Updated analysis of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>D</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>P</mml:mi><mml:mi>P</mml:mi><mml:mo>,</mml:mo><mml:mi>V</mml:mi><mml:mi>P</mml:mi></mml:mrow></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>V</mml:mi><mml:mi>V</mml:mi></mml:math> decays: Implications for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msubsup><mml:mi>K</mml:mi><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msubsup><mml:mo>−</mml:mo><mml:msubsup><mml:mi>K</mml:mi><mml:mi>L</mml:mi><mml:mn>0</mml:mn></mml:msubsup></mml:math> asymmetries and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>D</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:msup><mml:mover accent="true"><mml:mi>D</mml:mi><mml:mo stretchy="false">¯</mml:mo></mml:mover><mml:mn>0</mml:mn></mml:msup></mml:math> mixing

Abstract

An updated analysis of the two-body <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>D</a:mi><a:mo stretchy="false">→</a:mo><a:mi>P</a:mi><a:mi>P</a:mi><a:mo>,</a:mo><a:mi>V</a:mi><a:mi>P</a:mi></a:math>, and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi>V</d:mi><d:mi>V</d:mi></d:math> decays within the framework of the topological diagram approach is performed. A global fit to the Cabibbo-favored (CF) modes in the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi>V</f:mi><f:mi>P</f:mi></f:math> sector gives many solutions with similarly small local minima in <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:msup><h:mi>χ</h:mi><h:mn>2</h:mn></h:msup></h:math>. The solution degeneracy is lifted once we use them to predict for the singly-Cabibbo-suppressed modes. Topological amplitudes are extracted for the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>η</j:mi><j:mo>−</j:mo><j:msup><j:mi>η</j:mi><j:mo>′</j:mo></j:msup></j:math> mixing angles <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi>ϕ</l:mi><l:mo>=</l:mo><l:mn>40.4</l:mn><l:mi>°</l:mi></l:math> and 43.5°. The <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msubsup><n:mi>K</n:mi><n:mi>S</n:mi><n:mn>0</n:mn></n:msubsup><n:mo>−</n:mo><n:msubsup><n:mi>K</n:mi><n:mi>L</n:mi><n:mn>0</n:mn></n:msubsup></n:math> asymmetries in <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>D</p:mi><p:mo stretchy="false">→</p:mo><p:msubsup><p:mi>K</p:mi><p:mrow><p:mi>S</p:mi><p:mo>,</p:mo><p:mi>L</p:mi></p:mrow><p:mn>0</p:mn></p:msubsup><p:mi>M</p:mi></p:math> decays denoted by <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>R</s:mi><s:mo stretchy="false">(</s:mo><s:mi>D</s:mi><s:mo>,</s:mo><s:mi>M</s:mi><s:mo stretchy="false">)</s:mo></s:math> are studied. While the predicted <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>R</w:mi><w:mo stretchy="false">(</w:mo><w:msup><w:mi>D</w:mi><w:mn>0</w:mn></w:msup><w:mo>,</w:mo><w:mi>P</w:mi><w:mo stretchy="false">)</w:mo></w:math> for <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>P</ab:mi><ab:mo>=</ab:mo><ab:msup><ab:mi>π</ab:mi><ab:mn>0</ab:mn></ab:msup><ab:mo>,</ab:mo><ab:mi>η</ab:mi></ab:math> and <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msup><cb:mi>η</cb:mi><cb:mo>′</cb:mo></cb:msup></cb:math> agree with experiment, the calculated <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>R</eb:mi><eb:mo stretchy="false">(</eb:mo><eb:msup><eb:mi>D</eb:mi><eb:mo>+</eb:mo></eb:msup><eb:mo>,</eb:mo><eb:msup><eb:mi>π</eb:mi><eb:mo>+</eb:mo></eb:msup><eb:mo stretchy="false">)</eb:mo></eb:math>, <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mi>R</ib:mi><ib:mo stretchy="false">(</ib:mo><ib:msubsup><ib:mi>D</ib:mi><ib:mi>s</ib:mi><ib:mo>+</ib:mo></ib:msubsup><ib:mo>,</ib:mo><ib:msup><ib:mi>K</ib:mi><ib:mo>+</ib:mo></ib:msup><ib:mo stretchy="false">)</ib:mo></ib:math>, <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:mi>R</mb:mi><mb:mo stretchy="false">(</mb:mo><mb:msup><mb:mi>D</mb:mi><mb:mn>0</mb:mn></mb:msup><mb:mo>,</mb:mo><mb:mi>ω</mb:mi><mb:mo stretchy="false">)</mb:mo></mb:math>, and <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:mi>R</qb:mi><qb:mo stretchy="false">(</qb:mo><qb:msup><qb:mi>D</qb:mi><qb:mn>0</qb:mn></qb:msup><qb:mo>,</qb:mo><qb:mi>ϕ</qb:mi><qb:mo stretchy="false">)</qb:mo></qb:math> deviate from the data. We conjecture that the relative phase between the topological amplitudes (<ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:mrow><ub:mi>C</ub:mi><ub:mo>+</ub:mo><ub:mi>A</ub:mi></ub:mrow></ub:math>) and (<wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:mrow><wb:mi>T</wb:mi><wb:mo>+</wb:mo><wb:mi>C</wb:mi></wb:mrow></wb:math>) should be slightly smaller than 90° in order to explain the first two discrepancies and that additional singlet contributions due to the SU(3)-singlet nature of <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:mi>ω</yb:mi></yb:math> and <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:mi>ϕ</ac:mi></ac:math> are needed to account for the last two. For doubly-Cabibbo-suppressed (DCS) <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"><cc:mi>D</cc:mi><cc:mo stretchy="false">→</cc:mo><cc:mi>V</cc:mi><cc:mi>P</cc:mi></cc:math> decays, their topological amplitudes (double primed) cannot be all the same as the corresponding ones in the CF modes. The assumption of <fc:math xmlns:fc="http://www.w3.org/1998/Math/MathML" display="inline"><fc:msubsup><fc:mi>E</fc:mi><fc:mrow><fc:mi>V</fc:mi><fc:mo>,</fc:mo><fc:mi>P</fc:mi></fc:mrow><fc:mo>′′</fc:mo></fc:msubsup><fc:mo>=</fc:mo><fc:msub><fc:mi>E</fc:mi><fc:mrow><fc:mi>V</fc:mi><fc:mo>,</fc:mo><fc:mi>P</fc:mi></fc:mrow></fc:msub></fc:math> for the <hc:math xmlns:hc="http://www.w3.org/1998/Math/MathML" display="inline"><hc:mi>W</hc:mi></hc:math>-exchange amplitude leads to some inconsistencies with the experiment. Through the measured relative phases between CF and DCS channels, the relations of <jc:math xmlns:jc="http://www.w3.org/1998/Math/MathML" display="inline"><jc:msubsup><jc:mi>E</jc:mi><jc:mrow><jc:mi>V</jc:mi><jc:mo>,</jc:mo><jc:mi>P</jc:mi></jc:mrow><jc:mo>′′</jc:mo></jc:msubsup></jc:math> with <lc:math xmlns:lc="http://www.w3.org/1998/Math/MathML" display="inline"><lc:msub><lc:mi>E</lc:mi><lc:mrow><lc:mi>V</lc:mi><lc:mo>,</lc:mo><lc:mi>P</lc:mi></lc:mrow></lc:msub></lc:math> are determined. Long-distance contributions to the <nc:math xmlns:nc="http://www.w3.org/1998/Math/MathML" display="inline"><nc:mrow><nc:msup><nc:mrow><nc:mi>D</nc:mi></nc:mrow><nc:mrow><nc:mn>0</nc:mn></nc:mrow></nc:msup><nc:mo>−</nc:mo><nc:msup><nc:mrow><nc:mover accent="true"><nc:mrow><nc:mi>D</nc:mi></nc:mrow><nc:mrow><nc:mo stretchy="false">¯</nc:mo></nc:mrow></nc:mover></nc:mrow><nc:mrow><nc:mn>0</nc:mn></nc:mrow></nc:msup></nc:mrow></nc:math> mixing parameter <rc:math xmlns:rc="http://www.w3.org/1998/Math/MathML" display="inline"><rc:mi>y</rc:mi></rc:math> are evaluated in the exclusive approach. In particular, we focus on <tc:math xmlns:tc="http://www.w3.org/1998/Math/MathML" display="inline"><tc:mi>D</tc:mi><tc:mo stretchy="false">→</tc:mo><tc:mi>P</tc:mi><tc:mi>P</tc:mi></tc:math> and <wc:math xmlns:wc="http://www.w3.org/1998/Math/MathML" display="inline"><wc:mi>V</wc:mi><wc:mi>P</wc:mi></wc:math> decays where <yc:math xmlns:yc="http://www.w3.org/1998/Math/MathML" display="inline"><yc:mi>y</yc:mi></yc:math> can be reliably estimated. We conclude that <ad:math xmlns:ad="http://www.w3.org/1998/Math/MathML" display="inline"><ad:msub><ad:mi>y</ad:mi><ad:mrow><ad:mi>P</ad:mi><ad:mi>P</ad:mi></ad:mrow></ad:msub><ad:mo>∼</ad:mo><ad:mo stretchy="false">(</ad:mo><ad:mn>0.110</ad:mn><ad:mo>±</ad:mo><ad:mn>0.011</ad:mn><ad:mo stretchy="false">)</ad:mo><ad:mo>%</ad:mo></ad:math> and the lower bound on <ed:math xmlns:ed="http://www.w3.org/1998/Math/MathML" display="inline"><ed:msub><ed:mi>y</ed:mi><ed:mrow><ed:mi>V</ed:mi><ed:mi>P</ed:mi></ed:mrow></ed:msub></ed:math> is <gd:math xmlns:gd="http://www.w3.org/1998/Math/MathML" display="inline"><gd:mo stretchy="false">(</gd:mo><gd:mn>0.220</gd:mn><gd:mo>±</gd:mo><gd:mn>0.071</gd:mn><gd:mo stretchy="false">)</gd:mo><gd:mo>%</gd:mo></gd:math>. It is thus conceivable that at least half of the mixing parameter <kd:math xmlns:kd="http://www.w3.org/1998/Math/MathML" display="inline"><kd:mi>y</kd:mi></kd:math> can be accounted for by the two-body <md:math xmlns:md="http://www.w3.org/1998/Math/MathML" display="inline"><md:mi>P</md:mi><md:mi>P</md:mi></md:math> and <od:math xmlns:od="http://www.w3.org/1998/Math/MathML" display="inline"><od:mi>V</od:mi><od:mi>P</od:mi></od:math> modes. The main uncertainties arise from the yet-to-be-measured DCS channels and their phases relative to the CF ones. Published by the American Physical Society 2024

Topics & Concepts