Three-body unitary coupled-channel approach to radiative <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>J</mml:mi><mml:mo>/</mml:mo><mml:mi>ψ</mml:mi></mml:math> decays and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>η</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1405</mml:mn><mml:mo>/</mml:mo><mml:mn>1475</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>

Abstract

Recent BESIII data on radiative <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> decays from <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:mo>∼</c:mo><c:msup><c:mrow><c:mn>10</c:mn></c:mrow><c:mrow><c:mn>10</c:mn></c:mrow></c:msup></c:mrow></c:math> <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:mi>J</e:mi><e:mo>/</e:mo><e:mi>ψ</e:mi></e:mrow></e:math> samples should significantly advance our understanding of the controversial nature of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>η</g:mi><g:mo stretchy="false">(</g:mo><g:mn>1405</g:mn><g:mo>/</g:mo><g:mn>1475</g:mn><g:mo stretchy="false">)</g:mo></g:math>. This motivates us to develop a three-body unitary coupled-channel model for radiative <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>J</k:mi><k:mo>/</k:mo><k:mi>ψ</k:mi></k:math> decays to three-meson final states of any partial wave (<m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msup><m:mi>J</m:mi><m:mrow><m:mi>P</m:mi><m:mi>C</m:mi></m:mrow></m:msup></m:math>). Basic building blocks of the model are bare resonance states such as <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>η</o:mi><o:mo stretchy="false">(</o:mo><o:mn>1405</o:mn><o:mo>/</o:mo><o:mn>1475</o:mn><o:mo stretchy="false">)</o:mo></o:math> and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>f</s:mi><s:mn>1</s:mn></s:msub><s:mo stretchy="false">(</s:mo><s:mn>1420</s:mn><s:mo stretchy="false">)</s:mo></s:math>, and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>π</w:mi><w:mi>K</w:mi></w:math>, <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>K</y:mi><y:mover accent="true"><y:mi>K</y:mi><y:mo stretchy="false">¯</y:mo></y:mover></y:math>, and <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>π</cb:mi><cb:mi>η</cb:mi></cb:math> two-body interactions that generate resonances such as <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:msup><eb:mi>K</eb:mi><eb:mo>*</eb:mo></eb:msup><eb:mo stretchy="false">(</eb:mo><eb:mn>892</eb:mn><eb:mo stretchy="false">)</eb:mo></eb:math>, <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:msubsup><ib:mi>K</ib:mi><ib:mn>0</ib:mn><ib:mo>*</ib:mo></ib:msubsup><ib:mo stretchy="false">(</ib:mo><ib:mn>700</ib:mn><ib:mo stretchy="false">)</ib:mo></ib:math>, and <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:msub><mb:mi>a</mb:mi><mb:mn>0</mb:mn></mb:msub><mb:mo stretchy="false">(</mb:mo><mb:mn>980</mb:mn><mb:mo stretchy="false">)</mb:mo></mb:math>. This model reasonably fits <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:msub><qb:mi>K</qb:mi><qb:mi>S</qb:mi></qb:msub><qb:msub><qb:mi>K</qb:mi><qb:mi>S</qb:mi></qb:msub><qb:msup><qb:mi>π</qb:mi><qb:mn>0</qb:mn></qb:msup></qb:math> Dalitz plot pseudodata generated from the BESIII’s <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:msup><sb:mi>J</sb:mi><sb:mrow><sb:mi>P</sb:mi><sb:mi>C</sb:mi></sb:mrow></sb:msup><sb:mo>=</sb:mo><sb:msup><sb:mn>0</sb:mn><sb:mrow><sb:mo>−</sb:mo><sb:mo>+</sb:mo></sb:mrow></sb:msup></sb:math> amplitude for <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:mi>J</ub:mi><ub:mo>/</ub:mo><ub:mi>ψ</ub:mi><ub:mo stretchy="false">→</ub:mo><ub:mi>γ</ub:mi><ub:msub><ub:mi>K</ub:mi><ub:mi>S</ub:mi></ub:msub><ub:msub><ub:mi>K</ub:mi><ub:mi>S</ub:mi></ub:msub><ub:msup><ub:mi>π</ub:mi><ub:mn>0</ub:mn></ub:msup></ub:math>. The experimental branching ratios of <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" display="inline"><xb:mi>η</xb:mi><xb:mo stretchy="false">(</xb:mo><xb:mn>1405</xb:mn><xb:mo>/</xb:mo><xb:mn>1475</xb:mn><xb:mo stretchy="false">)</xb:mo><xb:mo