Nature of the Pc states from compositeness criteria

Based on a coupled-channel approach, we investigate the structures of four <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>P</a:mi><a:mi>c</a:mi></a:msub></a:math> states through compositeness criteria. Toward a more precise description of the states, we have obtained refined fit results of the LHCb data on the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>J</c:mi><c:mo>/</c:mo><c:mi>ψ</c:mi><c:mi>p</c:mi></c:math> invariant mass distribution of the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msubsup><e:mi mathvariant="normal">Λ</e:mi><e:mi>b</e:mi><e:mn>0</e:mn></e:msubsup><e:mo stretchy="false">→</e:mo><e:mi>J</e:mi><e:mo>/</e:mo><e:mi>ψ</e:mi><e:mi>p</e:mi><e:msup><e:mi>K</e:mi><e:mo>−</e:mo></e:msup></e:math> decay. Allowing for the fact that each of the four <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>P</i:mi><i:mi>c</i:mi></i:msub></i:math> states couples strongly to a nearby <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>S</k:mi></k:math>-wave channel, three criteria on the compositeness/elementariness are adopted in this study: the pole-counting rule, the spectral density function, and the Gamow wave function. Compositeness information is extracted from the scattering amplitudes and the pole parameters (pole positions and residues), without any preconceived assumptions on the nature of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>P</m:mi><m:mi>c</m:mi></m:msub></m:math> states and without any dependence on the model parametrization. Consistently within the framework of all the three methods, it has been found that the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>P</o:mi><o:mi>c</o:mi></o:msub><o:mo stretchy="false">(</o:mo><o:mn>4312</o:mn><o:mo stretchy="false">)</o:mo><o:mn>1</o:mn><o:mo>/</o:mo><o:msup><o:mn>2</o:mn><o:mo>−</o:mo></o:msup></o:math> is mainly composed by <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mover accent="true"><s:mi>D</s:mi><s:mo stretchy="false">¯</s:mo></s:mover><s:msub><s:mi mathvariant="normal">Σ</s:mi><s:mi>c</s:mi></s:msub></s:math>, <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:msub><x:mi>P</x:mi><x:mi>c</x:mi></x:msub><x:mo stretchy="false">(</x:mo><x:mn>4380</x:mn><x:mo stretchy="false">)</x:mo><x:mn>3</x:mn><x:mo>/</x:mo><x:msup><x:mn>2</x:mn><x:mo>−</x:mo></x:msup></x:math> by <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:mover accent="true"><bb:mi>D</bb:mi><bb:mo stretchy="false">¯</bb:mo></bb:mover><bb:msubsup><bb:mi mathvariant="normal">Σ</bb:mi><bb:mi>c</bb:mi><bb:mo>*</bb:mo></bb:msubsup></bb:math>, while the <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msub><gb:mi>P</gb:mi><gb:mi>c</gb:mi></gb:msub><gb:mo stretchy="false">(</gb:mo><gb:mn>4440</gb:mn><gb:mo stretchy="false">)</gb:mo><gb:mn>1</gb:mn><gb:mo>/</gb:mo><gb:msup><gb:mn>2</gb:mn><gb:mo>−</gb:mo></gb:msup></gb:math> and <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:msub><kb:mi>P</kb:mi><kb:mi>c</kb:mi></kb:msub><kb:mo stretchy="false">(</kb:mo><kb:mn>4457</kb:mn><kb:mo stretchy="false">)</kb:mo><kb:mn>3</kb:mn><kb:mo>/</kb:mo><kb:msup><kb:mn>2</kb:mn><kb:mo>−</kb:mo></kb:msup></kb:math> states both turn out as composite states of <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:msup><ob:mover accent="true"><ob:mi>D</ob:mi><ob:mo stretchy="false">¯</ob:mo></ob:mover><ob:mo>*</ob:mo></ob:msup><ob:msub><ob:mi mathvariant="normal">Σ</ob:mi><ob:mi>c</ob:mi></ob:msub></ob:math>. The upper limits of the values of their elementariness are estimated to be rather small. This paper provides an additional confirmation of the molecular interpretation for the <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"><tb:msub><tb:mi>P</tb:mi><tb:mi>c</tb:mi></tb:msub></tb:math> states in the literature.