Enhanced $Z$ boson decays as a new probe of first-order electroweak phase transition at future lepton colliders

We study phenomenological consequences of the strong first-order electroweak phase transition in an extension of the standard model with an inert doublet and vectorlike leptons motivated by the muon $g-2$ anomaly and dark matter. We find that a condition for the strong first-order electroweak phase transition inevitably induces a large logarithmic enhancement in $Z$ boson decays, which relegates the explanation of the anomalous muon $g-2$ at below $2\sigma$ level. Our analysis shows that future lepton collider experiments, especially the Giga-$Z$ at the International Linear Collider and Tera-$Z$ at the Circular Electron Positron Collider as well as Future Circular Collider have great capability to explore the nature of the electroweak phase transition, which is complementary to conventional approaches via measurements of the triple Higgs boson coupling and gravitational waves.

Figure 1

Representative Feynman diagrams for the one-loop corrections to $Z\to {\mu }^{+}{\mu }^{-}$.

Figure 2

Schematic Feynman diagrams for the one-loop contributions to $(g-2{)}_{\mu }$.

Figure 3

Deviation of the lepton flavor universality in the $Z$ boson decays $\mathrm{\Delta }{R}_{\mu /e}$ as a function of $m_A$ with $m_E=120\mathrm{GeV}$. The solid curves in red, blue and black correspond to the deviation $\mathrm{\Delta }{R}_{\mu /e}$ in the $Z$ boson decay for $|y_{\mu E}|=1.0$, 0.5 and 0.3, respectively. The upper, middle and lower horizontal dashed lines represent the current bound, future Giga-$Z$ sensitivity and Tera-$Z$ sensitivity of $\mathrm{\Delta }{R}_{\mu /e}$, respectively. The region of the SFOEWPT is the right-hand side of the vertical line. The greater $v_C/T_C$ corresponds to the larger enhancement of $Z\to {\mu }^{+}{\mu }^{-}$.

Figure 4

Corrections to the muon magnetic dipole moment anomaly $\delta a_{\mu }$ as a function of $m_A$. The shaded regions in green, blue and magenta correspond to the $(g-2{)}_{\mu }$ regions within $1\sigma$, $2\sigma$ and $3\sigma$, respectively. The solid curves in red, blue and black correspond to the deviation $\delta a_{\mu }$ for $|y_{\mu E}|=1.0$, 0.5 and 0.3, respectively.

Figure 5

Direct search for the vectorlike leptons via the process involving dimuon plus MET at the lepton colliders.

Figure 6

The cross section $\sigma (e^{+}{e}^{-}\to \gamma /Z\to E^{+}{E}^{-})$ as a function of $m_E$. We take $\sqrt{s}=240\mathrm{GeV}$ (blue) and 250 GeV (orange), respectively.