Trilepton and tetralepton bound and resonant states: The QED counterpart of multiquark states

This work presents the first prediction of tetralepton resonant states containing muons, extending beyond the simplest tetralepton system, dipositronium (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msub><a:mrow><a:mi>Ps</a:mi></a:mrow><a:mrow><a:mn>2</a:mn></a:mrow></a:msub></a:mrow></a:math>). With the rapid advancements in experimental facilities, the production and study of these intriguing states may be within reach. We perform a comprehensive analysis of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>S</c:mi></c:math>-wave trilepton and tetralepton systems within the framework of a QED Coulomb potential. We employ the Gaussian expansion method to solve the three- or four-body Schrödinger equation and utilize the complex scaling method to identify resonant states. We uncover a series of bound and resonant states in the trilepton systems <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi>e</e:mi><e:mo>+</e:mo></e:msup><e:msup><e:mi>e</e:mi><e:mo>+</e:mo></e:msup><e:msup><e:mi>e</e:mi><e:mo>−</e:mo></e:msup></e:math>, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msup><g:mi>μ</g:mi><g:mo>+</g:mo></g:msup><g:msup><g:mi>μ</g:mi><g:mo>+</g:mo></g:msup><g:msup><g:mi>μ</g:mi><g:mo>−</g:mo></g:msup></g:math>, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msup><i:mi>e</i:mi><i:mo>+</i:mo></i:msup><i:msup><i:mi>e</i:mi><i:mo>+</i:mo></i:msup><i:msup><i:mi>μ</i:mi><i:mo>−</i:mo></i:msup></i:math>, and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>μ</k:mi><k:mo>+</k:mo></k:msup><k:msup><k:mi>μ</k:mi><k:mo>+</k:mo></k:msup><k:msup><k:mi>e</k:mi><k:mo>−</k:mo></k:msup></k:math>, as well as the tetralepton systems <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msup><m:mi>e</m:mi><m:mo>+</m:mo></m:msup><m:msup><m:mi>e</m:mi><m:mo>+</m:mo></m:msup><m:msup><m:mi>e</m:mi><m:mo>−</m:mo></m:msup><m:msup><m:mi>e</m:mi><m:mo>−</m:mo></m:msup></m:math>, <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msup><o:mi>μ</o:mi><o:mo>+</o:mo></o:msup><o:msup><o:mi>μ</o:mi><o:mo>+</o:mo></o:msup><o:msup><o:mi>μ</o:mi><o:mo>−</o:mo></o:msup><o:msup><o:mi>μ</o:mi><o:mo>−</o:mo></o:msup></o:math>, and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msup><q:mi>μ</q:mi><q:mo>+</q:mo></q:msup><q:msup><q:mi>μ</q:mi><q:mo>+</q:mo></q:msup><q:msup><q:mi>e</q:mi><q:mo>−</q:mo></q:msup><q:msup><q:mi>e</q:mi><q:mo>−</q:mo></q:msup></q:math>. The energies of these states range from <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mo>−</s:mo><s:mn>30</s:mn><s:mtext> </s:mtext><s:mtext> </s:mtext><s:mi>eV</s:mi></s:math> to <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mo>−</u:mo><u:mn>1</u:mn><u:mtext> </u:mtext><u:mtext> </u:mtext><u:mi>eV</u:mi></u:math> below the total mass of three or four leptons, with their widths varying from less than 0.01 eV to approximately 0.07 eV. Additionally, we calculate the spin configurations and root mean square radii of these states, providing insight into their spatial structures. No bound or resonant states are found in the trilepton <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:msup><w:mi>e</w:mi><w:mo>+</w:mo></w:msup><w:msup><w:mi>μ</w:mi><w:mo>+</w:mo></w:msup><w:msup><w:mi>e</w:mi><w:mo>−</w:mo></w:msup></w:math>, <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msup><y:mi>μ</y:mi><y:mo>+</y:mo></y:msup><y:msup><y:mi>e</y:mi><y:mo>+</y:mo></y:msup><y:msup><y:mi>μ</y:mi><y:mo>−</y:mo></y:msup></y:math> systems, nor in the tetralepton <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msup><ab:mi>μ</ab:mi><ab:mo>+</ab:mo></ab:msup><ab:msup><ab:mi>e</ab:mi><ab:mo>+</ab:mo></ab:msup><ab:msup><ab:mi>μ</ab:mi><ab:mo>−</ab:mo></ab:msup><ab:msup><ab:mi>e</ab:mi><ab:mo>−</ab:mo></ab:msup></ab:math> system. A comparison with fully heavy tetraquark systems reveals that the additional color degree of freedom in QCD results in the absence of low-energy bound and resonant states. However, this extra degree of freedom allows for a broader range of <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mrow><cb:msup><cb:mrow><cb:mi>J</cb:mi></cb:mrow><cb:mrow><cb:mi>P</cb:mi><cb:mi>C</cb:mi></cb:mrow></cb:msup></cb:mrow></cb:math> quantum numbers to produce resonant states, highlighting the rich complexity of QCD systems.