Searching for compact pentaquark state within the bag model framework

The search for the compact limit of multi-quark states is a challenging issue. Within the framework of the MIT bag model, we propose an effective limit bag radius of $R_{c} = 5.615 \, \text{GeV}^{-1}$(or $1.11 \, \text{fm}$) for bound states . When the bag radius of a hadron falls below this value, the bag binding energy satisfies $E_{B} < 0$, indicating that the system has a compact intention. We consider various combinations of different numbers and ratios of heavy and light quarks, indicating that the bag radius of hadrons depending on the number of quarks is suppressed by the presence of heavy quarks. Focusing on five-quark combinations, we find that the average bag radius of $nncc\bar{c}$ is below the threshold $R_{c}$. We take into account color-magnetic interactions and calculate the mass, magnetic moment, binding energy, and relative strong decay width for the $nnQQ\bar{Q}$ system. We show that the binding energies of most states in the flavor combination $nncc\bar{c}$ are approximately $-20 \, \text{MeV}$, whereas states involving bottom quarks have binding energies around $-120 \, \text{MeV}$, with some decay widths suppressed by the decay constant. Additionally, the $nQQQ\bar{Q}$ system exhibits even deeper binding. Our results support the compact intention of $nnQQ\bar{Q}$ and suggest that the $nQQQ\bar{Q}$ configuration demonstrates even stronger compactness.