Pole trajectories from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math>- and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>P</mml:mi></mml:math>-wave <mml:math

In this work, we investigate the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>S</a:mi></a:math>- and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>P</c:mi></c:math>-wave interactions of 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> system within the framework of the quasipotential Bethe-Salpeter equation, with the aim of exploring possible molecular states and their corresponding pole trajectories. The interaction potentials are constructed using the one-boson-exchange model, incorporating the exchanges of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>π</i:mi></i:math>, <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>η</k:mi></k:math>, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>ρ</m:mi></m:math>, <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>ω</o:mi></o:math>, <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>σ</q:mi></q:math>, and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>J</s:mi><s:mo>/</s:mo><s:mi>ψ</s:mi></s:math> mesons, based on heavy quark effective Lagrangians. The poles of the scattering amplitude are analyzed and their evolution on two Riemann sheets is systematically traced as the cutoff parameter increases up to 5 GeV. We identify four molecular states arising from the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>S</u:mi></u:math>-wave <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>D</w:mi><w:msup><w:mover accent="true"><w:mi>D</w:mi><w:mo stretchy="false">¯</w:mo></w:mover><w:mo>*</w:mo></w:msup></w:math> interaction. Among them, the bound state with quantum numbers <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msup><ab:mi>I</ab:mi><ab:mi>G</ab:mi></ab:msup><ab:mo stretchy="false">(</ab:mo><ab:msup><ab:mi>J</ab:mi><ab:mrow><ab:mi>P</ab:mi><ab:mi>C</ab:mi></ab:mrow></ab:msup><ab:mo stretchy="false">)</ab:mo><ab:mo>=</ab:mo><ab:msup><ab:mn>0</ab:mn><ab:mo>+</ab:mo></ab:msup><ab:mo stretchy="false">(</ab:mo><ab:msup><ab:mn>1</ab:mn><ab:mrow><ab:mo>+</ab:mo><ab:mo>+</ab:mo></ab:mrow></ab:msup><ab:mo stretchy="false">)</ab:mo></ab:math> corresponds well to the experimentally observed <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi>X</gb:mi><gb:mo stretchy="false">(</gb:mo><gb:mn>3872</gb:mn><gb:mo stretchy="false">)</gb:mo></gb:math>, while its isovector partner with <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:msup><kb:mi>I</kb:mi><kb:mi>G</kb:mi></kb:msup><kb:mo stretchy="false">(</kb:mo><kb:msup><kb:mi>J</kb:mi><kb:mrow><kb:mi>P</kb:mi><kb:mi>C</kb:mi></kb:mrow></kb:msup><kb:mo stretchy="false">)</kb:mo><kb:mo>=</kb:mo><kb:msup><kb:mn>1</kb:mn><kb:mo>−</kb:mo></kb:msup><kb:mo stretchy="false">(</kb:mo><kb:msup><kb:mn>1</kb:mn><kb:mrow><kb:mo>+</kb:mo><kb:mo>+</kb:mo></kb:mrow></kb:msup><kb:mo stretchy="false">)</kb:mo></kb:math> is found to exist only as a virtual state. Additionally, a <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:msup><qb:mn>0</qb:mn><qb:mo>−</qb:mo></qb:msup><qb:mo stretchy="false">(</qb:mo><qb:msup><qb:mn>1</qb:mn><qb:mrow><qb:mo>+</qb:mo><qb:mo>−</qb:mo></qb:mrow></qb:msup><qb:mo stretchy="false">)</qb:mo></qb:math> state appears as a bound state. The isovector <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:msup><ub:mn>1</ub:mn><ub:mrow><ub:mo>+</ub:mo><ub:mo>−</ub:mo></ub:mrow></ub:msup></ub:math> state, which may be associated with the <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:msub><wb:mi>Z</wb:mi><wb:mi>c</wb:mi></wb:msub><wb:mo stretchy="false">(</wb:mo><wb:mn>3900</wb:mn><wb:mo stretchy="false">)</wb:mo></wb:math>, is observed to evolve from a bound state to a virtual state as the interaction strength decreases. For the <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:mi>P</ac:mi></ac:math>-wave <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"><cc:mi>D</cc:mi><cc:msup><cc:mover accent="true"><cc:mi>D</cc:mi><cc:mo stretchy="false">¯</cc:mo></cc:mover><cc:mo>*</cc:mo></cc:msup></cc:math> interaction, the structure <gc:math xmlns:gc="http://www.w3.org/1998/Math/MathML" display="inline"><gc:mi>G</gc:mi><gc:mo stretchy="false">(</gc:mo><gc:mn>3900</gc:mn><gc:mo stretchy="false">)</gc:mo></gc:math> recently observed at BESIII is likely connected to a <kc:math xmlns:kc="http://www.w3.org/1998/Math/MathML" display="inline"><kc:msup><kc:mn>0</kc:mn><kc:mo>−</kc:mo></kc:msup><kc:mo stretchy="false">(</kc:mo><kc:msup><kc:mn>1</kc:mn><kc:mrow><kc:mo>−</kc:mo><kc:mo>−</kc:mo></kc:mrow></kc:msup><kc:mo stretchy="false">)</kc:mo></kc:math> state. A <oc:math xmlns:oc="http://www.w3.org/1998/Math/MathML" display="inline"><oc:msup><oc:mn>0</oc:mn><oc:mo>+</oc:mo></oc:msup><oc:mo stretchy="false">(</oc:mo><oc:msup><oc:mn>0</oc:mn><oc:mrow><oc:mo>−</oc:mo><oc:mo>+</oc:mo></oc:mrow></oc:msup><oc:mo stretchy="false">)</oc:mo></oc:math> state is also predicted in this channel. Both can appear as either resonance or bound/virtual state depending on the interaction strength.