Low-Mass Dark Matter Search with CDMSlite
SuperCDMS Collaboration, R. Agnese, A.J. Anderson, T. Aralis, T. Aramaki, I.J. Arnquist, W. Baker, D. Balakishiyeva, D. Barker, R. Basu Thakur, D.A. Bauer, T. Binder, M.A. Bowles, P.L. Brink, R. Bunker, B. Cabrera, D.O. Caldwell, R. Calkins, C. Cartaro, D.G. Cerdeno, Y. Chang, H. Chagani, Y. Chen, J. Cooley, B. Cornell, P. Cushman, M. Daal, P.C.F. Di Stefano, T. Doughty, L. Esteban, E. Fascione, E. Figueroa-Feliciano, M. Fritts, G. Gerbier, M. Ghaith, G.L. Godfrey, S.R. Golwala, J. Hall, H.R. Harris, Z. Hong, E.W. Hoppe, L. Hsu, M.E. Huber, V. Iyer, D. Jardin, A. Jastram, C. Jena, M.H. Kelsey, A. Kennedy, A. Kubik, N.A. Kurinsky, A. Leder, B. Loer, E. Lopez Asamar, P. Lukens, D. MacDonell, R. Mahapatra, V. Mandic, N. Mast, E.H. Miller, N. Mirabolfathi, R.A. Moffatt, B. Mohanty, J.D. Morales Mendoza, J. Nelson, J.L. Orrell, S.M. Oser, K. Page, W.A. Page, R. Partridge, M. Pepin, M. Penalver Martinez, A. Phipps, S. Poudel, M. Pyle, H. Qiu, W. Rau, P. Redl, A. Reisetter, T. Reynolds, A. Roberts, A.E. Robinson, H.E. Rogers, T. Saab, B. Sadoulet, J. Sander, K. Schneck, R.W. Schnee, S. Scorza, K. Senapati, B. Serfass, D. Speller, M. Stein, J. Street, H.A. Tanaka, D. Toback, R. Underwood, A.N. Villano, B. von Krosigk, B. Welliver, et al. (8 additional authors not shown)
LLow-mass dark matter search with CDMSlite
R. Agnese, A.J. Anderson, T. Aralis, T. Aramaki, I.J. Arnquist, W. Baker, D. Balakishiyeva, D. Barker, R. Basu Thakur,
3, 25
D.A. Bauer, T. Binder, M.A. Bowles, P.L. Brink, R. Bunker, B. Cabrera, D.O. Caldwell, ∗ R. Calkins, C. Cartaro, D.G. Cerde˜no,
2, 18
Y. Chang, H. Chagani, Y. Chen, J. Cooley, B. Cornell, P. Cushman, M. Daal, P.C.F. Di Stefano, T. Doughty, L. Esteban, E. Fascione, E. Figueroa-Feliciano, M. Fritts, G. Gerbier, M. Ghaith, G.L. Godfrey, S.R. Golwala, J. Hall, H.R. Harris, Z. Hong, E.W. Hoppe, L. Hsu, M.E. Huber, V. Iyer, D. Jardin, A. Jastram, C. Jena, M.H. Kelsey, A. Kennedy, A. Kubik, N.A. Kurinsky, A. Leder, B. Loer, E. Lopez Asamar, P. Lukens, D. MacDonell,
19, 29
R. Mahapatra, V. Mandic, N. Mast, E.H. Miller, N. Mirabolfathi, R.A. Moffatt, B. Mohanty, J.D. Morales Mendoza, J. Nelson, J.L. Orrell, S.M. Oser,
19, 29
K. Page, W.A. Page,
19, 29
R. Partridge, M. Pepin, † M. Pe˜nalver Martinez, A. Phipps, S. Poudel, M. Pyle, H. Qiu, W. Rau, P. Redl, A. Reisetter, T. Reynolds, A. Roberts, A.E. Robinson, H.E. Rogers, T. Saab, B. Sadoulet,
20, 4
J. Sander, K. Schneck, R.W. Schnee, S. Scorza, K. Senapati, B. Serfass, D. Speller, M. Stein, J. Street, H.A. Tanaka, D. Toback, R. Underwood, A.N. Villano, B. von Krosigk,
19, 29
B. Welliver, J.S. Wilson, M.J. Wilson, D.H. Wright, S. Yellin, J.J. Yen, B.A. Young, X. Zhang, and X. Zhao (SuperCDMS Collaboration) Division of Physics, Mathematics, & Astronomy,California Institute of Technology, Pasadena, CA 91125, USA Institute for Particle Physics Phenomenology, Department of Physics, Durham University, Durham DH1 3LE, UK Fermi National Accelerator Laboratory, Batavia, IL 60510, USA Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni 752050, India Department of Physics & Astronomy, Northwestern University, Evanston, IL 60208-3112, USA Pacific Northwest National Laboratory, Richland, WA 99352, USA Department of Physics, Queen’s University, Kingston, ON K7L 3N6, Canada Department of Physics, Santa Clara University, Santa Clara, CA 95053, USA SLAC National Accelerator Laboratory/Kavli Institute for Particle Astrophysics and Cosmology, Menlo Park, CA 94025, USA SNOLAB, Creighton Mine Department of Physics, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA Department of Physics, Southern Methodist University, Dallas, TX 75275, USA Department of Physics, Stanford University, Stanford, CA 94305, USA Department of Physics, Syracuse University, Syracuse, NY 13244, USA Department of Physics and Astronomy, and the Mitchell Institute for Fundamental Physics and Astronomy,Texas A&M University, College Station, TX 77843, USA Instituto de F´ısica Te´orica UAM/CSIC, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain Department of Physics & Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1, Canada Department of Physics, University of California, Berkeley, CA 94720, USA Department of Physics, University of California, Santa Barbara, CA 93106, USA Departments of Physics and Electrical Engineering,University of Colorado Denver, Denver, CO 80217, USA Department of Physics, University of Evansville, Evansville, IN 47722, USA Department of Physics, University of Florida, Gainesville, FL 32611, USA Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA School of Physics & Astronomy, University of Minnesota, Minneapolis, MN 55455, USA Department of Physics, University of South Dakota, Vermillion, SD 57069, USA Department of Physics, University of Toronto, Toronto, ON M5S 1A7, Canada TRIUMF, Vancouver, BC V6T 2A3, Canada (Dated: January 19, 2018)The SuperCDMS experiment is designed to directly detect weakly interacting massive particles(WIMPs) that may constitute the dark matter in our Galaxy. During its operation at the SoudanUnderground Laboratory, germanium detectors were run in the CDMSlite mode to gather datasets with sensitivity specifically for WIMPs with masses <
10 GeV/ c . In this mode, a higherdetector-bias voltage is applied to amplify the phonon signals produced by drifting charges. Thispaper presents studies of the experimental noise and its effect on the achievable energy threshold,which is demonstrated to be as low as 56 eV ee (electron equivalent energy). The detector-biasingconfiguration is described in detail, with analysis corrections for voltage variations to the level of afew percent. Detailed studies of the electric-field geometry, and the resulting successful developmentof a fiducial parameter, eliminate poorly measured events, yielding an energy resolution ranging from a r X i v : . [ a s t r o - ph . C O ] J a n ∼ ee at 0 keV to 101 eV ee at ∼
10 keV ee . New results are derived for astrophysical uncertaintiesrelevant to the WIMP-search limits, specifically examining how they are affected by variations inthe most probable WIMP velocity and the Galactic escape velocity. These variations become moreimportant for WIMP masses below 10 GeV/ c . Finally, new limits on spin-dependent low-massWIMP-nucleon interactions are derived, with new parameter space excluded for WIMP masses (cid:46) c . PACS numbers: 95.35.+d, 14.80.Ly, 29.40.Wk, 95.55.Vj
I. INTRODUCTION
In the last few decades, astronomical observations haveconsistently indicated that most of the matter content ofthe Universe is nonluminous and nonbaryonic dark mat-ter [1, 2]. There is strong evidence that dark matter isdistributed in large halos encompassing the visible mat-ter in galaxies, including the Milky Way. If this darkmatter is composed of particles that interact with nor-mal matter through a nongravitational force, it may bepossible to directly detect it in laboratory experiments.The first generation of direct detection experimentssearched for dark matter in the form of weakly inter-acting massive particles (WIMPs), with particle massesspanning from a few GeV/ c to a few TeV/ c , and inter-action strengths with normal matter less than the weakforce [3, 4]. These searches were partly motivated by su-persymmetric theories in which the lightest neutral parti-cles are WIMPs and thus natural dark matter candidates.However, no confirmed WIMP signals have been found,and there is no evidence as yet for supersymmetry at theLHC [5, 6].Other theoretical models have been developed, moti-vated by possible symmetries between normal and darkmatter ( e.g. asymmetric dark matter [7]) or the pos-sibility of a parallel dark sector that may contain manydark matter particles [8]. These new models predict darkmatter particles with masses <
10 GeV/ c , stimulatingexperiments to search in this region.WIMPs are expected to scatter elastically and co-herently from atomic nuclei, producing nuclear recoils(NRs). Neutrons also produce nuclear recoils, but oftenscatter multiple times in a detector; WIMPs interact tooweakly to scatter more than once. Residual radioactivityin the experimental apparatus predominantly interactswith atomic electrons, causing electron recoils (ERs) thatare the dominant source of background. Experiments try to reduce the rate of all backgrounds using layers of ra-diopure shielding and through the detection of multipletypes of signals to discriminate between electron and nu-clear recoils.The nuclear-recoil energy spectrum expected fromsimple WIMP models is featureless and quasiexponen-tial [3, 9]. The differential nuclear-recoil rate isd R d E r = N T m T m χ µ T (cid:2) σ SI0 F ( E r ) + σ SD0 F ( E r ) (cid:3) I halo , (1)where m χ and m T are the masses of the WIMP and thetarget nucleus, respectively, µ T = m χ m T / ( m χ + m T )is the reduced mass of the WIMP-target system, N T isthe number of nuclei per target mass, and E r is the en-ergy of the recoiling nucleus. The spin-independent (SI)and spin-dependent (SD) cross sections for the WIMP-nucleus scattering are each factored into a total zero-energy cross section σ SI/SD0 and nuclear form factor F ( E r ).The rate’s dependence on the astrophysical descriptionof the WIMP halo is encompassed by the halo-model fac-tor I halo . This factor depends on the velocities of theWIMPs in the halo’s frame v and the velocity of theEarth with respect to the halo v E as I halo = ρ k (cid:90) v max v min f ( v , v E ) v d v , (2)where ρ is the local dark matter mass density, k is a nor-malization constant, and the halo’s velocity distributionwith respect to the Earth f ( v , v E ) is integrated from theminimum v min to the maximum v max WIMP velocitiesthat can cause a recoil of energy E r . The maximum ve-locity is related to the Galactic escape velocity v esc , whilethe minimum velocity is v min = (cid:112) m T E r / µ T . Assum-ing the standard Maxwellian velocity distribution witha characteristic velocity v (see Sec. VII A) gives an ex-pression for I halo as [10] I halo = k k ρ yv erf( x + y ) − erf( x − y ) − √ π ye − z < x < z − y erf( z ) − erf( x − y ) − √ π ( y + z − x ) e − z z − y < x < y + z y + z < x, (3) ∗ Deceased. † Corresponding author: [email protected] − Recoil Energy [keV] − − − W I M P R a t e [ k e V k g d a y ] − G e V / c G e V / c G e V / c → Threshold ← Figure 1. Differential rates for WIMP recoils on a ger-manium target as functions of recoil energy. WIMPs withWIMP-nucleon spin-independent cross section of 10 − cm and masses of 2, 5, and 10 GeV/ c are considered. The bandsencompassing each curve are computed by varying the astro-physical parameters of the dark matter halo within knownobservational uncertainties. The vertical lines designate ex-ample nuclear-recoil thresholds of 0.5 and 2 keV, respectively. where x = v min /v , y = v E /v , z = v esc /v , k = (cid:0) πv (cid:1) / , and k = k (cid:2) erf( z ) − (2 / √ π ) z exp (cid:0) − z (cid:1)(cid:3) . Thefinal case in this expression is set to zero to avoid un-physical negative rates.Figure 1 shows the predicted differential rates on agermanium target for three low-mass WIMPs with spin-independent WIMP-nucleon cross-sections of 10 − cm .Lowering the experimental energy threshold boosts thesignal-to-background ratio, assuming a flat backgroundspectrum, and reduces the dependence of the WIMP sig-nal on astrophysical uncertainties. A lower thresholdthus dramatically increases an experiment’s sensitivityto lower-mass WIMPs.The Cryogenic Dark Matter Search low ionizationthreshold experiment (CDMSlite) uses a technique de-veloped by the SuperCDMS Collaboration to reduce theexperiment’s energy threshold and increase sensitivity tolow-mass WIMPs [11, 12]. This paper presents furtherdetails of the published CDMSlite analyses and some newresults. The organization of the paper is as follows. Sec-tion II discusses the experimental technique, CDMSlitedata sets, and data-reduction improvements. Section IIIdiscusses the analysis and removal of noise. Section IVdiscusses an energy resolution model and energy thresh-olds. Section V discusses the effects of bias instability inthe analyses and the steps taken to account for those ef-fects. Section VI discusses the definition of a fiducial vol-ume and its effect on backgrounds. Finally, new WIMPresults are given in Sec. VII based on the effects of astro-physical uncertainties on the spin-independent WIMP-nucleon scattering limit presented in Ref. [12] and new spin-dependent WIMP-nucleon scattering limits. II. DESCRIPTION OF THE EXPERIMENT
The SuperCDMS Soudan experiment was located atthe Soudan Underground Laboratory and used the samecryogenics system, shielding, and electronics as the ear-lier CDMS II experiment [13, 14]. Five towers, each con-sisting of three germanium interleaved Z-sensitive ioniza-tion and phonon detectors (iZIPs), were operated from2011 to 2015 [15]. Each iZIP was roughly cylindrical witha ∼
