We investigate the capability of the future electron collider CEPC in probing the
parameter space of several dark matter models, including millicharged dark matter
models,

Article funded by SCOAP3

M_{Z'}$, the resonance in the photon energy spectrum exceeds the maximum energy of the final state photon and thus cannot be observed. \begin{figure} \centering \includegraphics[width=0.45\columnwidth]{figs/xsection_mq_4}\hfill \includegraphics[width=0.45\columnwidth]{figs/xsection_df_2}\\ \includegraphics[width=0.45\columnwidth]{figs/xsection_4fvv_1000}\hfill \includegraphics[width=0.45\columnwidth]{figs/xsection_4fss_1000} ]]>

0.1 \GeV$ and $|z_\gamma|<0.99$. ]]>

0.1$\,GeV and $|z_\gamma|<0.99$. The center-of-mass energy square is $\sqrt{s} = 240$ (91.2) GeV for the $H$-mode ($Z$-mode); for the $WW$-mode, we use $\sqrt{s}=160 \GeV$ as the benchmark point. In the millicharged DM model, the monophoton cross section increases when $\sqrt{s}$ decreases at CEPC, for DM mass lighter than 40\,GeV, as shown in the upper-left panel figure of figure~\ref{fig:mqxsections}. Thus the $Z$-mode has the better sensitivity than the other two modes in probing light millicharged DM. For the four-fermion effective operators, the monophoton cross section increases when $\sqrt{s}$ increases at CEPC, as shown in the lower two panel figures of figure~\ref{fig:mqxsections}. Thus the $H$-mode has the better sensitivity than the other two low energy modes in probing the DM effective operators with TeV suppression scale. For the $Z'$ portal DM model, we consider the case in which the mass of the $Z'$ boson is 150\,GeV; the monophoton cross section in the $WW$-mode is larger than the other two modes, for the case in which DM is lighter than $\sim 70$\,GeV, as shown in the upper-right panel figure of figure~\ref{fig:mqxsections}, where we consider the vector coupling only case. It is interesting to note that the monophoton cross section exhibits a resonance feature when the mass of the DM is in the vicinity of $M_{Z'}/2$. Thus we expect a better sensitivity in probing the $Z'$ portal DM models in the $WW$-mode, and enhanced constraints in the parameter space where the $Z'$ boson is nearly twice of the DM mass. We note that the different relations between the total production cross section and $\sqrt{s}$ in the three types of DM models are primarily due to the mass scale of the mediator. ]]>

0.1\GeV$ and $|\cos\theta_\gamma| < 0.99$. ]]>

0.1$\,GeV. For the photon energy resolution, we use the resolution in the dual-readout calorimeter~\cite{CEPCStudyGroup-ml-2018ghi} \be {\sigma(E) \over E} = \frac{10.1 \%}{\sqrt{E/{\rm GeV}}}\bigoplus0.4\%. \label{eq:resolution} \ee We use the above detector parameters in our simulations. The EMC positional resolution has been estimated as $\sim 0.1$ mm~\cite{Wang-ml--ml-2017}, which gives rise to a relative spatial resolution as $\sim 10^{-5}$, since the radius of the detector cylinder is about 2 meters. Thus we do not take the spatial resolution into consideration in our simulations due to its smallness. \begin{figure} \centering \includegraphics[width=0.5\columnwidth]{figs/fig4.pdf} ]]>

0.99$. For example, due to the collinear singularity, the $e^+e^-\to \gamma {e}^+ {e}^- $ process has large cross section when both final state electron and positron go along the beam directions; the corresponding Feynman diagrams are exhibited in figure~\ref{fig:feynman-rbg}. \begin{figure} \centering \includegraphics[width=0.32\columnwidth]{figs/fig7c}\hfill \includegraphics[width=0.32\columnwidth]{figs/fig7b}\hfill \includegraphics[width=0.32\columnwidth]{figs/fig7a_3} ]]>

10\GeV$ \& $|\cos\theta_\gamma| < 0.9$, which is different from the first two plots.]]>

