Search for Dark-Matter–Nucleon Interactions with a Dark Mediator in PandaX-4T

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From the perspective of particle physics, one popular construction involves a dark force carrier that mediates the interactions between DM and SM particles.In the traditional DM direct search experiments [1][2][3], a heavy mediator is usually assumed, leading to a mediator-massindependent DM-nucleon cross section.However, when the mediator mass is comparable to or even smaller than the momentum transfer in nuclear recoils (NRs), a softer recoil spectrum is expected, which leads to mediatormass-dependent search results [4][5][6][7].The dark mediator can also indirectly interact with SM particles by mixing with gauge or Higgs bosons.If it kinetically mixes with the ordinary photon, it is called a dark photon, which is a well-motivated vector boson arising from a hidden U(1) gauge symmetry.Dark photons can provide possible explanations of anomalies ranging from particle physics to cosmology, e.g., in muon g − 2 [8,9], the 8 Be nuclear transitions [10], and the so-called small-scale problems in galactic astronomy (see Ref. [11] for an overview).They have been searched extensively at dedicated fixed-target experiments [12][13][14][15][16][17] and at colliders [18].More dedicated experiments are planned [19,20].
In this Letter, we report a highly sensitive search for DM-nucleon interaction mediated by a dark mediator using optimized low-energy data from the commissioning run of the PandaX-4T experiment, and use the results to test a class of dark-photon-mediated DM models.PandaX-4T is the third generation DM direct detection experiment of the PandaX project, located at the China Jinping Underground Laboratory.The central apparatus of PandaX-4T is a dual-phase xenon time projection chamber (TPC), which contains 3.7 tonnes of liquid xenon in the sensitive volume.Particle interaction with xenon nuclei or electrons in the liquid xenon produces scintillation photons (S1 signal) and the ionized electrons (S2 signal).Both signals are detected by the top and bottom arrays of 368 Hamamatsu R11410-23 3inch photomultiplier tubes (PMTs).Analog signals from the PMTs are digitized and then read out in a triggerless scheme [21], using CAEN V1725B digitizers [22], which is crucial to enhance the sensitivity to low-energy recoils.More detailed descriptions of the PandaX-4T experiment can be found in Ref. [1].
DM-nucleon interaction with a dark mediator.-First, we consider the physics scenario in which a vector or scalar force mediator ϕ coherently mediates the interaction between the DM and nucleon, with the same effective couplings for the proton and neutron.The differential NR recoil rate (in the unit of events/day/kg/keV) for elastic scattering between the DM and xenon nucleus is given by [23,24] where q 2 is the four-momentum-transfer-squared, σ| q 2 =0 is the DM-nucleon cross section in the limit of zero momentum transfer, A is the xenon mass number, µ p is the DM-nucleon reduced mass, m ϕ (m χ ) is the mediator (DM) mass, F (q 2 ) is the nuclear form factor, ρ is the local DM density, f (v) is the DM velocity distribution relative to the detector, and v min is the minimum DM velocity that results in a NR energy E NR .All input parameters to Eq. ( 1) were set to the conventions given in Ref. [25].
Besides the pure NR process, we consider the NRinduced electron recoil (ER) signals by the Migdal effect [26], which has been employed by XENON, LZ, LUX, CRESST, SuperCDMS and CDEX experiments [27][28][29][30][31][32] to extend the reach for low-mass DM searches [33].When a DM particle scatters with a xenon atom, the nucleus undergoes an abrupt momentum change with respect to the orbital electrons, resulting in the excitation or ionization of the atomic electrons due to the lack of transient movement of the electron cloud.This effect leads to the possible generation of ER signals in the keV range that accompany the primary NR.Therefore, even if the NR energy deposition is below the detection threshold, the ER energy deposition due to the Migdal effect can still be detected, providing a way to probe low-mass DM particles that are otherwise not detectable in PandaX.In this Letter, we only consider the ionization process, as the excitation probabilities are negligible in the energy region of interest [26,34].The differential rate of ionization electron with energy E ER is given by folding the NR spectrum in Eq. ( 1) with the transition rate [26], where is the probability for an atomic electron with quantum numbers (n, l) and binding energy E nl to be ionized and receive a kinetic energy of E ER − E nl .It is related to the electron momentum relative to the struck nucleus, therefore depends on E NR .Similar to Refs.[27,29], we only take into account the contributions from the ionization of M-shell (n=3) and N-shell (n=4) electrons.The binding energies of inner shells (n=1, 2) are too strong to contribute significantly.The contribution from xenon valence electrons (n=5) has also been conservatively omitted, different from the treatment in Ref. [35], as the ambient atoms in the liquid may lead to large uncertainty in the ionization energy.We only consider DM masses up to 2 GeV, above which the contribution of NR signals becomes comparable or dominant.