76 mm diameter, ∼
25 mm height, and ∼
600 g mass.Particle interactions in these semiconductor crystals ex-cite electron-hole charge pairs as well as lattice vibrations(phonons). The top and bottom circular faces of an iZIPare instrumented with electrodes for sensing the chargesignal and tungsten transition edge sensors (TESs) formeasuring phonons. The electrons and holes are driftedto the electrodes by applying a bias voltage across thecrystal (nominally 4 V), while athermal phonons are ab-sorbed by Al fins that are coupled to the TESs. Duringdata taking, the output traces from the detectors wererecorded (“triggering” the experiment) if the analog sumof any detector’s raw phonon traces exceeded a user-sethardware threshold [16].Measuring both the charge and phonon signals allowsfor discrimination between NRs and ERs through the ion-ization yield Y : Y ( E r ) ≡ E Q E r , (4)where E Q is the charge signal, and, for electron recoils, E Q ≡ E r . The efficiency of producing electron-hole pairsis lower for nuclear recoils, leading to yields of Y ∼ . E r (cid:38)
10 keV. Below this energy, electronic noisecauses the widths of the ER and NR populations to in-crease until they largely overlap at ∼ A. CDMSlite
In 2012, SuperCDMS began running detectors in thealternate CDMSlite operating mode, where the detectorpotential difference was raised to 50–80 V. The standardiZIP electronics and biasing configuration were adaptedfor this higher-voltage operating mode; phonon and ion-ization sensors on one side of the detector were set tothe given bias, while all of the sensors on the oppositeface were held near ground potential. Figure 2 showsthe phonon sensor layout and biasing scheme of theCDMSlite detectors. The sensors on the grounded sideof the detector were then read out. The limitations of
Figure 2. Schematic showing the general coverage of the fourphonon read-out channels ( A – D ) overlaying the sensor pat-tern for the CDMSlite detector. The sensors on the bottomside are exclusively used for applying the high-voltage bias(HV) and are not read out. All sensors on the top side areheld at ground. the CDMS II electronics board prohibited two-sided op-eration as the board could not simultaneously be floatedto a potential and read out.The CDMSlite operating mode takes advantage ofphonon amplification via the Neganov-Trofimov-Luke(NTL) effect [18, 19]. Electrons and holes liberated bythe initial recoil drift across the detector, driven by theapplied electric potential. During transport, they col-lide with Ge atoms and reach a scattering-limited driftvelocity of O (cid:0) cm s − (cid:1) in (cid:46) (cid:38)
30 meV, high rates ofoptical and intervalley phonon scattering limit furtheracceleration and cause them to reach a terminal veloc-ity. The additional work done in drifting these chargecarriers, as they collide with the lattice (50–80 eV perelectron-hole pair at the biases under discussion), is emit-ted as phonons. The residual kinetic energy of ∼
30 meVper electron-hole pair, along with the band gap energy of0.74 eV [21], is eventually released as phonons, called re-laxation or recombination phonons, when the charge car-riers relax to the Fermi sea near detector boundaries. Thephonons emitted during charge transport are called NTLphonons, and the net energy in these phonons, E NTL , isthe work done by the electric field E NTL = N e/h e ∆ V. (5)Here, N e/h is the number of electron-hole pairs createdin the recoil, e is the elementary charge, and ∆ V is thepotential difference traversed by the pairs. ∆ V is nomi-nally the absolute value of the bias applied by the powersupply V b . The advantage of operating at relatively highbias potentials is an amplification of the charge signal (asobserved in the phonon signal) due to increased NTL-phonon production. This discussion of electron and hole transportation in a germa-nium crystal is taken from the rigorous calculations in Ref. [20].See, e.g. , Chaps. 2 and 4 of the reference for further details.
The total phonon energy in the crystal is thus the sumof ionization-associated NTL phonons, primary phononscreated at the initial recoil site, and relaxation phononscreated near detector surfaces. The sum of the primaryand relaxation phonons is E r and thus the total energyis E t = E r + E NTL = E r + N e/h e ∆ V. (6)The number of electron-hole pairs created by a recoildepends on the recoil type. For electron recoils in ger-manium, the average (photoexcitation) energy requiredto generate a single electron-hole pair is taken to be ε γ = 3 eV [22]. This gives N e/h = E Q /ε γ = Y ( E r ) E r /ε γ ,where Eq. 4 is used for the second equality. Substitutingthis last expression into Eq. 6 gives E t = E r (cid:18) Y ( E r ) e ∆ Vε γ (cid:19) . (7)As only one of two faces of an iZIP are read out inCDMSlite mode, the energy absorbed by the operablephonon sensors is half that of Eq. 7.The calibration of the measured phonon signal pro-ceeds in three steps, with three corresponding energyscales, using Eq. 7 assuming ∆ V = V b . The first stepis to convert the raw output to the “total phonon energyscale,” with units of keV t , using calibration data taken atthe standard operating bias of 4 V and the expectationfrom Eq. 7 (see Sec. V A). Converting the calibrated E t tothe interaction’s E r requires knowledge of the yield. Be-cause CDMSlite only measures phonons, the yield cannotbe constructed on an event-by-event basis and a modelfor Y ( E r ) is required. Two further energy scales are de-fined corresponding to the assumed ER/NR recoil type.The ER scale is stretched considerably compared to theNR scale with its smaller electron-hole production effi-ciency; this further increases the signal-to-backgroundratio for CDMSlite.The recoil energies are next calibrated assuming allevents are ERs, i.e. , Y ( E r ) = 1, called “electron-equivalent” energy in units of keV ee and denoted by E r,ee . This scale is useful for characterization of the back-grounds, which are primarily ERs. An ER calibration isavailable from electron-capture decays of Ge. Thermalneutron capture on Ge (20.6 % natural abundance) cre-ates Ge, which then decays by electron-capture witha half-life of 11.43 days [23]. The K -, L -, and M -shellbinding energies of the resulting Ga are 10.37, 1.30, and0.16 keV, respectively [24]. In the experiment, Ge wascreated in the detector by exposing it to a
Cf sourcetwo to five times per CDMSlite data set. The K -shellpeak, clearly visible in the data following such an acti-vation, is used to calibrate the energy scale to keV ee andto correct for any changes in the energy scale with time(see Sec. V).WIMP scatters are expected to be NRs; so a nuclear-recoil energy is ultimately constructed, called “nuclear-recoil equivalent” energy in units of keV nr and denotedby E r,nr . The calibration to keV nr is performed by com-paring Eq. 7, assuming the detector sees the full V b bias,for an ER and NR with the same E t , and solving for E r,nr , E r,nr = E r,ee (cid:18) eV b /ε γ Y ( E r,nr ) eV b /ε γ (cid:19) , (8)where Y ( E r,nr ) is the yield as a function of nuclear-recoilenergy, for which a model is needed. The model used isthat of Lindhard [25] Y ( E r,nr ) = k · g ( ε )1 + k · g ( ε ) , (9)where g ( ε ) = 3 ε . + 0 . ε . + ε , ε =11 . E r,nr (keV nr ) Z − / , and Z is the atomic num-ber of the material. For germanium, k = 0 . ∼
250 eV nr [26, 27],although measurements in this energy range are difficult,and relatively few exist [28–30]. The SuperCDMSCollaboration has a campaign planned to directlymeasure the nuclear-recoil energy scale for germanium(and silicon) down to very low energies, since this willbe required for the upcoming SuperCDMS SNOLABexperiment. B. Data Sets and Previous Results
A single detector was operated in CDMSlite mode dur-ing two operational periods, Run 1 in 2012 and Run 2 in2014. The initial analyses of these data sets, publishedin Refs. [11, 12], respectively, applied various selectioncriteria (cuts) to the data sets and used the remain-ing events to compute upper limits on the SI WIMP-nucleon interaction. These limits were computed usingthe optimum interval method [31], the nuclear form fac-tor of Helm [9, 32], and assuming that the SI interac-tion is isoscalar. Under this last assumption, the WIMP-nucleon cross section σ SI N is related to σ SI0 in Eq. 1 as σ SI0 = ( Aµ T /µ N ) σ SI N , where µ N is the reduced mass ofthe WIMP-nucleon system.CDMSlite Run 1 was a proof of principle and the firsttime WIMP-search data were taken in CDMSlite mode.For Run 1, the detector was operated at a nominal biasof −
69 V and an analysis threshold of 170 eV ee wasachieved. In an exposure of just 6.25 kg d (9.56 kg d raw),the experiment reached the SI sensitivity shown in Fig. 3(labeled “Run 1”), which was world leading for WIMPslighter than 6 GeV/ c at the time of publication [11]. Only a single detector was operated for each run due to limita-tions of the Soudan electronics and to preserve the live time forthe standard iZIP data taken concurrently. m WIMP
GeV /c $ − − − − − − σ S I N c m $ R un R un P a nd a X C R E SS T − − − − − − σ S I N [ pb ] Figure 3. Spin-independent WIMP-nucleon cross section 90 %upper limits from CDMSlite Run 1 (red dotted curve withred uncertainty band) [11] and Run 2 (black solid curve withorange uncertainty band) [12] compared to the other (morerecent) most sensitive results in this mass region: CRESST-II (magenta dashed curve) [33], which is more sensitive thanCDMSlite Run 2 for m WIMP (cid:46) . c , and PandaX-II(green dot-dashed curve) [34], which is more sensitive thanCDMSlite Run 2 for m WIMP (cid:38) c . The Run 1 un-certainty band gives the conservative bounding values due tothe systematic uncertainty in the nuclear-recoil energy scale.The Run 2 band additionally accounts for the uncertainty onthe analysis efficiency and gives the 95 % uncertainty on thelimit. The total efficiency and spectrum from Run 1 areshown in Figs. 4 and 5 respectively. In addition to the Ge-activation peaks, the K -shell activation peak from Zn is visible in the Run 1 spectrum at 8.89 keV ee [24].The Zn was created by cosmic-ray interactions, withproduction ceasing once the detector was brought under-ground in 2011, and decayed with a half-life of τ / ≈
244 d [35]. The analysis threshold was set at 170 eV ee tomaximize dark matter sensitivity while avoiding noise atlow energies (see Sec. III C). To compute upper limits, theconversion from keV ee to keV nr was performed using thestandard Lindhard-model k value (Eq. 9) of 0.157. Limitswere also computed using k = 0 . −
70 V, the analysis threshold was further reduced be-cause of improved noise rejection, and a novel fiducial-volume criterion was introduced to reduce backgrounds.The total efficiency and spectrum from this run are com-pared to those of the first run in Figs. 4 and 5. Becauseof the lower analysis threshold, decreased background,and a larger exposure of 70.10 kg d (80.25 kg d raw), theexperiment yielded even better sensitivity to the SI in-teraction than Run 1 [12], as shown in Fig. 3 (labeled
Energy [keV ee ] E ffi c i e n c y Run 1 E ff .Run 2 E ff . Figure 4. Total combined trigger and analysis efficienciesfor Run 1 (red dotted curve) and Run 2 (black solid curvewith orange 68 % uncertainty band). The implementation ofa fiducial-volume cut is primarily responsible for the reductionin efficiency at high recoil energies between the two analyses.Figure 5. Measured efficiency-corrected spectra for Run 1(red dotted curve) and Run 2 (gray shaded area). The Ge-activation peaks at 10.37 and 1.30 keV ee are prominent inboth spectra, and the peak at 0.16 keV ee is additionally visiblein the Run 2 spectrum. The Zn K -shell electron-capturepeak is also visible at 8.89 keV ee in the Run 1 spectrum. Inset: an enlargement of the spectra below 2 keV ee with binsfive times smaller and the runs’ analysis thresholds given bythe extended and labeled tick marks. “Run 2”). The second run was split into two distinctdata periods (see Sec. III C), labeled “Period 1” and “Pe-riod 2,” that had analysis thresholds of 75 and 56 eV ee ,respectively.For the Run 2 result, the uncertainties of the analysiswere propagated into the final limit by simulating 1000pseudoexperiments and setting a limit with each. The median and the central 95 % interval from the resultingdistribution of limits, at each WIMP mass, are taken asthe final result given in Fig. 3. For each pseudoexperi-ment, the keV ee energy of the events and thresholds wereconstant. The analysis efficiencies, as indicated by theband in Fig. 4, were sampled, as was the Lindhard-model k within a range of 0 . ≤ k ≤ .