E_B^{m}(\theta_\gamma)$ on the final state monophoton in our analysis to suppress the events arising from the SM reducible backgrounds, such as the $e^+e^-\rightarrow f\bar{f}\gamma$ and $e^+e^-\rightarrow \gamma \gamma \gamma$ processes. The right panel figure of figure~\ref{fig:3d} shows that the above cut is efficient in removing the reducible backgrounds. Thus, to suppress the contributions from SM, we apply the following detector cuts to the monophoton events in SM and in DM models: \begin{itemize} \item[(1)] $E_\gamma > 0.1 \GeV$, \item[(2)] $|\cos\theta_\gamma| < |\cos\theta_b| = 0.99$, \item[(3)] $E_\gamma < E_\chi^{m} = (s - 4 m_\chi^2)/(2 \sqrt{s})$, \item[(4)] veto $E_\gamma \in (E_\gamma^Z \pm 5 \Gamma_\gamma^Z)$, \item[(5)] $E_\gamma(\theta_\gamma) > E_{B}^m (\theta_\gamma) = \sqrt{s} (1+\sin\theta_\gamma/\sin\theta_b)^{-1}$. \end{itemize} We will collectively refer to the five detector cuts in the list as the ``basic detector cuts'' hereafter. Unlike the other detector cuts, the last detector cut in the list is a 2D cut which is applied to the two-dimension space spanned by $E_\gamma$ and $\theta_\gamma$. Both $E_\gamma^Z$ and $\Gamma_\gamma^Z$ in the 4th detector cut are functions of $\sqrt{s}$ in the CEPC running modes. ]]>

0.1$\,GeV and $|\cos\theta_\gamma| < 0.99$. For the millicharged models, we consider three different masses: $m_\chi=1$\,GeV, 40\,GeV, and 100\,GeV for $\varepsilon = 0.01$. The photon energy bin width on the left panel figure, $\sim$ 2.4\,GeV, is larger than the photon energy resolution which is $\delta E_\gamma \simeq 1$ (0.3, 0.1) GeV for $E_\gamma=100$ (10, 1) GeV, according to eq.~(\ref{eq:resolution}). The vertical line $E=0.12 \sqrt{s}$ indicates the boundary of the detector cut designed to remove the reducible backgrounds. ]]>

0.1$\,GeV and $|\cos\theta_\gamma| < 0.99$. Figure~\ref{fig:mqzhist} shows the normalized $E_\gamma$ distributions of the signal and background in the $Z$-mode and in the $WW$-mode, with the detector cuts: $E_\gamma > 0.1$\,GeV and $|\cos\theta_\gamma| < 0.99$. \begin{figure} \centering \includegraphics[width=0.45\columnwidth]{figs/mqzhist_e_2}\qquad \includegraphics[width=0.45\columnwidth]{figs/mqwhist_e_2} ]]>

0.1$ and $|\cos\theta_\gamma| < 0.99$. For the millicharged models, we consider $\varepsilon = 0.01$ for three different masses in each case; we consider $m_\chi=1$\,GeV, 25\,GeV, and 40\,GeV in the $Z$-mode, and $m_\chi=1$\,GeV, 25\,GeV, and 50\,GeV in the $WW$-mode.]]>

E_\gamma^Z(s) + 5 \Gamma_\gamma^Z(s)$ in addition to the basic detector cuts, for $m_\chi < 25$ (30) GeV. \begin{figure} \centering \includegraphics[width=0.55\columnwidth]{figs/mqrej_configs_2} ]]>

