Exploring for sub-MeV Boosted Dark Matter from Xenon Electron Direct Detection

Light dark matter (DM) candidate is well motivated and can be naturally realized when the DM candidate couples feebly to visible sector Hall et al. (2010); Chu et al. (2012); Essig et al. (2012a); Knapen et al. (2017); Bernal et al. (2017). In particular, it is difficult for a sub-MeV DM candidate to satisfy observed relic abundance through the thermal freeze-out mechanism Boehm et al. (2013); Nollett and Steigman (2014); Cao et al. (2019); therefore, freeze-in via annihilation of electron-positron pairs is a primary mechanism for DM production  Chu et al. (2012); Essig et al. (2012a); Dvorkin et al. (2019). The traditional direct detection of DM-nucleus scattering loses sensitivity rapidly for a DM candidate whose mass is below $\sim{\rm GeV}$ due to the threshold of recoil energy. An alternative way to search for a light DM candidate is through the scattering off electrons Essig et al. (2012a, b, 2017), which is not sensitive to a sub-MeV DM candidate neither. It is crucial to develop new approach to probe freeze-in DM in such mass range.

A certain fraction of DM candidates in the Galactic halo would be accelerated by energetic Cosmic-Ray (CR) particles as long as the DM candidate interacts with SM particles. The CR-boosted mechanism relaxes the threshold problem and improves the sensitivity of detecting a light DM candidate An et al. (2018); Bringmann and Pospelov (2019). It has been extensively discussed in DM-nucleus direct detections, neutrino experiments and CR observations for various DM models Cappiello et al. (2019); Ema et al. (2019); Alvey et al. (2020); Cappiello and Beacom (2019); Dent et al. (2019); Krnjaic and McDermott (2019); Bondarenko et al. (2019); Berger et al. (2019); Wang et al. (2019). In this Letter we investigate the CR-boosted effect on the DM-electron direct detection in the freeze-in scenario and show that the existing data from xenon experiments are able to probe a sub-MeV DM candidate.

For illustration, we consider a typical freeze-in DM model based on the vector-portal, in which the DM candidate is a Dirac fermion ($\chi$) that couples to the visible sector through an additional gauge boson $A^{\prime}_{\mu}$, named as “dark photon”. The Lagrangian is given by

where $m_{\chi}$ and $m_{A^{\prime}}$ denote the mass of DM candidate and the dark photon, respectively. $g_{\chi}$ and $g_{\rm SM}$ are the coupling strength of $A^{\prime}$ to the DM candidate and the electron, respectively. When the DM candidate scatters off an incident CR electron with a given kinetic energy ($T_{\rm CR}$), the distribution of the DM recoil energy $T_{\chi}$ is

where $\bar{\sigma}_{e}$ denotes the cross section of DM-free electron scattering for a fixed momentum transfer $q=\alpha m_{e}$ Essig et al. (2012a). The maximal recoil energy of the DM candidate is Goldstein et al. (2001)

Convoluting the $T_{\chi}$ distribution in Eq. (2) with the energy spectrum of incident CR electrons $d\Phi_{e}/dT_{\rm CR}$ yields the recoil flux of boosted DM candidate Bondarenko et al. (2019)

where $D_{\rm eff}\equiv\int\frac{d\Omega}{4\pi}\int_{l.o.s}dl$ is an effective diffusion distance. See supplement materials for details. For a homogeneous CR distribution and NFW DM halo profile Navarro et al. (1996, 1997) (scale radius $r_{s}=20$ kpc and local DM density $\rho^{\rm local}_{\chi}=0.4\,{\rm GeV}\,{\rm cm}^{-3}$), integrating along the line-of-sight to 10 kpc yields $D_{\rm eff}=8.02~{}{\rm kpc}$ Bringmann and Pospelov (2019). In order to produce a recoil energy $T_{\chi}$ after the DM and CR-electron scattering, the minimum kinetic energy ($T_{\rm CR}^{\rm min}$) of the incident CR electron is given by

where the plus and minus sign corresponds to $T_{\chi}>2m_{e}$ and $T_{\chi}<2m_{e}$, respectively.

Figure 1 plots the recoil flux $d\Phi_{\chi}/dT_{\chi}$ distributions as a function of $T_{\chi}$ for various $m_{A^{\prime}}$’s. Two simplified models are also plotted for comparison; one is the cross section $\sigma_{\chi e}$ being a constant (black-solid curve), the other is that the the squared matrix element of the DM-electron scattering ($\overline{|\mathcal{M}|^{2}}$), averaged over initial and summed over final spin states, is a constant (black-dashed curve), i.e.

