New physics search with the new gauge boson $Z'$ of the bestest little Higgs model at the muon collider

The Bestest Little Higgs Model (BLHM) has attracted increasing attention in recent years, mainly because it can explain the hierarchy problem without fine-tuning by introducing one-loop corrections to the Higgs boson mass through heavy top quark partners and heavy gauge bosons. In the context of this new model, we exhaustively investigated the impact of the BLHM parameters on the Higgs-strahlung production processes $\mu^{-} \mu^{+} \to Z'h_0$ and $\mu^{-} \mu^{+} \to Z'H_0$, where $Z'$ and $H_0$ represent a new heavy gauge boson and a new heavy Higgs boson, respectively. We consider the integrated luminosities of ${\cal L}=2000, 20 000$ ${\rm fb^{-1}}$ and the respective center-of-mass energies of $\sqrt{s}=3000, 10 000$ GeV projected for a future muon collider to estimate the number of expected Higgs boson production events $h_0$ or $H_0$ in association with a new gauge boson $Z'$ at the muon collider. In addition, we derive and present the Feynman rules for scalars ($h_0, H_0, \phi^{0}, \eta^{0}, \sigma, H^{\pm}, \phi^{\pm}, \eta^{\pm}, A_0$) and include the self-interactions of Higgs bosons.