Complementarity of the future $e^+ e^-$ colliders and gravitational waves in the probe of complex singlet extension to the Standard Model

In this work, we study the future probes of the complex singlet extension to the Standard Model (cxSM). This model is possible to realize a strongly first-order electroweak phase transition (SFOEWPT). The cxSM naturally provides dark matter (DM) candidate, with or without an exact $\mathbb{Z}_2$ symmetry in the scalar sector. The benchmark models which can realize the SFOEWPT are selected, and passed to the current observational constraints to the DM candidates, including the relic densities and the direct detection limits set by the latest XENON1T results. We then calculate the one-loop corrections to the SM-like Higgs boson decays and the precision electroweak parameters due to the cxSM scalar sector. We perform a global fit to the benchmark models and study the extent to which they can be probed by the future high-energy $e^+ e^-$ colliders, such as CEPC and FCC-ee. Besides, the gravitational wave (GW) signals generated by the benchmark models are also evaluated. We further find that the future GW detector, such as LISA, is complementary in probing the benchmark models that are beyond the sensitivity of the future precision tests at the $e^+ e^-$ colliders.