Constraining the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>D</mml:mi><mml:mi>D</mml:mi><mml:msup><mml:mi>D</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:math> three-body bound state via the <mml:math xmlns:mml="ht

In this study, we propose using the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>Z</a:mi><a:mi>c</a:mi></a:msub><a:mo stretchy="false">(</a:mo><a:mn>3900</a:mn><a:mo stretchy="false">)</a:mo></a:math> pole position to constrain the existence of the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>D</e:mi><e:mi>D</e:mi><e:msup><e:mi>D</e:mi><e:mo>*</e:mo></e:msup></e:math> three-body bound state within the one-boson-exchange (OBE) model. The existence of the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>D</g:mi><g:mi>D</g:mi><g:msup><g:mi>D</g:mi><g:mo>*</g:mo></g:msup></g:math> bound state remains uncertain due to significant variations in the OBE interaction, particularly in the strength of scalar-meson-exchange interactions, which can differ by a factor about 20 between two commonly used OBE models. This discrepancy renders the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>D</i:mi><i:mi>D</i:mi><i:msup><i:mi>D</i:mi><i:mo>*</i:mo></i:msup></i:math> system highly model-dependent. To address this issue, we constrain the scalar-meson-exchange interaction using the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>Z</k:mi><k:mi>c</k:mi></k:msub><k:mo stretchy="false">(</k:mo><k:mn>3900</k:mn><k:mo stretchy="false">)</k:mo></k:math> pole position, where the pseudoscalar-meson coupling is well-determined, and the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>ρ</o:mi></o:math>- and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>ω</q:mi></q:math>-exchange interactions nearly cancel each other out, leaving the coupling constant of the <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>σ</s:mi></s:math>-exchange as the only unknown parameter. Our results indicate that the isospin-<u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mfrac><u:mn>1</u:mn><u:mn>2</u:mn></u:mfrac></u:math> <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>D</w:mi><w:mi>D</w:mi><w:msup><w:mi>D</w:mi><w:mo>*</w:mo></w:msup></w:math> bound states exist when <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msub><y:mi>Z</y:mi><y:mi>c</y:mi></y:msub><y:mo stretchy="false">(</y:mo><y:mn>3900</y:mn><y:mo stretchy="false">)</y:mo></y:math> is a virtual state of <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mrow><cb:mi>D</cb:mi><cb:msup><cb:mrow><cb:mover accent="true"><cb:mrow><cb:mi>D</cb:mi></cb:mrow><cb:mrow><cb:mo stretchy="false">¯</cb:mo></cb:mrow></cb:mover></cb:mrow><cb:mrow><cb:mo>*</cb:mo></cb:mrow></cb:msup><cb:mo>/</cb:mo><cb:mover accent="true"><cb:mrow><cb:mi>D</cb:mi></cb:mrow><cb:mrow><cb:mo stretchy="false">¯</cb:mo></cb:mrow></cb:mover><cb:msup><cb:mrow><cb:mi>D</cb:mi></cb:mrow><cb:mrow><cb:mo>*</cb:mo></cb:mrow></cb:msup></cb:mrow></cb:math> located within approximately <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mo>−</ib:mo><ib:mn>15</ib:mn><ib:mtext> </ib:mtext><ib:mtext> </ib:mtext><ib:mi>MeV</ib:mi></ib:math> of the threshold. However, the three-body bound state is gone when the <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:msub><kb:mi>Z</kb:mi><kb:mi>c</kb:mi></kb:msub><kb:mo stretchy="false">(</kb:mo><kb:mn>3900</kb:mn><kb:mo stretchy="false">)</kb:mo></kb:math> virtual state pole is more than 20 MeV away from the threshold. Each experimental progress, either on the <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:mi>D</ob:mi><ob:mi>D</ob:mi><ob:msup><ob:mi>D</ob:mi><ob:mo>*</ob:mo></ob:msup></ob:math> state or the <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:msub><qb:mi>Z</qb:mi><qb:mi>c</qb:mi></qb:msub><qb:mo stretchy="false">(</qb:mo><qb:mn>3900</qb:mn><qb:mo stretchy="false">)</qb:mo></qb:math>, can shed light on the nature of the other state. Another significant outcome is a refined set of OBE model parameters calibrated using the pole positions of <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:mi>X</ub:mi><ub:mo stretchy="false">(</ub:mo><ub:mn>3872</ub:mn><ub:mo stretchy="false">)</ub:mo></ub:math>, <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:msub><yb:mi>T</yb:mi><yb:mrow><yb:mi>c</yb:mi><yb:mi>c</yb:mi></yb:mrow></yb:msub><yb:mo stretchy="false">(</yb:mo><yb:mn>3875</yb:mn><yb:mo stretchy="false">)</yb:mo></yb:math>, and <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"><cc:msub><cc:mi>Z</cc:mi><cc:mi>c</cc:mi></cc:msub><cc:mo stretchy="false">(</cc:mo><cc:mn>3900</cc:mn><cc:mo stretchy="false">)</cc:mo></cc:math>, rigorously addressing the cutoff dependence. These parameters provide a valuable resource for more accurate calculations of systems involving few-body <gc:math xmlns:gc="http://www.w3.org/1998/Math/MathML" display="inline"><gc:mi>D</gc:mi></gc:math>, <ic:math xmlns:ic="http://www.w3.org/1998/Math/MathML" display="inline"><ic:msup><ic:mi>D</ic:mi><ic:mo>*</ic:mo></ic:msup></ic:math>, and their antiparticles. Additionally, we find no evidence of the <kc:math xmlns:kc="http://www.w3.org/1998/Math/MathML" display="inline"><kc:mi>D</kc:mi><kc:mi>D</kc:mi><kc:msup><kc:mi>D</kc:mi><kc:mo>*</kc:mo></kc:msup></kc:math> three-body resonances after extensive search using a combination of the Gaussian expansion method and the complex scaling method.