γ and J/ψ production through the initial state radiation process in electron-positron annihilation at B factories

We study the initial state radiation (ISR) process, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msup><a:mrow><a:mi>e</a:mi></a:mrow><a:mrow><a:mo>−</a:mo></a:mrow></a:msup><a:msup><a:mrow><a:mi>e</a:mi></a:mrow><a:mrow><a:mo>+</a:mo></a:mrow></a:msup><a:msup><a:mrow><a:mo stretchy="false">→</a:mo></a:mrow><a:mrow><a:mo>*</a:mo></a:mrow></a:msup><a:mi>γ</a:mi><a:mo stretchy="false">→</a:mo><a:mi>γ</a:mi><a:mo>+</a:mo><a:mi>J</a:mi><a:mo>/</a:mo><a:mi>ψ</a:mi></a:mrow></a:math> observed at high luminosity <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:math> colliders such as <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>B</g:mi></g:math> factories, with the ISR photon being emitted through the initial electron or positron, prior to their annihilation. The emitted <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>J</i:mi><i:mo>/</i:mo><i:mi mathvariant="normal">Ψ</i:mi></i:math> is produced through the photon fragmentation process, with the produced <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi>J</l:mi><l:mo>/</l:mo><l:mi>ψ</l:mi></l:math> in turn decaying into a pair of leptons, <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msup><n:mi>μ</n:mi><n:mo>−</n:mo></n:msup><n:mo>+</n:mo><n:msup><n:mi>μ</n:mi><n:mo>+</n:mo></n:msup></n:math> through the leptonic decay process, <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mrow><p:mi>J</p:mi><p:mo>/</p:mo><p:mi mathvariant="normal">Ψ</p:mi><p:mo stretchy="false">→</p:mo><p:msup><p:mrow><p:mi>γ</p:mi></p:mrow><p:mrow><p:mo>*</p:mo></p:mrow></p:msup><p:mo stretchy="false">→</p:mo><p:mi>γ</p:mi><p:msup><p:mrow><p:mi>μ</p:mi></p:mrow><p:mrow><p:mo>−</p:mo></p:mrow></p:msup><p:msup><p:mrow><p:mi>μ</p:mi></p:mrow><p:mrow><p:mo>+</p:mo></p:mrow></p:msup></p:mrow></p:math>. The cross section for the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>J</u:mi><u:mo>/</u:mo><u:mi mathvariant="normal">Ψ</u:mi></u:math> production, and the branching ratio for the leptonic decay of <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mi>J</x:mi><x:mo>/</x:mo><x:mi mathvariant="normal">Ψ</x:mi><x:mo stretchy="false">→</x:mo><x:msup><x:mi>μ</x:mi><x:mo>−</x:mo></x:msup><x:msup><x:mi>μ</x:mi><x:mo>+</x:mo></x:msup></x:math>, both calculated in the framework of Bethe-Salpeter equation are used to work out the cross section for the complete process, <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:msup><bb:mi>e</bb:mi><bb:mo>−</bb:mo></bb:msup><bb:msup><bb:mi>e</bb:mi><bb:mo>+</bb:mo></bb:msup><bb:mo stretchy="false">→</bb:mo><bb:mi>γ</bb:mi><bb:mo>+</bb:mo><bb:mi>J</bb:mi><bb:mo>/</bb:mo><bb:mi>ψ</bb:mi><bb:mo stretchy="false">→</bb:mo><bb:mi>γ</bb:mi><bb:mo>+</bb:mo><bb:msup><bb:mi>μ</bb:mi><bb:mo>−</bb:mo></bb:msup><bb:mo>+</bb:mo><bb:msup><bb:mi>μ</bb:mi><bb:mo>+</bb:mo></bb:msup></bb:math>. Our results are compared with the data, and other models. This study is then extended to production and decays of <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:mi mathvariant="normal">ϒ</fb:mi><fb:mo stretchy="false">(</fb:mo><fb:mn>1</fb:mn><fb:mi>S</fb:mi><fb:mo stretchy="false">)</fb:mo></fb:math> through the ISR process.