Hidden-charm tetraquarks in a mixture model: Coupled-channel analysis with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>c</mml:mi><mml:mover accent="true"><mml:mi>c</mml:mi><mml:mo stretchy="false">¯</mml:mo></mml

The nature of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>X</a:mi><a:mo stretchy="false">(</a:mo><a:mn>3872</a:mn><a:mo stretchy="false">)</a:mo></a:math> and other exotic hadrons has been a subject of extensive investigation since the first observation of the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>X</e:mi><e:mo stretchy="false">(</e:mo><e:mn>3872</e:mn><e:mo stretchy="false">)</e:mo></e:math> in 2003. While various theoretical models have been proposed, including hadronic molecular and compact tetraquark interpretations, some experimental evidence suggests that the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>X</i:mi><i:mo stretchy="false">(</i:mo><i:mn>3872</i:mn><i:mo stretchy="false">)</i:mo></i:math> may be a mixture state of a hadronic molecule and a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>c</m:mi><m:mover accent="true"><m:mi>c</m:mi><m:mo stretchy="false">¯</m:mo></m:mover></m:math> core. In this work, we perform a systematic study of the hidden-charm tetraquark candidates <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>X</q:mi><q:mo stretchy="false">(</q:mo><q:mn>3860</q:mn><q:mo stretchy="false">)</q:mo></q:math>, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>X</u:mi><u:mo stretchy="false">(</u:mo><u:mn>3872</u:mn><u:mo stretchy="false">)</u:mo></u:math>, and <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>Z</y:mi><y:mo stretchy="false">(</y:mo><y:mn>3930</y:mn><y:mo stretchy="false">)</y:mo></y:math> using a coupled-channel model that incorporates both <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>c</cb:mi><cb:mover accent="true"><cb:mi>c</cb:mi><cb:mo stretchy="false">¯</cb:mo></cb:mover></cb:math> states and <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mrow><gb:msup><gb:mrow><gb:mi>D</gb:mi></gb:mrow><gb:mrow><gb:mo stretchy="false">(</gb:mo><gb:mo>*</gb:mo><gb:mo stretchy="false">)</gb:mo></gb:mrow></gb:msup><gb:msup><gb:mrow><gb:mover accent="true"><gb:mrow><gb:mi>D</gb:mi></gb:mrow><gb:mrow><gb:mo stretchy="false">¯</gb:mo></gb:mrow></gb:mover></gb:mrow><gb:mrow><gb:mo stretchy="false">(</gb:mo><gb:mo>*</gb:mo><gb:mo stretchy="false">)</gb:mo></gb:mrow></gb:msup></gb:mrow></gb:math> hadronic molecular components. The <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:mi>c</ob:mi><ob:mover accent="true"><ob:mi>c</ob:mi><ob:mo stretchy="false">¯</ob:mo></ob:mover></ob:math> sector is described based on the constituent quark model predictions for the <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:msub><sb:mi>χ</sb:mi><sb:mrow><sb:mi>c</sb:mi><sb:mi>J</sb:mi></sb:mrow></sb:msub><sb:mo stretchy="false">(</sb:mo><sb:mn>2</sb:mn><sb:mi>P</sb:mi><sb:mo stretchy="false">)</sb:mo></sb:math> (<wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:mi>J</wb:mi><wb:mo>=</wb:mo><wb:mn>0</wb:mn></wb:math>, 1, 2) states, while the meson-meson interactions are modeled using pseudoscalar and vector-meson exchange potentials. The model parameters are fixed to reproduce the masses of the <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:mi>X</yb:mi><yb:mo stretchy="false">(</yb:mo><yb:mn>3872</yb:mn><yb:mo stretchy="false">)</yb:mo></yb:math> and <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"><cc:mi>Z</cc:mi><cc:mo stretchy="false">(</cc:mo><cc:mn>3930</cc:mn><cc:mo stretchy="false">)</cc:mo></cc:math>, and the resulting framework is used to predict the mass and structure of the <gc:math xmlns:gc="http://www.w3.org/1998/Math/MathML" display="inline"><gc:msup><gc:mi>J</gc:mi><gc:mrow><gc:mi>P</gc:mi><gc:mi>C</gc:mi></gc:mrow></gc:msup><gc:mo>=</gc:mo><gc:msup><gc:mn>0</gc:mn><gc:mrow><gc:mo>+</gc:mo><gc:mo>+</gc:mo></gc:mrow></gc:msup></gc:math> state associated with the <ic:math xmlns:ic="http://www.w3.org/1998/Math/MathML" display="inline"><ic:mi>X</ic:mi><ic:mo stretchy="false">(</ic:mo><ic:mn>3860</ic:mn><ic:mo stretchy="false">)</ic:mo></ic:math>. Our results support the mixture interpretation of these exotic hadrons, exhibiting strong attractions from the transition potential between <mc:math xmlns:mc="http://www.w3.org/1998/Math/MathML" display="inline"><mc:mi>c</mc:mi><mc:mover accent="true"><mc:mi>c</mc:mi><mc:mo stretchy="false">¯</mc:mo></mc:mover></mc:math> and <qc:math xmlns:qc="http://www.w3.org/1998/Math/MathML" display="inline"><qc:msup><qc:mi>D</qc:mi><qc:mrow><qc:mo stretchy="false">(</qc:mo><qc:mo>*</qc:mo><qc:mo stretchy="false">)</qc:mo></qc:mrow></qc:msup><qc:msup><qc:mover accent="true"><qc:mi>D</qc:mi><qc:mo stretchy="false">¯</qc:mo></qc:mover><qc:mrow><qc:mo stretchy="false">(</qc:mo><qc:mo>*</qc:mo><qc:mo stretchy="false">)</qc:mo></qc:mrow></qc:msup></qc:math> components. The molecular component is found to dominate in the <yc:math xmlns:yc="http://www.w3.org/1998/Math/MathML" display="inline"><yc:mi>X</yc:mi><yc:mo stretchy="false">(</yc:mo><yc:mn>3872</yc:mn><yc:mo stretchy="false">)</yc:mo></yc:math>, while the <cd:math xmlns:cd="http://www.w3.org/1998/Math/MathML" display="inline"><cd:mi>c</cd:mi><cd:mover accent="true"><cd:mi>c</cd:mi><cd:mo stretchy="false">¯</cd:mo></cd:mover></cd:math> component plays a more prominent role in the <gd:math xmlns:gd="http://www.w3.org/1998/Math/MathML" display="inline"><gd:mi>X</gd:mi><gd:mo stretchy="false">(</gd:mo><gd:mn>3860</gd:mn><gd:mo stretchy="false">)</gd:mo></gd:math> and <kd:math xmlns:kd="http://www.w3.org/1998/Math/MathML" display="inline"><kd:mi>Z</kd:mi><kd:mo stretchy="false">(</kd:mo><kd:mn>3930</kd:mn><kd:mo stretchy="false">)</kd:mo></kd:math>.