<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> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>χ</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3872</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> with the complex scaling method and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>D</mml:mi> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mover accent="true"> <mml:mi>D</mml:mi> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> <mml:mo stretchy="false">)</mml:mo> <mml:mi>π</mml:mi> </mml:math> three-body effect

Abstract

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. Published by the American Physical Society 2024

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