2. The uncertainty in theenergy conversion dominates the band in Fig. 3, with thenext-largest uncertainty being that of the fiducial-volumeacceptance efficiency (Sec. VI B).
C. Pulse fitting and energy measurement
Several improvements were made in the analysis ofRun 2 data, compared to that of the Run 1 data, bythe introduction of a new data-reduction algorithm usedto extract energy and position information about scat-ters in the detector. To motivate and understand thisnew algorithm, the dynamics of phonon detection andthe older algorithms, which are still used for many partsof the analyses, are first discussed.The phonon sensors cover only ∼ ∼
40 % probability ofabsorption when they strike an aluminum sensor fin butare reflected when striking an uninstrumented surface.The phonons continue to rebound between surfaces of thecrystal until they are absorbed by, or become lost to, thesensors [36]. Phonons become undetectable by the sen-sors either by falling below the aluminum superconduct-ing gap energy or by being absorbed through nonsensormaterials ( e.g. , stabilizing clamps). The small fractionof phonons striking a fin at the first surface interactionproduces an early absorption signal that is concentratedclose to the location of the interaction, while the major-ity of the phonons contribute to a later absorption sig-nal that is mostly homogeneous throughout the detector.The phonon pulse shape thus contains both position andenergy information about the initial scatter in the earlierand later portions of the signal trace, respectively.The CDMSlite analyses employ three algorithms basedon optimal filter theory (see Appendix B of Ref. [37])to extract the position and energy information of theunderlying event based on the measured pulse shapes andamplitudes. For these algorithms, the signal trace S ( t ) isgenerally modeled as a template, or linear combination oftemplates, A ( t − t ), which can be shifted by some timedelay t , and Gaussian noise n ( t ) as S ( t ) = aA ( t − t ) + n ( t ) , (10)where the template is scaled by some amplitude a . Theoptimal values of a and t are then found by minimizing, This value of 40 % is determined by tuning a phonon simula-tion in a detector to match recorded pulses. Specifically, howquickly pulses return to their baseline values is sensitive to thisabsorption probability.
Time [ms] A m p l.[ a r b . un i t ] OF/NSOF/2T Slow2T Fast
Figure 6. Templates used for the standard OF, NSOF, and2T-fit algorithms for CDMSlite analysis. The green solidcurve is the single trace used for the OF, NSOF, and 2T-fitslow templates, which is derived from averaging high-energytraces. In the 2T fit, the slow template’s amplitude carriesthe main energy information. The 2T-fit fast template (or-ange dotted), is derived by considering the differences betweenthe slow template and the traces used in the slow template’sderivation. In the 2T fit, the fast template’s amplitude cap-tures the position information from the signal trace. Themaxima of the amplitudes (Ampl.) are scaled to unity in thefigure. in frequency space, the χ between the left- and right-hand sides of Eq. 10. The amplitude, time delay, andgoodness-of-fit χ value are returned by the algorithms.The first algorithm is called the “standard” optimalfilter (OF). The OF algorithm fits a single template toa trace, as in Eq. 10, without attempting to account forthe position dependence in the early portion of the trace.The template was created by averaging a large number ofhigh-energy traces taken from the Ge K -shell capturepeak and can be seen in Fig. 6. The energy estimate fromthis fit, the amplitude a in Eq. 10, has poor resolutionbecause of the position dependence. The position of anevent’s initial scatter in the detector can be estimatedby fitting the traces from each individual channel of agiven event and comparing the fit amplitudes among thechannels: channels of which the sensors are nearer to theinteraction will have a larger amplitude than those ofwhich the sensors are farther away.The second algorithm is called the “nonstationary” op-timal filter (NSOF) (see Appendix E of Ref. [38]), andit produces an energy estimator that is less affected bythe early-trace position dependence. The NSOF uses thesame single template as in the OF fit but treats the resid-ual deviations between the trace and the template as non-stationary noise. This procedure deweights the parts ofthe trace that show larger variance and results in a moreaccurate energy estimator. Additionally, the NSOF fit iscalculated only for the summed trace of each individualdetector, which also serves to reduce, but does not com-pletely eliminate, the effect of position dependence on theenergy estimate. The NSOF is not useful for computingposition information about the initial scatter.The third algorithm, utilized for the first time withCDMSlite Run 2 data, is called the “two-template” op-timal filter (2T fit) (see Appendix E of Ref. [38] and A m p li t ud e [ n A ] Time [ms] A m p li t ud e [ n A ] Time [ms]
Channel D
Figure 7. Results of the 2T-fit algorithm for an example eventchosen from the Ge L -shell capture peak in Run 2. Thetraces and fits from all four phonon channels, labeled A–D(where channel A is the outer ring) are given. For each chan-nel, the raw trace (blue solid) is compared to the final totalfit (black dashed) which is a linear combination of the slow(green solid) and fast (orange dotted) templates. The channelwith the largest fast-template amplitude, channel B for thisevent, is the channel of which the sensors are closest to theinitial recoil. Chap. 10 of Ref. [39]). The 2T fit uses a linear com-bination of two different templates, replacing aA ( t − t )with (cid:80) i = s,f a i A i ( t − t ). The two templates are shownin Fig. 6 and are labeled the “slow” and “fast” templates.The slow template is the same template used in the OFand NSOF fits. The fast template is derived by consid-ering the differences between the slow template and thetraces used to define it, termed the residual traces. Tocalculate this template, the residuals with negative am-plitude are inverted before all residuals are averaged. Theinversion conserves the shape and is needed because theaverage of the residuals without the inversion is zero bydefinition. The 2T fit returns an energy estimator—theamplitude of the slow template—which, like the NSOF,is less affected by the position of the initial scatter thanthe OF fit, but it also returns the amplitude of the fasttemplate which encodes position information. The 2T fitis applied to each individual channel’s trace as well asthe summed trace. An example of this fit is shown inFig. 7. Negative fast-template amplitudes are expectedin fit results and indicate greater distance from the initialscatter.In the Run 1 analysis, the energy estimator from theNSOF algorithm was used without any further correc-tions for position dependence. For the Run 2 analysis,the NSOF energy estimator was again used, but an addi-tional position correction was applied based on the 2T fitinformation. As shown in Fig. 8, a correlation betweenthe fitted NSOF energy estimate and 2T-fit fast-templateamplitude is observed. The linear fit to this correlation is -4 -2 0 2 . unit] N S O F E n e r g y [ k e V t ] Figure 8. NSOF-fit energy estimator as a function of the2T-fit fast-template amplitude from the summed trace. Thehigh-density band of events is the Ge K -shell activation line.Residual position dependence is reflected in the slope of theband. This dependence is corrected according to the straight-line fit shown by the solid line. The location of the peak at ∼ t is discussed in Sec. V A. used for the correction. In the Run 2 analysis, a cut wasplaced to remove events for which the NSOF fit returnedlarge χ values to ensure that the energy estimator wasreliable. Such a cut removes events that have more thanone pulse in the trace, or that exhibit a distorted pulseshape due to TES saturation. The signal efficiency forthe cut is near 100 % as computed via a pulse simulationthat is described in Sec. III C 2. No poorly fit events wereobserved above threshold in the smaller Run 1 WIMP-search data set, and thus such a cut was unnecessary. III. STUDY AND REMOVAL OF NOISE
Understanding the noise in the readout wave forms iscrucial for optimizing the low-energy analysis and achiev-ing the desired low-energy thresholds using the CDMSlitetechnique. Studies from both runs showed that the noisedepended on both bias voltage and time. Most crucially,cryocooler-induced low-frequency noise was present andlimited the Run 1 threshold. A combination of timingcorrelations with the cryocooler and pulse-shape fittingwas used in Run 2 to reject this background.
A. Dependence of noise on bias potential
The operating potential difference for each run wasdetermined by studying the noise as a function of theapplied potential difference. The baseline resolution as The energy estimator extracted from the slow-template ampli-tude of the 2T fit has more position dependence than that of theNSOF, manifesting itself in a stronger correlation with the 2T-fit fast-template amplitude. After correcting for this correlation,the performance is very similar with a marginally better resolu-tion of the NSOF-based algorithm in the Ge K -shell peak. R e s o l u t i o n [ e V t ]
50 55 60 65 70 75
Applied Bias Potential [V] S N R Figure 9. Baseline resolution (top) and the correspondingSNR (bottom) as a function of the applied bias potential.Each point represents a single 3 h long data set taken priorto Run 2. The resolution and SNR increase and decrease,respectively, past ∼
70 V in applied bias. The average uncer-tainty for each point is 3.6 eV t for the resolution and 0.39for the SNR. The additional variation seen at a given bias islikely a result of time dependence of the noise. For reference,1 keV t ≈
66 eV ee . a function of this potential difference is shown in Fig. 9for data taken prior to Run 2. The resolution slowlyincreased until the potential difference passed ∼
70 V,where a larger increase was observed. Taking the po-tential difference up to 85 V resulted in greatly increasednoise signaling the start of detector breakdown. A recoil-energy-independent signal-to-noise ratio (SNR) was alsoconsidered by comparing the measured signal and noiseto the V b = 0 V case. The signal, according to Eq. 7(assuming a yield of unity), was then 1 + eV b /ε γ . Thenoise was the measured resolution in Fig. 9 divided byan assumed zero-volt resolution of 120 eV t . The SNR isalso shown in Fig. 9, with a peak SNR at ∼
70 V. Thesestudies were used to determine the operating potentialdifferences of 69 and 70 V for the two runs respectively.o
B. Time dependence of noise
For iZIP detectors, the charge collection efficiency de-teriorated after being biased and operated for longer than ∼ Time [min] -1012 E n e r g y [ k e V t ] -1 0 1 2 3 4 5 Time [ms] A m p li t ud e [ n A ] Frequency [kHz] − − P S D h n A / √ H z i Start of Series TracesEnd of Series Traces
Figure 10.
Top: total phonon energy, or noise, as a function oftime since biasing in Run 1. The noise decays quasiexponen-tially with time; four example events are given by noncircularmarkers. The first and last 500 traces are highlighted in lightand dark orange, respectively. The noise distribution is offsetfrom 0 keV t as the energy-estimating algorithm tends to fit toupward noise fluctuations. For reference, 1 keV t ≈
66 eV ee . Middle: raw traces of the events marked in the top panel.Traces are shifted by 100 nA with respect to each other forclarity.
