It was previously shown that unlike the ratios <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msubsup><a:mi>R</a:mi><a:mi>K</a:mi><a:mrow><a:mi>μ</a:mi><a:mi>e</a:mi></a:mrow></a:msubsup><a:mo>≡</a:mo><a:msub><a:mi>R</a:mi><a:mi>K</a:mi></a:msub><a:mo>≡</a:mo><a:mi mathvariant="normal">Γ</a:mi><a:mo stretchy="false">(</a:mo><a:mi>B</a:mi><a:mo stretchy="false">→</a:mo><a:mi>K</a:mi><a:msup><a:mi>μ</a:mi><a:mo>+</a:mo></a:msup><a:msup><a:mi>μ</a:mi><a:mo>−</a:mo></a:msup><a:mo stretchy="false">)</a:mo><a:mo>/</a:mo><a:mi mathvariant="normal">Γ</a:mi><a:mo stretchy="false">(</a:mo><a:mi>B</a:mi><a:mo stretchy="false">→</a:mo><a:mi>K</a:mi><a:msup><a:mi>e</a:mi><a:mo>+</a:mo></a:msup><a:msup><a:mi>e</a:mi><a:mo>−</a:mo></a:msup><a:mo stretchy="false">)</a:mo></a:math> and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msubsup><k:mi>R</k:mi><k:msup><k:mi>K</k:mi><k:mo>*</k:mo></k:msup><k:mrow><k:mi>μ</k:mi><k:mi>e</k:mi></k:mrow></k:msubsup><k:mo>≡</k:mo><k:msub><k:mi>R</k:mi><k:msup><k:mi>K</k:mi><k:mo>*</k:mo></k:msup></k:msub><k:mo>≡</k:mo><k:mi mathvariant="normal">Γ</k:mi><k:mo stretchy="false">(</k:mo><k:mi>B</k:mi><k:mo stretchy="false">→</k:mo><k:msup><k:mi>K</k:mi><k:mo>*</k:mo></k:msup><k:msup><k:mi>μ</k:mi><k:mo>+</k:mo></k:msup><k:msup><k:mi>μ</k:mi><k:mo>−</k:mo></k:msup><k:mo stretchy="false">)</k:mo><k:mo>/</k:mo><k:mi mathvariant="normal">Γ</k:mi><k:mo stretchy="false">(</k:mo><k:mi>B</k:mi><k:mo stretchy="false">→</k:mo><k:msup><k:mi>K</k:mi><k:mo>*</k:mo></k:msup><k:msup><k:mi>e</k:mi><k:mo>+</k:mo></k:msup><k:msup><k:mi>e</k:mi><k:mo>−</k:mo></k:msup><k:mo stretchy="false">)</k:mo></k:math>, the ratios <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msubsup><u:mi>R</u:mi><u:mi>K</u:mi><u:mrow><u:mi>τ</u:mi><u:mi>μ</u:mi></u:mrow></u:msubsup></u:math> and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:msubsup><w:mi>R</w:mi><w:msup><w:mi>K</w:mi><w:mo>*</w:mo></w:msup><w:mrow><w:mi>τ</w:mi><w:mi>μ</w:mi></w:mrow></w:msubsup></w:math> can deviate from their Standard Model (SM) predictions even with universal new physics couplings. This observation highlights the critical need to identify and establish genuine lepton flavor universality violating (LFUV) observables in the <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>τ</y:mi><y:mo>−</y:mo><y:mi>μ</y:mi></y:math> sector. This work embarks on establishing genuine LFUV ratio observables in <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>B</ab:mi><ab:mo stretchy="false">→</ab:mo><ab:mi>K</ab:mi><ab:mo>ℓ</ab:mo><ab:mo>ℓ</ab:mo></ab:math> and <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:mi>B</db:mi><db:mo stretchy="false">→</db:mo><db:msup><db:mi>K</db:mi><db:mo>*</db:mo></db:msup><db:mo>ℓ</db:mo><db:mo>ℓ</db:mo></db:math> decays through comprehensive analysis of their angular distributions. We find that like <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msubsup><gb:mi>R</gb:mi><gb:msup><gb:mi>K</gb:mi><gb:mo>*</gb:mo></gb:msup><gb:mrow><gb:mi>τ</gb:mi><gb:mi>μ</gb:mi></gb:mrow></gb:msubsup></gb:math>, the ratios <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:msubsup><ib:mi>R</ib:mi><ib:msub><ib:mi>A</ib:mi><ib:mrow><ib:mi>F</ib:mi><ib:mi>B</ib:mi></ib:mrow></ib:msub><ib:mrow><ib:mi>τ</ib:mi><ib:mi>μ</ib:mi></ib:mrow></ib:msubsup></ib:math> and <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:msubsup><kb:mi>R</kb:mi><kb:msub><kb:mi>f</kb:mi><kb:mi>L</kb:mi></kb:msub><kb:mrow><kb:mi>τ</kb:mi><kb:mi>μ</kb:mi></kb:mrow></kb:msubsup></kb:math> do not qualify as genuine LFUV observables, whereas the ratios of all optimized observables in <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:mi>B</mb:mi><mb:mo stretchy="false">→</mb:mo><mb:msup><mb:mi>K</mb:mi><mb:mo>*</mb:mo></mb:msup><mb:mo>ℓ</mb:mo><mb:mo>ℓ</mb:mo></mb:math> decays within the <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"><pb:mi>τ</pb:mi><pb:mo>−</pb:mo><pb:mi>μ</pb:mi></pb:math> sector definitively do. In the case of <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"><rb:mi>B</rb:mi><rb:mo stretchy="false">→</rb:mo><rb:mi>K</rb:mi><rb:mo>ℓ</rb:mo><rb:mo>ℓ</rb:mo></rb:math> decays, similar to <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:msubsup><ub:mi>R</ub:mi><ub:mi>K</ub:mi><ub:mrow><ub:mi>τ</ub:mi><ub:mi>μ</ub:mi></ub:mrow></ub:msubsup></ub:math>, the ratio <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:msub><wb:mi>R</wb:mi><wb:msub><wb:mi>F</wb:mi><wb:mi>H</wb:mi></wb:msub></wb:msub></wb:math> is influenced by mass effects and therefore cannot be considered a genuine LFUV observable in the <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:mi>τ</yb:mi><yb:mo>−</yb:mo><yb:mi>μ</yb:mi></yb:math> sector. However, the ratio <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:msub><ac:mi mathvariant="normal">Γ</ac:mi><ac:mi>τ</ac:mi></ac:msub><ac:mo stretchy="false">(</ac:mo><ac:mn>1</ac:mn><ac:mo>−</ac:mo><ac:msubsup><ac:mi>F</ac:mi><ac:mi>H</ac:mi><ac:mi>τ</ac:mi></ac:msubsup><ac:mo stretchy="false">)</ac:mo><ac:mo>/</ac:mo><ac:msub><ac:mi mathvariant="normal">Γ</ac:mi><ac:mi>μ</ac:mi></ac:msub><ac:mo stretchy="false">(</ac:mo><ac:mn>1</ac:mn><ac:mo>−</ac:mo><ac:msubsup><ac:mi>F</ac:mi><ac:mi>H</ac:mi><ac:mi>μ</ac:mi></ac:msubsup><ac:mo stretchy="false">)</ac:mo></ac:math> stands as the sole genuine LFUV observable in <ic:math xmlns:ic="http://www.w3.org/1998/Math/MathML" display="inline"><ic:mi>B</ic:mi><ic:mo stretchy="false">→</ic:mo><ic:mi>K</ic:mi><ic:mo>ℓ</ic:mo><ic:mo>ℓ</ic:mo></ic:math> decays. Furthermore, by making use of new physics Lorentz structures which provide a better fit to the current <lc:math xmlns:lc="http://www.w3.org/1998/Math/MathML" display="inline"><lc:mi>b</lc:mi><lc:mo stretchy="false">→</lc:mo><lc:mi>s</lc:mi><lc:mo>ℓ</lc:mo><lc:mo>ℓ</lc:mo></lc:math> data as compared to the SM, we demonstrate how the nongenuine LFUV ratios <oc:math xmlns:oc="http://www.w3.org/1998/Math/MathML" display="inline"><oc:msubsup><oc:mi>R</oc:mi><oc:msub><oc:mi>A</oc:mi><oc:mrow><oc:mi>F</oc:mi><oc:mi>B</oc:mi></oc:mrow></oc:msub><oc:mrow><oc:mi>τ</oc:mi><oc:mi>μ</oc:mi></oc:mrow></oc:msubsup></oc:math> and <qc:math xmlns:qc="http://www.w3.org/1998/Math/MathML" display="inline"><qc:msubsup><qc:mi>R</qc:mi><qc:msub><qc:mi>f</qc:mi><qc:mi>L</qc:mi></qc:msub><qc:mrow><qc:mi>τ</qc:mi><qc:mi>μ</qc:mi></qc:mrow></qc:msubsup></qc:math> can be employed to distinguish between framework with solely universal lepton couplings and those with both universal and nonuniversal couplings.
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Alok et al. (2024) studied this question.
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