Implication of a negative effective range on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>D</mml:mi><mml:msup><mml:mover accent="true"><mml:mi>D</mml:mi><mml:mo stretchy="false">¯</mml:mo></mml:mover><mml:mo>*

A recent analysis of the LHCb data [Esposito , ] obtained a sizable negative effective range for 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>. This has attracted intensive discussions on whether <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> can be deemed as a <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>D</i:mi><i:msup><i:mover accent="true"><i:mi>D</i:mi><i:mo stretchy="false">¯</i:mo></i:mover><i:mo>*</i:mo></i:msup></i:math> molecular state. This work explicitly demonstrates that the negative effective range of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>X</m:mi><m:mo stretchy="false">(</m:mo><m:mn>3872</m:mn><m:mo stretchy="false">)</m:mo></m:math> does not contradict the molecular picture, adopting an effective field theory formulation of the <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>D</q:mi><q:msup><q:mover accent="true"><q:mi>D</q:mi><q:mo stretchy="false">¯</q:mo></q:mover><q:mo>*</q:mo></q:msup></q:math> interaction that can simultaneously reproduce the binding energy and effective range of the <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>. We elaborate on the implications of the large negative effective range of <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>X</y:mi><y:mo stretchy="false">(</y:mo><y:mn>3872</y:mn><y:mo stretchy="false">)</y:mo></y:math> and the small binding energy on the underlying <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>D</cb:mi><cb:msup><cb:mover accent="true"><cb:mi>D</cb:mi><cb:mo stretchy="false">¯</cb:mo></cb:mover><cb:mo>*</cb:mo></cb:msup></cb:math> interaction. Such results are relevant for a better understanding of hadronic molecules and their binding mechanism.