Decoding the near-threshold <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">

To decode the near-threshold dynamics of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>X</a:mi> <a:mrow> <a:mn>0</a:mn> <a:mo>,</a:mo> <a:mn>1</a:mn> </a:mrow> </a:msub> <a:mo stretchy="false">(</a:mo> <a:mn>4140</a:mn> <a:mo stretchy="false">)</a:mo> </a:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>X</e:mi> <e:mn>1</e:mn> </e:msub> <e:mo stretchy="false">(</e:mo> <e:mn>4685</e:mn> <e:mo stretchy="false">)</e:mo> </e:math> states, we investigate the OZI-suppressed <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mo stretchy="false">{</i:mo> <i:msub> <i:mi>D</i:mi> <i:mi>s</i:mi> </i:msub> <i:msub> <i:mover accent="true"> <i:mi>D</i:mi> <i:mo stretchy="false">¯</i:mo> </i:mover> <i:mi>s</i:mi> </i:msub> <i:mo>,</i:mo> <i:mi>J</i:mi> <i:mo>/</i:mo> <i:mi>ψ</i:mi> <i:mi>ϕ</i:mi> <i:mo>,</i:mo> <i:msubsup> <i:mi>D</i:mi> <i:mi>s</i:mi> <i:mo>*</i:mo> </i:msubsup> <i:msubsup> <i:mover accent="true"> <i:mi>D</i:mi> <i:mo stretchy="false">¯</i:mo> </i:mover> <i:mi>s</i:mi> <i:mo>*</i:mo> </i:msubsup> <i:mo stretchy="false">}</i:mo> </i:math> coupled-channel scattering in <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>B</q:mi> <q:mo stretchy="false">→</q:mo> <q:msub> <q:mi>D</q:mi> <q:mi>s</q:mi> </q:msub> <q:msub> <q:mover accent="true"> <q:mi>D</q:mi> <q:mo stretchy="false">¯</q:mo> </q:mover> <q:mi>s</q:mi> </q:msub> <q:mi>K</q:mi> </q:math> decays using the effective range expansion. We demonstrate that the <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:msub> <v:mi>X</v:mi> <v:mn>0</v:mn> </v:msub> <v:mo stretchy="false">(</v:mo> <v:mn>4140</v:mn> <v:mo stretchy="false">)</v:mo> </v:math> , associated with a dip in the line shape, corresponds to a dynamically generated pole near the <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:mi>J</z:mi> <z:mo>/</z:mo> <z:mi>ψ</z:mi> <z:mi>ϕ</z:mi> </z:math> threshold. The single-channel <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:mi>J</bb:mi> <bb:mo>/</bb:mo> <bb:mi>ψ</bb:mi> <bb:mi>ϕ</bb:mi> </bb:math> scattering length is extracted to be <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"> <db:mn>1.11</db:mn> <db:mo>±</db:mo> <db:mn>0.65</db:mn> <db:mtext> </db:mtext> <db:mtext> </db:mtext> <db:mi>fm</db:mi> </db:math> , yielding an effective scattering length of <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"> <fb:msubsup> <fb:mn>0.12</fb:mn> <fb:mrow> <fb:mo>−</fb:mo> <fb:mn>0.10</fb:mn> </fb:mrow> <fb:mrow> <fb:mo>+</fb:mo> <fb:mn>0.20</fb:mn> </fb:mrow> </fb:msubsup> <fb:mo>+</fb:mo> <fb:mi>i</fb:mi> <fb:mn>0.7</fb:mn> <fb:msubsup> <fb:mn>8</fb:mn> <fb:mrow> <fb:mo>−</fb:mo> <fb:mn>0.40</fb:mn> </fb:mrow> <fb:mrow> <fb:mo>+</fb:mo> <fb:mn>0.20</fb:mn> </fb:mrow> </fb:msubsup> <fb:mtext> </fb:mtext> <fb:mtext> </fb:mtext> <fb:mi>fm</fb:mi> </fb:math> when coupled channels are included. By treating the spin-spin interaction as a subleading effect, we predict a <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"> <hb:msup> <hb:mi>J</hb:mi> <hb:mrow> <hb:mi>P</hb:mi> <hb:mi>C</hb:mi> </hb:mrow> </hb:msup> <hb:mo>=</hb:mo> <hb:msup> <hb:mn>1</hb:mn> <hb:mrow> <hb:mo>+</hb:mo> <hb:mo>+</hb:mo> </hb:mrow> </hb:msup> </hb:math> virtual state near the <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"> <jb:mi>J</jb:mi> <jb:mo>/</jb:mo> <jb:mi>ψ</jb:mi> <jb:mi>ϕ</jb:mi> </jb:math> threshold, which naturally resolves the empirical ambiguities surrounding the <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"> <lb:msub> <lb:mi>X</lb:mi> <lb:mn>1</lb:mn> </lb:msub> <lb:mo stretchy="false">(</lb:mo> <lb:mn>4140</lb:mn> <lb:mo stretchy="false">)</lb:mo> </lb:math> width. Extending this framework via heavy quark spin symmetry, we further interpret the <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"> <pb:msub> <pb:mi>X</pb:mi> <pb:mn>1</pb:mn> </pb:msub> <pb:mo stretchy="false">(</pb:mo> <pb:mn>4685</pb:mn> <pb:mo stretchy="false">)</pb:mo> </pb:math> as a <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"> <tb:mi>ψ</tb:mi> <tb:mo stretchy="false">(</tb:mo> <tb:mn>2</tb:mn> <tb:mi>S</tb:mi> <tb:mo stretchy="false">)</tb:mo> <tb:mi>ϕ</tb:mi> </tb:math> hadronic molecule. Ultimately, these findings highlight how the <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" display="inline"> <xb:mi>X</xb:mi> <xb:mo stretchy="false">(</xb:mo> <xb:mn>4140</xb:mn> <xb:mo stretchy="false">)</xb:mo> </xb:math> family and <bc:math xmlns:bc="http://www.w3.org/1998/Math/MathML" display="inline"> <bc:msub> <bc:mi>X</bc:mi> <bc:mn>1</bc:mn> </bc:msub> <bc:mo stretchy="false">(</bc:mo> <bc:mn>4685</bc:mn> <bc:mo stretchy="false">)</bc:mo> </bc:math> serve as unique theoretical windows into the Fierz rearrangement and OZI suppression mechanisms in low-energy strong interactions.