E_\gamma^Z-5\Gamma_Z$ on top of the basic detector cuts. For the $H$ mode, we always select events in the $Z'$ resonance if it is present; thus for $m_\chi \leq 75 \GeV$ in the $H$ mode, we require $147 \GeV < M_\gamma < 153 \GeV$ where $M_\gamma = \sqrt{s-2 \sqrt{s} E_\gamma}$. We can also search for the $Z'$ boson via its visible decay channels at CEPC (see e.g.~\cite{Karliner-ml-2015tga,He-ml-2017zzr} for previous studies on this topic). We take the $\mu^+\mu^-$ final state as the visible channel to probe the $Z'$ portal DM models in this section. We adopt the muon momentum resolution as follows \be {\delta p_T \over p_T} = {p_T \over 10^5\GeV} \bigoplus 0.1\% ~{\rm for}~ |\eta| < 1.0 \label{eq:mu} \ee and 10 times greater for $1.0<|\eta|<3.0$, which has been implemented in the CEPC card in Delphes~\cite{deFavereau-ml-2013fsa}. The muon momentum resolution is much better than the photon energy resolution, as given in eq.~(\ref{eq:resolution}). \begin{figure} \centering \includegraphics[width=0.45\columnwidth]{figs/delphes_histo_egamma.pdf}\qquad \includegraphics[width=0.45\columnwidth]{figs/delphes_histo_mmumu.pdf} ]]>

75$\,GeV. In our $Z'$ portal DM model, we assume that the $Z'$ boson has a universal coupling to both charged leptons and to quarks. Thus, one expects a recoil signal arising from the $Z'$ portal DM model in dark matter direct detection experiments that look for weakly interacting massive particles (WIMPs). Below we consider two different cases. First, we consider the case where the $Z'$ boson couples both to SM fermions and to DM fermion via vector couplings; in this case, the dark matter direct detection cross section is dominated by the spin-independent (SI) cross section which is given by \be \sigma^{\rm SI}_{n\chi} = \sigma^{\rm SI}_{p\chi} = {9 \over \pi} {(g_V^fg_V^\chi \mu_{n\chi})^2 \over M_{Z^\prime}^4} \ee where $\mu_{n\chi}$ is the reduced mass of the DM and nucleon. We also consider the case where the $Z'$ boson couples both to SM fermions and to DM fermion via axial-vector couplings; in this case, the dark matter direct detection cross section is dominated by the spin-dependent (SD) cross section which is given by~\cite{Aprile-ml-2019dbj,Boveia-ml-2016mrp} \be \sigma^{\rm SD}_{n\chi} = \sigma^{\rm SD}_{p\chi} \simeq {0.31 \over \pi} {(g_A^fg_A^\chi\mu_{n\chi})^2 \over M_{Z^\prime}^4}. \ee \begin{figure} \centering \includegraphics[width=0.45\columnwidth]{figs/dfrej_3.pdf}\qquad \includegraphics[width=0.45\columnwidth]{figs/dfrejaxial_3.pdf} ]]>

75\GeV$, the visible decays of the $Z'$ boson become the better channel to study the $Z'$ model considered where the monophoton channel quickly loses its sensitivity, as shown on both plots in figure~\ref{fig:zpRes}. Interestingly, there is a sudden increase in the sensitivity in the monophoton channel when the dark matter mass approaches 75\,GeV from below so that a very narrow dip structure is shown near $m_\chi=75$\,GeV, in the vector coupling only case. There is also an increased sensitivity in the di-muon channel when $m_\chi$ becomes larger than 75\,GeV, due to the change of the detector cuts as the $Z'$ width turns narrower for $m_\chi$ crosses 75\,GeV. We also compute the upper bound from Xenon1T experiment, including the SI limit~\cite{Aprile-ml-2018dbl} for the vector coupling case, and the SD limit~\cite{Aprile-ml-2019dbj} for the axial-vector coupling case. The Xenon1T limit is stronger than the CEPC limit for $m_\chi \geq 5$\,GeV in the vector only case. For the axial-vector only case, however, the CEPC limits are typically better than the current Xenon1T limits. We apply Xenon1T limits to the $Z'$ portal DM models based on the assumption that the $Z'$ boson couples to both charged leptons and quarks with equal coupling strength. If the $Z'$ boson only interacts with electrons in the SM sector, the Xenon1T limits analyzed is no longer applicable. \begin{figure} \centering \includegraphics[width=0.4\columnwidth]{figs/dfrej_couplings4.pdf} ]]>

95 \GeV$ in addition to the basic detector cuts, to improve the CEPC sensitivity. \begin{figure} \centering \includegraphics[width=0.45\columnwidth]{figs/efthist_vv_e_2.pdf} ]]>