The former case is commonly used in the study of non-relativistic DM candidates, the later one takes the energy dependence from phase space into account. However, the both treatments are not appropriate for an energetically boosted DM candidate whose kinetic energy is much larger than its mass such that the momentum transfer $q$ cannot be neglected. We consider the relativistic kinematics throughout this work. As shown in Fig. 1, the flux distribution exhibits a significant enhancement at the large $T_{\chi}$ range with increasing $m_{A^{\prime}}$. Note that various recoil flux curves intersect at $T_{\chi}=(\alpha m_{e})^{2}/(2m_{\chi})$, and the recoil flux distribution of the constant $\overline{|\mathcal{M}|^{2}}$ slightly deviates from that of the constant $\sigma_{\chi e}$ when $2m_{\chi}T_{\mathrm{CR}}>(m_{e}+m_{\chi})^{2}$.

It is worth mentioning that the recoil flux distribution is independent of $m_{A^{\prime}}$ when the dark photon is very heavy ($m_{A^{\prime}}\gg\sqrt{2m_{\chi}T_{\chi}}$) or ultralight ($m_{A^{\prime}}\ll\alpha m_{e}$). See the red and blue boundaries of the contour. The recoil flux distributions in the above two limits exhibit distinct dependence on $T_{\chi}$; for example, the recoil flux of ultralight dark photons drops rapidly with $T_{\chi}$ while the recoil flux of heavy dark photons mildly decreases with $T_{\chi}$. The heavy dark photon represents the so-called $Z^{\prime}$-portal model while the ultralight dark photon the milli-charged DM model Holdom (1986).

Equipped with the boosted DM flux, we now discuss the DM direct detection through the DM interaction with the electron in xenon atoms. For the ionization process of $\chi+A\rightarrow\chi+A^{+}+e^{-}$ with the atom $A$ in the $(n,\,l)$ atomic shell, the velocity-averaged differential cross section with respect to the electron recoil energy $E_{R}$ is given by Essig et al. (2012a, 2016)

where $F_{DM}$ is the DM form factor, $\eta$ denotes the mean inverse speed function and $\left|f_{ion}^{nl}(k^{\prime},q)\right|^{2}$ represents the ionization form factor for an electron with initial state $(n,l)$ and final state with momentum $k^{\prime}=\sqrt{2m_{e}E_{R}}$. In the case of boosted DM, the DM form factor $F_{DM}$ is

In the non-relativistic limit, $T_{\chi},E_{R}\ll m_{e}$, it reproduces the form factor without CR-boost effects, i.e.,

The mean inverse speed function $\eta$ is replaced by An et al. (2018)

where $\Phi_{\rm halo}=n_{\chi}\bar{v}_{\chi}$ is the background DM flux in Galactic halo with $\bar{v}_{\chi}$ being the corresponding average velocity. Here, $E_{\min}$ is the minimal DM energy to trigger electron with recoil energy $E_{R}$. Similarly, Eq. (10) reproduces the conventional expression

in the non-relativistic limit. The ionization form factor $\left|f_{ion}^{nl}(k^{\prime},q)\right|^{2}$ is calculated by using the Roothaan-Hartree-Fock radial wavefunction Bunge et al. (1993) for initial electron state and applying plane wave approximation for final state. For initial electron state, we take into account contributions from $(5p^{6},5s^{2},4d^{10},4p^{6},4s^{2})$ xenon electron shells. The differential ionization rate is obtained by multiplying Eq. (7) with background DM flux $\Phi_{\rm halo}$, the number of target atoms $N_{T}$, and sum over different electron shells,

Figure 2 shows the ionization rate as a function of the electron recoil energy $E_{R}$ (in unit of ${\rm tonne}^{-1}~{}{\rm year}^{-1}$) for both ultralight (red) and heavy (blue) dark photons with the choices of $m_{\chi}=1~{}{\rm keV}$ and $\bar{\sigma}_{e}=10^{-30}~{}\mathrm{cm}^{2}$. The vertical band represents the order of magnitude of energy coverage for current xenon experiments. The ultralight dark photon prefers to produce electrons with small recoil energy; however, the heavy dark photon is likely to generate electrons with large recoil energy. The distinct difference follows from the energy dependence in the distribution of $d\sigma_{\chi e}/dT_{\chi}$ and the DM form factor $F_{DM}(q)$. It implies that one might distinguish between the dark photon and the heavy dark photon from the recoil energy spectrum of the ionized electron when the background is well understood.

The recoiling electron are then converted into scintillation ($S1$) and ionization ($S2$) signal in liquid xenon experiments, and the observable is the number of photonelectrons (PE). We consider $S2$ signal hereafter as the XENON100 and XENON1T collaborations release the data sets that are based only on the ionization signal Aprile et al. (2016, 2019a). The event spectrum can be schematically written as following:

