Searching for the light leptophilic gauge boson $Z_x$ via four-lepton final states at the CEPC

We investigate the possibility of detecting the leptophilic gauge boson $Z_x$ predicted by the $U(1{)}_{L_e-L_{\mu }}$ model via the processes $e^{+}{e}^{-}\to {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}{Z}_x(Z_x\to {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-})$ and $e^{+}{e}^{-}\to {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}{Z}_x(Z_x\to {\nu }_{\mathrm{\ell }}\overline{{\nu }_{\mathrm{\ell }}})$ at the Circular Electron Positron Collider (CEPC) with the center of mass energy $\sqrt{s}=240$ GeV and the luminosity $\mathcal{L}=5.6$ $\mathrm{a}{\mathrm{b}}^{-1}$. We give the expected sensitivities of the CEPC to the parameter space at $1\sigma$, $2\sigma$, $3\sigma$ and $5\sigma$ levels.