Constraining dark matter models with a light mediator from the CDEX-10 experiment at China Jinping Underground Laboratory

Various cosmological and astronomical observations provide compelling evidence of the existence of dark matter (DM) in the Universe [1]. The weakly interacting massive particle (WIMP) is a well-motivated DM candidate that can naturally explain the relic abundance of DM [2−4], but decades of direct detection experiments have not found positive signals yet [5−9]. However, introducing a light force carrier that mediates the DM-nucleus interaction can expand the regions of direct detection experiments.

If a mediator couples to both DM particles and standard model (SM) particles, the DM particle could interact with a nucleus via the exchange of the mediator. In this case, the nuclear recoil spectrum depends sensitively on both the DM and mediator masses. Furthermore, the low mediator mass causes a significant enhancement of direct detection cross sections; thus, an observation of DM scattering is possible despite the small couplings [10]. Light mediators have been widely proposed in many different DM models, such as hidden sector models [11], light DM models [12] and self-interacting DM (SIDM) models [13, 14].

SIDM models, in which DM particles scatter elastically with each other are partly motivated to solve the small-scale challenges to the ΛCDM Paradigm [14−16]. DM self-interaction can change the inner structure of DM halos and better explain astrophysical observations in galaxies while maintaining all the successes of ΛCDM on larger scales. For an observable effect on DM halos over cosmological time scales, the cross section of DM self-scattering per unit mass must be of order [13]

where $ m_\chi $ is the DM particle mass. This value of $ \sigma/m_\chi $, as shown in Eq. (1), significantly exceeds the expectation from weak-scale physics. For a typical WIMP, the cross section per unit mass is $ \sigma/m_\chi $$ \sim $10$ ^{-38} $ $ \mathrm {cm^2/GeV} $. Naturally, a light mediator is introduced to participate in the DM self-interaction [17]. The mediator mass is comparable to or lower than the typical momentum transfer ($ \mathcal{O} $(10) MeV) to yield the required cross section of DM self-scattering. Coupling with SM particles, light mediator particles have been directly searched in SIDM models [18−20].

In this work, we report the limits on the zero-momentum DM-nucleon interaction cross section through a light mediator based on the 205.4 kg-day exposure data from the CDEX-10 experiment. We further apply our results to constrain a well-motivated SIDM model with a light mediator.

The CDEX-10 experiment, aiming to detect light DM, operates a 10 kg p-type point-contact germanium (PPCGe) detector array in the China Jinping Underground Laboratory (CJPL) with a rock overburden of about 2400 m [21]. The detector array, consisting of three triple-element PPCGe detector strings (C10-A, B, C) encapsulated in copper vacuum tubes, is directly immersed in liquid nitrogen (LN$ _2 $) for cooling. A passive shield composed of 20-cm-thick high-purity oxygen-free copper surrounds the detector array in the LN$ _2 $ cryostat against environmental radioactivity. The LN$ _2 $cryostat and data acquisition (DAQ) system operate in a shielding room with 1-m-thick polyethylene walls at CJPL-I. The detailed configuration of the CDEX-10 experiment was described in our previous paper [22].

A 205.4 kg-day dataset from C10-B1 in the nine detectors was used for the DM search in this study. C10-B1, with the dead layer thicknesses of 0.88$ \pm $0.12 mm and fiducial mass of 939 g, was operated for data collection from February 2017 to August 2018. The data analysis followed several procedures described in previous papers [22, 23], including pedestal cut, physics-noise events cut, and bulk or surface events cut. The dead time of the DAQ system caused by the reset of the charge-sensitive preamplifier was measured to be 5.7% using random trigger events. The detector achieved an analysis threshold of 160 electron equivalent energy (eVee), with the combined efficiency of 4.5% and event rate of 2.5 counts $ \mathrm{kg^{-1} \;keV^{-1} \;day^{-1}} $ in the 2$-$4 keV energy range after all event selections and efficiency correction [22, 24].

A minimal-$ \chi^2 $ analysis method [8] was applied to the C10-B1 energy spectrum from 0.16 to 2.16 keV to search for nuclear recoil signals of DM. $ \chi^2 $ is defined as

where $ n_i $ is the measured event rate at the $i^{\rm th}$ energy bin, and $ S_i(m_\chi, m_\phi, \sigma_{\chi N}) $ is the expected DM event rate corresponding to DM mass $ m_\chi $, mediator mass $ m_\phi $, and DM-nucleus interaction cross section $ \sigma_{\chi N} $. Uncertainties $ \sigma_{\mathrm {stat},i} $ and $ \sigma_{\mathrm {syst},i} $ represent the statistical and systematical components, respectively. $ B_i $ denotes the background event rate. The background contribution of C10-B1 is assumed to be a flat continuum ($ p_{\mathrm 0} $) plus known M-shell and L-shell X-ray peaks [8] and can be constructed as

where $ E_{\mathrm {M/L}} $ is the peak energy of the M/L-shell X-rays listed in Table 1, and σ is the energy resolution. I is the intensity of the M/L-shell peak and is constrained by the intensity of the corresponding K-shell peak, which was measured as shown in Fig. 1.

