Likely existence of bound states and the Efimov effect in the triple-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>J</mml:mi><mml:mo>/</mml:mo><mml:mi>ψ</mml:mi></mml:math> system

The ground-breaking discovery of the first fully charmed tetraquark state <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>X</a:mi><a:mo stretchy="false">(</a:mo><a:mn>6900</a:mn><a:mo stretchy="false">)</a:mo></a:math> in the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>J</e:mi><e:mo>/</e:mo><e:mi>ψ</e:mi><e:mi>J</e:mi><e:mo>/</e:mo><e:mi>ψ</e:mi></e:math> invariant mass distribution by the LHCb collaboration has inspired intensive theoretical studies. Various interpretations, such as molecular states, compact tetraquark states, and coupled-channel effects, have been proposed for these states. Of particular interest is the ongoing search for the triple-<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>J</g:mi><g:mo>/</g:mo><g:mi>ψ</g:mi></g:math> state–a fully-charmed hexaquark state. To deepen our understanding of the triple-<i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>J</i:mi><i:mo>/</i:mo><i:mi>ψ</i:mi></i:math> state and to guide future experimental searches, we study the triple-<k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>J</k:mi><k:mo>/</k:mo><k:mi>ψ</k:mi></k:math> system in this work employing the Gaussian expansion method and the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>J</m:mi><m:mo>/</m:mo><m:mi>ψ</m:mi><m:mi>J</m:mi><m:mo>/</m:mo><m:mi>ψ</m:mi></m:math> potential parametrized to yield a shallow bound state, as suggested in several theoretical works. Our results support a triple-<o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>J</o:mi><o:mo>/</o:mo><o:mi>ψ</o:mi></o:math> bound state, even in cases where the attractive interaction between the two <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>J</q:mi><q:mo>/</q:mo><q:mi>ψ</q:mi></q:math> mesons is very weak. Moreover, our analysis implies the Efimov effect in the triple-<s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>J</s:mi><s:mo>/</s:mo><s:mi>ψ</s:mi></s:math> system. In addition, we extend our investigation to the triple-<u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:mi mathvariant="normal">ϒ</u:mi><u:mo stretchy="false">(</u:mo><u:mn>1</u:mn><u:mi>S</u:mi><u:mo stretchy="false">)</u:mo></u:mrow></u:math> system and obtain results similar to those for the triple-<z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mi>J</z:mi><z:mo>/</z:mo><z:mi>ψ</z:mi></z:math> system. Published by the American Physical Society 2024