Bottom: power spectral densities (PSDs) for the noiseat the start (light orange) and end (dark orange) of the series.The earlier traces have more power below ∼
10 kHz. exposure increased the temperature of the detectors, anda 10 min cool-down period was required before beginningthe next series. In detectors operated in CDMSlite mode,trapped charges resulted in excess noise, and steps weredeveloped to minimize this effect.During Run 1 operation, the noise in the CDMSlitedetector was seen to be excessively high immediately af-ter the detector was biased to its fixed operating pointat the start of a series. The noise decayed quasiexpo-nentially with time, presumably due to the tunneling oftrapped charges, until an asymptotic level was achieved(see Appendix B of Ref. [38]). Noise-trace data froma typical series are shown in Fig. 10, where the recon-structed energy has higher rms earlier in the series. Theexcess noise amplitude decayed with an exponential timeconstant τ ∼
10 min. In Run 1, the data taken duringthe first 4 τ following the application of the bias voltagewere discarded, as a balance between live time and opti-mal baseline resolution. Thus, in Run 1, only ∼
70 % ofthe data collected could be used for the analysis. In Run 2, the high initial noise was avoided by holdingthe detector at a larger potential difference than the oper-ating voltage prior to the start of each series, after whichthe bias was dropped to the operating voltage. Underthe assumption that the initial noise is due to the releaseof trapped charges, this initial bias at higher potentialdifference allows for all traps accessible at the lower po-tential difference to be cleared. This operational proce-dure is termed “prebiasing” and the SuperCDMS dataacquisition system(DAQ) was configured to prebias be-fore each data series in Run 2. The prebiasing procedurewas as follows: • At the end of each series, ground the detector whileit is exposed to the photons from the light emittingdiodes. • During the necessary 10 min cool-down period,hold the detector at a potential difference of −
80 V. • After the cooldown, lower the potential differenceto the −
70 V operating voltage, and begin datataking for the next series.The effectiveness of prebiasing can be seen in Fig. 11,which compares the baseline noise distributions for se-ries which were, or were not, prebiased. The series weretaken during the bias scan prior to Run 2, described inSec. III A, and were thus taken at various biases (the datain Fig. 9 were prebiased). The widths of the distributionswhich were prebiased are smaller than those which werenot, as shown by the values in the figure.
C. Low-frequency noise
In Run 1, the baseline noise resolution was 14 eV ee and the detector had 50 % trigger efficiency at 108 eV ee .The analysis threshold was set at 170 eV ee to avoid be-ing overwhelmed by a source of ∼ kHz noise (labeled“low-frequency”) that dominated the triggered-event ratebelow ∼
200 eV ee . The primary source of this low-frequency noise was identified as vibrations from theGifford-McMahon cryocooler used to intercept heat trav-eling down the electronics stem via the readout cables.The cryocooler cycled at ∼ . t , and the low-frequencynoise distribution is dominant from 0.5–1.5 keV t . Theseevents were identified as noise by studying their pulseshape compared to the OF algorithm template as shownin the middle panel of Fig. 12. In comparing the noisepower spectral densities from 500 events (each) of low-frequency and electronic noise (bottom panel of Fig. 12),the low-frequency noise events have more power below ∼ Figure 11. Baseline noise distribution for series that wereprebiased (gray area) and series that were not prebiased(red curve) taken at potential differences of 51/60/66 V(top/middle/bottom). The Gaussian-equivalent widths (seeSec. IV A) of the distributions with σ w and without σ wo prebiasing are also given, in keV t . The thinner distribu-tion widths for prebiased series compared to nonprebiasedseries demonstrates the effect of prebiasing. For reference,1 keV t ≈
66 eV ee . The push to reject low-frequency noise, and subse-quently reach a lower analysis threshold, for Run 2 oc-curred in two steps. The first step was to characterizethe low-frequency noise with regard to the timing of thecryocooler and identify blocks of calendar time that hadsimilar low-frequency noise behavior (Sec. III C 1). Thesecond step was to define a rejection criterion based onthe pulse shape of individual events and to tune the po-sition of the rejection threshold individually between thedifferent calendar blocks (Sec. III C 2).
1. Cryocooler timing characterization
For Run 2, two accelerometers were placed on and nearthe cryocooler to monitor vibrations. Custom processingelectronics were also installed to record the cryocooler cy-cle in the DAQ [38, 39]. Comparing the time stamps ofrecorded events to those of the cryocooler gives, for eachevent, the time since the start of the previous cryocoolercycle ˆ t − . The precision of ˆ t − is 3 ms and is dictatedby the precision of the accelerometer read-out. The cry-ocooler cycle ( ∼
830 ms) starts with a compression event,which causes the largest amount of vibrational noise, andincludes an expansion phase, ∼
400 ms after the compres-sion, which also causes noise. These two parts of the cry-ocooler cycle are distinctly observed in Fig. 13, which his-tograms the number of low-energy triggered events (dom-inated by low-frequency noise) in both ˆ t − and calendar -0.5 0 0.5 1 1.5 Energy [keV t ] − − − C o un t s [ k e V t ] − Time [ms] A m p li t ud e [ n A ] Raw LF-noiseFiltered LF-noiseStandard Template Frequency [kHz] − − P S D h n A / √ H z i Low-Frequency NoiseElectronic Noise
Figure 12.
Top:
Run 2 noise distribution. The electronic-noise distribution is centered at ∼ t while the low-frequency noise distribution dominates from 0.5–1.5 keV t .For reference, 1 keV t ≈
66 eV ee . Middle: raw (thin lightblue solid) and filtered (thick black dotted) trace from a typi-cal low-frequency noise event compared to the standard-eventtemplate (thick green solid), derived from high-energy Ge K -shell events. The difference in pulse shape is most evidentbetween 0 and 2 ms. Bottom: power spectral densities (PSDs)for 500 low-frequency (light blue) and electronic (dark blue)noise traces. The low-frequency noise population has morepower below ∼ time.During the course of Run 2, the cryocooler degradedfurther, and the rate of events triggered by low-frequencynoise greatly increased. The rate increase was accompa-nied by a change in the low-frequency noise inductionpattern as seen on the right side of Fig. 13. During thispart of the run, low-frequency noise appeared throughoutthe entirety of the cryocooler cycle. This obvious dete-rioration demanded a room-temperature warm-up of theexperiment for servicing of the cryocooler cold head, anddivided the run into the aforementioned Periods 1 and 2.The low-frequency noise induction was characterizedby developing and applying a smoothing filter to the his-togram in Fig. 13 [39]. As the average number of particleinteractions expected in each bin is O (cid:0) − (cid:1) , bins with10 –10 counts are clear outliers due to low-frequencynoise. Correlations between neighboring bins are also in-dicators of low-frequency noise, as the noise typically oc-curs in bursts in calendar time and cryocooler time. Ap-plying a smoothing filter then deemphasizes true noise1 Figure 13. Number of low-energy triggered events for Run 2Period 1 in the two-dimensional plane of cryocooler time, ˆ t − ,and calendar time in 2014. The color scale is logarithmic withempty bins mapped to black. The rate of low-frequency noiseinjection evolved throughout the run because of the deterio-ration of the cryocooler, ranging from 0 to > fluctuations, high-count bins surrounded by low-countbins, and allows better identification of times with a highlow-frequency noise rate. Using the filtered data, eightblocks in calendar time were defined such that the low-frequency noise behavior within each block was roughlyconsistent. These time blocks are indicated at the top ofFig. 13.In Period 2 of Run 2, the accelerometers were not con-figured in the DAQ. This oversight was not discovereduntil after the end of the run and thus the cryocoolertiming information was not available in Period 2. In-stead, four time blocks were defined in Period 2 based onshifts in the energy scale and general noise environment.The first two blocks occurred during the end of Septem-ber and the beginning of October. The energy scale no-ticeably shifted between these periods (see Sec. V C andFig. 21). The last two blocks, taken at the end of Octo-ber and beginning of November, each contained a smallamount of live time and coincided with a number of un-related calibration and noise studies. Small shifts in thenoise environment were observed between these blocks.In total, Run 2 was divided into 12 nonoverlapping timeblocks.
2. Pulse-shape discrimination
The criterion that was ultimately used to remove low-frequency noise from the data set was based on pulseshape, tailored to the different time blocks. A newtrace template was created by averaging a large numberof low-frequency noise events; these traces were identi-fied as those which triggered the detector, were in the
Time [ms] A m p l.[ a r b . un i t ] Standard OF TemplateLow-Frequency Noise Template
Figure 14. Template traces for the standard OF (green solid)and low-frequency noise (orange dotted) fits. The templateswere generated by averaging many events’ pulse shapes, whichremoved uncorrelated noise. Details of the low-frequencynoise template generation are discussed in the text, and thestandard OF template definition is discussed in Sec. II C. Themaxima of the amplitudes (Ampl.) are scaled to unity in thefigure. energy range characteristic of low-frequency noise, andtook longer than 1 ms to reach their maximum value.This template is compared to the standard OF templatein Fig. 14. This new template was then fit to everytrace using the single-template OF algorithm described inSec. II C ( i.e. , using the new template for A ( t ) in Eq. 10),returning a goodness-of-fit parameter χ . A discrimi-nation parameter ∆ χ was then defined as∆ χ ≡ χ − χ , (11)where χ is the goodness-of-fit parameter from thesingle-template OF algorithm using the standard tem-plate.Example planes of ∆ χ versus energy are given inFig. 15 for time blocks 2 and 7, both from Period 1. Pulseshapes that better fit the standard OF template havenegative ∆ χ and lie on a downward opening parabola,while those which better fit the low-frequency noise shapehave positive ∆ χ . The cut was tuned piecewise withthree components. The first is a flat portion tuned to re-ject the worst (based on ∆ χ ) ∼
10 % of the electronicnoise distribution. The second component was tuned onthe good-event parabola, where the mean µ and width σ of the ∆ χ distribution in a number of energy binsextending to 400 keV t were computed and the thresh-old fit to the µ + 5 σ points from each bin. The µ + σ values were used to ensure a loose cut at high energieswhere no low-frequency noise is expected. However, inorder for the threshold to be tight enough to exclude thelow-frequency noise distribution at low energies, an ad-ditional constraint of an upper bound on the y -interceptwas also required. The third component was based on atwo-dimensional kernel-density estimate [41] of the ∆ χ and energy of low-energy triggers (dominated by the low-frequency noise). The threshold was taken as a convexhull around the largest nσ contour from the estimate,where n varied from 2.5–5 in steps of 0.5. The tuningof this position was set individually for each time block2based on a manual scan of borderline traces; i.e. , if anytrace that appeared to be contaminated by low-frequencynoise was found, n was increased. Thus, the cut wastighter in time blocks of greater low-frequency noise rateand looser in time blocks with a lower low-frequency noiserate. The time blocks shown in Fig. 15 represent exam-ples of low and high cryocooler-induced triggered noiserates, with looser and tighter cut thresholds, respectively.The joint efficiency of three pulse-shape-based cuts,including the low-frequency noise cut, was determinedby generating simulated traces, applying the same pulse-fitting techniques as the experimental data, and comput-ing the fraction of simulated events that pass the cutsas a function of energy. Efficiency was also assessed forcuts that remove events with high NSOF-returned χ val-ues and electronic-glitch events, which are events withpulses that have uncharacteristically fast fall times. Thesimulated traces were constructed by combining a mea-sured noise trace, selected from those recorded routinelythroughout the WIMP search, and a noiseless templatescaled to a desired amplitude. The procedure was re-peated using three templates of different shapes to assessthe systematic uncertainty of the efficiency due to pulseshape. The templates were the standard OF-fit templateand two new templates defined as T ± = T s ± αT f , where T s/f are the slow and fast templates from the 2T fit(Fig. 6). α was chosen to be 0.125 to encompass the ob-served fast-to-slow template ratio of events in the Ge K -shell peak. The efficiency of these cuts is shown inFig. 16, including the uncertainty from varying the tem-plate shape. The loss in efficiency due to the non-low-frequency noise cuts is < ee is where the kernel-density-estimate portions of the low-frequency noise cutare active. The sharp onset of this decrease differs bytime block, while the more gradual decrease seen in thefigure (particularly for Period 1) is due to averaging overall time blocks. Also note that, while the cut thresh-olds, such as those shown in Fig. 15, are defined in thekeV t energy scale, the efficiency must be evaluated in theenergy scale used in the final analysis, keV ee . IV. RUN 2 ENERGY RESOLUTION ANDTHRESHOLD
The low-frequency noise cut described in the previ-ous section allowed the event selection in Run 2 to avoidevents resulting from known noise sources. The remain-ing noise distribution was studied to measure the base-line resolution of the detector, which in turn was usedto model the detector’s energy resolution. The analy-sis threshold, however, was constrained by the detector’sefficiency for triggering on low-energy events, i.e. , thetrigger threshold.
