Decoding the role of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>ρ</mml:mi></mml:math> mesonic states for elucidating the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:m

Recently, the BESIII Collaboration observed a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>ρ</a:mi></a:math>-like structure <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>Y</c:mi><c:mo stretchy="false">(</c:mo><c:mn>2044</c:mn><c:mo stretchy="false">)</c:mo></c:math> in <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msup><g:mi>e</g:mi><g:mo>+</g:mo></g:msup><g:msup><g:mi>e</g:mi><g:mo>−</g:mo></g:msup><g:mo stretchy="false">→</g:mo><g:msub><g:mi>a</g:mi><g:mn>2</g:mn></g:msub><g:mo stretchy="false">(</g:mo><g:mn>1320</g:mn><g:mo stretchy="false">)</g:mo><g:mi>π</g:mi></g:math>, suggesting that <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi>Y</l:mi><l:mo stretchy="false">(</l:mo><l:mn>2044</l:mn><l:mo stretchy="false">)</l:mo></l:math> may be a candidate of vector meson <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>ρ</p:mi><p:mo stretchy="false">(</p:mo><p:mn>2</p:mn><p:mi>D</p:mi><p:mo stretchy="false">)</p:mo></p:math> by comparing resonance parameters. However, the theoretical prediction for the combined branching ratio <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:msub><t:mi mathvariant="normal">Γ</t:mi><t:mrow><t:msup><t:mi>e</t:mi><t:mo>+</t:mo></t:msup><t:msup><t:mi>e</t:mi><t:mo>−</t:mo></t:msup></t:mrow></t:msub><t:msub><t:mi mathvariant="script">B</t:mi><t:mrow><t:msub><t:mi>a</t:mi><t:mn>2</t:mn></t:msub><t:mo stretchy="false">(</t:mo><t:mn>1320</t:mn><t:mo stretchy="false">)</t:mo><t:mi>π</t:mi></t:mrow></t:msub></t:math> for the pure <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mi>ρ</z:mi><z:mo stretchy="false">(</z:mo><z:mn>2</z:mn><z:mi>D</z:mi><z:mo stretchy="false">)</z:mo></z:math> state is about 2 orders of magnitude smaller than the experimental value. To resolve this discrepancy and decipher the nature of <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:mi>Y</db:mi><db:mo stretchy="false">(</db:mo><db:mn>2044</db:mn><db:mo stretchy="false">)</db:mo></db:math>, this work proposes an <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:mrow><hb:mi>S</hb:mi><hb:mtext>−</hb:mtext><hb:mi>D</hb:mi></hb:mrow></hb:math> mixing scheme to reanalyze the cross section of <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"><jb:msup><jb:mi>e</jb:mi><jb:mo>+</jb:mo></jb:msup><jb:msup><jb:mi>e</jb:mi><jb:mo>−</jb:mo></jb:msup><jb:mo stretchy="false">→</jb:mo><jb:msub><jb:mi>a</jb:mi><jb:mn>2</jb:mn></jb:msub><jb:mo stretchy="false">(</jb:mo><jb:mn>1320</jb:mn><jb:mo stretchy="false">)</jb:mo><jb:mi>π</jb:mi></jb:math>, and finds that the aforementioned branching ratio discrepancy can be resolved. Our results show that the <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:mi>Y</ob:mi><ob:mo stretchy="false">(</ob:mo><ob:mn>2044</ob:mn><ob:mo stretchy="false">)</ob:mo></ob:math> structure can be reproduced by introducing four theoretically predicted <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:mrow><sb:mi>S</sb:mi><sb:mtext>−</sb:mtext><sb:mi>D</sb:mi></sb:mrow></sb:math> mixing <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:mi>ρ</ub:mi></ub:math> meson states <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:msubsup><wb:mi>ρ</wb:mi><wb:mrow><wb:mn>3</wb:mn><wb:mi>S</wb:mi><wb:mo>−</wb:mo><wb:mn>2</wb:mn><wb:mi>D</wb:mi></wb:mrow><wb:mo>′</wb:mo></wb:msubsup></wb:math>, <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:msubsup><yb:mi>ρ</yb:mi><yb:mrow><yb:mn>3</yb:mn><yb:mi>S</yb:mi><yb:mo>−</yb:mo><yb:mn>2</yb:mn><yb:mi>D</yb:mi></yb:mrow><yb:mrow><yb:mo>′′</yb:mo></yb:mrow></yb:msubsup></yb:math>, <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:msubsup><ac:mi>ρ</ac:mi><ac:mrow><ac:mn>4</ac:mn><ac:mi>S</ac:mi><ac:mo>−</ac:mo><ac:mn>3</ac:mn><ac:mi>D</ac:mi></ac:mrow><ac:mo>′</ac:mo></ac:msubsup></ac:math>, and <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"><cc:msubsup><cc:mi>ρ</cc:mi><cc:mrow><cc:mn>4</cc:mn><cc:mi>S</cc:mi><cc:mo>−</cc:mo><cc:mn>3</cc:mn><cc:mi>D</cc:mi></cc:mrow><cc:mrow><cc:mo>′′</cc:mo></cc:mrow></cc:msubsup></cc:math> as intermediate resonances, in which dominant contribution arises from <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:msubsup><ec:mi>ρ</ec:mi><ec:mrow><ec:mn>3</ec:mn><ec:mi>S</ec:mi><ec:mo>−</ec:mo><ec:mn>2</ec:mn><ec:mi>D</ec:mi></ec:mrow><ec:mrow><ec:mo>′′</ec:mo></ec:mrow></ec:msubsup></ec:math> and their inference effect is also significant. Furthermore, we reanalyzed five additional isospin vector processes <gc:math xmlns:gc="http://www.w3.org/1998/Math/MathML" display="inline"><gc:msup><gc:mi>e</gc:mi><gc:mo>+</gc:mo></gc:msup><gc:msup><gc:mi>e</gc:mi><gc:mo>−</gc:mo></gc:msup><gc:mo stretchy="false">→</gc:mo><gc:mi>ω</gc:mi><gc:msup><gc:mi>π</gc:mi><gc:mn>0</gc:mn></gc:msup></gc:math>, <jc:math xmlns:jc="http://www.w3.org/1998/Math/MathML" display="inline"><jc:msup><jc:mi>e</jc:mi><jc:mo>+</jc:mo></jc:msup><jc:msup><jc:mi>e</jc:mi><jc:mo>−</jc:mo></jc:msup><jc:mo stretchy="false">→</jc:mo><jc:msub><jc:mi>f</jc:mi><jc:mn>1</jc:mn></jc:msub><jc:mo stretchy="false">(</jc:mo><jc:mn>1285</jc:mn><jc:mo stretchy="false">)</jc:mo><jc:msup><jc:mi>π</jc:mi><jc:mo>+</jc:mo></jc:msup><jc:msup><jc:mi>π</jc:mi><jc:mo>−</jc:mo></jc:msup></jc:math>, <oc:math xmlns:oc="http://www.w3.org/1998/Math/MathML" display="inline"><oc:msup><oc:mi>e</oc:mi><oc:mo>+</oc:mo></oc:msup><oc:msup><oc:mi>e</oc:mi><oc:mo>−</oc:mo></oc:msup><oc:mo stretchy="false">→</oc:mo><oc:msup><oc:mi>π</oc:mi><oc:mo>+</oc:mo></oc:msup><oc:msup><oc:mi>π</oc:mi><oc:mo>−</oc:mo></oc:msup></oc:math>, <rc:math xmlns:rc="http://www.w3.org/1998/Math/MathML" display="inline"><rc:msup><rc:mi>e</rc:mi><rc:mo>+</rc:mo></rc:msup><rc:msup><rc:mi>e</rc:mi><rc:mo>−</rc:mo></rc:msup><rc:mo stretchy="false">→</rc:mo><rc:mi>ρ</rc:mi><rc:mi>η</rc:mi></rc:math>, and <uc:math xmlns:uc="http://www.w3.org/1998/Math/MathML" display="inline"><uc:msup><uc:mi>e</uc:mi><uc:mo>+</uc:mo></uc:msup><uc:msup><uc:mi>e</uc:mi><uc:mo>−</uc:mo></uc:msup><uc:mo stretchy="false">→</uc:mo><uc:msup><uc:mi>η</uc:mi><uc:mo>′</uc:mo></uc:msup><uc:msup><uc:mi>π</uc:mi><uc:mo>+</uc:mo></uc:msup><uc:msup><uc:mi>π</uc:mi><uc:mo>−</uc:mo></uc:msup></uc:math> based on the same <xc:math xmlns:xc="http://www.w3.org/1998/Math/MathML" display="inline"><xc:mrow><xc:mi>S</xc:mi><xc:mtext>−</xc:mtext><xc:mi>D</xc:mi></xc:mrow></xc:math> mixing framework, and simultaneously reproduced their experimental cross section data. This work provides a unified framework to elucidate all observed <zc:math xmlns:zc="http://www.w3.org/1998/Math/MathML" display="inline"><zc:mi>ρ</zc:mi></zc:math>-like structures near 2 GeV in the <bd:math xmlns:bd="http://www.w3.org/1998/Math/MathML" display="inline"><bd:msup><bd:mi>e</bd:mi><bd:mo>+</bd:mo></bd:msup><bd:msup><bd:mi>e</bd:mi><bd:mo>−</bd:mo></bd:msup></bd:math> annihilation processes, and suggests that the <dd:math xmlns:dd="http://www.w3.org/1998/Math/MathML" display="inline"><dd:mrow><dd:mi>S</dd:mi><dd:mtext>−</dd:mtext><dd:mi>D</dd:mi></dd:mrow></dd:math> mixing effect may be crucial for understanding the mass spectrum and decay behaviors of the higher <fd:math xmlns:fd="http://www.w3.org/1998/Math/MathML" display="inline"><fd:mi>ρ</fd:mi></fd:math> meson states.