$Z_H \rightarrow H^0 \gamma$ decay within the littlest Higgs model at $\mathcal{O}(\alpha_{{\rm ew}}^{3}\alpha_s)$ accuracy

We study the rare decay $Z_H \rightarrow H^0 \gamma$ at $\mathcal{O}(\alpha_{{\rm ew}}^{3}\alpha_s)$ accuracy including two-loop QCD corrections in the context of the littlest Higgs model (LHM) without $T$-parity. We revisit analytically and numerically the leading-order (LO) contributions of the one-loop diagrams induced by massive fermions, scalars and charged gauge bosons in the LHM, and further study the NLO QCD correction to this decay process. We perform the numerical calculation by taking the LHM input parameters $f=3,~4~\text{TeV}$ and $0.1< c <0.6$, and discuss the numerical results of the decay width up to the QCD NLO within the recent experimentally constrained LHM parameter space region. Our results show that the two-loop QCD correction always reduces the LO decay width and the top-induced QCD correction is the dominant contribution at the QCD NLO. For $f=4~\text{TeV}$ and $c=0.3$, the NLO QCD corrected decay width reaches $75.099~\text{keV}$ and the NLO QCD relative correction is about $-11.0\%$.