Search for Possible Stable Structures in the <inline-formula><math display="inline"><semantics><msub><mi mathvariant="bold-italic">T</mi><mrow><mi mathvariant="bold-italic">c</mi><mi mathvariant="bold-italic">c</mi><mover accent="true"><mi mathvariant="bold-italic">q</mi><mo mathvariant="bold">¯</mo></mover><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo mathvariant="bold">¯</mo></mover></mrow></msub></semantics></math></inline-formula> System

Inspired by the well-known experimental connections between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Z</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mrow><mo>(</mo><mn>4220</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Y</mi><mo>(</mo><mn>4620</mn><mo>)</mo></mrow></semantics></math></inline-formula>, we systematically study the recently reported strange partner of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub></semantics></math></inline-formula>, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>1</mn><mo>+</mo></msup></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mi>c</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mover accent="true"><mi>s</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula> system, and its orbital excitation state <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>1</mn><mo>−</mo></msup></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mi>c</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mover accent="true"><mi>s</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula>. A chiral quark model incorporating SU(3) symmetry is considered to study these two systems. To better investigate their spatial structure, we introduce a precise few-body calculation method, the Gaussian Expansion Method (GEM). In our calculations, we include all possible physical channels, including molecular states and diquark structures, and consider channel coupling effects. To identify the stable structures in the system (bound states and resonance states) we employ a powerful resonance search method, the Real-Scaling Method (RSM). According to our results, in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>1</mn><mo>+</mo></msup></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mi>c</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mover accent="true"><mi>s</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula> system, we obtain two bound states with energies of 3890 MeV and 3940 MeV, as well as two resonance states with energies of 3975 MeV and 4090 MeV. The decay channels of these two resonance states are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><msubsup><mi>D</mi><mi>s</mi><mo>∗</mo></msubsup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>D</mi><mo>∗</mo></msup><msub><mi>D</mi><mi>s</mi></msub></mrow></semantics></math></inline-formula>, respectively. In the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>1</mn><mo>−</mo></msup></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mi>c</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mover accent="true"><mi>s</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula> system, we obtain only one resonance state, with an energy of 4570 MeV, and two main decay channels: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><msubsup><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow><mo>∗</mo></msubsup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>D</mi><mo>∗</mo></msup><msubsup><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow><mo>′</mo></msubsup></mrow></semantics></math></inline-formula>. We strongly suggest that experimental groups use our predictions to search for these stable structures.