Since the pentaquark discovery in the LHCb collaboration, physicists have tried to describe its structure and possible decay processes, resulting in the determination of the quantum numbers of the pentaquark. In this research, we constructed the pentaquark wave functions by using the quark model under the compact pentaquark picture. The wave function was derived from the combination of 3 light quarks and heavy quark-antiquark pair ($c cbar$). There are two possible color singlets for pentaquarks, which are the combination of color singlet-singlet ($[111]_{{qqq}}otimes [111]_{{c cbar}}$) and color octet-octet ($[21]_{{qqq}}otimes [21]_{{c cbar}}$). The possible pentaquark configurations can be in 17 states. The transition amplitudes and partial width ratios were calculated between the pentaquark states and the possible decay channel states. We found that the decay channels $pJ/psi$ remained dominant when compared with other decay channels. Meanwhile, the partial width ratio showed that if there is no mixing among the $I=frac{1}{2}$ and $J=frac{3}{2}$ states as well as among the $I=frac{1}{2}$ and $J=frac{1}{2}$ states, the two states in $pJ/psi$ channel have the same decay widths, which indicates that $P_c(4440)$ may not be a compact pentaquark state since its decay width is much larger than others. Our results suggested that $P_c(4312)$ might be a spin-$frac{1}{2}$ particle while the spin-$frac{3}{2}$ may be assigned to $P_c(4457)$. Moreover, this work constructs all possible pentaquark states by using group theory, and the pentaquark states in other decay channels discussed in this thesis can be possibly searched in the future.

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