Benchmark calculations of fully heavy compact and molecular tetraquark states

Abstract

We calculate the mass spectrum of the S-wave fully heavy tetraquark systems <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>Q</a:mi><a:mi>Q</a:mi><a:mover accent="true"><a:mrow><a:mi>Q</a:mi></a:mrow><a:mrow><a:mo stretchy="false">¯</a:mo></a:mrow></a:mover><a:mover accent="true"><a:mrow><a:mi>Q</a:mi></a:mrow><a:mrow><a:mo stretchy="false">¯</a:mo></a:mrow></a:mover><a:mo stretchy="false">(</a:mo><a:mi>Q</a:mi><a:mo>=</a:mo><a:mi>c</a:mi><a:mo>,</a:mo><a:mi>b</a:mi><a:mo stretchy="false">)</a:mo></a:mrow></a:math> with both normal <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mo stretchy="false">(</i:mo><i:msup><i:mi>J</i:mi><i:mrow><i:mi>P</i:mi><i:mi>C</i:mi></i:mrow></i:msup><i:mo>=</i:mo><i:msup><i:mn>0</i:mn><i:mrow><i:mo>+</i:mo><i:mo>+</i:mo></i:mrow></i:msup><i:mo>,</i:mo><i:msup><i:mn>1</i:mn><i:mrow><i:mo>+</i:mo><i:mo>−</i:mo></i:mrow></i:msup><i:mo>,</i:mo><i:msup><i:mn>2</i:mn><i:mrow><i:mo>+</i:mo><i:mo>+</i:mo></i:mrow></i:msup><i:mo stretchy="false">)</i:mo></i:math> and exotic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mo stretchy="false">(</m:mo><m:msup><m:mi>J</m:mi><m:mrow><m:mi>P</m:mi><m:mi>C</m:mi></m:mrow></m:msup><m:mo>=</m:mo><m:msup><m:mn>0</m:mn><m:mrow><m:mo>+</m:mo><m:mo>−</m:mo></m:mrow></m:msup><m:mo>,</m:mo><m:msup><m:mn>1</m:mn><m:mrow><m:mo>+</m:mo><m:mo>+</m:mo></m:mrow></m:msup><m:mo>,</m:mo><m:msup><m:mn>2</m:mn><m:mrow><m:mo>+</m:mo><m:mo>−</m:mo></m:mrow></m:msup><m:mo stretchy="false">)</m:mo></m:math> C-parities using three different quark potential models (AL1, AP1, BGS). The exotic C-parity systems refer to the ones that cannot be composed of two S-wave ground heavy quarkonia. We incorporate the molecular dimeson and compact diquark-antidiquark spatial correlations simultaneously, thereby discerning the actual configurations of the states. We employ the Gaussian expansion method to solve the four-body Schrödinger equation, and the complex scaling method to identify the resonant states. The mass spectra in three different models qualitatively agree with each other. We obtain several resonant states with <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msup><q:mi>J</q:mi><q:mrow><q:mi>P</q:mi><q:mi>C</q:mi></q:mrow></q:msup><q:mo>=</q:mo><q:msup><q:mn>0</q:mn><q:mrow><q:mo>+</q:mo><q:mo>+</q:mo></q:mrow></q:msup><q:mo>,</q:mo><q:msup><q:mn>1</q:mn><q:mrow><q:mo>+</q:mo><q:mo>−</q:mo></q:mrow></q:msup><q:mo>,</q:mo><q:msup><q:mn>2</q:mn><q:mrow><q:mo>+</q:mo><q:mo>+</q:mo></q:mrow></q:msup><q:mo>,</q:mo><q:msup><q:mn>1</q:mn><q:mrow><q:mo>+</q:mo><q:mo>+</q:mo></q:mrow></q:msup></q:math> in the mass region (6.92,7.30) GeV, some of which are good candidates of the experimentally observed <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>X</s:mi><s:mo stretchy="false">(</s:mo><s:mn>6900</s:mn><s:mo stretchy="false">)</s:mo></s:math> and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>X</w:mi><w:mo stretchy="false">(</w:mo><w:mn>7200</w:mn><w:mo stretchy="false">)</w:mo></w:math>. We also obtain several exotic C-parity zero-width states with <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><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>=</ab:mo><ab:msup><ab:mn>0</ab:mn><ab:mrow><ab:mo>+</ab:mo><ab:mo>−</ab:mo></ab:mrow></ab:msup></ab:math> and <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msup><cb:mn>2</cb:mn><cb:mrow><cb:mo>+</cb:mo><cb:mo>−</cb:mo></cb:mrow></cb:msup></cb:math>. These zero-width states have no corresponding S-wave diquarkonium threshold and can only decay strongly to final states with P-wave quarkonia. With the notation <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:msub><eb:mi>T</eb:mi><eb:mrow><eb:mn>4</eb:mn><eb:mi>Q</eb:mi><eb:mo>,</eb:mo><eb:mi>J</eb:mi><eb:mo stretchy="false">(</eb:mo><eb:mi>C</eb:mi><eb:mo stretchy="false">)</eb:mo></eb:mrow></eb:msub><eb:mo stretchy="false">(</eb:mo><eb:mi>M</eb:mi><eb:mo stretchy="false">)</eb:mo></eb:math>, we deduce from the root mean square radii that the <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:mi>X</kb:mi><kb:mo stretchy="false">(</kb:mo><kb:mn>7200</kb:mn><kb:mo stretchy="false">)</kb:mo></kb:math> candidates <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:msub><ob:mi>T</ob:mi><ob:mrow><ob:mn>4</ob:mn><ob:mi>c</ob:mi><ob:mo>,</ob:mo><ob:mn>0</ob:mn><ob:mo stretchy="false">(</ob:mo><ob:mo>+</ob:mo><ob:mo stretchy="false">)</ob:mo></ob:mrow></ob:msub><ob:mo stretchy="false">(</ob:mo><ob:mn>7173</ob:mn><ob:mo stretchy="false">)</ob:mo><ob:mo>,</ob:mo><ob:msub><ob:mi>T</ob:mi><ob:mrow><ob:mn>4</ob:mn><ob:mi>c</ob:mi><ob:mo>,</ob:mo><ob:mn>2</ob:mn><ob:mo stretchy="false">(</ob:mo><ob:mo>+</ob:mo><ob:mo stretchy="false">)</ob:mo></ob:mrow></ob:msub><ob:mo stretchy="false">(</ob:mo><ob:mn>7214</ob:mn><ob:mo stretchy="false">)</ob:mo></ob:math> and the state <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:msub><yb:mi>T</yb:mi><yb:mrow><yb:mn>4</yb:mn><yb:mi>c</yb:mi><yb:mo>,</yb:mo><yb:mn>1</yb:mn><yb:mo stretchy="false">(</yb:mo><yb:mo>−</yb:mo><yb:mo stretchy="false">)</yb:mo></yb:mrow></yb:msub><yb:mo stretchy="false">(</yb:mo><yb:mn>7191</yb:mn><yb:mo stretchy="false">)</yb:mo></yb:math> look like molecular states although most of the resonant and zero-width states are compact states. Published by the American Physical Society 2024

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