<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>Z</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>3900</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> in a hadronic mo

Studying exotic hadrons is a challenge against the conventional quark model, providing us with a good platform to deepen our understanding of the strong interaction. An inclusive study of the exotic hadrons in vacuum and at finite temperature is an intriguing approach to shed light on their nature. As a first step, we study the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>Z</a:mi><a:mi>c</a:mi></a:msub><a:mo stretchy="false">(</a:mo><a:mn>3900</a:mn><a:mo stretchy="false">)</a:mo></a:math> in both the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>D</e:mi><e:msup><e:mover accent="true"><e:mi>D</e:mi><e:mo stretchy="false">¯</e:mo></e:mover><e:mo>*</e:mo></e:msup></e:math> hadronic molecular and the triangle singularity pictures, and discuss the behaviors of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>Z</i:mi><i:mi>c</i:mi></i:msub><i:mo stretchy="false">(</i:mo><i:mn>3900</i:mn><i:mo stretchy="false">)</i:mo></i:math> at finite temperature in these two different pictures. As a result, we show the properties that its mass becomes smaller and its width becomes larger when the temperature increases, which are seen commonly in the hadronic molecular picture and in the triangle singularity picture. The enhanced widths in hot medium indicate that the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>Z</m:mi><m:mi>c</m:mi></m:msub><m:mo stretchy="false">(</m:mo><m:mn>3900</m:mn><m:mo stretchy="false">)</m:mo></m:math> will be dissociated at a sufficiently high temperature. This feature is also reflected by a decrease of the effective couplings in the hadronic molecular picture. It is concluded that, not only in vacuum but also in hot medium, the behaviors of <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>Z</q:mi><q:mi>c</q:mi></q:msub><q:mo stretchy="false">(</q:mo><q:mn>3900</q:mn><q:mo stretchy="false">)</q:mo></q:math> are similar in the two different interpretations of <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>Z</u:mi><u:mi>c</u:mi></u:msub><u:mo stretchy="false">(</u:mo><u:mn>3900</u:mn><u:mo stretchy="false">)</u:mo></u:math>.