In this Letter, the DM candidates are selected in two complementary ways, with and without requiring the presence of the S1 signal in each event.The former (denoted as S1-S2 below), has lower background contamination but a higher energy threshold, while the latter (S2-only) is the opposite.
For the S1-S2 data, we follow the same procedure for the solar 8 B neutrino and low-mass DM search as in Ref. [36], using the commissioning data with an exposure of 0.48 tonne-year.Each event requires one pair of physically correlated S1 and S2 signals within the fiducial volume of the TPC.The S1 signal is required to have 2 or 3 coincident PMT hits (corresponding to a 1%acceptance threshold of 0.95 keV nr ), and the S2 signal needs to be 65-230 PE for 2-hit S1 and 65-190 PE for 3hit S1.A boosted decision tree (BDT) algorithm-based selection is applied to suppress the dominated accidental coincidence background from randomly paired S1 and S2 signals.After the BDT cut, there is one event left in the 2-hit S1 data (Fig. 1).The total number of background events, dominated by the accidental background and the 8 B neutrino, is estimated to be 2.9 ± 1.0.No 3-hit S1 events survive and the background prediction is 0.4 ± 0.1 events.
For the S2-only data, we used the same datasets and procedure as in Ref. [37] that correspond to an effec- tive exposure of 0.55 tonne-year.The dataset, the background models, and the conversion response matrix from a given energy deposition to a distribution of S2 are provided in a public repository [38].In total, there are 105 events with unpaired S2 (with no accompanying S1 greater than 2 PE) between 60 and 200 PE [37], and the detection energy threshold is approximately 0.8 keV nr corresponding to a 14% acceptance.The background is mainly composed of electrons from TPC cathode, microdischarge noises (MD) from the electrodes, and solar neutrinos.Figure 2 shows the S2 distribution in data, in comparison with the background from the backgroundonly best fit.
For the statistical inference of DM signals, the binned profile likelihood ratio [43,44] is constructed as the test statistics, including the same treatments for the systematic uncertainties as in Refs.[36,37].The left panel of Fig. 3 shows the zero-momentum DM-nucleon cross section limits for mediator mass of 1 MeV/c 2 or 1 GeV/c 2 , using both S1-S2 data and S2-only data for DM masses ranging from 3 GeV/c 2 to 10 GeV/c 2 .As expected, the limits obtained from S2-only data are more stringent than those from S1-S2 data at smaller DM masses.At 5 GeV/c 2 , our limits are 1 order of magnitude stronger than the previous results from PandaX-II experiment.The right panel of Fig. 3 shows the limits using S2only data with Migdal effect for DM masses between 30 MeV/c 2 and 2 GeV/c 2 .The limits for m ϕ = 1 GeV/c 2 is within 1% of the presented heavy mediator limit.Our results represent a significant improvement over other ex- FIG. 3. Left: the 90% C.L. upper limits on the zero-momentum DM-nucleon cross section for light mediator DM models for mediator masses 1 MeV/c 2 and ≥ 1 GeV/c 2 .The red lines represent the exclusion limit of this work using S1-S2 data.The orange lines and green shaded region represent the limits and ±1σ sensitivity (for m ϕ ≥ 1 GeV/c 2 ) with S2-only data.For comparison, the limits from our previous analysis based on the full dataset of PandaX-II data release in 2021 [5] (green) and the results of XENON1T S2-only [39] (blue), LZ S1-S2 [2] (brown), and DarkSide-50 [40] with quenching fluctuations (violet) are shown.Right: upper limits on the DM-nucleon cross section at 90% C.L. obtained with a signal including the Migdal effect for mediator masses 0.1 and 1 MeV/c 2 and ≥ 1 GeV/c 2 (red lines), together with ±1σ sensitivity for m ϕ ≥ 1 GeV/c 2 (green shaded area).Also shown are limits under the heavy mediator case from CRESST-III [30] (green), XENON1T [27] (blue), LUX [29] (cyan), LZ [28] (brown), CDMSlite [41] (yellow), SuperCDMS [31] (magenta) , DarkSide-50 [42] (violet), and CDEX-10 [32] (orange).