Figure 15. ∆ χ as a function of total phonon energy for timeblocks 2 (top) and 7 (bottom) showing the three portions ofthe low-frequency noise rejection cut (dotted) with the defin-ing portion at any given energy darkened. Low-frequencynoise events cluster near ∼ t , while good events fall ona downward opening parabola. The major difference betweenthe two subplots is the difference in low-frequency noise: timeblock 2 shows low noise, while time block 7 is more noisy.Events above any portion of the cut are rejected (light blue),while those below are retained (dark blue). Time block 2is relatively less noisy, while time block 7 is relatively morenoisy. The contour portion in block 2 cuts more loosely (2 . σ )than in block 7 (5 σ ) because of the changing low-frequencynoise environment throughout the run. A preselection cut re-moving events with unusually high NSOF χ values has beenapplied in these figures and, for reference, 1 keV t ≈
66 eV ee . E ffi c i e n c y P e r i o d Energy [keV ee ] P e r i o d Figure 16. Efficiency of the pulse-shape based cuts for Run 2Period 1 (top) and Period 2 (bottom) as a function of electron-equivalent energy. Almost all loss in efficiency is due to thelow-frequency noise cut, with the sharp drop in efficiency be-low 100 eV ee due to the kernel-density-estimate portion ofthat cut. The insets give an enlargement in the O (100 eV ee )range, where the systematic uncertainty from varying thepulse shape, shown by the error bars, is largest. The aver-age statistical uncertainty for each bin, due to the number oftraces simulated, is 1.2 %. A. Run 2 energy resolution model
The total energy resolution σ T ( E r,ee ) for the detectorwas modeled as σ T ( E r,ee ) = (cid:113) σ + σ ( E r,ee ) + σ ( E r,ee ) (12)= (cid:113) σ + BE r,ee + ( AE r,ee ) , (13)where σ E is the baseline resolution caused by electronicnoise, σ F ( E r,ee ) describes the additional width due toelectron-hole pair statistics including the Fano factor [42],and σ PD ( E r,ee ) is the broadening due to position de-pendence. The electronic noise is energy independent.The variance due to electron-hole pair statistics can bewritten as F ε γ E r,ee ≡ BE r,ee , where F is the Fano fac-tor. Previous measurements at higher temperatures give F = 0 .
13 [43], and using ε γ (cid:39) B = 0 .
39 eV ee . Finally,variations due to position dependence are expected to beproportional to energy; this final term may also includeother effects that scale with energy.The baseline resolution can be measured using the re-constructed energy of noise-only events taken throughoutthe run. When applied to noise traces, the algorithmsdescribed in Sec. II C tend to fit to the largest noise fluc-tuation, which biases the fit toward nonzero amplitudes.This is undesirable for characterizing the baseline noisedistribution; for this study, the time delay is forced tobe zero, and the corresponding energy distribution for -0.1 -0.05 0 0.05 0.1 Energy [keV ee ] P r o b a b ili t y D i s t r i bu t i o n F un c t i o n C u mm u l a t i v e D i s t r i bu t i o n F un c t i o n Noise PDFNoise CDF1 σ Equiv
Figure 17. Reconstructed energy probability distributionfunction (PDF) of noise-only events in Run 2 (blue solid,left vertical axis) with the corresponding cumulative distri-bution function (CDF) (orange dotted, right vertical axis).The 1 σ -equivalent is taken as half the distance between the15.87th and 84.13th percentiles (dark purple dashed) and is9 . ± .
11 eV ee . Run 2 is shown in Fig. 17. To avoid efficiency effects,no cut against low-frequency noise was applied, and thusthe distribution is slightly skewed to positive energy. Asimple Gaussian fit would not be representative of thedistribution; the resolution is determined via a Gaussian-equivalent computation: the 1 σ -equivalent is taken asone-half the energy between the 15.87th and 84.13th per-centiles (the µ ± σ values for a normal distribution). Re-peating the procedure for a variety of histogram bin sizesgives an estimate of the uncertainty. The baseline reso-lution determined in this way is 9 . ± .
11 eV ee .The resolution model of Eq. 13 with parameters σ E , B , and A was fit to the peaks, weighted by their un-certainties, at four different energies: the zero-energybaseline distribution and the three Ge-activation peaksat 10.37 keV ee ( K shell), 1.30 keV ee ( L shell), and0.16 keV ee ( M shell). The resolution of each of thesepeaks is given in Table I. The final fit is given inFig. 18 with a goodness-of-fit per degree of freedom χ / dof = 1 .
22. Because of the small uncertainty on thebaseline resolution, and the weighting of the fit, σ E =9 . ± .
11 eV ee is very similar to the measured value.The best-fit Fano coefficient is B = 0 . ± .
11 eV ee , whilethe position-dependence coefficient is A = (5 . ± . × − . The last two parameters are strongly anticorre-lated with a Pearsons product-moment correlation coef-ficient of ρ AB = − . B fixed tothe expected value gives A = (7 . ± . × − , witha goodness-of-fit per degree of freedom of χ / dof = 3 . Peak Energy [keV ee ] Resolution [eV ee ]Baseline 0 . . ± . M Shell 0 .
16 18 . ± . L Shell 1 .
30 31 ± K Shell 10 .
37 101 ± Ge-activation peaks. ee ]050100 R e s o l u t i o n [ e V ee ] MeasurementsFit1 σ Fit Uncert. ee ]02040 R e s o l u t i o n [ e V ee ] Figure 18. Width of four points in the Run 2 energy spectrum(red points), the best-fit curve (black), and 68 % uncertaintyband (orange). The bottom panel is an enlargement of thetop panel below 1.5 keV ee . factor and any other unaccounted effects. B. Run 2 trigger efficiency and threshold
During WIMP-search data taking, the traces from alldetectors were recorded when the experiment triggered.For calibration data, only the detectors in the same toweras the triggering detector were recorded. Recall that theexperiment triggered if the analog sum of any detector’sphonon traces exceeded a user-set hardware threshold.In anticipation of better low-frequency noise rejection,the hardware trigger threshold was lowered for Run 2compared to Run 1, and again within Run 2, betweenPeriod 1 and Period 2.For Run 2, the analysis thresholds were defined as theenergy at which the detector’s trigger efficiency reached50 %. The trigger efficiency for a given detector D wasdetermined using events that triggered one of the otherdetectors and may or may not have deposited energy indetector D . The efficiency at a given energy E was thengiven by the fraction out of all events with energy E indetector D that also generated a trigger in detector D .The Cf calibration data set, which has more recordedevents than the WIMP-search data set, was used to mea- E ffi c i e n c y Period 1
Energy [keV ee ] E ffi c i e n c y Period 2
Energy [keV ee ] Figure 19. Binned trigger efficiency without (top) and with(bottom) a cut on cryocooler timing for Run 2 Periods 1 (left)and 2 (right). Using the cryocooler information noticeably im-proved the Period 1 measurement while marginally improvingthat for Period 2. The best-fit error function (black dashedcurve) and its 68 % uncertainty (gray shaded) are given in thebottom row for each period. sure trigger efficiency, with strict cuts applied to removenonparticle interactions that also caused triggers, i.e. ,due to noise or detector cross-talk.Two cuts were used to remove low-frequency noise,which triggered the detector at a high rate and couldbias the trigger efficiency calculation, from the calibra-tion data. The first was a pulse-shape cut based on the∆ χ parameter defined in Sec. III C 2, and the sec-ond was based on the cryocooler timing discussed inSec. III C 1. The ∆ χ -based cut was independent ofenergy and tighter than the energy-independent portionsof the WIMP-search-data specific cut of Sec. III C 2. Atighter cut was used to be particularly cautious againstusing low-frequency noise in the calculation.The binned trigger efficiency shown in the top row ofFig. 19 is the result of using the pulse-shape-based cutalone. The highest-energy nonunity bin in Period 1 is at95 eV ee . The highest-energy events that failed to triggerthe detector in Period 1 were found to coincide with thehigh-rate periods of the cryocooler cycle; i.e. , they werecontaminated with low-frequency noise and therefore arenot representative of true physical events. The secondrow in Fig. 19 shows the binned efficiency after apply-ing the second cut against low-frequency noise, removingthe high-rate periods of the cryocooler cycle. After thissecond cut, the highest-energy nonunity bin in Period 1shifts to 82 eV ee .The absence of accelerometer data in Period 2 was dis-covered very soon after the end of the run. Given theutility of the cryocooler timing information in determin-ing the Period 1 trigger efficiency, a dedicated Period 25 Cf calibration was performed with the accelerometersproperly configured. The binned Period 2 trigger effi-ciency is shown in the right panels of Fig. 19. The dif-ference between applying the cryocooler timing or notis marginal, retrospectively unsurprising considering thebetter state of the cryocooler following the repair. Thehighest-energy nonunity bin for the final Period 2 cal-culation is at 62 eV ee . As a verification, the computa-tion was repeated, for both Period 1 and Period 2, usingthe lower-rate WIMP-search data, and consistent resultswere found.The final 50 % trigger efficiency points come from fit-ting the resulting events’ energy to an error function bymaximizing an unbinned log-likelihood function whichcontains a rising error function for events that do trig-ger the CDMSlite detector and a falling error functionfor those that do not. Both functions are needed as theevent energies themselves are used in the fit as opposedto a binned passage fraction. The log-likelihood functionis ln L ( µ, σ ) = N + (cid:88) i ln f + ( E i ; µ, σ ) + N − (cid:88) j ln f − ( E j ; µ, σ ) , (14)where N ± is the number of events passing/failing thetrigger condition on the CDMSlite detector and f ± ( E i ; µ, σ ) = 0 . (cid:20) ± erf (cid:18) E i − µ √ σ (cid:19)(cid:21) , (15)where E i is the total phonon energy of the given eventand µ and σ are the 50 % point and width of the er-ror function, respectively. A Markov chain Monte Carlosimulation was used to scan the parameter space, with alog-normal prior on σ and flat prior on µ . The prior on σ was required as the turn on is very sharp in Period 1; thelog-normal prior inputs knowledge of the detector’s res-olution to prevent fits with an unphysical turn on. Thebest-fit values give thresholds of µ = 75 +4 − and 56 +6 − eV ee for the two periods with the corresponding curves and68 % uncertainty bands shown in the bottom panel ofFig. 19. V. EFFECTS OF BIAS VOLTAGE VARIATION
The bias applied at the detector, and therefore theNTL amplification, varied with time because of the pres-ence of parasitic resistances in the biasing-electronicschain. This variation affected the calibration of the ERand NR energy scales, which thus required empirical cor-rection. Additionally, the observed energy scale of Run 2calls the assumed bias potential of Run 1 into question,though the effect on the Run 1 result is found to be smallcompared to other uncertainties.
A. Total phonon energy scale
The measured scale for total phonon energy E t is deter-mined by calibrating the TES-readout units of amperesto keV t using calibration data taken at the standard iZIPoperating bias of 4 V. In Run 1, the location of the strong Ge K -shell activation peak at ∼
120 keV t , close to theexpected 124 keV t , was taken as confirmation of this pro-cedure, and E t was then converted to E r,ee using Eq. 7with an assumed −
69 V bias.However, this procedure did not match the expectationin Run 2, both for the final −
70 V, data as well as ini-tial −
60 V data taken during Run 2 commissioning. Thepeak appears at 135 and 154 keV t for −
60 and −
70 V re-spectively, both of which are ∼
23 % higher than expected.This is now understood as the effect of a bias-dependentionization extraction and collection efficiency. For thesedetectors, the collection efficiency is <
100 % at 4 V, whilebeing at or above 100 % at CDMSlite biases ( >
100 % ispossible because of impact ionization [44]). These ef-fects were not well understood at the time of Run 1. ForRun 2, the calibration from E t to E r,ee was thus per-formed empirically by scaling the energy such that the K -shell peak appeared at the expected 10.37 keV ee (seeSec. V C).The Run 2 study thus implies a problem with the in-terpretation of the data from the first run, as the ob-served NTL amplification in the second run was notice-ably higher than in the first run though the nominal biasvoltages were similar at −
69 and −
70 V. In Run 2, thehigh-voltage power-supply current was measured, verify-ing that the bias at the detector was close to the nominal70 V. However, such a measurement was not done dur-ing Run 1, and postrun inspections of the high-voltagebiasing board indicated deterioration of a sealant epoxy,originally applied to the biasing electronics to preventhumidity-related effects. Thus, it is possible that a sig-nificant leakage current across the bias resistor, whichwould have reduced the effective bias voltage at the de-tector, went undetected. Assuming that the ionizationcollection efficiency was the same for both runs, and us-ing the energy calibration from Run 2, the Run 1 peaklocation indicated that the effective bias potential wasapproximately −
55 V. This ∼
20 % difference in NTLgain affected the final Run 1 results, and is considered inthe next section.
