Constrain the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>χ</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>J</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">→</mml:mo> <mml:msup> <mml:mi>D</

Hidden-charm decays provide irreplaceable platforms for probing the structures of charmoniumlike states, such as <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> , <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>Y</e:mi> <e:mo stretchy="false">(</e:mo> <e:mn>4260</e:mn> <e:mo stretchy="false">)</e:mo> </e:math> , and <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> , and their heavy-quark-symmetry partners. In the hadronic molecular scenario, these hidden-charm decays are dominated by intermediate meson loops (IMLs), and the couplings of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi>χ</m:mi> <m:mrow> <m:mi>c</m:mi> <m:mi>J</m:mi> </m:mrow> </m:msub> <m:mo stretchy="false">→</m:mo> <m:msup> <m:mi>D</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mo>*</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msup> <m:msup> <m:mover accent="true"> <m:mi>D</m:mi> <m:mo stretchy="false">¯</m:mo> </m:mover> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mo>*</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msup> </m:math> are essential building blocks of the amplitudes for the pionic and radiative transitions of the charmoniumlike states to the <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:msub> <v:mi>χ</v:mi> <v:mrow> <v:mi>c</v:mi> <v:mi>J</v:mi> </v:mrow> </v:msub> </v:math> and <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:msub> <x:mi>h</x:mi> <x:mi>c</x:mi> </x:msub> </x:math> states, e.g., <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:mi>X</z:mi> <z:mo stretchy="false">(</z:mo> <z:mn>3872</z:mn> <z:mo stretchy="false">)</z:mo> <z:mo stretchy="false">→</z:mo> <z:msup> <z:mi>π</z:mi> <z:mn>0</z:mn> </z:msup> <z:msub> <z:mi>χ</z:mi> <z:mrow> <z:mi>c</z:mi> <z:mi>J</z:mi> </z:mrow> </z:msub> <z:mo>,</z:mo> <z:mi>π</z:mi> <z:mi>π</z:mi> <z:msub> <z:mi>χ</z:mi> <z:mrow> <z:mi>c</z:mi> <z:mi>J</z:mi> </z:mrow> </z:msub> <z:mo>,</z:mo> <z:mi>γ</z:mi> <z:msub> <z:mi>χ</z:mi> <z:mrow> <z:mi>c</z:mi> <z:mi>J</z:mi> </z:mrow> </z:msub> </z:math> , and <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mi>Y</eb:mi> <eb:mo stretchy="false">(</eb:mo> <eb:mn>4260</eb:mn> <eb:mo stretchy="false">)</eb:mo> <eb:mo stretchy="false">→</eb:mo> <eb:msup> <eb:mi>π</eb:mi> <eb:mn>0</eb:mn> </eb:msup> <eb:msub> <eb:mi>h</eb:mi> <eb:mi>c</eb:mi> </eb:msub> <eb:mo>,</eb:mo> <eb:mi>η</eb:mi> <eb:msub> <eb:mi>h</eb:mi> <eb:mi>c</eb:mi> </eb:msub> </eb:math> . These couplings cannot be extracted from the partial decay widths of the <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"> <jb:msub> <jb:mi>χ</jb:mi> <jb:mrow> <jb:mi>c</jb:mi> <jb:mi>J</jb:mi> </jb:mrow> </jb:msub> </jb:math> directly and only have estimated values from the vector meson dominance (VMD) model. Utilizing the recent precise determination of the pole position and the isospin breaking properties of the <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"> <lb:mi>X</lb:mi> <lb:mo stretchy="false">(</lb:mo> <lb:mn>3872</lb:mn> <lb:mo stretchy="false">)</lb:mo> </lb:math> , we give an estimation on the upper bounds of the absolute values of the <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"> <pb:msub> <pb:mi>χ</pb:mi> <pb:mrow> <pb:mi>c</pb:mi> <pb:mi>J</pb:mi> </pb:mrow> </pb:msub> <pb:mo stretchy="false">→</pb:mo> <pb:msup> <pb:mi>D</pb:mi> <pb:mrow> <pb:mo stretchy="false">(</pb:mo> <pb:mo>*</pb:mo> <pb:mo stretchy="false">)</pb:mo> </pb:mrow> </pb:msup> <pb:msup> <pb:mover accent="true"> <pb:mi>D</pb:mi> <pb:mo stretchy="false">¯</pb:mo> </pb:mover> <pb:mrow> <pb:mo stretchy="false">(</pb:mo> <pb:mo>*</pb:mo> <pb:mo stretchy="false">)</pb:mo> </pb:mrow> </pb:msup> </pb:math> couplings. Our results show that the VMD model may overestimate the <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"> <yb:msub> <yb:mi>χ</yb:mi> <yb:mrow> <yb:mi>c</yb:mi> <yb:mi>J</yb:mi> </yb:mrow> </yb:msub> <yb:mo stretchy="false">→</yb:mo> <yb:msup> <yb:mi>D</yb:mi> <yb:mrow> <yb:mo stretchy="false">(</yb:mo> <yb:mo>*</yb:mo> <yb:mo stretchy="false">)</yb:mo> </yb:mrow> </yb:msup> <yb:msup> <yb:mover accent="true"> <yb:mi>D</yb:mi> <yb:mo stretchy="false">¯</yb:mo> </yb:mover> <yb:mrow> <yb:mo stretchy="false">(</yb:mo> <yb:mo>*</yb:mo> <yb:mo stretchy="false">)</yb:mo> </yb:mrow> </yb:msup> </yb:math> couplings considering the <hc:math xmlns:hc="http://www.w3.org/1998/Math/MathML" display="inline"> <hc:mi>X</hc:mi> <hc:mo stretchy="false">(</hc:mo> <hc:mn>3872</hc:mn> <hc:mo stretchy="false">)</hc:mo> </hc:math> as a <lc:math xmlns:lc="http://www.w3.org/1998/Math/Mat