Production of leptonium in heavy quarkonium decays

Lepton pairs with opposite charges can form bound states known as “leptonium” through quantum electrodynamic interactions. Heavy quarkonia such as <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>J</a:mi><a:mo>/</a:mo><a:mi>ψ</a:mi></a:math> are abundantly produced at facilities like BESIII and the future Super Tau-Charm Facility (STCF). In this work, we investigate leptonium production in heavy quarkonium decays, specifically focusing on the processes <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:mi mathvariant="script">Q</c:mi><c:mo stretchy="false">⟶</c:mo><c:mo stretchy="false">(</c:mo><c:msubsup><c:mrow><c:mi>l</c:mi></c:mrow><c:mrow><c:mn>1</c:mn></c:mrow><c:mrow><c:mo>+</c:mo></c:mrow></c:msubsup><c:msubsup><c:mrow><c:mi>l</c:mi></c:mrow><c:mrow><c:mn>2</c:mn></c:mrow><c:mrow><c:mo>−</c:mo></c:mrow></c:msubsup><c:mo stretchy="false">)</c:mo><c:mo stretchy="false">[</c:mo><c:mi>n</c:mi><c:mo stretchy="false">]</c:mo><c:mo>+</c:mo><c:mi>γ</c:mi></c:mrow></c:math> (<k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>l</k:mi><k:mrow><k:mn>1</k:mn><k:mo>,</k:mo><k:mn>2</k:mn></k:mrow></k:msub><k:mo>=</k:mo><k:mi>τ</k:mi></k:math>, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:mi>μ</m:mi></m:mrow></m:math>, <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mrow><o:mi>e</o:mi></o:mrow></o:math>) and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi mathvariant="script">Q</q:mi><q:mo stretchy="false">⟶</q:mo><q:mo stretchy="false">(</q:mo><q:msubsup><q:mi>l</q:mi><q:mn>1</q:mn><q:mo>+</q:mo></q:msubsup><q:msubsup><q:mi>l</q:mi><q:mn>2</q:mn><q:mo>−</q:mo></q:msubsup><q:mo stretchy="false">)</q:mo><q:mo stretchy="false">[</q:mo><q:mi>n</q:mi><q:mo stretchy="false">]</q:mo><q:mo>+</q:mo><q:msubsup><q:mi>l</q:mi><q:mn>1</q:mn><q:mo>−</q:mo></q:msubsup><q:msubsup><q:mi>l</q:mi><q:mn>2</q:mn><q:mo>+</q:mo></q:msubsup></q:math>. Here, <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi mathvariant="script">Q</y:mi></y:math> denotes the heavy quarkonium <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:mi>J</bb:mi><bb:mo>/</bb:mo><bb:mi>ψ</bb:mi></bb:math> or <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:mi mathvariant="normal">ϒ</db:mi></db:math>, while <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi>n</gb:mi><gb:mo>=</gb:mo><gb:msub><gb:mrow><gb:mmultiscripts><gb:mrow><gb:mi>S</gb:mi></gb:mrow><gb:mprescripts/><gb:none/><gb:mrow><gb:mn>1</gb:mn></gb:mrow></gb:mmultiscripts></gb:mrow><gb:mrow><gb:mn>0</gb:mn></gb:mrow></gb:msub></gb:math> or <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mrow><ib:mmultiscripts><ib:mrow><ib:msub><ib:mrow><ib:mi>S</ib:mi></ib:mrow><ib:mrow><ib:mn>1</ib:mn></ib:mrow></ib:msub></ib:mrow><ib:mprescripts/><ib:none/><ib:mrow><ib:mn>3</ib:mn></ib:mrow></ib:mmultiscripts></ib:mrow></ib:math> corresponds to para-leptonium and ortho-leptonium, respectively. With an annual production of <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:mn>3.4</kb:mn><kb:mo>×</kb:mo><kb:msup><kb:mn>10</kb:mn><kb:mn>12</kb:mn></kb:msup><kb:mtext> </kb:mtext><kb:mtext> </kb:mtext><kb:mi>J</kb:mi><kb:mo>/</kb:mo><kb:mi>ψ</kb:mi></kb:math> events at STCF, there is significant potential to observe positronium <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:mo stretchy="false">(</mb:mo><mb:msup><mb:mi>e</mb:mi><mb:mo>+</mb:mo></mb:msup><mb:msup><mb:mi>e</mb:mi><mb:mo>−</mb:mo></mb:msup><mb:mo stretchy="false">)</mb:mo></mb:math>, muonium <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:mo stretchy="false">(</qb:mo><qb:msup><qb:mi>μ</qb:mi><qb:mo>+</qb:mo></qb:msup><qb:msup><qb:mi>e</qb:mi><qb:mo>−</qb:mo></qb:msup><qb:mo stretchy="false">)</qb:mo></qb:math>, and dimuonium <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:mo stretchy="false">(</ub:mo><ub:msup><ub:mi>μ</ub:mi><ub:mo>+</ub:mo></ub:msup><ub:msup><ub:mi>μ</ub:mi><ub:mo>−</ub:mo></ub:msup><ub:mo stretchy="false">)</ub:mo></ub:math>. In particular, the ortho-dimuonium <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:mo stretchy="false">(</yb:mo><yb:msup><yb:mi>μ</yb:mi><yb:mo>+</yb:mo></yb:msup><yb:msup><yb:mi>μ</yb:mi><yb:mo>−</yb:mo></yb:msup><yb:mo stretchy="false">)</yb:mo><yb:msup><yb:mo stretchy="false">[</yb:mo><yb:mn>3</yb:mn></yb:msup><yb:msub><yb:mi>S</yb:mi><yb:mn>1</yb:mn></yb:msub><yb:mo stretchy="false">]</yb:mo></yb:math> may be discovered at the future STCF, with an inclusive branching fraction of <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:mrow><ec:mstyle mathvariant="italic"><ec:mi>Br</ec:mi></ec:mstyle><ec:mo stretchy="false">(</ec:mo><ec:mi>J</ec:mi><ec:mo>/</ec:mo><ec:mi>ψ</ec:mi><ec:mo stretchy="false">⟶</ec:mo><ec:mo stretchy="false">(</ec:mo><ec:msup><ec:mrow><ec:mi>μ</ec:mi></ec:mrow><ec:mrow><ec:mo>+</ec:mo></ec:mrow></ec:msup><ec:msup><ec:mrow><ec:mi>μ</ec:mi></ec:mrow><ec:mrow><ec:mo>−</ec:mo></ec:mrow></ec:msup><ec:mo stretchy="false">)</ec:mo><ec:msup><ec:mrow><ec:mo stretchy="false">[</ec:mo></ec:mrow><ec:mrow><ec:mn>3</ec:mn></ec:mrow></ec:msup><ec:msub><ec:mrow><ec:mi>S</ec:mi></ec:mrow><ec:mrow><ec:mn>1</ec:mn></ec:mrow></ec:msub><ec:mo stretchy="false">]</ec:mo><ec:mo>+</ec:mo><ec:mi>X</ec:mi><ec:mo stretchy="false">)</ec:mo><ec:mo>=</ec:mo><ec:mn>1.5</ec:mn><ec:mo>×</ec:mo><ec:msup><ec:mrow><ec:mn>10</ec:mn></ec:mrow><ec:mrow><ec:mo>−</ec:mo><ec:mn>12</ec:mn></ec:mrow></ec:msup></ec:mrow></ec:math>.