B. Effect of gain variation on nuclear-recoil energyscale in Run 1
The NTL-amplification gain was measured by trackingvariations of the total phonon energy of the 10.37 keVactivation line with time. The line’s intensity decreasedexponentially with an 11.43 d half-life [23] and increasedwhenever a
Cf calibration was performed. This acti-vation line is shown as a function of time during Run 1in Fig. 20. The measured energy of this line shows vari-6
Contiguous Time [hr] E n e r g y [ k e V t ] Figure 20. Phonon energy as a function of run time for Run 1.The overdensity around 120 keV t is from the 10.37 keV K -shell electron-capture products. Gaps exist because of unsta-ble conditions. The different colors/orientations of the tri-angles indicate the four time periods which were fit to inde-pendent polynomials in the gain-correcting piecewise fit. Thehorizontal line indicates the peak’s expected location (underthe assumptions made for the Run 1 analysis; see text) withdepartures of 5 and 10 % indicated by the bands. The mea-sured energy of the line shows up to 15 % variation over thecourse of the run. ations up to 15 %. In the Run 1 analysis, this variationwas corrected for by an empirical piecewise polynomialfit to the K -shell peak. The different colors in Fig. 20 in-dicate the parts of the run that were fit with independentpolynomials.These variations of the total phonon energy scale, fromthe inferred 20 % correction due to calibration and theobserved time dependence, necessarily affect the nuclear-recoil energy scale, and hence the threshold and finallimit. As described Sec. I, the effect of varying the thresh-old can be non-negligible. Thus, it is imperative to under-stand what a 10 %–20 % variation in total phonon energyimplies for the nuclear-recoil energy scale.The effect of reducing the potential difference, com-pared to the assumed 69 V, is estimated by consideringthe relation between the reconstructed energies E r,nr and E r,ee as given by Eq. 8. At any given E r,ee , E r,nr is cal-culated, assuming the standard Lindhard yield model,for both the original 69 V and at the reduced poten-tial difference. A 10 %–20 % reduction in potential dif-ference has minimal effect on the nuclear-recoil energyscale. The maximum fractional change at the Run 1threshold for gain drops of 10 %, 15 %, and 20 % are | δE r,nr | /E r,nr (170 eV ee ,
69 V) = 1 . , . . < nr at threshold. Reeval-uating the Run 1 result assuming a −
55 V bias, as in-dicated in the previous section, leads to a 2.7 % drop in threshold, which in turn leads to an improvement of thesensitivity for lower-mass WIMPs of up to 12 %, whilethe sensitivity to higher-mass WIMPs decreases by about2 %. This is less than the uncertainty due to the ioniza-tion yield model as shown in Fig. 3. In conclusion, a10 %–20 % drop in gain, even if unaccounted for, doesnot significantly impact the interpretation of the Run 1result in terms of the sensitivity to low-mass WIMPs.
C. Gain correction in Run 2
Laboratory testing after Run 1 revealed that the biasvariations were likely due to humidity on the high-voltagebiasing board, leading to varying parasitic resistances R p ∼ O (10 MΩ), parallel to a biasing resistance of R b ∼
400 MΩ. A new circuit was designed with a bi-asing resistance of R b ∼
200 MΩ. The board was spe-cially treated in an ultrasonic bath, baked, and layeredwith HumiSeal (cid:114) (HumiSeal, Westwood, MA), reducingthe effects of parasitic resistances under humid condi-tions to R p (cid:38) O (1 GΩ). See Appendix A of Ref. [38] fordetails of the biasing board.For Run 2, the DAQ was configured to record the bias V b and current I b of the high-voltage power supply foreach event. Changes in the current are indicative ofchanges in total resistance encountered by the power sup-ply, i.e. some combination of R b and R p . The recordedcurrent was then used to correct the energy scale on anevent-by-event basis as E Corrt = E t · eV b /ε γ e ( V b − I b R ) /ε γ , (16)where R is the encountered resistance. A fit of E t vs. I b demonstrated that R ≈ R b ; i.e. , R p is much greater than R b , is parallel to the detector, and is downstream of R b .Based on this fit and a measured bias current I b (cid:46)
10 nA,a (cid:46) (cid:46) Ge K -shellpeak was consistent in time throughout Period 1. How-ever, there were two distinct populations in Period 2, onelower than Period 1 by 2.87 %, and the other higher thanPeriod 1 by 0.81 %. The origin of these shifts was notidentified. They were corrected for by scaling the meansof the activation peak distributions to match that of Pe-riod 1. A comparison of the initial to final keV t energyscale over the duration of Run 2 is given in Fig. 21. The7 E n e r g y [ k e V t ] RawFeb Mar Apr May Jun Jul Aug Sep Oct Nov Dec140150160 E n e r g y [ k e V t ] Corrected
Figure 21. K -shell activation peak (cluster at 150–160 keV t )in Run 2 as a function of time without (top) and with (bot-tom) corrections for gain variations. Cf calibrations oc-curred in February, May, and September/October. The hori-zontal lines indicate the means of the two peak distributions. mean of the final distribution was then used to scale tothe E r,ee energy scale. VI. CDMSLITE BACKGROUNDS
CDMSlite is an ER background-limited search becauseit cannot discriminate between ER and NR events. How-ever, efforts have been made to understand and reducethe overall background rate in order to extend sensitiv-ity to smaller WIMP scattering cross sections. Oper-ating a SuperCDMS iZIP detector in CDMSlite moderequired grounding one side of the detector, which cre-ated an asymmetric electric-field geometry. This geom-etry was studied in simulation to understand how it af-fects ER background modeling. Motivated by this un-derstanding of the electric field, a fiducial volume wasdefined in Run 2 to remove areas of the detector wherethe electric-field configuration led to reduced signal am-plification and therefore a higher background rate at lowenergies. Defining a fiducial volume thus significantlyreduced the background rate in Run 2.
A. Run 2 radial fiducial-volume cut motivation
The two primary reasons to apply a radial fiducial-volume cut are to remove events of which the energyreconstruction is inaccurate and to remove low-energybackground events ( e.g. , Rn daughters on the detec-tor surfaces and surrounding material). Such a cut wasnot applied in the Run 1 analysis as the small data setdid not allow the impact of the cut to be properly as-sessed. With the larger Run 2 exposure, however, a ra- dial fiducial-volume study became possible. The Run 2cut was particularly motivated by further study of theCDMSlite electric-field configuration and an unexpectedinstrumental background population.
1. Improved understanding of electric-field effects
A copper detector housing enclosed the crystal radi-ally with a small gap between the detector edge and thegrounded housing. Such an arrangement, coupled withthe asymmetric biasing configuration, led to an inhomo-geneous electric field. The field geometry was modeledby finite-element simulation using
COMSOL Multi-physics (cid:114) software (COMSOL, Inc., Burlington, MA).The simulation only included a single detector, and thusany effects from the biased detectors above and belowthe CDMSlite detector were not included. The result-ing electric field showed in which parts of the detectorfreed charges were attracted to the sidewall, and thegrounded housing outside, rather than the grounded flatface. These regions experienced reduced NTL phononemission and therefore a reduced reconstructed energycompared to events of the same initial-energy depositionin the bulk of the detector.To further quantify the position-dependent effectivebias voltage due to field inhomogeneities, a Monte Carlosimulation was performed of the detector crystal con-sidering the calculated field map. In this simulation,electron-hole pairs were placed at various points through-out the detector volume and allowed to propagate accord-ing to the electric-field map. The difference in electricpotential at the final positions of the charge carriers wasrecorded for each pair, allowing for the construction of apotential difference map δV = f ( x, y, z ). A slice of thismap is given in Fig. 22 and shows the region of reducedpotential near the sidewall and the biased face.The reduced NTL phonon emission at the edge of thedetector has the effect of smearing the energy responseto lower energies. Of particular interest is the effect onthe Ge K -shell peak, which has visible smearing in thenonfiducialized Run 2 data as shown in Fig. 23. To es-timate this smearing, sample events were drawn from aflat spectrum to model the Compton background, plusa Gaussian peak distribution, with the rate, mean, andwidth of the distributions chosen to match the observedspectrum. Next, a position was uniformly selected inthe crysta,l and the corresponding potential drop from δV = f ( x, y, z ) was used. For every sample from theinitial spectrum, E init i , the energy E final i expected to bemeasured for an interaction at the respective position in The electrons travel along the direction of the field at high biasvoltages. Thus, oblique propagation and internally scatteringmechanisms were disabled in order to increase the efficiency ofthe simulation. Figure 22. Difference in electric potential between the finallocations of electrons and holes (color map), after propagat-ing through the crystal, as a function of their initial positionin the detector. A single vertical slice of the detector, perpen-dicular to the circular top and bottom faces (see Fig. 2) andalong an arbitrary radius ( R coordinate, with 0 at the centerof the detector) is shown. To uniformly cover the crystal, thesquared radius is sampled, and thus R is plotted. The topof the crystal (along the Z coordinate) is at 70 V, and thebottom is at 0 V. The copper housing (not shown at high R )surrounding the detector is also at 0 V, and a small gap existsbetween it and the sidewall. This causes the total potentialdifference experienced by drifting charges to be <
70 V in re-gions where field lines terminate on the sidewall. Radii with R <
800 mm experience the full 70 V potential differenceand are not shown. the detector was calculated as E final i = E init i × eδV i /ε γ eV b /ε γ , (17)where V b is the applied 70 V bias. The result of thissmearing is also shown in Fig. 23. The asymmetric peakobserved in the data, as expected from the reduced NTLgain, is matched by the smeared simulation. The smear-ing also partially explains the rise in counts below thepeak.The Run 1 analysis did not apply a cut to removeevents from this region of the detector; nor did it ac-count for this smearing in the assumed WIMP-recoilspectrum used for deriving the published upper limit.The effect on the Run 1 result was studied postpublica-tion by considering the fractional change of the cumu-lative above-threshold WIMP spectrum due to smearingthe spectrum. The smear decreased the expected above-threshold WIMP spectrum by (cid:46) c . The change to the published resultswould thus be well within the uncertainty associated withthe ionization yield model shown in Fig. 3.The simulation and study performed here are sufficientto identify the electric field as the source of the observedspectral smearing. They are insufficient, however, for usein the analysis of the measured data, as they cannot in- ee ]10 C o un t s Non-Smeared ModelMC SmearedData
Figure 23. Ge K -shell peak in the Run 2 data, with nofiducial-volume cut, compared to the results of the electric-field study. The study simulates peak events on top of a flatCompton background before applying a smearing function.The smeared low-energy tail observed in the data is replicatedin the simulation. form how to remove the low-gain events. Regions at highradius are clearly seen to be most affected. However, amap of the true physical location as derived from accessi-ble position-dependent analysis parameters is not known a priori , requiring an in-depth simulation of the phononpropagation and signal formation in the detector. Sucha simulation is under development by SuperCDMS [45].The underlying physics is understood and implementedin these simulations, but work is still needed to matchsimulated pulses to data. Thus, these simulations couldnot be used for the studies presented here.
2. Localized instrumental background
In Period 2 of Run 2, an instrumental background ap-peared at 100–200 eV ee . These events are identifiable asbackground as they are localized in time, only occurringduring Period 2, and position. This position localiza-tion can be seen in an x - y -plane representation shownin Fig. 24, where the positions X OF and Y OF are com-puted by the partition of energy between the three innerchannels as X OF = cos (30 ° ) D OF + cos (150 ° ) B OF + cos (270 ° ) C OF B OF + C OF + D OF (18) Y OF = sin (30 ° ) D OF + sin (150 ° ) B OF + sin (270 ° ) C OF B OF + C OF + D OF , (19)where B OF , C OF , and D OF are the OF fit amplitudesfor the three inner channels and the angles correspond totheir relative locations ( cf. Fig. 2); events at the corners9 -0.3 -0.2 -0.1 0 0.1 0.2 x Energy Partition [arb. unit] -0.25-0.2-0.15-0.1-0.0500.050.10.15 y E n e r g y P a r t i t i o n [ a r b . un i t ] Full Energy RangeLow Energy Cluster
Figure 24. Position of Run 2 events using the energy partitioncoordinates. Events in the full energy range are gray, whilethose between 100 and 200 eV ee are highlighted in black. Thepopulation at low energy is clearly clustered in position. of the triangle correspond to events that are predomi-nately underneath a single channel’s sensors. The eventsin the energy range of the low-energy cluster are high-lighted and localized near the top left corner, implyingthat they are localized in a single channel. The exactsource of these events is unknown, but their localizationin time and position identifies them as an instrumentalbackground that can be removed, as shown in the nextsection. B. Run 2 radial fiducial volume cut implementation
A fiducial-volume algorithm was developed based onthe position information from the 2T fit (defined inSec. II C). The channel nearest the event has the highestfast-amplitude contribution (see Fig. 7) and the earliestpulse onset. These features are used to define a new ra-dial parameter with improved position resolution, whichis used to exclude events at high radius [39]. The param-eter was derived in several steps:1. Correct for time variations: correct the energy-carrying slow-template amplitude for each chan-nel in the same manner as described in Sec. V C.Derive the corrected fast amplitude N Corr f (where N stands for the channel labels A – D ) by apply-ing these same correction factors to the fitted fast-template amplitude. Similar instrumental backgrounds have been observed duringearly CDMSlite testing of other detectors.