where $T_{\rm exp}$ is the exposure of detector and $\varepsilon_{S2}$ is the efficiency of triggering and accepting the $S2$ signal. For a given deposit energy $\Delta E_{e}=E_{R}+|E^{nl}_{B}|$ with $|E^{nl}_{B}|$ the binding energy of $(n,l)$ shell, the conversion probability of $S2$ is ${\rm pdf}\left(S2|\Delta E_{e}\right)$, which is modeled as follows Essig et al. (2012b, 2017). The number of primary quanta produced at the interaction point is $n^{(1)}_{Q}={\rm Floor}(E_{R}/W)$ with $W=13.8$ eV, and $n^{(1)}_{Q}$ is divided into $n_{e}$ observable ionized electrons escaping from interaction point and $n_{\gamma}$ unobservable scintillation photons. The fiducial value of the fraction of primary quanta identified as electrons is chosen as $f_{e}\simeq 0.83$. In addition, in the case of the DM candidate ionizes an inner shell electron, the secondary quanta is produced by subsequent electron transitions from outer to inner shell. The number of the secondary quanta is $n^{(2)}_{Q}={\rm Floor}((E_{i}-E_{j})/W)$ where $E_{i}$ denotes the binding energy of the $i$th shell. The production number of secondary electrons follows a binomial distribution with $n_{Q}^{(1)}+n_{Q}^{(2)}$ trials and the success probability $f_{e}$. Finally, the number of PE converted from electrons (with total number $n_{e}=n^{(1)}_{e}+n^{(2)}_{e}$) is described by a gaussian distribution with mean value $n_{e}\mu$ and width $\sqrt{n_{e}}\sigma$. The parameters are chosen as $\mu=19.7~{}(11.4)$ and $\sigma=6.9~{}(2.8)$ Aprile et al. (2016, 2019b).

We derive the limits of $\bar{\sigma}_{e}$ imposed by the XENON100 data Aprile et al. (2016) ($T_{\rm exp}=30~{}{\rm kg-years}$) and by the XENON1T data Aprile et al. (2019a) (effective $T_{\rm exp}=22~{}{\rm tonne-days}$), using the same bin steps. We choose the detection efficiency as $\varepsilon_{S2}=1$ for simplicity and obtain the limits by demanding that signal does not exceed $1\sigma$ upper bound in each bin. Figure 3 presents benchmark signal spectra versus PE for the ultralight and heavy mediator cases.

Figure 4(A) shows the exclusion limits in the $m_{\chi}$-$\bar{\sigma}_{e}$ plane for the case of a ultralight mediator, derived from the XENON100 data (red) and the XENON1T data (green). The acceleration mechanism greatly enhances the discovery potential of direct detection experiments on a light DM candidate. For comparison we also plot the parameter region for the freeze-in DM (brown curve) Essig et al. (2012a); Dvorkin et al. (2019). Even though the parameter space of freeze-in DM is well below the current direct detection sensitivity, it can be reached when large experimental exposures are achieved. For example, the experimental exposure of 30 tonne-years can probe the signal region of freeze-in DM with $m_{\chi}\sim 1$ eV when the background is fully controlled. In addition, the DM with a ultralight mediator (or equivalent milli-charged DM) can also be constrained by astrophysical observations from supernova cooling and stellar energy loss Chang et al. (2018); Vogel and Redondo (2014). The bounds from the direct detection experiments are comparable to those astrophysical constraints.

Figure 4(B) displays the exclusion limit of $\bar{\sigma}_{e}$ for the case of a heavy mediator. We also plot the limits from Super-Kamiokande neutrino experiment Cappiello and Beacom (2019), solar reflection An et al. (2018), and the direct detection without CR-DM scattering effect Essig et al. (2017). After considering the CR-DM effect, the direct detection experiments have a better sensitivity in the sub-keV mass region.

In summary, we studied the effect of boosted DM on DM-electron direct detections and demonstrate that the current data from liquid noble gas experiments is sensitive to light DM candidates in the range of sub-MeV. More importantly, the energy dependence in cross section plays a crucial role in improving the exclusion limits, e.g., the recoil spectra increase with recoil energy for heavy mediator case while decrease with recoil energy for ultralight mediator. Such opposite energy dependences imply that the neutrino experiments such as Super-K are more powerful for heavy mediator due to their much larger acceptance volume and higher energy coverage Bays et al. (2012). On the other hand, direct detection has more advantage on ultralight mediator. Such two kind of experiments are complementary to each other.

The CR boosted DM mechanism has very rich phenomenologies. For example, it is interesting to investigate boosted DM flux coming from the Galactic center which possesses high DM density and CR flux. One also expects that the morphology of signal resulted from the Galactic center is different from that originated from local interstellar Carlson and Profumo (2015). Moreover, light DM with significant CR acceleration and heavy DM ($m_{\chi}\gtrsim 10~{}\mathrm{MeV}$) with negligible CR acceleration could potentially produce degenerate signal; therefore, discrimination of such two kinds of scenarios in both model independent and model specific way is an intriguing issue Cao et al. . The boosted mechanism might explain or be constrained by the recoiled energy spectrum of electrons recently reported by the XENON1T collaboration Aprile et al. (2020).

SUPPLEMENTAL MATERIALS

The supplemental materials provide additional details to various results presented in the main text. Some of the results can be applied to other light DM models.