Isospin violation effect and three-body decays of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msubsup><mml:mi>T</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msubsup></

In this work, we make a study of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msubsup><a:mi>T</a:mi><a:mrow><a:mi>c</a:mi><a:mi>c</a:mi></a:mrow><a:mo>+</a:mo></a:msubsup></a:math> state observed by the LHCb collaboration in 2021. In obtaining the effective potentials using the one-boson-exchange potential model we use an exponential form factor and find that, in the short and medium range, the contributions of the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>π</c:mi></c:math>, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>ρ</e:mi></e:math>, and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>ω</g:mi></g:math> exchanges are comparable while in the long range the pion-exchange contribution is dominant. Based on the assumption that <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msubsup><i:mi>T</i:mi><i:mrow><i:mi>c</i:mi><i:mi>c</i:mi></i:mrow><i:mo>+</i:mo></i:msubsup></i:math> is a loosely bound state of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>D</k:mi><k:mo>*</k:mo></k:msup><k:mi>D</k:mi></k:math>, we focus on its three-body decay using the meson-exchange method. Considering that the difference between the thresholds of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msup><m:mi>D</m:mi><m:mrow><m:mo>*</m:mo><m:mo>+</m:mo></m:mrow></m:msup><m:msup><m:mi>D</m:mi><m:mn>0</m:mn></m:msup></m:math> and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msup><o:mi>D</o:mi><o:mrow><o:mo>*</o:mo><o:mn>0</o:mn></o:mrow></o:msup><o:msup><o:mi>D</o:mi><o:mo>+</o:mo></o:msup></o:math> is even larger than the binding energy of <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msubsup><q:mi>T</q:mi><q:mrow><q:mi>c</q:mi><q:mi>c</q:mi></q:mrow><q:mo>+</q:mo></q:msubsup></q:math>, the isospin-breaking effect is amplified by the small binding energy of <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msubsup><s:mi>T</s:mi><s:mrow><s:mi>c</s:mi><s:mi>c</s:mi></s:mrow><s:mo>+</s:mo></s:msubsup></s:math>. Explicitly including such an isospin-breaking effect we obtain, by solving the Schrödinger equation, that the probability of the isoscalar component is about 91% while that of the isovector component is around 9% for <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msubsup><u:mi>T</u:mi><u:mrow><u:mi>c</u:mi><u:mi>c</u:mi></u:mrow><u:mo>+</u:mo></u:msubsup></u:math>. Using the experimental value of the mass of <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:msubsup><w:mi>T</w:mi><w:mrow><w:mi>c</w:mi><w:mi>c</w:mi></w:mrow><w:mo>+</w:mo></w:msubsup></w:math> as an input, we obtain the wave function of <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msubsup><y:mi>T</y:mi><y:mrow><y:mi>c</y:mi><y:mi>c</y:mi></y:mrow><y:mo>+</y:mo></y:msubsup></y:math> and further obtain its width via the three-body hadronic as well as the radiative decays. The total width we obtain is in agreement with the experimental value of the LHCb measurement with a unitarized Breit-Wigner profile. Conversely, the current results support the conclusion that <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msubsup><ab:mi>T</ab:mi><ab:mrow><ab:mi>c</ab:mi><ab:mi>c</ab:mi></ab:mrow><ab:mo>+</ab:mo></ab:msubsup></ab:math> is a hadronic molecule of <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msup><cb:mi>D</cb:mi><cb:mo>*</cb:mo></cb:msup><cb:mi>D</cb:mi></cb:math>. Published by the American Physical Society 2024