2. Correct for spatial variations: for channel N calcu-late a relative calibration coefficient ξ N, by nor-malizing the average of the slow-template ampli-tude over all good pulses in the energy region of in-terest to the respective average of channel A . Thisensures that the energy scale is the same in all sen-sors.3. Determine a weight factor for each channel. Thisis done in three steps:(a) Determine peakiness: For channel N , thepeakiness P N is given by the corrected fastamplitude N Corr f scaled by the relative cali-bration factor ξ N, of that channel normal-ized by the total energy of the event E r,ee asdefined in Sec. V C: P N = ξ N, · N Corr f /E r,ee (20) P N will be high for channels close to the in-teraction point.(b) Determine the delay: For channel N , the de-lay ∆ N is given by the difference of the 2T-fitdelay parameters for that channel, δ N, andfor the total phonon pulse, δ tot, :∆ N = δ N, − δ tot, (21)∆ N will be low for channels close to the inter-action point.(c) The weight factor W N for channel N is nowdefined as the difference between the delay andthe peakiness: W N = ∆ N − P N (22) W N will be low for channels close to the inter-action point.4. Construct a preliminary radial parameter R , asthe difference between the weight of the outer chan-nel and that of the inner channel that is closest tothe interaction point: R , = min( W B , W C , W D ) − W A (23) R , is low for events in the center of the detectorand high for events near the edge.5. Construct x - and y -positions X and Y in thesame manner as the numerators of Eqs. 18 and 19using the weights derived here instead of the OF-fitted amplitudes.6. Derive the final radial parameter R by correctingfor a systematic dependence on angular position,reflecting the threefold symmetry of the sensor lay-out, that is observed in the X vs. Y plane.0 -15-10-505 R a d i a l P a r a m e t e r [ a r b . un i t ] P e r i o d − Energy [keV ee ]-15-10-505 P e r i o d Threshold
Figure 25. 2T-fit-based radial parameter as a function of en-ergy for Run 2 Period 1 (top) and Period 2 (bottom). Thevertical clusters are the Ge-activation lines, and the horizon-tal band at high radius contains reduced-amplification events.The radial cut thresholds are indicated by the blue dashedline, effectively removing events at high radius, including thelow-energy cluster seen in Period 2.
Figure 25 shows the final R as a function of re-constructed energy. A higher density of events is seenat higher radius, and the Ge-activation peaks arevisible as vertically oriented populations at 1.30 and10.37 keV ee . The low-energy instrumental background inPeriod 2 is also visible, localized at high radial parameter.Note that events from within the cluster were not usedin defining the radial parameter. It is obvious that R is a nonlinear function of the true radius; the event den-sity in the activation lines (particularly the L -shell peak)shows a clear decrease with increasing radius and thenrises when the edge events begin to contribute. The cutthreshold in the radial parameter, given by the dashedhorizontal lines in Fig. 25, was chosen empirically on thefalling edge of the radial distribution of the inner eventsof the L -shell peak, maximizing the efficiency while re-moving the low-energy cluster along with essentially theentire edge-event distribution. The radial distributionsof the two periods differ somewhat, leading to slightlydifferent choices of cut threshold values between the pe-riods.The signal efficiency of the radial cut was determinedusing the known 11.43 day half-life [23] of the Ge pro-duced in situ during neutron calibrations, together witha pulse-simulation technique. The expected distributionof events from a monoenergetic and uniformly distributedsource in the plane of radial parameter vs. reconstructedenergy is sketched out in Fig. 26. The population is di-vided into two groups: events with reduced NTL am-plification due to field variation ( R ) and those with fullamplification that appear in the peak ( P ). The peak pop-ulation is further split into two sub-groups: inner eventsthat pass the radial cut ( P i ) and outer events that do not Figure 26. Diagram showing the morphology of the expectedevent distribution in the radial-parameter vs. reconstructed-energy plane from a monoenergetic homogeneously dis-tributed source. The distribution is split (vertical solid lines)into nonpeak events R , with reduced NTL amplification, andpeak events P . The latter group is further separated into in-ner peak events P i , that pass the cut threshold (horizontaldotted line), and outer peak events P o , that do not. In prac-tice, the Ge-activation peaks were considered and can beseparated from background because of the known half-life ofthe isotope. ( P o ). The signal efficiency E of the radial cut is definedby the probability that an individual event of the popu-lation passes the cut and appears at the expected energy: E = P i R + P = PR + P · P i P . (24)The second step separately calculates the fractionof events that have full NTL amplification, E = P/ ( R + P ), and the fraction of events with full ampli-fication that pass the radial cut, E = P i /P . These twofactors are determined separately, taking into accountthe presence of background events that are not associ-ated with the Ge decay.To compute E , the plane spanned by the radialand energy parameters was separated into several two-dimensional bins with notably different concentrationsof K -shell capture events. The event distribution as afunction of time was then fit, within each of these bins,with the sum of a constant and an exponential with an11.43 day half-life, to separate the background from the Ge contribution. The known ratio of K - to L -captureevents, together with the assumption that the energy re-duction is based on the electric-field geometry and thusproportional to the recoil energy, was used to identifythe distribution of K -capture events at energies belowthe L -capture line. Following the steps outlined in thisparagraph gives E = 86 ± . vs. energy bin. For the chosen cut position,more than 90 % of the events with reduced energy are re-moved. This calculation also provides E for the K -shellactivation line as E = 54 . ± . . ± . K -shell peak is then E = 47 . ± . . ± . E at lower energies, a pulse-simulationmethod was implemented. All events from the L -peakwere converted to quasi-noise-free pulses by combiningthe fast and slow templates from the 2T fit accordingto their respective fit amplitudes for each of the phononchannels. The K -peak would have provided considerablymore events; however, because of saturation of the 2T-fit–fast-template amplitude in the outer channel above ∼ ee , these were not a good representation of thelow-energy events, and thus could not be used for thisstudy. The noise-free pulses were then scaled to each of13 different energies between 0.04 and 1.30 keV ee beforemeasured noise traces were added. The full L -shell pop-ulation was scaled to each energy, as opposed to usingsubpopulations for each, because of the limited numberof peak events. In each case, the measured noise wastaken from the same time period as the original pulse.At each scaled energy, the same combination of the L -peak event and noise event was used. By using the mea-sured 2T-fit fast/slow amplitude ratio for the simulatedpulses, the radial distribution of the L -shell peak eventswas simulated at each energy.The cut efficiency was then measured by applying thechosen radial cut to the distribution of artificial eventsat each energy, accounting for the radial distribution ofsignal and background as measured in and around the L -peak. At lower scaled energies, some events whichwere close to, and on one side of, the cut threshold inthe original L -shell sample moved to the other side be-cause of the added noise. However, threshold crossingoccurred in both directions; therefore, the overall cut ef-ficiency stayed almost constant down to the lowest ener-gies tested, as shown in Fig. 27. The uncertainty on E contains statistical uncertainty due to the limited numberof L -shell peak events (same for each energy simulated),statistical uncertainty due to the number of simulatedevents that passed the cut (different for each energy sim-ulated), and a systematic uncertainty on the estimateof nonpeak background events simulated (same for eachenergy simulated). C. Effect of the delay parameter in the radialefficiency calculation
As discussed in the previous section, the radial pa-rameter was constructed from a combination of 2T-fitamplitude differences and relative delay of the outer andprimary inner phonon channels. The pulse simulationused to compute the radial cut efficiency, described inthe previous section and implemented for the originalpublication of the Run 2 data [12], only considered therelative amplitude of the input L -shell events without in-cluding the relative delay. In order to confirm that this The onset of this saturation was used to determine the upperenergy threshold for events used in the final WIMP results. E ffi c i e n c y P e r i o d Energy [keV ee ] P e r i o d E ffi ciency in Peak E Total E ffi ciency E Figure 27. Radial fiducial-volume cut efficiency below 2 keV ee for Period 1 (top) and Period 2 (bottom). The efficiency atfull NTL amplification E (orange triangles) as well as thetotal efficiency E (blue circles) are shown along with theirrespective uncertainties. The error bars on E encompass sta-tistical uncertainty due to the available number of L -shellpeak events used as simulation inputs (same for each energysimulated), statistical uncertainty due to the number of sim-ulated events passing the cut (different for each energy sim-ulated), and a systematic uncertainty due to the estimate ofnonpeak background events simulated (same for each energysimulated). The error bars on E additionally contain a smallstatistical uncertainty from the computation of the efficiencyto have full NTL amplification (same for each energy simu-lated). omission did not introduce any significant systematic un-certainty, a new version of the pulse simulation that in-cluded this relative delay of the input pulses was tested.The largest change between the original implementationand the improved version of the pulse simulation is seenat 60 eV ee , just above threshold in Period 2, where thecentral value of the efficiency drops by about 6%. How-ever, all changes are well within the statistical uncertain-ties (typically ±
10 %–15 %). Given the lack of statisticalsignificance, this modification was not propagated intoany final results.
D. Background rates and energy dependence
The effectiveness of the Run 2 radial fiducial-volumecut in reducing the background rate can be seen by com-paring the resulting spectrum to that of Run 1 (Fig. 5).These spectra show the energy of events that scatteronly in the CDMSlite detector, called “single scatters.”Single-scatter events are of interest as WIMPs are ex-pected to scatter extremely rarely, whereas photons andelectrons often scatter multiple times in the detector ar-ray giving “multiple scatters.” Multiple-scatter eventswere removed from the analysis of both data sets to re-duce the background rate, with a loss of < Range Run 1 Rate Run 2 Rate [keV ee kg d] − [keV ee ] [keV ee kg d] − Full Period 1 Period 20 . .
14 - 16 ± . ± . ± . . . ± . . ± . . ± . . ± . . . . ± . . ± .
07 0 . ± .
08 1 . ± . . ± . . ± .
03 0 . ± .
03 0 . ± . ±√ N counting uncertainties, and the Run 2 values additionally in-clude uncertainty from the analysis efficiency (negligible inRun 1). For Run 2 Period 1, the first energy bin cuts off atthat period’s threshold of 75 eV ee . See the text for discussionon the various rates. efficiency for both analyses.In both spectra, the germanium activation lines areseen to be on top of a continuous background, primarilyfrom Compton scattering γ ’s. The average rate betweenthe various activation peaks and analysis thresholds aregiven in Table II for both analyses. The Run 2 rate abovethe K -shell peak is reduced by a factor of 6 from theRun 1 rate by the fiducial-volume cut. The Run 2 ratesare also significantly reduced at lower energies comparedto those of Run 1, though some energy dependence isseen.Previous measurements of the Compton back-ground at higher energies indicated a flat rate of ∼ ee kg d] − [46]. As shown in Table II,this rate was confirmed above the K -shell activation linein Run 1. Additionally, the measurements show that,below this peak, the overall background rate increasedtoward lower energy in both analyses. The increase inrate going from above to below the K -shell peak can beexplained by the decay of cosmogenic isotopes within thedetector and, for the Run 1 spectrum, Ge events withreduced NTL amplification (see Sec. VI A 1).The Run 1 spectrum shows a further increase in ratebelow the L -shell peak. A statistical test to compare thesingle- and multiple-scatter spectra was performed to un-derstand this energy region. The Run 1 multiple-scatterspectrum is shown together with the single-scatter spec-trum below 2 keV ee in Fig. 28. These two spectra werecompared by performing a Kolmogorov-Smirnov (KS)test using the energies for events between the L -shell peakand threshold. The test accepts the hypothesis that thesetwo spectra are drawn from the same underlying proba-bility distribution functions, giving a p-value of 79 .
