Constraints on Lightly Ionizing Particles from CDMSlite

The Cryogenic Dark Matter Search low ionization threshold experiment (CDMSlite) achieved efficient detection of very small recoil energies in its germanium target, resulting in sensitivity to Lightly Ionizing Particles (LIPs) in a previously unexplored region of charge, mass, and velocity parameter space. We report first direct-detection limits calculated using the optimum interval method on the vertical intensity of cosmogenically-produced LIPs with an electric charge smaller than $e/(3\times10^5$), as well as the strongest limits for charge $\leq e/160$, with a minimum vertical intensity of $1.36\times10^{-7}$\,cm$^{-2}$s$^{-1}$sr$^{-1}$ at charge $e/160$. These results apply over a wide range of LIP masses (5\,MeV/$c^2$ to 100\,TeV/$c^2$) and cover a wide range of $\beta\gamma$ values (0.1 -- $10^6$), thus excluding non-relativistic LIPs with $\beta\gamma$ as small as 0.1 for the first time.

The Cryogenic Dark Matter Search low ionization threshold experiment (CDMSlite) achieved efficient detection of very small recoil energies in its germanium target, resulting in sensitivity to Lightly Ionizing Particles (LIPs) in a previously unexplored region of charge, mass, and velocity parameter space. We report first direct-detection limits calculated using the optimum interval method on the vertical intensity of cosmogenically-produced LIPs with an electric charge smaller than e/(3 × 10 5 ), as well as the strongest limits for charge ≤ e/160, with a minimum vertical arXiv:2011.09183v4 [hep-ex] 19 Feb 2022 intensity of 1.36 × 10 −7 cm −2 s −1 sr −1 at charge e/160. These results apply over a wide range of LIP masses (5 MeV/c 2 to 100 TeV/c 2 ) and cover a wide range of βγ values (0.1 -10 6 ), thus excluding non-relativistic LIPs with βγ as small as 0.1 for the first time.
Introduction: The strong CP problem [1], observation of neutrino oscillations [2], matter-antimatter asymmetry [3], evidence for dark matter [4], and evidence for dark energy [5] all suggest that the Standard Model (SM) provides an incomplete framework and motivate searches for physics beyond the SM. A promising avenue of exploration is the search for particles with a fractional electric charge. Fractionally-charged particles (FCPs) have charge q = ± f e, where e is the elementary charge and f has a value between 0 and 1. Many extensions to the SM [6][7][8][9][10] contain unconfined ('free') FCPs. A nonrelativistic FCP has been proposed to explain the annual modulation signal observed by the DAMA/LIBRA [11] and CoGeNT [12] detectors [13,14]. If particles with fractional charge exist, the lightest FCP must be stable, motivating these searches [15].
Constraints on FCP parameter space arise from astrophysical observations and laboratory experimentation [16,17]. Figure 1 shows the constraints for free FCPs in the mass-charge plane. Free FCPs with small electric charge are known as Lightly Ionizing Particles (LIPs), because their mean energy loss per unit length dE/dx is suppressed as f 2 [18] compared to particles with electron charge. Direct-detection experiments for LIPs are of particular interest, because they are sensitive to cosmogenically produced LIPs with both smaller f and larger mass than any other experimental searches (see Fig. 1).
Based on data from the SuperCDMS experiment, this paper describes the first direct search for LIPs with a variety of incident βγ values (0.1-10 6 ) and for f as small as 10 −8 , where β = v/c, γ = 1/ 1 − β 2 , and v is the LIP velocity. This is the first work to set limits on non-relativistic LIPs with βγ as small as 0.1 (still ∼55 times larger than that expected of galactically bound [34] LIPs). The analysis described herein searches for LIPs in an unexplored parameter space for masses between 5 MeV/c 2 and 100 TeV/c 2 .
