Searching for compact pentaquark states within the bag model framework

The search for the compact limit of multiquark states is a challenging issue. Within the framework of the MIT bag model, we propose an effective limit bag radius of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>R</a:mi><a:mi>c</a:mi></a:msub><a:mo>=</a:mo><a:mn>5.615</a:mn><a:mtext> </a:mtext><a:mtext> </a:mtext><a:msup><a:mi>GeV</a:mi><a:mrow><a:mo>−</a:mo><a:mn>1</a:mn></a:mrow></a:msup></a:math> (or 1.11 fm) for bound states. When the bag radius of a hadron falls below this value, the bag binding energy satisfies <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>E</c:mi><c:mi>B</c:mi></c:msub><c:mo>&lt;</c:mo><c:mn>0</c:mn></c:math>, 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>n</e:mi><e:mi>n</e:mi><e:mi>c</e:mi><e:mi>c</e:mi><e:mover accent="true"><e:mi>c</e:mi><e:mo stretchy="false">¯</e:mo></e:mover></e:math> is below the threshold <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>R</i:mi><i:mi>c</i:mi></i:msub></i:math>. We take into account color-magnetic interactions and calculate the mass, magnetic moment, binding energy, and relative strong decay width for the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>n</k:mi><k:mi>n</k:mi><k:mi>Q</k:mi><k:mi>Q</k:mi><k:mover accent="true"><k:mi>Q</k:mi><k:mo stretchy="false">¯</k:mo></k:mover></k:math> system. We show that the binding energies of most states in the flavor combination <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>n</o:mi><o:mi>n</o:mi><o:mi>c</o:mi><o:mi>c</o:mi><o:mover accent="true"><o:mi>c</o:mi><o:mo stretchy="false">¯</o:mo></o:mover></o:math> are approximately <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mo>−</s:mo><s:mn>20</s:mn><s:mtext> </s:mtext><s:mtext> </s:mtext><s:mi>MeV</s:mi></s:math>, whereas states involving bottom quarks have binding energies around <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mo>−</u:mo><u:mn>120</u:mn><u:mtext> </u:mtext><u:mtext> </u:mtext><u:mi>MeV</u:mi></u:math>, with some decay widths suppressed by the decay constant. Additionally, the <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>n</w:mi><w:mi>Q</w:mi><w:mi>Q</w:mi><w:mi>Q</w:mi><w:mover accent="true"><w:mi>Q</w:mi><w:mo stretchy="false">¯</w:mo></w:mover></w:math> system exhibits even deeper binding. Our results support the compact intention of <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>n</ab:mi><ab:mi>n</ab:mi><ab:mi>Q</ab:mi><ab:mi>Q</ab:mi><ab:mover accent="true"><ab:mi>Q</ab:mi><ab:mo stretchy="false">¯</ab:mo></ab:mover></ab:math> and suggest that the <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>n</eb:mi><eb:mi>Q</eb:mi><eb:mi>Q</eb:mi><eb:mi>Q</eb:mi><eb:mover accent="true"><eb:mi>Q</eb:mi><eb:mo stretchy="false">¯</eb:mo></eb:mover></eb:math> configuration demonstrates even stronger compactness.