stretchy="false">→</xb:mo><xb:mi>η</xb:mi><xb:mi>π</xb:mi><xb:mi>π</xb:mi></xb:math> and <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"><cc:mi>η</cc:mi><cc:mo stretchy="false">(</cc:mo><cc:mn>1405</cc:mn><cc:mo>/</cc:mo><cc:mn>1475</cc:mn><cc:mo stretchy="false">)</cc:mo><cc:mo stretchy="false">→</cc:mo><cc:mi>γ</cc:mi><cc:mi>ρ</cc:mi></cc:math> relative to that of <hc:math xmlns:hc="http://www.w3.org/1998/Math/MathML" display="inline"><hc:mi>η</hc:mi><hc:mo stretchy="false">(</hc:mo><hc:mn>1405</hc:mn><hc:mo>/</hc:mo><hc:mn>1475</hc:mn><hc:mo stretchy="false">)</hc:mo><hc:mo stretchy="false">→</hc:mo><hc:mi>K</hc:mi><hc:mover accent="true"><hc:mi>K</hc:mi><hc:mo stretchy="false">¯</hc:mo></hc:mover><hc:mi>π</hc:mi></hc:math> are simultaneously fitted. Our <oc:math xmlns:oc="http://www.w3.org/1998/Math/MathML" display="inline"><oc:msup><oc:mn>0</oc:mn><oc:mrow><oc:mo>−</oc:mo><oc:mo>+</oc:mo></oc:mrow></oc:msup></oc:math> amplitude is analytically continued to find three poles, two of which correspond to <qc:math xmlns:qc="http://www.w3.org/1998/Math/MathML" display="inline"><qc:mi>η</qc:mi><qc:mo stretchy="false">(</qc:mo><qc:mn>1405</qc:mn><qc:mo stretchy="false">)</qc:mo></qc:math> on different Riemann sheets of the <uc:math xmlns:uc="http://www.w3.org/1998/Math/MathML" display="inline"><uc:msup><uc:mi>K</uc:mi><uc:mo>*</uc:mo></uc:msup><uc:mover accent="true"><uc:mi>K</uc:mi><uc:mo stretchy="false">¯</uc:mo></uc:mover></uc:math> channel, and the third one for <yc:math xmlns:yc="http://www.w3.org/1998/Math/MathML" display="inline"><yc:mi>η</yc:mi><yc:mo stretchy="false">(</yc:mo><yc:mn>1475</yc:mn><yc:mo stretchy="false">)</yc:mo></yc:math>. This is the first pole determination of <cd:math xmlns:cd="http://www.w3.org/1998/Math/MathML" display="inline"><cd:mi>η</cd:mi><cd:mo stretchy="false">(</cd:mo><cd:mn>1405</cd:mn><cd:mo>/</cd:mo><cd:mn>1475</cd:mn><cd:mo stretchy="false">)</cd:mo></cd:math> and, furthermore, the first-ever pole determination from analyzing experimental Dalitz plot distributions with a manifestly three-body unitary coupled-channel framework. Process-dependent <gd:math xmlns:gd="http://www.w3.org/1998/Math/MathML" display="inline"><gd:mi>η</gd:mi><gd:mi>π</gd:mi><gd:mi>π</gd:mi></gd:math>, <id:math xmlns:id="http://www.w3.org/1998/Math/MathML" display="inline"><id:mi>γ</id:mi><id:msup><id:mi>π</id:mi><id:mo>+</id:mo></id:msup><id:msup><id:mi>π</id:mi><id:mo>−</id:mo></id:msup></id:math>, and <kd:math xmlns:kd="http://www.w3.org/1998/Math/MathML" display="inline"><kd:mi>π</kd:mi><kd:mi>π</kd:mi><kd:mi>π</kd:mi></kd:math> lineshapes of <md:math xmlns:md="http://www.w3.org/1998/Math/MathML" display="inline"><md:mi>J</md:mi><md:mo>/</md:mo><md:mi>ψ</md:mi><md:mo stretchy="false">→</md:mo><md:mi>γ</md:mi><md:mo stretchy="false">(</md:mo><md:msup><md:mn>0</md:mn><md:mrow><md:mo>−</md:mo><md:mo>+</md:mo></md:mrow></md:msup><md:mo stretchy="false">)</md:mo><md:mo stretchy="false">→</md:mo><md:mi>γ</md:mi><md:mo stretchy="false">(</md:mo><md:mi>η</md:mi><md:mi>π</md:mi><md:mi>π</md:mi><md:mo stretchy="false">)</md:mo></md:math>, <ud:math xmlns:ud="http://www.w3.org/1998/Math/MathML" display="inline"><ud:mi>γ</ud:mi><ud:mo stretchy="false">(</ud:mo><ud:mi>γ</ud:mi><ud:mi>ρ</ud:mi><ud:mo stretchy="false">)</ud:mo></ud:math>, and <yd:math xmlns:yd="http://www.w3.org/1998/Math/MathML" display="inline"><yd:mi>γ</yd:mi><yd:mo stretchy="false">(</yd:mo><yd:mi>π</yd:mi><yd:mi>π</yd:mi><yd:mi>π</yd:mi><yd:mo stretchy="false">)</yd:mo></yd:math> are predicted, and are in reasonable agreement with data. A triangle singularity is shown to play a crucial role to cause the large isospin violation of <ce:math xmlns:ce="http://www.w3.org/1998/Math/MathML" display="inline"><ce:mi>J</ce:mi><ce:mo>/</ce:mo><ce:mi>ψ</ce:mi><ce:mo stretchy="false">→</ce:mo><ce:mi>γ</ce:mi><ce:mo stretchy="false">(</ce:mo><ce:mi>π</ce:mi><ce:mi>π</ce:mi><ce:mi>π</ce:mi><ce:mo stretchy="false">)</ce:mo></ce:math>. Published by the American Physical Society 2024

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