Experimental setup and data: The SuperCDMS experiment employed five vertical stacks of detectors in the Soudan Underground Laboratory with each stack comprised of three germanium detectors [35]. Each detector was a ∼600 g cylindrical crystal with a 3.8 cm radius and 2.5 cm height, instrumented on each face (top and bottom) with four phonon and two ionization sensors. One of the detectors located in the middle of a stack was operated in CDMSlite mode [35], with a bias of 70 V applied between its two faces. All others were biased at 4 V. The detector operated at higher bias voltage amplifies the phonon signal via the Neganov-Trofimov-Luke (NTL) effect [36], allowing it to achieve a < 100 eV energy threshold. For information about detector opera-  [19], CDMS II [20], Majorana [21], TEXONO [22] and CDMSlite (this search) constrain the intensity of cosmogenic FCPs; other constraints are adapted from Refs. [23,24] and include those from accelerator-based experiments (AC) [25,26], Ar-goNeut (AG) [24], the search for the invisible decay of orthopositronium (OP) [27], the SLAC millicharged particle search (SLAC) [28], the Lamb shift (L) [29], Big Bang nucleosynthesis (BBN) [23], plasmon decay in red giants (RG) [30], plasmon decay in white dwarfs (WD) [30], the cosmic microwave background (CMB) [31] and Supernova 1987A (SN) [32]. The CDMSlite experimental constraints extend to the greatest value of f −1 permitted such that the cosmological density of relic FCP does not exceed the total density of our universe [33]. The constraints shown are for βγ (see definition in text) of 0.1 which gives the least restrictive upper bounds on masses. This analysis is the first direct detection experiment to probe the impact of mass on the signal model. tion, readout, and response, see Ref. [35].
This analysis uses data from the first period (February through July 2014) of the second CDMSlite run [35]. CDMSlite Run 2 Period 1 had a live time of 97.81 days, which was 84.6 % of the full Run 2 live time. Using only Period 1 data simplified the analysis with only a marginal reduction in sensitivity. This analysis was performed in an effectively "blind" fashion: although the CDMSlite Run 2 spectrum based on both Period 1 and Period 2 data is published [35], the analyzers did not use the Period 1 data to develop the limit-setting framework, including selection criteria, or to project sensitivities. Reconstructed energy depositions in the CDMSlite detector between 100 eV and 2 keV were analyzed with the 2 keV upper limit chosen for the same reason as the CDMSlite Run 2 WIMP search [35]. Energy-deposition FIG. 2. Simulated energy-deposition distributions averaged over incident angle dP dE (f, βγ) for LIPs incident on the detector with two different values of βγ and various f −1 between 10 2 and 2 × 10 3 , before the application of selection criteria, and after convolution with the detector energy resolution. The solid lines show the energy-deposition distributions assuming an isotropic incident LIP distribution, and the dotted lines show the distribution assuming a cos 2 θ incident distribution. The figure also shows the total probability (p) of energy deposition within the analysis energy range for the isotropic distribution. The atomic L-shell peaks at 1.3 keV can be seen. For f −1 > 2 × 10 3 , the shape of dP/dE does not change but merely scales down as f 2 . The distributions are independent of mass in the range considered. spectra were simulated using the CDMSlite Run 2 background model [37] and were used to develop the analysis framework and make limit projections.
Signal model : The LIP flux is attenuated by the atmosphere and rock overburden before reaching the experimental site. This can introduce an angular dependence in the LIP distribution. As in Ref. [21], we consider two limiting cases: 1) an isotropic angular distribution and 2) a cos 2 θ angular distribution, where θ is the angle of an incident LIP relative to zenith. The former case corresponds to minimal attenuation for small f , while the latter case corresponds to muon-like attenuation for large f .
Expected energy-deposition probability distribution functions for LIPs passing through the CDMSlite detector are obtained using Geant4 [38] simulations. The simulation incorporates several processes including ionization, bremsstrahlung, pair production, and scattering (single and multiple). The Geant4 Photo Absorption Ionization (G4PAI 1 ) [39] model is used for the simula-tion of energy loss via ionization. The G4PAI model is typically used to model energy depositions in situations where a paucity of interactions is expected.
The simulated energy-deposition distributions are convolved with the detector resolution [35] and are calculated for a range of values of the LIP parameters: f , mass, and βγ for both angular distributions. Figure 2 shows convolved energy-deposition distributions dP/dE for various f and βγ of LIPs incident on the detector. Example distributions for both minimum-ionizing (βγ ∼ 3.1) and non-relativistic (βγ ∼ 0.1) LIPs are shown to illustrate the impact of LIP velocity on the scattering probability for a given fractional charge.
The LIP mass impacts the expected energy-deposition distribution through the bremsstrahlung process. The number of bremsstrahlung interactions is proportional to the inverse square of the LIP mass. Simulations show that the bremsstrahlung contribution to dP/dE is negligible within the analyzed energy window and the chosen LIP mass range 2 (5 MeV/c 2 -100 TeV/c 2 ); consequently, dP/dE is found to be effectively independent of the LIP mass in our analysis.
