Molecular nature of the Ωc(3120) and its analogy with the Ω(2012<mml:mo stretchy="false…

We make a study of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi mathvariant="normal">Ω</a:mi><a:mi>c</a:mi></a:msub><a:mo stretchy="false">(</a:mo><a:mn>3120</a:mn><a:mo stretchy="false">)</a:mo></a:math>, one of the five <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:msub><f:mi mathvariant="normal">Ω</f:mi><f:mi>c</f:mi></f:msub></f:math> states observed by the LHCb Collaboration, which is well reproduced as a molecular state from the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msubsup><i:mi mathvariant="normal">Ξ</i:mi><i:mi>c</i:mi><i:mo>*</i:mo></i:msubsup><i:mover accent="true"><i:mi>K</i:mi><i:mo stretchy="false">¯</i:mo></i:mover></i:math> and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msubsup><n:mi mathvariant="normal">Ω</n:mi><n:mi>c</n:mi><n:mo>*</n:mo></n:msubsup><n:mi>η</n:mi></n:math> channels mostly. The state with <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msup><q:mi>J</q:mi><q:mi>P</q:mi></q:msup><q:mo>=</q:mo><q:mn>3</q:mn><q:mo>/</q:mo><q:msup><q:mn>2</q:mn><q:mo>−</q:mo></q:msup></q:math> decays to <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi mathvariant="normal">Ξ</s:mi><s:mi>c</s:mi></s:msub><s:mover accent="true"><s:mi>K</s:mi><s:mo stretchy="false">¯</s:mo></s:mover></s:math> in the <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mi>D</x:mi></x:math> wave, and we include this decay channel in our approach, as well as the effect of the <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:msubsup><z:mi mathvariant="normal">Ξ</z:mi><z:mi>c</z:mi><z:mo>*</z:mo></z:msubsup></z:math> width. With all these ingredients, we determine the fraction of the <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mi mathvariant="normal">Ω</cb:mi><cb:mi>c</cb:mi></cb:msub><cb:mo stretchy="false">(</cb:mo><cb:mn>3120</cb:mn><cb:mo stretchy="false">)</cb:mo></cb:math> width that goes into <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:msub><hb:mi mathvariant="normal">Ξ</hb:mi><hb:mi>c</hb:mi></hb:msub><hb:mi>π</hb:mi><hb:mover accent="true"><hb:mi>K</hb:mi><hb:mo stretchy="false">¯</hb:mo></hb:mover></hb:math>, which could be a measure of the <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:msubsup><mb:mi mathvariant="normal">Ξ</mb:mi><mb:mi>c</mb:mi><mb:mo>*</mb:mo></mb:msubsup><mb:mover accent="true"><mb:mi>K</mb:mi><mb:mo stretchy="false">¯</mb:mo></mb:mover></mb:math> molecular component, but due to a relatively big binding, compared to its analogous <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"><rb:mi mathvariant="normal">Ω</rb:mi><rb:mo stretchy="false">(</rb:mo><rb:mn>2012</rb:mn><rb:mo stretchy="false">)</rb:mo></rb:math> state, we find only a small fraction of about 3%, which makes this measurement difficult with present statistics. As an alternative, we evaluate the scattering length and effective range of the <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:msubsup><wb:mi mathvariant="normal">Ξ</wb:mi><wb:mi>c</wb:mi><wb:mo>*</wb:mo></wb:msubsup><wb:mover accent="true"><wb:mi>K</wb:mi><wb:mo stretchy="false">¯</wb:mo></wb:mover></wb:math> and <bc:math xmlns:bc="http://www.w3.org/1998/Math/MathML" display="inline"><bc:msubsup><bc:mi mathvariant="normal">Ω</bc:mi><bc:mi>c</bc:mi><bc:mo>*</bc:mo></bc:msubsup><bc:mi>η</bc:mi></bc:math> channels, which, together with the binding and width of the <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:msub><ec:mi mathvariant="normal">Ω</ec:mi><ec:mi>c</ec:mi></ec:msub><ec:mo stretchy="false">(</ec:mo><ec:mn>3120</ec:mn><ec:mo stretchy="false">)</ec:mo></ec:math> state, could give us an answer to the issue of the compositeness of this state when these magnitudes are determined experimentally, something feasible nowadays, for instance, measuring correlation functions.