Dipion transitions from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>X</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3872</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> to <mml:math xmlns:mml="ht

In this work, we investigate the dipion transition processes <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:mo stretchy="false">→</a:mo> <a:mi>π</a:mi> <a:mi>π</a:mi> <a:msub> <a:mi>χ</a:mi> <a:mrow> <a:mi>c</a:mi> <a:mi>J</a:mi> </a:mrow> </a:msub> </a:math> ( <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mrow> <f:mi>J</f:mi> <f:mo>=</f:mo> <f:mn>0</f:mn> </f:mrow> </f:math> , 1, 2) within the framework of heavy hadron chiral perturbation theory, treating <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mi>X</h:mi> <h:mo stretchy="false">(</h:mo> <h:mn>3872</h:mn> <h:mo stretchy="false">)</h:mo> </h:math> as a molecular state composed of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mi>D</l:mi> <l:msup> <l:mover accent="true"> <l:mi>D</l:mi> <l:mo stretchy="false">¯</l:mo> </l:mover> <l:mo>*</l:mo> </l:msup> <l:mo>+</l:mo> <l:mi mathvariant="normal">H</l:mi> <l:mo>.</l:mo> <l:mi mathvariant="normal">c</l:mi> <l:mo>.</l:mo> </l:math> components. By analyzing the box and triangle loop diagrams with the nonrelativistic effective field theory power-counting rule, we demonstrate that box diagrams dominate these dipion transition processes. Branching ratios are calculated as functions of the mixing angle <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:mi>θ</r:mi> </r:math> , which parametrizes the neutral and charged meson compositions of the <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi>X</t:mi> <t:mo stretchy="false">(</t:mo> <t:mn>3872</t:mn> <t:mo stretchy="false">)</t:mo> </t:math> . Our results indicate that the branching fractions for <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:mi>X</x:mi> <x:mo stretchy="false">(</x:mo> <x:mn>3872</x:mn> <x:mo stretchy="false">)</x:mo> <x:mo stretchy="false">→</x:mo> <x:mi>π</x:mi> <x:mi>π</x:mi> <x:msub> <x:mi>χ</x:mi> <x:mrow> <x:mi>c</x:mi> <x:mn>0</x:mn> </x:mrow> </x:msub> </x:math> , <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mi>X</cb:mi> <cb:mo stretchy="false">(</cb:mo> <cb:mn>3872</cb:mn> <cb:mo stretchy="false">)</cb:mo> <cb:mo stretchy="false">→</cb:mo> <cb:mi>π</cb:mi> <cb:mi>π</cb:mi> <cb:msub> <cb:mi>χ</cb:mi> <cb:mrow> <cb:mi>c</cb:mi> <cb:mn>1</cb:mn> </cb:mrow> </cb:msub> </cb:math> , and <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"> <hb:mi>X</hb:mi> <hb:mo stretchy="false">(</hb:mo> <hb:mn>3872</hb:mn> <hb:mo stretchy="false">)</hb:mo> <hb:mo stretchy="false">→</hb:mo> <hb:mi>π</hb:mi> <hb:mi>π</hb:mi> <hb:msub> <hb:mi>χ</hb:mi> <hb:mrow> <hb:mi>c</hb:mi> <hb:mn>2</hb:mn> </hb:mrow> </hb:msub> </hb:math> are of the orders of <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"> <mb:msup> <mb:mn>10</mb:mn> <mb:mrow> <mb:mo>−</mb:mo> <mb:mn>4</mb:mn> </mb:mrow> </mb:msup> </mb:math> , <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"> <ob:msup> <ob:mn>10</ob:mn> <ob:mrow> <ob:mo>−</ob:mo> <ob:mn>3</ob:mn> </ob:mrow> </ob:msup> </ob:math> , and <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"> <qb:msup> <qb:mn>10</qb:mn> <qb:mrow> <qb:mo>−</qb:mo> <qb:mn>5</qb:mn> </qb:mrow> </qb:msup> </qb:math> , respectively. We also predict the ratios <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"> <sb:mrow> <sb:mi mathvariant="script">B</sb:mi> <sb:mo stretchy="false">[</sb:mo> <sb:mi>X</sb:mi> <sb:mo stretchy="false">(</sb:mo> <sb:mn>3872</sb:mn> <sb:mo stretchy="false">)</sb:mo> <sb:mo stretchy="false">→</sb:mo> <sb:mi>π</sb:mi> <sb:mi>π</sb:mi> <sb:msub> <sb:mi>χ</sb:mi> <sb:mrow> <sb:mi>c</sb:mi> <sb:mn>0</sb:mn> <sb:mo>/</sb:mo> <sb:mn>2</sb:mn> </sb:mrow> </sb:msub> <sb:mo stretchy="false">]</sb:mo> </sb:mrow> <sb:mo>/</sb:mo> <sb:mrow> <sb:mi mathvariant="script">B</sb:mi> <sb:mo stretchy="false">[</sb:mo> <sb:mi>X</sb:mi> <sb:mo stretchy="false">(</sb:mo> <sb:mn>3872</sb:mn> <sb:mo stretchy="false">)</sb:mo> <sb:mo stretchy="false">→</sb:mo> <sb:mi>π</sb:mi> <sb:mi>π</sb:mi> <sb:msub> <sb:mi>χ</sb:mi> <sb:mrow> <sb:mi>c</sb:mi> <sb:mn>1</sb:mn> </sb:mrow> </sb:msub> <sb:mo stretchy="false">]</sb:mo> </sb:mrow> </sb:math> and <gc:math xmlns:gc="http://www.w3.org/1998/Math/MathML" display="inline"> <gc:mrow> <gc:mi mathvariant="script">B</gc:mi> <gc:mo stretchy="false">[</gc:mo> <gc:mi>X</gc:mi> <gc:mo stretchy="false">(</gc:mo> <gc:mn>3872</gc:mn> <gc:mo stretchy="false">)</gc:mo> <gc:mo stretchy="false">→</gc:mo> <gc:msup> <gc:mi>π</gc:mi> <gc:mo>+</gc:mo> </gc:msup> <gc:msup> <gc:mi>π</gc:mi> <gc:mo>−</gc:mo> </gc:msup> <gc:msub> <gc:mi>χ</gc:mi> <gc:mrow> <gc:mi>c</gc:mi> <gc:mi>J</gc:mi> </gc:mrow> </gc:msub> <gc:mo stretchy="false">]</gc:mo> </gc:mrow> <gc:mo>/</gc:mo> <gc:mrow> <gc:mi mathvariant="script">B</gc:mi> <gc:mo stretchy="false">[</gc:mo>