Hidden heavy flavor tetraquarks in the Born-Oppenheimer approximation

The Born-Oppenheimer approximation is one of the very successful tools for solving the hydrogen atom problem. The experimental discovery of hidden heavy flavor tetraquarks <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>Q</a:mi><a:mover accent="true"><a:mi>Q</a:mi><a:mo stretchy="false">¯</a:mo></a:mover><a:mi>q</a:mi><a:mover accent="true"><a:mi>q</a:mi><a:mo stretchy="false">¯</a:mo></a:mover></a:math> (<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:mi>Q</g:mi><g:mo>=</g:mo><g:mi>c</g:mi></g:mrow></g:math>, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mrow><i:mi>b</i:mi></i:mrow></i:math> and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>q</k:mi><k:mo>=</k:mo><k:mi>u</k:mi></k:math>, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:mi>d</m:mi></m:mrow></m:math>, <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mrow><o:mi>s</o:mi></o:mrow></o:math>) provides great possibilities for the hydrogen-bond-like structure of the quantum chromodynamics version. In this work, considering that the colors of <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>Q</q:mi><q:mover accent="true"><q:mi>Q</q:mi><q:mo stretchy="false">¯</q:mo></q:mover></q:math> and <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>q</u:mi><u:mover accent="true"><u:mi>q</u:mi><u:mo stretchy="false">¯</u:mo></u:mover></u:math> are both 8, the tetraquark <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>Q</y:mi><y:mover accent="true"><y:mi>Q</y:mi><y:mo stretchy="false">¯</y:mo></y:mover><y:mi>q</y:mi><y:mover accent="true"><y:mi>q</y:mi><y:mo stretchy="false">¯</y:mo></y:mover></y:math> system is formed by color coupling <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mn>8</eb:mn><eb:mo stretchy="false">⊗</eb:mo><eb:mn>8</eb:mn><eb:mo stretchy="false">→</eb:mo><eb:mn>1</eb:mn></eb:math>. In order to study the mass splitting caused by the color-spin hyperfine interaction, the color-spin basis vectors of the <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mi>S</ib:mi></ib:math>-wave tetraquark states are appropriately constructed. Then, we use the Born-Oppenheimer approximation to calculate the mass spectra of the <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:mi>S</kb:mi></kb:math>-wave hidden heavy flavor tetraquark states. The results show that some of the hidden heavy flavor exotic hadrons discovered experimentally can be well explained as this type of hydrogen-bond-like tetraquark structure. In addition, some candidates for tetraquark bound states are predicted and may be compact tetraquark states.