The successful operation of LHC provides a great opportunity to study the processes where heavy baryons are involved. {In this work we mainly study} the weak transitions of $Sigma_bto Sigma_c$. Assuming the reasonable quark-diquark structure where the two light quarks constitute an axial vector, we calculate the widths of semi-leptonic decay $Sigma_{b}toSigma_c e u_e$ and non-leptonic decay modes $Sigma_{b}toSigma_c +M$ (light mesons) in terms of the light front quark model. We first construct the vertex function for the concerned baryons and then deduce the form factors which are related to two Isgur-Wise functions for the $Sigma_{b}toSigma_c$ transition under the heavy quark limit. Our numerical results indicate that $Gamma(Sigma_{b}toSigma_c e u_e)$ is about $1.38times10^{10}{rm s}^{-1}$ and $Gamma(Sigma_{b}toSigma_c +M)$ is slightly below $1times10^{10}{rm s}^{-1}$ which may be accessed at the LHCb detector. By the flavor SU(3) symmetry we estimate the rates of $Omega_btoOmega_c$. We suggest to measure weak decays of $Omega_btoOmega_c$, because $Omega_b$ does not decay via strong interaction, the advantage is obvious.
Backtracking is a basic strategy to solve constraint satisfaction problems (CSPs). A satisfiable CSP instance is backtrack-free if a solution can be found without encountering any dead-end during a backtracking search, implying that the instance is easy to solve. We prove an exact phase transition of backtrack-free search in some random CSPs, namely in Model RB and in Model RD. This is the first time an exact phase transition of backtrack-free search can be identified on some random CSPs. Our technical results also have interesting implications on the power of greedy algorithms, on the width of random hypergraphs and on the exact satisfiability threshold of random CSPs.
