Dark photon effects with the kinetic and mass mixing in Z -boson decay processes

Motivated by the most recent measurement of tau polarization in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>Z</a:mi><a:mo stretchy="false">→</a:mo><a:msup><a:mi>τ</a:mi><a:mo>+</a:mo></a:msup><a:msup><a:mi>τ</a:mi><a:mo>−</a:mo></a:msup></a:math> by CMS, we have introduced a new <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi>U</d:mi><d:mo stretchy="false">(</d:mo><d:mn>1</d:mn><d:msub><d:mo stretchy="false">)</d:mo><d:mi>X</d:mi></d:msub></d:math> gauge boson field <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>X</h:mi></h:math>, which can have renormalizable kinetic mixing with the standard model <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mrow><j:mi>U</j:mi><j:mo stretchy="false">(</j:mo><j:mn>1</j:mn><j:msub><j:mrow><j:mo stretchy="false">)</j:mo></j:mrow><j:mrow><j:mi>Y</j:mi></j:mrow></j:msub></j:mrow></j:math> gauge boson field <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>Y</n:mi></n:math>. In addition to the kinetic mixing of the dark photon, denoted as <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>σ</p:mi></p:math>, there may also be mass mixing introduced by the additional Higgs doublet with a vacuum expectation value participating in <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi>U</r:mi><r:mo stretchy="false">(</r:mo><r:mn>1</r:mn><r:msub><r:mo stretchy="false">)</r:mo><r:mi>X</r:mi></r:msub></r:math> and electroweak symmetry breaking simultaneously. The interaction of the <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mi>Z</v:mi></v:math> boson with the SM leptons is modified by the introduction of the mixing ratio parameter <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mi>ε</x:mi></x:math>, which quantifies the magnitude of both the mass and kinetic mixing of the dark photon. Initially, we use the tau lepton as an example to explore the <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mi>Z</z:mi></z:math>-boson phenomenology of the dark photon model with both kinetic and mass mixing. The goal is to determine the allowed parameter regions by taking into account constraints from the vector and axial-vector couplings <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:mrow><bb:msubsup><bb:mrow><bb:mi>g</bb:mi></bb:mrow><bb:mrow><bb:mi>V</bb:mi><bb:mo>,</bb:mo><bb:mi>A</bb:mi></bb:mrow><bb:mrow><bb:mi>τ</bb:mi></bb:mrow></bb:msubsup></bb:mrow></bb:math>, the decay branching ratio <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:mrow><db:mi>Br</db:mi><db:mo stretchy="false">(</db:mo><db:mi>Z</db:mi><db:mo stretchy="false">→</db:mo><db:msup><db:mrow><db:mi>τ</db:mi></db:mrow><db:mrow><db:mo>−</db:mo></db:mrow></db:msup><db:msup><db:mrow><db:mi>τ</db:mi></db:mrow><db:mrow><db:mo>+</db:mo></db:mrow></db:msup><db:mo stretchy="false">)</db:mo></db:mrow></db:math> and tau-lepton polarization in <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mi>Z</ib:mi><ib:mo stretchy="false">→</ib:mo><ib:msup><ib:mi>τ</ib:mi><ib:mo>−</ib:mo></ib:msup><ib:msup><ib:mi>τ</ib:mi><ib:mo>+</ib:mo></ib:msup></ib:math>. We found that the mixing ratio plays important role in the <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"><lb:mi>Z</lb:mi></lb:math>-boson features by choosing different <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"><nb:mi>ε</nb:mi></nb:math> values. Furthermore, we aim to generalize our analysis from the tau-lepton case to include all fermions by conducting global fits. This allows us to identify viable regions by incorporating relevant fermion constraints and the <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"><pb:mi>W</pb:mi><pb:mo>/</pb:mo><pb:mi>Z</pb:mi></pb:math> mass ratio. Correspondingly, we obtain the fit results with the kinetic mixing parameter <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"><rb:mi>σ</rb:mi><rb:mo>=</rb:mo><rb:mn>0.074</rb:mn><rb:mo>±</rb:mo><rb:mn>0.021</rb:mn></rb:math>, mixing ratio <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"><tb:mi>ε</tb:mi><tb:mo>=</tb:mo><tb:mo>−</tb:mo><tb:mn>1.37</tb:mn><tb:mo>±</tb:mo><tb:mn>0.46</tb:mn></tb:math>, and dark photon mass <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" display="inline"><vb:msub><vb:mi>m</vb:mi><vb:mi>X</vb:mi></vb:msub><vb:mo>=</vb:mo><vb:mn>275</vb:mn><vb:mo>±</vb:mo><vb:mn>39</vb:mn><vb:mtext> </vb:mtext><vb:mtext> </vb:mtext><vb:mi>GeV</vb:mi></vb:math>. Our global analysis indicates a preference for a dark photon mass larger than <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" display="inline"><xb:msub><xb:mi>m</xb:mi><xb:mi>Z</xb:mi></xb:msub></xb:math>. Published by the American Physical Society 2024