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Recently, the CMS and ATLAS collaborations have announced the results for$H\rightarrow Z[\rightarrow \ell^{+}\ell^{-}]\gamma$ with $\ell=e$ or $\mu$\cite{CMS:2022ahq,CMS:2023mku}, where $H\rightarrow Z\gamma$ is a sub-processof $H\rightarrow \ell^{+} \ell^{-} \gamma$. This semi-leptonic Higgs decayreceives loop induced resonant $H\rightarrow Z[\rightarrow\ell^{+}\ell^{-}]\gamma$ as well as non-resonant contributions. % as discussedin \cite{Kachanovich:2021pvx}.  To probe further features coming from these contributions to $H\rightarrow\ell^{+} \ell^{-} \gamma$, we argue that the polarization of the final stateleptons is also an important parameter. We show that the contribution from theinterference of resonant and non-resonant terms plays an important role whenthe polarization of final state lepton is taken into account, which isnegligible in the case of unpolarized leptons. For this purpose, we havecalculated the polarized decay rates and the longitudinal ($P_L$), normal($P_N$) and transverse ($P_T$) polarization asymmetries. We find that theseasymmetries purely come from the loop contributions and are helpful to furtherinvestigate the resonant and non-resonant nature of $H\rightarrow Z[\rightarrow\ell^{+}\ell^{-}]\gamma$ decay. We observe that for $\ell=e,\mu$, thelongitudinal decay rate is highly suppressed around $m_{\ell\ell}\approx 60$GeVwhen the final lepton spin is $-\frac{1}{2}$, dramatically increasing thecorresponding lepton polarization asymmetries. Furthermore, we analyze anotherobservable, the ratio of decay rates $R^{\ell\ell'}_{i\pm}$, where $\ell$ and$\ell'$ refer to different final state lepton generations. Precise measurementsof these observables at the HL-LHC and the planned $e^{+}e^{-}$ can provide afertile ground to test not only the SM but also to examine the signatures ofpossible NP beyond the SM.

Analysis of final state lepton polarization-dependent observables in $H\to \ell^{+}\ell^{-} \gamma$ in the SM at loop level