However, dP/dE is dependent on βγ due to the ionization process, which is the dominant LIP energy-loss mechanism in the detector. The ionization cross section is a function of f and βγ. The assumption of minimumionizing (βγ ∼ 3.1) LIPs leads to the least restrictive limits for LIPs with f −1 550 as will be shown later. LIPs with smaller f −1 and/or smaller βγ ( 1) have substantial probability of depositing energy above the largest energy deposition considered (2 keV), resulting in a reduced LIP sensitivity. This dependence of LIP sensitivity on βγ motivates our consideration of a range of LIP βγ values.
Selection criteria and efficiency: All data selection criteria used in the CDMSlite Run 2 WIMP search [35] including the single-detector-hit criterion and the fiducialvolume criterion are applied to this LIP search. However, for LIPs the efficiencies of the single-detector-hit and fiducial-volume selections tend to be lower than those for WIMPs; correction factors to account for these relative inefficiencies are calculated using Monte Carlo simulation. The product of efficiency correction factors for the single-detector-hit ( sdh (f, βγ)) and fiducial-volume ( fv (f, βγ)) criteria is taken as the combined efficiency correction factor ( corr (f, βγ)).
The single-detector-hit criterion requires the CDMSlite detector to be the only detector from all five stacks with a reconstructed energy deposition greater than its energy threshold [35]. This selection criterion reduces background sources capable of depositing energy in multiple detectors. The single-detector-hit criterion is relatively efficient for LIPs with small f , because other detectors have energy thresholds ∼10 times larger ( 1 keV ee ) than the CDMSlite detector. However, LIPs with large f may deposit energy in multiple detectors and hence can be rejected by this selection criterion. To account for lost sensitivity, sdh (f, βγ) is estimated as where N CDMSlite (f, βγ) is the number of simulated LIP events depositing energy in the CDMSlite detector within the analyzed energy window (0.1-2 keV), and N md (f, βγ) is the number of LIP events that also deposit energy in at least one other detector above its threshold ( 1 keV). Because the non-uniform electric field at high radius in the CDMSlite detector results in an inaccurate reconstruction of deposited energy, a fiducial-volume selection criterion was applied to remove events with energy depositions located at relatively high radius [35]. While calculating fv (f, βγ), we conservatively assume that the position reconstruction of all events with more than one interaction point in the CDMSlite detector is such that they are rejected. Hence, where N total (f, βγ) is the number of simulated LIP events depositing energy in the CDMSlite detector, and . Also shown is the analysis efficiency (based on CDM-Slite Run 2 WIMP-search [35] and depicted by the red dashed curve labeled on right axis) with 1σ uncertainty (red band), before the correction for additional LIP-selection inefficiency (see Fig. 3). The energy depositions are measured in electron equivalent units (keVee) where it is assumed that all energy depositions in the detector are due to electron recoils [35].
N m (f, βγ) is the number of these interacting at more than one location in the same detector. Figure 3 shows the combined efficiency correction factor corr as a function of f −1 for various values of βγ. It is smallest for f −1 = 10 2 as these LIPs have a higher probability of interaction, and it rapidly approaches unity as the value of f −1 increases. The efficiency correction was made under the approximation that the cuts were uncorrelated, which was checked to produce less than a 10 % inaccuracy. The correction factor is usually lower ( 15 %) for an isotropic angular distribution than for a cos 2 θ angular distribution; an isotropic angular distribution results in a higher average LIP path length within the CDMSlite detector, which increases the fraction of LIPs capable of interacting more than once. The most ionizing LIPs considered (βγ 0.3 and f −1 300) have a substantial probability of depositing above-threshold energy in the detector immediately above or below the CDMSlite detector, causing them to fail the single-detector-hit criterion. As a result, corr is ∼ 3.5 times larger for the most ionizing LIPs for an isotropic distribution. The CDMSlite Run 2 Period 1 analysis efficiency (Fig. 4) is multiplied by corr to obtain the final LIP-selection efficiency, (f, βγ, E). Intensity limit calculation: The upper limit at 90 % confidence level on the LIP vertical intensity, I 90 v (f, βγ), for an isotropic incident angular distribution is given by where N 90 (f, βγ) is the 90 % confidence upper limit on the expected number of observed LIPs, τ is the live time of the detector, dP dE (f, βγ, θ) is the LIP energy-deposition distribution at LIP incident angle θ and (f, βγ, E) is the LIP-selection efficiency. The effective cross-sectional area of the detector surface at θ is A(θ) = πr 2 cos θ +2rh sin θ, where r and h are the detector radius and height, respectively. To compute I 90 v (f, βγ) for a cos 2 θ angular distribution, dP dE (f, βγ, θ) is weighted by a cos 2 θ factor. We calculate N 90 (f, βγ) using the Optimum Interval (OI) method [40] under the conservative assumption that all observed events in the energy-deposition distribution could be due to LIP interactions. This method does not provide any discovery potential. The values of N 90 (f ) obtained are between 41 and 79.
