Semileptonic decays of doubly charmed baryons with bag model

We study the semileptonic decays of $B_{cc}$ ${\rightarrow}$ $B_c\ell^+\nu_\ell$ with the bag model, where $\ell$ = $(e, \mu)$, $B_{cc}$ = $(\Xi_{cc}^{++}$, $\Xi_{cc}^+$, $\Omega_{cc}^+$), and $B_c$ are the singly charmed baryons with $J^P= 1/2^+$. We obtain the decay widths of $\Gamma(\Xi_{cc}^{++}{\rightarrow}\Xi_c^+e^+\nu_e, \Xi_c^{\prime+}e^+\nu_e, \Lambda_c^+e^+\nu_e, \Omega_c^+ e^+\nu_e) =(5.1\pm 0.1 , 11\pm 1, 0.34\pm 0.06, 0.76\pm 0.06)\times 10^{-14}$~GeV, $\Gamma(\Xi_{cc}^+\rightarrow \Xi_c^0e^+\nu_e, \Xi_c^{\prime0}e^+\nu_e , \Sigma_c^0e^+\nu_e) = (5.1\pm 0.6, 11\pm 1, 1.5\pm 0.1) \times 10^{-14}$~GeV, and $\Gamma(\Omega_{cc}^+\rightarrow \Omega_c^0 e^+\nu_e, \Xi_c^0e^+\nu_e , \Xi_c^{\prime0} e^+\nu_e) = (22\pm 2, 0.32 \pm 0.04, 0.77\pm 0.06)\times 10^{-14}$~GeV. We also get that $\Gamma$($B_{cc}$ ${\rightarrow}$ $B_c\mu^+\nu_\mu$)/$\Gamma$($B_{cc}$ ${\rightarrow}$ $B_ce^+\nu_e$) = $0.97\sim 1.00$. In addition, we discuss the $SU(3)$ flavor breaking effects, classified into three aspects: phase space differences, spectator quarks, and overlappings of the transited quarks. In particular, we show that the breaking effects are dominated by the phase space differences, which can be as large as 25\%. Explicitly, we find that $\Gamma(\Xi_{cc}^{++} \to \Lambda_c ^+ e^+ \nu _e) V_{cs}^2/\Gamma(\Xi_{cc}^{++} \to \Xi_c ^+ e^+ \nu_e )V_{cd}^2 = 1.24$, which is expected as $1$ under the exact $SU(3)$ flavor symmetry.