Exploring Possible Triangle Singularities in the <inline-formula> <mml:math id="mm9999" display="block"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mo>Ξ</mml:mo> <mml:mi>b</mml:mi> <mml:mo>−</mml:mo> </mml:msubsup> <mml:mspace width="3.33333pt"/> <mml:mo>→</mml:mo> <mml:mspace width="3.33333pt"/> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>−</mml:mo> </mml:msup> <mml:mi>J</mml:mi> <mml:mo>/</mml:mo> <mml:mi>ψ</mml:mi> <mml:mo>Λ</mml:mo> </mml:mrow> </mml:semantics> </mml:math> </inline-formula> Decay

• Main Authors: Chao-Wei Shen, Hao-Jie Jing, Feng-Kun Guo, Jia-Jun Wu
• Format: Article
• Language: English
• Published: MDPI AG 2020-09-01
• Series: Symmetry
• Subjects: triangle singularity; pentaquarks; exotic hadrons
• Online Access: https://www.mdpi.com/2073-8994/12/10/1611

We analyze possible singularities in the <inline-formula><math display="inline"><semantics><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>Λ</mo></mrow></semantics></math></inline-formula> invariant mass distribution of the <inline-formula><math display="inline"><semantics><mrow><msubsup><mo>Ξ</mo><mi>b</mi><mo>−</mo></msubsup><mspace width="3.33333pt"></mspace><mo>→</mo><mspace width="3.33333pt"></mspace><msup><mi>K</mi><mo>−</mo></msup><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>Λ</mo></mrow></semantics></math></inline-formula> process via triangle loop diagrams. Triangle singularities in the physical region are found in 18 different triangle loop diagrams. Among those with <inline-formula><math display="inline"><semantics><msup><mo>Ξ</mo><mo>*</mo></msup></semantics></math></inline-formula>-charmonium-<inline-formula><math display="inline"><semantics><mo>Λ</mo></semantics></math></inline-formula> intermediate states, the one from the <inline-formula><math display="inline"><semantics><mrow><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo>Ξ</mo><mrow><mo>(</mo><mn>2120</mn><mo>)</mo></mrow><mo>Λ</mo></mrow></semantics></math></inline-formula> loop, which is located around 4628 MeV, is found the most likely to cause observable effects. One needs <i>S</i>- and <i>P</i>-waves in <inline-formula><math display="inline"><semantics><mrow><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo>Λ</mo></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>Λ</mo></mrow></semantics></math></inline-formula> systems, respectively, when the quantum numbers of these systems are <inline-formula><math display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>+</mo></msup></mrow></semantics></math></inline-formula> or <inline-formula><math display="inline"><semantics><mrow><mn>3</mn><mo>/</mo><msup><mn>2</mn><mo>+</mo></msup></mrow></semantics></math></inline-formula>. When the quantum numbers of the <inline-formula><math display="inline"><semantics><mrow><mo>Ξ</mo><mo>(</mo><mn>2120</mn><mo>)</mo></mrow></semantics></math></inline-formula> are <inline-formula><math display="inline"><semantics><mrow><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>+</mo></msup></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup></mrow></semantics></math></inline-formula> or <inline-formula><math display="inline"><semantics><mrow><mn>3</mn><mo>/</mo><msup><mn>2</mn><mo>+</mo></msup></mrow></semantics></math></inline-formula>, the peak structure should be sharper than the other <inline-formula><math display="inline"><semantics><msup><mi>J</mi><mi>P</mi></msup></semantics></math></inline-formula> choices. This suggests that although the whole strength is unknown, we should pay attention to the contributions from the <inline-formula><math display="inline"><semantics><msup><mo>Ξ</mo><mo>*</mo></msup></semantics></math></inline-formula>-charmonium-<inline-formula><math display="inline"><semantics><mo>Λ</mo></semantics></math></inline-formula> triangle diagram if structures are observed in the <inline-formula><math display="inline"><semantics><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>Λ</mo></mrow></semantics></math></inline-formula> invariant mass spectrum experimentally. In addition, a few triangle diagrams with the <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow><mo>*</mo></msubsup><mrow><mo>(</mo><mn>2700</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> as one of the intermediate particles can also produce singularities in the <inline-formula><math display="inline"><semantics><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>Λ</mo></mrow></semantics></math></inline-formula> distribution, but at higher energies above 4.9 GeV.