Expected sensitivity and uncertainty: The LIPs projected sensitivity is determined by computing the mean expected upper limit from background alone, based on 200 different energy-deposition spectra simulated using the CDMSlite Run 2 background model [37]. Each simulated distribution contains a random number of events that is statistically consistent with that predicted for the Period 1 live time. For each sensitivity calculation, the analysis efficiency and the energy thresholds are varied within their uncertainties. The resulting 1σ uncertainty in the sensitivity is ∼32 %.
It is difficult to estimate the systematic uncertainty due to possible deviation of the true dP/dE from that given by Geant4. We estimate this uncertainty by comparing the sensitivity obtained herein with that resulting from dP/dE obtained using the CDMS II convolution method [20,41]. Our Geant4-based sensitivity is ∼24 % less restrictive, which we take as an estimate of the systematic uncertainty on dP/dE for the entire range of βγ values considered. We estimate the total uncertainty (∼40 %) on the expected sensitivity by combining this dP/dE systematic uncertainty in quadrature with the estimate for the other sources of uncertainty. The uncertainty on the final LIP vertical intensity limit is ∼37 %; it includes the analysis-efficiency and energy-threshold uncertainties, and the dP/dE systematic uncertainty.
Unblinding and results: We examined the LIP-search data for the first time only after finalizing the eventselection criteria and their efficiencies, the systematic uncertainties, and the procedure for calculating the LIP vertical intensity. The measured spectrum contains 180 events after application of all selection criteria and is shown in Fig. 4. The most prominent features in the spectrum are the L-and M-shell peaks from decays of intrinsic Ge radioisotopes, as described in Ref. [35]. A general agreement was observed between the data spec-FIG. 5. The CDMSlite 90 % upper confidence limit on the LIP vertical intensity (solid black) under the assumptions of an isotropic distribution for minimum-ionizing LIPs. The red band shows the 1σ level uncertainty in the limit. The result is compared to those from all prior searches for cosmogenic LIPs, including LSD [42] (brown ), Kamiokande [43] (purple ∆), MACRO [19] (yellow dot-dashed), CDMS-II [20] (blue dashed), Majorana [21] (green dotted), and TEX-ONO [22] (magenta dot-dashed). The expected sensitivity generally lies on top of the final limit to within the resolution of the plot except for a small range of larger fractional charges (160 < f −1 < 500), where the limits are slightly less restrictive.
trum and the simulated background spectrum.
Due to the f 2 suppression of the interaction rate in the rock overburden, the expected intensity of energetic LIPs (βγ ≥ 3.1) at the experiment is minimally reduced relative to that at the surface for the range of LIP charges considered. LIP βγ is reduced by 10 % for LIPs with f −1 > 10 4 , and for LIPs with βγ ≥ 1 and f −1 > 10 3 . LIPs with mass 1 GeV/c 2 , lower values of βγ, and lower values of f −1 may be attenuated or have their value of βγ reduced by the overburden. The vertical-intensity limit I 90 v (f, βγ) is shown in Fig. 5 for a minimum-ionizing LIP (βγ = 3.1) with an isotropic incident distribution and is compared to limits from prior direct searches for cosmogenic LIPs. This result sets the strongest constraint on LIPs with f −1 > 160, including a minimum verticalintensity limit of 1.36 × 10 −7 cm −2 s −1 sr −1 at f −1 = 160. The final limit agrees with the expected sensitivity to within about 2 σ for 160 < f −1 < 500 and within 1 σ elsewhere. Figure 6 shows the limits for a variety of βγ values. The results are valid for the entire mass range considered: 5 MeV/c 2 to 100 TeV/c 2 . The intensity limit computed for a cos 2 θ angular distribution is nearly three times weaker than that for an isotropic angular distribution for most values of f . Summary: Utilizing a SuperCDMS detector operated in CDMSlite mode, this work presents the first directdetection limits on the vertical intensity of cosmogenic LIPs with charge less than e/(3 × 10 5 ) for values of incident βγ ranging from 0.1 to 10 6 . Although the OI limit-setting method used does not have discovery potential, the result reported herein represents a significant step towards searching for dark matter with fractional charge [13,14] by setting the first limit on non-relativistic LIPs with βγ values as small as 0.1. Future searches extending to yet lower values of βγ may probe galactically bound LIPs.