About the magnitude of the γ*N→N(1520) transverse amplitudes near <…

The <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>γ</a:mi><a:mo>*</a:mo></a:msup><a:mi>N</a:mi><a:mo stretchy="false">→</a:mo><a:mi>N</a:mi><a:mo stretchy="false">(</a:mo><a:mn>1520</a:mn><a:mo stretchy="false">)</a:mo></a:math> transition has a property that differs from the other low-lying nucleon resonance amplitudes: the magnitude of the transverse helicity amplitudes. The transition helicity amplitudes are defined in terms of square-transfer momentum <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:msup><f:mi>q</f:mi><f:mn>2</f:mn></f:msup></f:math>, or <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:msup><h:mi>Q</h:mi><h:mn>2</h:mn></h:msup><h:mo>=</h:mo><h:mo>−</h:mo><h:msup><h:mi>q</h:mi><h:mn>2</h:mn></h:msup></h:math>. Near the photon point (<j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:msup><j:mi>Q</j:mi><j:mn>2</j:mn></j:msup><j:mo>=</j:mo><j:mn>0</j:mn></j:math>) there is a significant difference in the magnitude of the transverse amplitudes: <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msub><l:mi>A</l:mi><l:mrow><l:mn>3</l:mn><l:mo>/</l:mo><l:mn>2</l:mn></l:mrow></l:msub></l:math> is very large and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msub><n:mi>A</n:mi><n:mrow><n:mn>1</n:mn><n:mo>/</n:mo><n:mn>2</n:mn></n:mrow></n:msub></n:math> is very small. This atypical behavior contrasts with the relation between the amplitudes at the pseudothreshold [the limit where the nucleon and the <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>N</p:mi><p:mo stretchy="false">(</p:mo><p:mn>1520</p:mn><p:mo stretchy="false">)</p:mo></p:math> are both at rest and <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:msup><t:mi>Q</t:mi><t:mn>2</t:mn></t:msup><t:mo><</t:mo><t:mn>0</t:mn></t:math>], where <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:msub><v:mi>A</v:mi><v:mrow><v:mn>3</v:mn><v:mo>/</v:mo><v:mn>2</v:mn></v:mrow></v:msub><v:mo>=</v:mo><v:msub><v:mi>A</v:mi><v:mrow><v:mn>1</v:mn><v:mo>/</v:mo><v:mn>2</v:mn></v:mrow></v:msub><v:mo>/</v:mo><v:msqrt><v:mn>3</v:mn></v:msqrt></v:math>, and also in the large-<x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:msup><x:mi>Q</x:mi><x:mn>2</x:mn></x:msup></x:math> region, where theory and data suggest that <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:msub><z:mi>A</z:mi><z:mrow><z:mn>3</z:mn><z:mo>/</z:mo><z:mn>2</z:mn></z:mrow></z:msub></z:math> is suppressed relative to <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:msub><bb:mi>A</bb:mi><bb:mrow><bb:mn>1</bb:mn><bb:mo>/</bb:mo><bb:mn>2</bb:mn></bb:mrow></bb:msub></bb:math>. In the present work, we look for the source of the suppression of the <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:msub><db:mi>A</db:mi><db:mrow><db:mn>1</db:mn><db:mo>/</db:mo><db:mn>2</db:mn></db:mrow></db:msub></db:math> amplitude at <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:msup><fb:mi>Q</fb:mi><fb:mn>2</fb:mn></fb:msup><fb:mo>=</fb:mo><fb:mn>0</fb:mn></fb:math>. The result is easy to understand in first approximation, when we look into the relation between the transverse amplitudes and the elementary form factors, defined by a gauge-invariant parametrization of the <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:msup><hb:mi>γ</hb:mi><hb:mo>*</hb:mo></hb:msup><hb:mi>N</hb:mi><hb:mo stretchy="false">→</hb:mo><hb:mi>N</hb:mi><hb:mo stretchy="false">(</hb:mo><hb:mn>1520</hb:mn><hb:mo stretchy="false">)</hb:mo></hb:math> transition current, near <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:msup><mb:mi>Q</mb:mi><mb:mn>2</mb:mn></mb:msup><mb:mo>=</mb:mo><mb:mn>0</mb:mn></mb:math>. There is a partial cancellation between contributions of two elementary form factors near <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:msup><ob:mi>Q</ob:mi><ob:mn>2</ob:mn></ob:msup><ob:mo>=</ob:mo><ob:mn>0</ob:mn></ob:math>. We conclude, however, that the correlation between the two elementary form factors at <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:msup><qb:mi>Q</qb:mi><qb:mn>2</qb:mn></qb:msup><qb:mo>=</qb:mo><qb:mn>0</qb:mn></qb:math> is not sufficient to explain the transverse amplitude data below <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:msup><sb:mi>Q</sb:mi><sb:mn>2</sb:mn></sb:msup><sb:mo>=</sb:mo><sb:mn>1</sb:mn><sb:mtext> </sb:mtext><sb:mtext> </sb:mtext><sb:msup><sb:mrow><sb:mi>GeV</sb:mi></sb:mrow><sb:mrow><sb:mn>2</sb:mn></sb:mrow></sb:msup></sb:math>. The description of the dependence of the transverse amplitudes on <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:msup><ub:mi>Q</ub:mi><ub:mn>2</ub:mn></ub:msup></ub:math> requires the determination of the scale of variation of the elementary form factors in the range <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:mrow><wb:msup><wb:mrow><wb:mi>Q</wb:mi></wb:mrow><wb:mrow><wb:mn>2</wb:mn></wb:mrow></wb:msup><wb:mo>=</wb:mo><wb:mn>0</wb:mn><wb:mo>…</wb:mo><wb:mn>0.5</wb:mn><wb:mtext> </wb:mtext><wb:mtext> </wb:mtext><wb:msup><wb:mrow><wb:mi>GeV</wb:mi></wb:mrow><wb:mrow><wb:mn>2</wb:mn></wb:mrow></wb:msup></wb:mrow></wb:math>, a region with almost nonexistent data. We conclude at the end that the low-<yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:msup><yb:mi>Q</yb:mi><yb:mn>2</yb:mn></yb:msup></yb:math> data for the transverse amplitudes can be well described when we relate the scale of variation of the elementary form factors with the nucleon dipole form factor. Published by the American Physical Society 2024