine the effective electro-weak mixing angles is that the doubly heavy flavor(s) and the out-going direction of the produced doubly-heavy hadron can be experimental determined precisely, i.e. there are no such errors caused by missing identification of the heavy flavor(s) and by determining the so-called thrust axis of the produced jets, which can not be avoided in the early determination at LEP-I and SLC.