Complete determination of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>F</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> amplitudes and strong phase in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">Ξ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math>

Abstract

The BESIII collaboration has recently reported the first time measurement of the decay asymmetry <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>α</a:mi><a:mo stretchy="false">(</a:mo><a:msubsup><a:mrow><a:mi mathvariant="normal">Λ</a:mi></a:mrow><a:mrow><a:mi>c</a:mi></a:mrow><a:mrow><a:mo>+</a:mo></a:mrow></a:msubsup><a:mo stretchy="false">→</a:mo><a:msup><a:mrow><a:mi mathvariant="normal">Ξ</a:mi></a:mrow><a:mrow><a:mn>0</a:mn></a:mrow></a:msup><a:msup><a:mrow><a:mi>K</a:mi></a:mrow><a:mrow><a:mo>+</a:mo></a:mrow></a:msup><a:mo stretchy="false">)</a:mo><a:mo>=</a:mo><a:mn>0.01</a:mn><a:mo>±</a:mo><a:mn>0.16</a:mn><a:mo stretchy="false">(</a:mo><a:mi>stat</a:mi><a:mo stretchy="false">)</a:mo><a:mo>±</a:mo><a:mn>0.03</a:mn><a:mo stretchy="false">(</a:mo><a:mi>syst</a:mi><a:mo stretchy="false">)</a:mo></a:mrow></a:math> and also a sizable phase shift of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msub><l:mi>δ</l:mi><l:mi>P</l:mi></l:msub><l:mo>−</l:mo><l:msub><l:mi>δ</l:mi><l:mi>S</l:mi></l:msub><l:mo>=</l:mo><l:mo>−</l:mo><l:mn>1.55</l:mn><l:mo>±</l:mo><l:mn>0.25</l:mn></l:math> or <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mn>1.59</n:mn><n:mo>±</n:mo><n:mn>0.25</n:mn></n:math> between S- and P-wave amplitudes. This implies significant strong phase shifts in the decay amplitudes. The strong phases indicate the existence of rescattering or loop effects, which are challenging to calculate due to nonperturbative effects. By employing the flavor <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>S</p:mi><p:mi>U</p:mi><p:mo stretchy="false">(</p:mo><p:mn>3</p:mn><p:msub><p:mo stretchy="false">)</p:mo><p:mi>F</p:mi></p:msub></p:math> symmetry and applying the Körner-Pati-Woo theorem to reduce the number of parameters, we find that the current data already allow us to obtain, for the first time, model-independent decay amplitudes and their strong phases. The establishment of the existence of sizable strong phases opens a window for future investigations into <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mi>C</t:mi><t:mi>P</t:mi></t:math> violation. In our fit, a notable discrepancy emerges in the branching ratio of <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:msubsup><v:mi mathvariant="normal">Ξ</v:mi><v:mi>c</v:mi><v:mn>0</v:mn></v:msubsup><v:mo stretchy="false">→</v:mo><v:msup><v:mi mathvariant="normal">Ξ</v:mi><v:mo>−</v:mo></v:msup><v:msup><v:mi>π</v:mi><v:mo>+</v:mo></v:msup></v:math>. The direct relationship between <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi mathvariant="normal">Γ</ab:mi><ab:mo stretchy="false">(</ab:mo><ab:msubsup><ab:mi mathvariant="normal">Λ</ab:mi><ab:mi>c</ab:mi><ab:mo>+</ab:mo></ab:msubsup><ab:mo stretchy="false">→</ab:mo><ab:mi mathvariant="normal">Λ</ab:mi><ab:msup><ab:mi>e</ab:mi><ab:mo>+</ab:mo></ab:msup><ab:msub><ab:mi>ν</ab:mi><ab:mi>e</ab:mi></ab:msub><ab:mo stretchy="false">)</ab:mo></ab:math> and <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mi mathvariant="normal">Γ</ib:mi><ib:mo stretchy="false">(</ib:mo><ib:msubsup><ib:mi mathvariant="normal">Ξ</ib:mi><ib:mi>c</ib:mi><ib:mn>0</ib:mn></ib:msubsup><ib:mo stretchy="false">→</ib:mo><ib:msup><ib:mi mathvariant="normal">Ξ</ib:mi><ib:mo>−</ib:mo></ib:msup><ib:msup><ib:mi>e</ib:mi><ib:mo>+</ib:mo></ib:msup><ib:msub><ib:mi>ν</ib:mi><ib:mi>e</ib:mi></ib:msub><ib:mo stretchy="false">)</ib:mo></ib:math>, along with newly discovered <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:mi>S</qb:mi><qb:mi>U</qb:mi><qb:mo stretchy="false">(</qb:mo><qb:mn>3</qb:mn><qb:msub><qb:mo stretchy="false">)</qb:mo><qb:mi>F</qb:mi></qb:msub></qb:math> relations, collectively suggests an underestimation of <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:mi mathvariant="script">B</ub:mi><ub:mo stretchy="false">(</ub:mo><ub:msubsup><ub:mi mathvariant="normal">Ξ</ub:mi><ub:mi>c</ub:mi><ub:mn>0</ub:mn></ub:msubsup><ub:mo stretchy="false">→</ub:mo><ub:msup><ub:mi mathvariant="normal">Ξ</ub:mi><ub:mo>−</ub:mo></ub:msup><ub:msup><ub:mi>π</ub:mi><ub:mo>+</ub:mo></ub:msup><ub:mo stretchy="false">)</ub:mo></ub:math> in experimental findings. Published by the American Physical Society 2024

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