Nuclide	$ T_{1/2} $	X-ray energy/keV		K/L ratio
		K-shell	M/L-shell
$ ^{68} $Ge	270.9 d	10.37	1.30(L)	0.12
			0.16(M)	0.03
$ ^{68} $Ga	68.1 min	9.66	1.19(L)	0.11
$ ^{65} $Zn	243.9 d	8.98	1.10(L)	0.12
$ ^{55} $Fe	2.7 yr	6.54	0.76(L)	0.11
$ ^{54} $Mn	312.2 d	5.99	0.70(L)	0.11
$ ^{49} $V	330.0 d	4.97	0.56(L)	0.11

Table 1.  Characteristic X-rays from cosmogenic radionuclides in germanium [8, 26].

Figure 1.  (color online) Spectrum fitting of the K-shell X-ray peaks of cosmogenic radionuclides in C10-B1 data. The fitting energy range is from 4.5 to 11 keV, with an energy bin width of 100 eV.

The best estimator of the DM-nucleon interaction cross section at a certain mediator mass is probed using $ \chi^2 $ minimization of the right hand-side of Eq. (2), with the background model shown in Eq. (3). Upper limits at the 90% confidence level (C.L.) are computed using the Feldman-Cousins method [25].

We first explore a general scenario where the interaction between DM and nucleons is mediated by a force carrier, ϕ. The light mediator is further assumed to possess equal effective couplings to the proton and neutron, motivated by the standard WIMP search. The cross section of DM–nucleon scattering via a light mediator can be expressed as [13]

where $ \sigma(q^2=0) $ is the DM-nucleon interaction cross section with zero momentum transfer $ (q^2=0) $, A is the mass number of the target nucleus, $ \mu(\mu_p) $ is the DM-nucleus (nucleon) reduced mass, $ m_\phi $ is the mediator mass, and $ F(q^2) $ is the Helm form factor of the target nucleus [27]. $ \sigma_{\chi N} $ is momentum dependent and negatively correlated with momentum transfer, which benefits detectors with a lighter target nucleus. When $ m_\phi $ $ \gg $ q, $ \sigma_{\chi N} $ converges to the standard WIMP case.

The expected energy spectrum of the DM signal is described as [28]

where local DM density $ \rho_{\chi} $ is set to 0.3 $ \mathrm {GeV/cm^3} $, $ v_{\mathrm {min}}(E_R) $ is the minimum DM velocity at a given recoil energy $ E_R $, and $ f(v,t) $ is the time-dependent DM velocity distribution relative to the detector [29].

Fig. 2 shows the expected nuclear recoil spectra in the C10-B1 detector for 10 GeV DM particles with $ m_\phi=1 $ MeV (red) and $ m_\phi=10 $ MeV (blue), calculated using Eqs. (4) and (5). These spectra fall steeply in the low-energy region and are also affected by the mediator mass. Thus, low background detectors with low energy thresholds, such as PPCGe detectors, are advantageous in searching for this type of DM signals.

Figure 2.  (color online) Expected spectra in the C10-B1 detector for 10 GeV DM particles with $ m_\phi=1 $ MeV (red) and $ m_\phi=10 $ MeV (blue). The energy resolution is fitted by the zero energy (defined by the random trigger events) and cosmogenic X-ray peaks, as shown in Fig. 1. The black dot represents the C10-B1 spectrum from 0.16 to 2.16 keV, with an energy bin width of 100 eV. The gray dash indicates the analysis threshold.

