We study the some topics on the $b\bar{b}$ events in the $e^+e^-$ collision at Z resonance. According to the Standard Model, b quarks are produced almost completely longitudinally polarized at $\sqrt{s}\simeqM_Z$. Unfortunately, a measurement of $P_b$ is not as much straightforward. This is mainly due to hadronization and fragmentation properties b quarks that largely dilute and obscure b polarization effects in the observed final states. We suggest a strategy for detecting b polarization effects in the $e^+e^-$ collision on Z resonance. We select dilepton events in the $Z → b\bar{b}$ events, in which both b hadrons decay semileptonically. Then we catalog these events into the 'like-sign' events in which leptons have same charge and the 'unlike-sign' events in which leptons have different charge. Clearly the existence of the 'like-sign' events is a signal of b-$\bar{B}$ mixing. In the 'like-sign' events, one of the b hadrons must be a $B_s$ or $B_d$ meson. However this is not the case in the 'unlike-sign' events. So the fractions of $∧_b$ would be different for each types of events. Then we have two types of inclusive leptonic samples with different $∧_b$ fractions. With these dileptonic events, the several polarization-dependent variables are calculated. Using this method, the b polarization effects may be detected without large uncertainties of b fragmentation and $∧_b$ decay model.
The inclusive semi-leptonic decays of b mesons are attracting a lot of attention as testing ground for hadronic models and as a source for the weak parameters $V_ub$ and $V_cb$. In general there are two kinds of models which predict the lepton energy spectrum of inclusive decays of b meson. In the ACCMM model, the free quark decay picture is improved by taking into account the Fermi motion of the spectator in the b-meson. The second approach uses the parton model at an infinite momentum frame(IMF). It pictures the mesonic decay as the decay of the partons. The probability of finding a b-quark in a b-meson carrying a fraction x of the mesons momentum in the IMF is given by the distribution function f(x). We study both of the model by comparing their own distribution functions. The distribution function of the ACCMM model is obtained in the infinite momentum frame to show the results consistently. We show that the definitions of b-quark mass which is consistent with the result of HQET are equivalent between both models. So the specific choice adopted in ACCMM model for the weight function φ(p) corresponds to a particular form of the distribution function f(x) in the parton model. Also we compare the behaviors of the distribution functions and obtain the heavy quark effective theory parameters in terms of each model parameters.