periments.For example, for DM masses ranging from 100 to 500 MeV/c 2 , our limits are 1 to 2 orders of magnitude stronger than those of XENON1T in the case of a heavy mediator.Note that we have assumed the socalled constant-W model for the charge yield all the way to zero energy, to be consistent with our earlier work in Refs.[37,45], whereas in the XENON1T treatment in Ref. [27], the charge yield was truncated below 186 eV.This is partly responsible for our tighter constraint in this Letter.For the upper limits in this Letter, we have checked that the Earth attenuation effect [46] is negligible.For example, considering a scattering cross section of 10 −30 cm 2 , along with m ϕ at 0.1 MeV/c 2 and m χ at both 0.04 and 2 GeV/c 2 , the impact on event rates was found to be 1.6% and 1.9%, respectively.
Constraints to dark-photon-mediated DM model.-Next,we consider the dark mediator as a dark photon, with its kinetic mixing strength to ordinary photon denoted as ϵ γ .Such a mixing provides an important connection between DM in the hidden sector and SM particles in the visible sector [12,[47][48][49][50] in the early Universe.One typical case is that the DM abundance is set by its annihilation into the SM particles, dominated by the process with an s-channel dark photon mediator [13].For scalar DM [51], the annihilation cross section scales as ⟨σv⟩ ∼ ϕ for a mediator with m ϕ ≫ 2m χ and a negligible decay width [13].Here, α EM and α χ are the fine structure constants in the visible and dark sectors, respectively.As we can see, the aforementioned cross section depends on m χ and a dimensionless parameter y = ϵ 2 γ α χ (m χ /m ϕ ) 4 [53].The observed DM relic abundance requires ⟨σv⟩ ∼ ⟨σv⟩ relic , which in turn fixes y for a given m χ (see Fig. 4).
On the other hand, the zero-momentum DM-nucleon cross section in direct detection experiments is given by [24] where Z is the proton number of the xenon nuclei.Therefore, the cross section σ| q 2 =0 scales with y×µ 2 p /m 4 χ , which provides a constraint in y versus m χ .
It is worth noting that for direct detection experiment, when m ϕ > ∼ 0.05 m χ , the q 2 contribution is negligible in Eq. (1).For m χ < 1 GeV/c 2 , our results using S2-only data with the Migdal effect and with m ϕ = 1 GeV/c 2 (in Fig. 3, right) clearly satisfy this condition.Figure 4 illustrates the resulting upper limit on the parameter y.Therefore, our results provide a direct test of darkphoton-mediated DM as the thermal relic, and rule out the aforementioned scalar DM model in the mass range between 30 MeV/c 2 and 1 GeV/c 2 .Similarly, recent constraints on DM-electron cross sections (F DM = 1) from PandaX-4T [37]  FIG. 4. The 90% C.L. exclusion limit (red) on the dimensionless interaction strength from PandaX-4T S2-only data for DM masses ranging from 30 MeV/c 2 to 1 GeV/c 2 , assuming m ϕ > ∼ 0.05 mχ.The limit from PandaX-4T DM electron [37] is also shown as the dark green curve.The black curve refers to the constraint from the observed relic DM density for the scalar elastic DM [13].Also included are results of NA64 [14] (azure), MiniBooNE [15] (brown) , LSND [16] (orange), E137 [17] (cyan), and BABAR [18] (green) experiments, where αχ = 0.5 and m ϕ = 3 mχ are assumed.The projected limits from the LDMX experiment [52] (phase 1) and DarkSHINE experiment [20] (9 × 10 14 electron on target) are shown as the green dashed line and blue dot-dashed line, respectively.Note that exclusions and sensitivity curves from the dark photon experiments scale with αχ × (mχ/m ϕ ) 4 .