We scanned the DM mass in the range of 2$ - $10 GeV for $ m_\phi $ = 1 MeV and 1 GeV, and no significant signal was observed. An example of the background model plus an expected DM signal compared with the C10-B1 data is shown in Fig. 3. By minimizing the $ \chi^2 $ values of Eq. (2), the upper limits at 90% C.L. were derived and shown in Fig. 4. For $ m_\phi $ = 1 MeV, the upper limit is $ \sigma(q^2=0) = \mathrm{1.6\times10^{-36} cm^2} $ at $ m_\chi $ = 2 GeV. When $ m_\phi \geq $ 100 MeV, the momentum transfer q in Eq. (4) becomes negligible, and the upper limits approach those derived for the WIMP case [20]. For $ m_\phi $ = 1 GeV, the upper limit is $\sigma(q^2=0) = \mathrm{5.7\times10^{-40} cm^2}$ at $ m_\chi $ = 2 GeV. Compared with the results from PandaX-4T [30], this study used a high-purity germanium detector with lower energy threshold and excluded new parameter space in lower DM mass ranging from 2 to 3 GeV.

Figure 3.  (color online) Example of the background model plus an expected DM signal compared with the C10-B1 data. The best-fit background model (red) consists of the flat part (orange) and contributions from X-ray peaks (gray). The DM signal for ($ m_\chi $, $ m_\phi $, $ \sigma(q^2=0) $) = (5 GeV, 10 MeV, $ \mathrm{3.3\times10^{-42} cm^2} $) is depicted in blue.

Figure 4.  (color online) C10-B1 90% C.L. upper limits (red) on the DM-nucleon interaction cross section for DM models with mediator mass $ m_\phi $ = 1 MeV (dash line) and 1 GeV (solid line). The limits from PandaX-4T [30] are also shown for comparison (black and blue).

We consider a specific case, where DM is assumed to be a Dirac fermion and it couples to a light vector mediator through gauge mixing [13, 20]. We further assume that the mediator couples to a photon through kinetic mixing [31]. The DM-nucleon interaction cross section with zero momentum transfer $ (q^2=0) $ can be expressed as [20]

where $ \alpha_{\mathrm {EM}} $ and $ \alpha_{\chi} $ are the fine structure constants in the visible and dark sectors, respectively. We set $\alpha_{\rm EM}$ = 1/137 and take the value of $ \alpha_{\chi} $ from Ref. [32]. $ \epsilon_{\gamma} $ is the kinetic mixing parameter, and Z is the proton number of target nucleus.

The 90% C.L. lower limits in the $ m_\phi $-$ m_\chi $ plane for two $ \epsilon_{\gamma} $ values ($ \mathrm{10^{-8}} $ and $ \mathrm{10^{-9}} $) of Eq. (6) are derived. Our results can constrain a large portion of the SIDM parameter space favored by astrophysical observations. As shown in the top panel of Fig. 5, the sensitivity improves as $ \epsilon_{\gamma} $ increases. For $ \epsilon_{\gamma} = \mathrm{10^{-8}} $, the region with $ m_\chi $ > 2 GeV is excluded. For $ \epsilon_{\gamma} = \mathrm{10^{-9}} $, the region with $ m_\chi $ > 50 GeV is excluded. Alternatively, we can derive upper limits in the $ \epsilon_{\gamma} $-$ m_{\phi} $ plane for given $ m_\chi $, shown in the bottom panel of Fig. 5 with $ m_\chi $ = 2 and 10 GeV for instance.

Figure 5.  (color online) Top: CDEX-10 90% C.L. lower limits (red) on the mediator mass for the kinetic mixing parameter $ \epsilon_{\gamma} $ = 10$ ^{-8} $ (dash line) and 10$ ^{-9} $ (solid line). The blue band indicates the SIDM parameter region favored by astrophysical observations of DM halos [32]. The limits from PandaX-II [20] are shown for comparison (gray solid line). Bottom: CDEX-10 90% C.L. upper limits (red) on the kinetic mixing parameter for DM mass $ m_\chi $ = 2 GeV (dash line) and 10 GeV (solid line). The vertical band indicates the mediator mass range for SIDM.

We used the 205.4 kg-day dataset from CDEX-10 experiment to constrain DM models with a light mediator. With an analysis threshold of 160 eVee, we set new limits on the DM-nucleon interaction cross section for the DM mass ranging from 2 to 3 GeV when the mediator mass is comparable to or less than the typical momentum transfer ($ \mathcal{O} $(10) MeV). We further applied our results to constrain the parameter space under a SIDM model with a light mediator coupling to the photon through kinetic mixing. With the kinetic mixing parameter of $ \mathrm{10^{-8}} $, we excluded the parameter region favored by astrophysical observations with $ m_\chi $ > 2 GeV. Upper limits on the kinetic mixing parameter for DM mass of 2 GeV and 10 GeV were also derived.

We would like to thank CJPL and its staff for hosting and supporting the CDEX project. CJPL is jointly operated by Tsinghua University and Yalong River Hydropower Development Company.