Analysis of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">Ξ</mml:mi><mml:mo>(</mml:mo><mml:mn>1620</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> resonance and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mover accent="true"><mml:mi>K</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:math> scattering length with a chiral unitary approach

<jats:p>We study the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi mathvariant="normal">Ξ</a:mi><a:mo>(</a:mo><a:mn>1620</a:mn><a:mo>)</a:mo></a:mrow></a:math> resonance near the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mover accent="true"><c:mi>K</c:mi><c:mo>¯</c:mo></c:mover><c:mi mathvariant="normal">Λ</c:mi></c:mrow></c:math> threshold in the light of recent experimental constraints. The Belle Collaboration have found a resonance peak of <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi mathvariant="normal">Ξ</f:mi><f:mo>(</f:mo><f:mn>1620</f:mn><f:mo>)</f:mo></f:mrow></f:math> slightly below the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:msup><h:mover accent="true"><h:mi>K</h:mi><h:mo>¯</h:mo></h:mover><h:mn>0</h:mn></h:msup><h:mi mathvariant="normal">Λ</h:mi></h:mrow></h:math> threshold in the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:msup><k:mi>π</k:mi><k:mo>+</k:mo></k:msup><k:msup><k:mi mathvariant="normal">Ξ</k:mi><k:mo>−</k:mo></k:msup></k:mrow></k:math> invariant mass spectrum, and the ALICE Collaboration have determined the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msup><m:mi>K</m:mi><m:mo>−</m:mo></m:msup><m:mi mathvariant="normal">Λ</m:mi></m:mrow></m:math> scattering length from the measurement of the momentum correlation functions in heavy ion collisions. Using the effective range expansion, we classify the nature of the pole of the near-threshold eigenstate in terms of the scattering length, in the presence of the decay channel. It is shown that the quasibound state below the threshold can be described by only the scattering length, while the description of the resonance above the threshold requires the contribution from the effective range. Based on the chiral unitary approach, we construct a theoretical model which generates the pole of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:mrow><o:mi mathvariant="normal">Ξ</o:mi><o:mo>(</o:mo><o:mn>1620</o:mn><o:mo>)</o:mo></o:mrow></o:math> below the <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:mrow><q:mover accent="true"><q:mi>K</q:mi><q:mo>¯</q:mo></q:mover><q:mi mathvariant="normal">Λ</q:mi></q:mrow></q:math> threshold with relatively narrow width, as reported by the Belle Collaboration. It is quantitatively demonstrated that the spectrum of the <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"><t:mrow><t:mi mathvariant="normal">Ξ</t:mi><t:mo>(</t:mo><t:mn>1620</t:mn><t:mo>)</t:mo></t:mrow></t:math> quasibound state is distorted by the effect of the nearby <v:math xmlns:v="http://www.w3.org/1998/Math/MathML"><v:mrow><v:mover accent="true"><v:mi>K</v:mi><v:mo>¯</v:mo></v:mover><v:mi mathvariant="normal">Λ</v:mi></v:mrow></v:math> threshold. We then construct another model which reproduces the <y:math xmlns:y="http://www.w3.org/1998/Math/MathML"><y:mrow><y:msup><y:mi>K</y:mi><y:mo>−</y:mo></y:msup><y:mi mathvariant="normal">Λ</y:mi></y:mrow></y:math> scattering length by the ALICE Collaboration. In this case, the eigenstate pole does not appear in the physically relevant Riemann sheet, and the spectrum shows a cusp structure at the <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML"><ab:mrow><ab:mover accent="true"><ab:mi>K</ab:mi><ab:mo>¯</ab:mo></ab:mover><ab:mi mathvariant="normal">Λ</ab:mi></ab:mrow></ab:math> threshold. We finally examine the compatibility of the value of the <db:math xmlns:db="http://www.w3.org/1998/Math/MathML"><db:mrow><db:mover accent="true"><db:mi>K</db:mi><db:mo>¯</db:mo></db:mover><db:mi mathvariant="normal">Λ</db:mi></db:mrow></db:math> scattering length and the subthreshold pole of <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML"><gb:mrow><gb:mi mathvariant="normal">Ξ</gb:mi><gb:mo>(</gb:mo><gb:mn>1620</gb:mn><gb:mo>)</gb:mo></gb:mrow></gb:math> including the experimental uncertainties.</jats:p> <jats:sec> <jats:title/> <jats:supplementary-material> <jats:permissions> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2024</jats:copyright-year> </jats:permissions> </jats:supplementary-material> </jats:sec>