Rare\begin{document}${{\Lambda_b \rightarrow \Lambda l^+ l^- }}$\end{document} ![]()
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Abstract:
We study the rare decays
\begin{document}$\Lambda_b \rightarrow \Lambda l^+ l^-~(l=e,\mu, \tau)$\end{document} in the Bethe-Salpeter equation approach. We find that the branching ratio is
${\rm Br}(\Lambda_b \rightarrow \Lambda \mu^+ \mu^-)\times 10^{6} = 1.051 \sim 1.098$ in our model. This result agrees with the experimental data well. In the same parametric region, we find that the branching ratio is
${\rm Br}(\Lambda_b \rightarrow \Lambda e^+ e^-(\tau^+ \tau^-) )\times 10^{6} = 0.252 \sim 0.392 ~(0.286 \sim 0.489)$ .
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Key words:
- charmonium /
- decay width /
- lattice QCD
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Figure 1. The BS equation for
$ b(ud)_{00} $ system in the momentum space (K is the interaction kernel).
Figure 2. (color online) The BS wave functions for
$ \Lambda $ with$ E_0 = -0.19 $ GeV.
Figure 3. (color online) The BS wave functions for
$ \Lambda $ with$ E_0=-0.14 $ GeV.
Figure 4. (color online) The BS wave functions for
$ \Lambda $ with$ E_0=-0.09 $ GeV.
Figure 5. (color online) The BS wave functions for
$ \Lambda $ with$ \kappa = -0.05 $ GeV3.
Figure 6. (color online) The BS wave function for
$ \Lambda_b $ with$ E_0 = -0.19 $ GeV,$ E_0 = -0.14 $ GeV.
Figure 7. (color online) The BS wave function for
$ \Lambda_b $ with$ E_0 = -0.09 $ GeV, and$ \kappa = -0.05 $ GeV3.
Figure 8. (color online) The values of
$ R(\omega) $ for different binding energies$ E_0 $ and$ \kappa $ (the value of R decreases with increasing$ \kappa $ , and with increasing$ \kappa $ the line thickens ($ \kappa $ from$ 0.040 $ to$ 0.060 $ ) for the same color line).
Figure 9. (color online) The differential decay width of
$ \Lambda_b \rightarrow \Lambda l^+ l^- $ with the binding energy$ E_0 = -0.19 $ GeV (the decay width increases with increasing$ \kappa $ from$ 0.040 $ to$ 0.060 $ GeV3 for the same color line).
Figure 10. (color online) The differential decay width of
$ \Lambda_b \rightarrow \Lambda l^+ l^- $ with the binding energy$ E_0=-0.14 $ GeV (the decay width increases with increasing$ \kappa $ from$ 0.040 $ to$ 0.060 $ GeV3 for the same color line).
Figure 11. (color online) The differential decay width of
$ \Lambda_b \rightarrow \Lambda l^+ l^- $ with the binding energy$ E_0 = -0.09 $ GeV (the decay width increases with increasing$ \kappa $ from$ 0.040 $ to$ 0.060 $ GeV3 for the same color line).Table 1. The values of
$\alpha_{\rm seff}$ for$ \Lambda $ (the units of$ E_0 $ and$ \kappa $ are GeV and GeV3, respectively).$ E_0 $ $\alpha_{\rm seff}$ −0.19 0.616 0.611 0.661 0.606 0.601 0.596 0.592 0.588 0.584 0.580 0.577 −0.14 0.576 0.570 0.566 0.561 0.557 0.553 0.549 0.546 0.542 0.539 0.536 −0.09 0.521 0.517 0.513 0.509 0.506 0.503 0.500 0.497 0.495 0.492 0.490 $ \kappa \times 10^{3} $ 40 42 44 46 48 50 52 54 56 58 60 
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Table 2. The values of
$\alpha_{\rm seff}$ for$ \Lambda_b $ (the units of$ E_0 $ and$ \kappa $ are GeV and GeV3, respectively).$ E_0 $ $\alpha_{\rm s}$ −0.19 0.806 0.808 0.809 0.796 0.811 0.812 0.814 0.815 0.817 0.818 0.819 −0.14 0.770 0.772 0.774 0.776 0.777 0.779 0.781 0.783 0.785 0.786 0.788 −0.09 0.729 0.732 0.735 0.737 0.713 0.740 0.742 0.744 0.747 0.749 0.751 $ \kappa \times 10^{3} $ 40 42 44 46 48 50 52 54 56 58 60
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Table 3. The values of the branching ratios for
$ \Lambda_b\rightarrow \Lambda l^+ l^- $ , and comparison with other models.$ -E_0 $( $ \times 10^2 $ GeV)
$ \kappa $( $ \times 10^3 $ GeV3)present work 1450±5 present work 14±550 HQET [55] QCD sum rules [32] Exp. [54] $ Br( \Lambda_b\rightarrow \Lambda e^+ e^- ) \times 10^{6} $ 0.464−1.144 0.611−0.867 2.23−3.34 4.6±1.6 − $ Br( \Lambda_b\rightarrow \Lambda \mu^+ \mu^- )\times 10^{6} $ 0.602−1.482 0.856−1.039 2.08−3.19 4.0±1.2 1.08±0.28 $ Br( \Lambda_b\rightarrow \Lambda \tau^+ \tau^- )\times 10^{6} $ 0.177−0.437 0.233−0.331 0.179−0.276 0.8±0.3 −
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