Measurement of the branching fraction of ψ(2S)→γπ0

Based on <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mo stretchy="false">(</a:mo><a:mn>2712.4</a:mn><a:mo>±</a:mo><a:mn>14.1</a:mn><a:mo stretchy="false">)</a:mo><a:mo>×</a:mo><a:msup><a:mn>10</a:mn><a:mn>6</a:mn></a:msup><a:mtext> </a:mtext><a:mtext> </a:mtext><a:mi>ψ</a:mi><a:mo stretchy="false">(</a:mo><a:mn>2</a:mn><a:mi>S</a:mi><a:mo stretchy="false">)</a:mo></a:math> events, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mn>7.9</g:mn><g:mtext> </g:mtext><g:mtext> </g:mtext><g:msup><g:mi>fb</g:mi><g:mrow><g:mo>−</g:mo><g:mn>1</g:mn></g:mrow></g:msup><g:mtext> </g:mtext><g:mtext> </g:mtext><g:mi>ψ</g:mi><g:mo stretchy="false">(</g:mo><g:mn>3773</g:mn><g:mo stretchy="false">)</g:mo></g:math> data, and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mn>0.8</k:mn><k:mtext> </k:mtext><k:mtext> </k:mtext><k:msup><k:mi>fb</k:mi><k:mrow><k:mo>−</k:mo><k:mn>1</k:mn></k:mrow></k:msup></k:math> off-resonance data samples collected with the BESIII detector, we measure the branching fraction of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>ψ</m:mi><m:mo stretchy="false">(</m:mo><m:mn>2</m:mn><m:mi>S</m:mi><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">→</m:mo><m:mi>γ</m:mi><m:msup><m:mi>π</m:mi><m:mn>0</m:mn></m:msup></m:math> and <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:msup><r:mi>e</r:mi><r:mo>+</r:mo></r:msup><r:msup><r:mi>e</r:mi><r:mo>−</r:mo></r:msup><r:mo stretchy="false">→</r:mo><r:mi>γ</r:mi><r:msup><r:mi>π</r:mi><r:mn>0</r:mn></r:msup></r:math> form factor at momentum transfers <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msup><u:mi>Q</u:mi><u:mn>2</u:mn></u:msup><u:mo>∼</u:mo><u:mn>13</u:mn><u:mtext> </u:mtext><u:mtext> </u:mtext><u:msup><u:mrow><u:mi>GeV</u:mi></u:mrow><u:mrow><u:mn>2</u:mn></u:mrow></u:msup></u:math>. The <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>e</w:mi><w:mo>−</w:mo></w:msup><w:mo stretchy="false">→</w:mo><w:mi>γ</w:mi><w:msup><w:mi>π</w:mi><w:mn>0</w:mn></w:msup></w:math> cross section is fitted with considering the interference between the <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mi>ψ</z:mi><z:mo stretchy="false">(</z:mo><z:mn>2</z:mn><z:mi>S</z:mi><z:mo stretchy="false">)</z:mo></z:math> and continuum amplitudes and two solutions are found, <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:mi mathvariant="script">B</db:mi><db:mo>=</db:mo><db:mn>3.74</db:mn><db:mo>×</db:mo><db:msup><db:mn>10</db:mn><db:mrow><db:mo>−</db:mo><db:mn>7</db:mn></db:mrow></db:msup></db:math> with <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi>ϕ</gb:mi><gb:mo>=</gb:mo><gb:mn>3.93</gb:mn><gb:mtext> </gb:mtext><gb:mtext> </gb:mtext><gb:mi>rad</gb:mi></gb:math> and <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mi mathvariant="script">B</ib:mi><ib:mo>=</ib:mo><ib:mn>7.87</ib:mn><ib:mo>×</ib:mo><ib:msup><ib:mn>10</ib:mn><ib:mrow><ib:mo>−</ib:mo><ib:mn>7</ib:mn></ib:mrow></ib:msup></ib:math> with <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"><lb:mi>ϕ</lb:mi><lb:mo>=</lb:mo><lb:mn>2.08</lb:mn><lb:mtext> </lb:mtext><lb:mtext> </lb:mtext><lb:mi>rad</lb:mi></lb:math>. Here, <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"><nb:mi mathvariant="script">B</nb:mi></nb:math> is the branching fraction of <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:mi>ψ</qb:mi><qb:mo stretchy="false">(</qb:mo><qb:mn>2</qb:mn><qb:mi>S</qb:mi><qb:mo stretchy="false">)</qb:mo><qb:mo stretchy="false">→</qb:mo><qb:mi>γ</qb:mi><qb:msup><qb:mi>π</qb:mi><qb:mn>0</qb:mn></qb:msup></qb:math> and <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" display="inline"><vb:mi>ϕ</vb:mi></vb:math> is the relative phase angle between the <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" display="inline"><xb:mi>ψ</xb:mi><xb:mo stretchy="false">(</xb:mo><xb:mn>2</xb:mn><xb:mi>S</xb:mi><xb:mo stretchy="false">)</xb:mo></xb:math> and continuum amplitudes. Due to insufficient off-resonance data, the branching fraction <bc:math xmlns:bc="http://www.w3.org/1998/Math/MathML" display="inline"><bc:mi mathvariant="script">B</bc:mi><bc:mo stretchy="false">(</bc:mo><bc:mi>ψ</bc:mi><bc:mo stretchy="false">(</bc:mo><bc:mn>2</bc:mn><bc:mi>S</bc:mi><bc:mo stretchy="false">)</bc:mo><bc:mo stretchy="false">→</bc:mo><bc:mi>γ</bc:mi><bc:msup><bc:mi>π</bc:mi><bc:mn>0</bc:mn></bc:msup><bc:mo stretchy="false">)</bc:mo></bc:math> is determined to be in the range <jc:math xmlns:jc="http://www.w3.org/1998/Math/MathML" display="inline"><jc:mo stretchy="false">[</jc:mo><jc:mn>2.7</jc:mn><jc:mo>,</jc:mo><jc:mn>9.7</jc:mn><jc:mo stretchy="false">]</jc:mo><jc:mo>×</jc:mo><jc:msup><jc:mn>10</jc:mn><jc:mrow><jc:mo>−</jc:mo><jc:mn>7</jc:mn></jc:mrow></jc:msup></jc:math> within 1 standard deviation of the contour region.