Investigation of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1232</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> resonance substructure in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>p</mml:mi> <mml:msup> <mml:mi>γ</mml:mi> <mml:mo>*</mml:mo> </mml:msup> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1232</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> process through helicity amplitudes

Abstract

The substructure of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="normal">Δ</a:mi> <a:mo stretchy="false">(</a:mo> <a:mn>1232</a:mn> <a:mo stretchy="false">)</a:mo> </a:math> resonance is studied in the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>p</f:mi> <f:msup> <f:mi>γ</f:mi> <f:mo>*</f:mo> </f:msup> <f:mo stretchy="false">→</f:mo> <f:mi mathvariant="normal">Δ</f:mi> <f:mo stretchy="false">(</f:mo> <f:mn>1232</f:mn> <f:mo stretchy="false">)</f:mo> </f:math> transition via helicity amplitudes within a quark-model framework that treats baryons as three-quark systems and includes both quark-core and meson-cloud contributions. In the calculation of the transition amplitudes, the proton wave function is Lorentz boosted from the proton rest frame to the <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mi mathvariant="normal">Δ</l:mi> <l:mo stretchy="false">(</l:mo> <l:mn>1232</l:mn> <l:mo stretchy="false">)</l:mo> </l:math> resonance rest frame. The comparison of the calculated amplitudes <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msub> <q:mi>A</q:mi> <q:mrow> <q:mn>1</q:mn> <q:mo>/</q:mo> <q:mn>2</q:mn> </q:mrow> </q:msub> </q:math> , <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msub> <s:mi>A</s:mi> <s:mrow> <s:mn>3</s:mn> <s:mo>/</s:mo> <s:mn>2</s:mn> </s:mrow> </s:msub> </s:math> , and <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:msub> <u:mi>S</u:mi> <u:mrow> <u:mn>1</u:mn> <u:mo>/</u:mo> <u:mn>2</u:mn> </u:mrow> </u:msub> </u:math> with experimental data indicates a non-negligible <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:mi>L</w:mi> <w:mo>=</w:mo> <w:mn>2</w:mn> </w:math> admixture in the <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mi mathvariant="normal">Δ</y:mi> <y:mo stretchy="false">(</y:mo> <y:mn>1232</y:mn> <y:mo stretchy="false">)</y:mo> </y:math> , about 80% of the <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"> <db:mi mathvariant="normal">Δ</db:mi> <db:mo stretchy="false">(</db:mo> <db:mn>1232</db:mn> <db:mo stretchy="false">)</db:mo> </db:math> wave function in the S wave and 20% in the D wave. In particular, the <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"> <ib:msub> <ib:mi>S</ib:mi> <ib:mrow> <ib:mn>1</ib:mn> <ib:mo>/</ib:mo> <ib:mn>2</ib:mn> </ib:mrow> </ib:msub> </ib:math> amplitude is found to be dominated by the <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"> <kb:mi>L</kb:mi> <kb:mo>=</kb:mo> <kb:mn>2</kb:mn> </kb:math> quark-core component rather than the <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"> <mb:mi>L</mb:mi> <mb:mo>=</mb:mo> <mb:mn>0</mb:mn> </mb:math> part. The results challenge the traditional understanding that the <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"> <ob:mi mathvariant="normal">Δ</ob:mi> <ob:mo stretchy="false">(</ob:mo> <ob:mn>1232</ob:mn> <ob:mo stretchy="false">)</ob:mo> </ob:math> baryon is dominantly an <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"> <tb:mi>L</tb:mi> <tb:mo>=</tb:mo> <tb:mn>0</tb:mn> </tb:math> state.