  We show model-independently that the negative like-sign charge asymmetry
$(-A^b_{s\ell})$ is less than $ 3.16\times 10^{-3}$ when the constraints from
the $B_q-\bar B_q$ mixings and the time-dependent CP asymmetries (CPAs) for
$B_q\to J/\Psi M_q$ with $M_q=K,\phi$ and $q=d,s$ are taken into account.
Although the result is smaller than the measured value by the D{\O}
Collaboration at Fermilab, there is still plenty of room to have new physics,
which is sensitive to new CP violating effects, as the standard model (SM)
prediction is $(2.3_{-0.5}^{+0.6})\times 10^{-4}$. To illustrate the potential
large $|A^{b}_{s\ell}|$, we show the influence of new $SU(2)_L$ singlet exotic
quarks in the vector-like quark model, where the $Z$-mediated flavor changing
neutral currents (FCNCs) are generated at tree level. In particular, we
demonstrate that (a) the like-sign charge asymmetry could be enhanced by a
factor of two in magnitude; (b) the CPA of $\sin2\beta^{J/\Psi \phi}_s$ could
reach to $-15%$; (c) the CPA of $\sin2\beta_{\phi K_S}$ could be higher than
$\sin2\beta_{J/\Psi K_S}$ when $|A^b_{s\ell}|$ is larger than the SM
prediction; and (d) the branching ratio for $B_s\to \mu^+ \mu^-$ could be as
large as $0.6\times 10^{-8}$.