DM-electron reduced mass, also overlaid on Fig. 4. We display this limit separately, as it bears no theoretical uncertainty associated with the Migdal effect.The limit is even stronger than that from DM-nucleon interaction for DM masses ranging from 38 to 100 MeV/c 2 .On the other hand, dedicated fixed target experiments for dark photon search (e.g NA64 [14], LSND [16], E137 [17]) can set upper limits on y, but usually assuming a particular value of α χ and a fixed ratio between m ϕ and m χ .Future experiments such as LDMX [52] and DarkSHINE [20] are expected to probe low-mass region that has not yet been accessible by PandaX.
Summary.-In summary, we present a search for the interactions between DM particles and nucleons via a dark mediator in the PandaX-4T experiment.The analysis used both low-energy S1-S2 and S2-only data from the commissioning run.The results from S2-only data with Migdal effect are further used to constrain the interaction strength between dark photon and ordinary photon.The result provides the most stringent constraint for the thermal scalar DM model, in which the s-channel dark photon decays into DM pairs in the early Universe, for the DM masses ranging from 30 MeV/c 2 to 1 GeV/c 2 .
We would like to thank Kun Liu and Tran Van Que for their useful discussions.This project is supported in part by grants from the National Natural Science Introduction.-Oneparticularly important question on the dark matter (DM) is how it interacts with standard arXiv:2308.01540v3[hep-ex] 19 Dec 2023 model (SM) particles beyond the gravitational effect.

FIG. 1 .
FIG.1.Selected DM candidate event in the S1-S2 plane after the BDT cut with the requirement of 2-hit S1.The cyan and violet lines represent the 95% contour of the signals for a DM mass of 5 GeV/c 2 with mediator masses of 1 MeV/c 2 and 1 GeV/c 2 , respectively.

FIG. 2 .
FIG. 2.Distribution of selected candidates in S2-only data and background components.The expected signals in PandaX-4T with and without the Migdal effect for DM masses of 0.5 and 5 GeV/c 2 are shown in red solid and blue dashed lines, respectively.In both cases, m ϕ is set to be 1 MeV/c 2 , and zero-momentum-transfer DM-nucleon cross section of 3 × 10 −38 cm 2 and 8 × 10 −40 cm 2 (close to the later exclusion limits) are assumed, respectively.

4 χ 16παEMµ 2 e
can be as well translated into limits on y by the relation y = σ e m[54], where µ e is the Foundation of China (No. 12090060, No. 12090061, No. 11875190, No. 12005131, No. 12105052, No. 11905128, No. 11925502, No. 11835005), a grant from Office of Science and Technology, Shanghai Municipal Government (Grant No. 22JC1410100), and Chinese postdoctoral Science Foundation (Grant No. 2021M700859).We thank Double First Class Plan of the Shanghai Jiao Tong University and the Tsung-Dao Lee Institute Experimental Platform Development Fund for support.We also thank the sponsorship from the Hongwen Foundation in Hong Kong, Tencent Foundation in China, and Yangyang Development Fund.Finally, we thank the China Jinping Underground Laboratory administration and the Yalong River Hydropower Development Company Ltd. for indispensable logistical support and other help.