Tcc+ and χc1(3872) with the complex…

We use the leading-order contact interactions and one-pion-exchange potentials to investigate the newly observed double-charm state <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>. We employ the complex scaling method to search for the poles on the first Riemann sheet with respect to the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>D</c:mi><c:msup><c:mi>D</c:mi><c:mo>*</c:mo></c:msup></c:math> threshold and the second Riemann sheet with respect to 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:mi>π</e:mi></e:math> threshold. 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:mi>π</g:mi></g:math> three-body effect is important in this system since the intermediate states can go on shell. We involve the three-body effect through an energy-dependent one-pion-exchange potential, which results in a unitary cut at 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:mi>π</i:mi></i:math> three-body threshold. We find a pole corresponding to the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msubsup><k:mi>T</k:mi><k:mrow><k:mi>c</k:mi><k:mi>c</k:mi></k:mrow><k:mo>+</k:mo></k:msubsup></k:math> on the physical Riemann sheet. Its width is around 80 keV and nearly independent of the choice of the cutoff. Assuming the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>D</m:mi><m:mover accent="true"><m:mi>D</m:mi><m:mo stretchy="false">¯</m:mo></m:mover><m:mi>π</m:mi></m:math> and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>D</q:mi><q:msup><q:mover accent="true"><q:mi>D</q:mi><q:mo stretchy="false">¯</q:mo></q:mover><q:mo>*</q:mo></q:msup></q:math> channels as the main decay channels, we apply the similar calculations to the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>χ</u:mi><u:mrow><u:mi>c</u:mi><u:mn>1</u:mn></u:mrow></u:msub><u:mo stretchy="false">(</u:mo><u:mn>3872</u:mn><u:mo stretchy="false">)</u:mo></u:math>, and find its width is even smaller. The isospin breaking effect is so significant for the <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msub><y:mi>χ</y:mi><y:mrow><y:mi>c</y:mi><y:mn>1</y:mn></y:mrow></y:msub><y:mo stretchy="false">(</y:mo><y:mn>3872</y:mn><y:mo stretchy="false">)</y:mo></y:math> that it is mainly a molecular state of <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msup><cb:mi>D</cb:mi><cb:mn>0</cb:mn></cb:msup><cb:msup><cb:mover accent="true"><cb:mi>D</cb:mi><cb:mo stretchy="false">¯</cb:mo></cb:mover><cb:mrow><cb:mo>*</cb:mo><cb:mn>0</cb:mn></cb:mrow></cb:msup></cb:math>. Furthermore, we introduce a revised Schrödinger equation for unstable particles to include the contribution of the <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msup><gb:mi>D</gb:mi><gb:mo>*</gb:mo></gb:msup></gb:math> width, which is also revised by the three-body effect.