24 %that is considerably above the standard 5 % hypothesisacceptance limit for a KS test. This shows that the shapeof the single-scatter spectrum is consistent with that ofthe WIMP-free multiple-scatter spectrum, and thus theincrease at low energy cannot be taken as indication of aWIMP signal. This is further supported by the fact thatthe single-scatter rates above and below the L -shell peakin the Run 2 spectrum are statistically compatible with Figure 28. Run 1 low-energy spectrum showing both single-(gray shaded) and multiple-scatter (red line) events. Belowthe L -shell peak, the shape of the multiple-scatter spectrumis statistically compatible with the shape of the single-scatterspectrum. each other.The Run 2 spectrum shows an increase in rate goingfrom above to below the M -shell peak. Comparing thetwo periods of Run 2 in this energy range gives insightinto this excess. For all energy regions above the M -shell peak, the two periods’ rates are statistically con-sistent. Below the M -shell peak, however, the rate inPeriod 2 is dramatically higher compared to Period 1.This indicates that the increase in rate is likely due tobackground events leaking past the selection cuts. Suchleakage is generally expected at lower energies, and leak-age of the localized instrumental background in Period 2(Sec. VI A 2) can explain the difference between the pe-riods.Further studies of the rate require a detailed knowledgeof the shape of all expected background distributions.The spectral shape of Compton recoils at very low ener-gies is actively being studied. A recent simulation studyof the effects of atomic shell structure using G eant e.g. , Pb daughters) will additionallymodify the expected spectral shape, and are still beingstudied with simulations. Future analyses will attemptto take this information into account.
VII. NEW RUN 2 DARK MATTER RESULTS
This section presents new results based on the Run 2analysis, including the effect of varying astrophysical pa-rameters on the spin-independent limit, as well as limitson spin-dependent interactions.3
A. Effects of varying astrophysical parameters
The astrophysical description of the WIMP halo de-scribed in Sec. I enters the differential WIMP-rate ex-pression through the halo-model factor I halo , which de-pends on the velocities of the WIMPs v , the velocity ofthe Earth with respect to the halo v E , and the local darkmatter mass density ρ . As defined in Eq. 2, this factoris an integral over the assumed velocity distribution ofthe halo with respect to the Earth f ( v , v E ).The limits computed for both Runs 1 and 2 assumethe standard halo model (SHM) for the dark matterspatial and velocity distributions. The SHM assumesan isotropic, isothermal, and nonrotating sphere of darkmatter in which the Galaxy is embedded. The velocitydistribution associated with this model is a Maxwelliandistribution boosted to the lab frame of the Earth as f ( v , v E ) ∝ exp (cid:16) − | v + v E | / σ v (cid:17) , (25)where the proportionality constant has already been sub-sumed into Eq. 2 and the velocity dispersion is σ v = v / v is the large-radius asymptotic Galactic circularvelocity. It is typically assumed that this asymptoticvalue has been reached at the Sun’s position [10], giv-ing v = Θ ≡ | Θ | . Θ is the Galactic local stan-dard of rest (LSR), corresponding to the average cir-cular orbital velocity at the Sun’s distance from theGalactic center. The Earth’s velocity is decomposedas v E = Θ + v (cid:12) + v ⊕ , where the other velocities are v (cid:12) , the solar peculiar velocity with respect to neighbor-ing stars, and v ⊕ , the Earth’s orbital velocity around theSun. The Earth’s orbital velocity is assumed to averageto zero over a year. Integrating this distribution overthe range of velocities described in Sec. I gives Eq. 3.Note that the maximum velocity used in the integration,which is related to the Galactic escape velocity v esc , trun-cates the theoretical distribution which would otherwiseextend to infinite velocities.The direct-detection experimental community hasbeen using a uniform set of measurements foreach of these parameters in its analyses: ρ =0 . − cm − [1], Θ = 220 ±
20 km s − in the direc-tion of Galactic rotation [49], v esc = 544 +64 − km s − [50],and v (cid:12) = (11 . ± . , . ± . , . ± .
1) km s − ,where the first component is the radial velocity towardthe Galactic center, the second component is in the di-rection of Galactic rotation, and the third component isthe vertical velocity (out of the Galactic plane) [51]. It iswell known that the uncertainties in these values, in par-ticular Θ and v esc , can have significant effects on com-puted WIMP exclusion limits [52], and thus astrophysical The LSR is of interest to astronomers regardless of whether thisassumption is true, and thus the Θ notation, common in theastrophysical literature, is used for the LSR and its equality to v only taken when specifically referring to the SHM. m WIMP
GeV /c $ − − − − − − σ S I N c m $ v esc = 492 km/s v esc = 533 km/s v esc = 544 km/s v esc = 587 km/s − − − − − − σ S I N [ pb ] − − − − − Figure 29. Effect on the Run 2 best-fit limit from varying theGalactic escape velocity v esc in the Maxwellian halo modelwhile keeping all other parameters constant. Curves shownare the median values of the 2007 and 2014 RAVE surveyresults at 544 km s − (black solid) and 533 km s − (red dot-ted), respectively, as well as the 90 % confidence bounds ofthe 2014 result at 492 km s − (green dashed) and 587 km s − (purple dot-dashed). The inset shows an enlargement belowWIMP masses of 2 GeV/ c . Varying v esc changes the lowestWIMP mass that can produce recoils above threshold, whilethe impact on the limit at higher masses is negligible. uncertainties are also expected on the CDMSlite Run 2spin-independent result. Although the local dark matterdensity is also uncertain [53], all experiments are equallyaffected by its value, so the effect of its uncertainty onthe Run 2 limit is not considered further.For this astrophysical-parameter discussion, the Run 2analysis uncertainties are not considered. Upper limitsare computed using the central efficiency curve in Fig. 4and the standard Lindhard model with k = 0 . All other assumptionsabout the rate discussed in Secs. I and II B are left un-changed, and the optimum interval method [31] is againused to compute limits.The SHM value of v esc comes from the median and 90 %confidence region of the 2007 RAVE survey study [50].The RAVE survey collaboration released an updatedstudy of the escape velocity in 2014 [54] in which theyfound a slightly lower median and reduced uncertaintyspan of v esc = 533 +54 − km s − . Varying the escape ve-locity changes the lower edge of the WIMP-mass range,as a higher maximum halo velocity allows lower-massWIMPs to deposit energy above threshold. The effecton the Run 2 limit of varying the escape velocity whilekeeping all other SHM parameters constant can be seenin Fig. 29. The difference between the 2007 and 2014 Calling this the “best fit” is a slight misnomer as no actual fittingwas performed to obtain the values. m WIMP
GeV /c $ − − − − − − σ S I N c m $ v = 196 km/s v = 220 km/s v = 270 km/s − − − − − − σ S I N [ pb ] Figure 30. Effect on the Run 2 best-fit limit from varyingthe most probable WIMP velocity Θ in the Maxwellian halomodel while keeping all other parameters constant. Curvesshown are for the SHM value of 220 km s − (black solid)and the upper and lower bounds of the measured valuesat 270 km s − (green dashed) and 196 km s − (purple dot-dashed). Varying Θ changes where the most sensitive partof the curve lies in addition to slight changes in the lowestaccessible WIMP mass. The effect is largest for the lowestWIMP masses, vertically shifting the limit by up to an orderof magnitude in either direction. RAVE medians is negligible at all but the lowest WIMPmasses.Recent measurements of the magnitude of the LSRΘ are numerous [55] and include different approachesin measurement technique, galactic modeling, and priorassumptions. The range that the collection of resultsspans, 196–270 km s − , is broader than any individualuncertainty, which indicates possible systematic uncer-tainties between the measurements and models. The ef-fect of varying Θ on the Run 2 limit, keeping all otherhalo parameters at their standard values, can be seen inFig. 30. Varying Θ , and therefore the most probablevelocity in the distribution v , changes where the mostsensitive part of the curve lies in addition to changingthe lowest accessible WIMP mass. This uncertainty hasa large effect at the lowest WIMP masses, shifting thelimit on σ SI N by up to an order of magnitude in eitherdirection.The effect of jointly varying Θ and v esc is consideredby computing the limit 1000 times, each time selecting adifferent set of velocity parameters from their respectivedistributions. For Θ , a conservative flat distribution be-tween the bounding measurements, 196–270 km s − , issampled. For v esc , the probability distribution of v esc from the 2014 RAVE study (distribution graciously pro-vided by the study authors) is directly sampled. The95 % central interval from the 1000 limit curves is shownin Fig. 31 around the SHM-value curve. The size of theuncertainty band is comparable to the uncertainty band m WIMP
GeV /c $ − − − − − − σ S I N c m $ SHM 95% Uncert.SHM (2007 v esc )Alt. v Dist.SHM (2014 v esc ) − − − − − − σ S I N [ pb ] Figure 31. The 95 % (orange) uncertainty band on the best-fitRun 2 spin-independent limit (black solid) due to the uncer-tainties in the most probable WIMP velocity ( v ) and theGalactic escape velocity ( v esc ) used in the SHM. The 2014RAVE survey v esc distribution is sampled, and thus the best-fit curve substituting the 2014 median value into the SHM isgiven for consistency (red dotted). The black and red-dottedcurves are the same as in Fig. 29, where an enlargement atlow WIMP mass is given. The best-fit limit computed usingthe alternative velocity distribution of Eq. 26 is also presented(blue dashed). on the analysis uncertainties given in Fig. 3. Note alsothat Ref. [54] demonstrates an anticorrelation betweenΘ and v esc , meaning that the computed uncertaintyband, which samples the velocity values independently,is an overestimate of the combined uncertainty.Finally, an alternative WIMP velocity distribution isalso considered in Fig. 31. The model is that of Mao etal. [56], which gives, in the rest frame of the dark matter, f ( v ) ∝ e − v/v a (cid:0) v − v (cid:1) p , (26)where v a and p are parameters of the model. Fits toa Milky-Way-like simulation with baryons give p = 2 . v a /v esc = 0 . v → v + Θ + v (cid:12) + v ⊕ ,where the SHM values for these astrophysical velocitiesare used. This model naturally tends to v = 0 at theescape velocity, which explains the reduced sensitivity atthe lightest WIMP masses seen in the limit curve. B. Spin-dependent limits on WIMPs
While the SuperCDMS technology is most sensitiveto spin-independent WIMP-nucleon scattering, the pres-ence of a neutron-odd isotope, Ge ( N = 41) with anabundance in natural Ge of 7.73 %, yields competi-tive limits for spin-dependent scattering at low WIMPmasses [58].5The differential elastic-scattering cross section for afermionic WIMP with respect to the momentum trans-ferred to the nucleus q is given byd σ SD d q = 8 G (2 J + 1) v S T ( q ) , (27)where G F is Fermi’s constant, J is the total nuclearspin of the target nucleus, and S T ( q ) is the momentum-transfer-dependent spin-structure function. S T ( q ) can beparametrized into isoscalar S , isovector S , and inter-ference S terms as S T ( q ) = a S ( q ) + a S ( q ) + a a S ( q ) , (28)where the isoscalar and isovector coupling coefficients arerelated to the proton and neutron couplings as a = a p + a n and a = a p − a n . Explicit forms of S T ( q ) are obtainedfrom detailed nuclear models for specific isotopes.The scattering cross section is typically written in aform similar to the spin-independent case as dσ SD dq = 8 G (2 J + 1) v S T (0) F ( q ) , (29)where F ( q ) ≡ S T ( q ) /S T (0) is the form factor of Eq. 1,which is normalized to unity at zero momentum transfer( q → S T (0) = (2 J + 1) ( J + 1)4 πJ × | ( a + a (cid:48) ) (cid:104) S p (cid:105) + ( a − a (cid:48) ) (cid:104) S n (cid:105)| , (30)where a (cid:48) = a (1 + δa (0)) includes contributions fromtwo-body current scattering as given by Klos et al. inRef. [59]. In two-body current scattering, the WIMP ef-fectively interacts with two nucleons in the nucleus, viathe δa (0) term. The expectation values of the protonand neutron groups within the nucleus (cid:104) S p (cid:105) and (cid:104) S n (cid:105) arecomputed from nuclear theory and usually (cid:104) S p (cid:105) (cid:29) (cid:104) S n (cid:105) for proton-odd nuclei and vice versa for neutron-odd nu-clei. Note that, although the spin-coupling to the even-nucleon species is weak, the inclusion of two-body cur-rents allows for WIMP-proton-neutron effective interac-tions. Thus, the odd-nucleon-species coupling dominatesthe scattering calculations for any coupling type.The standard cross section σ SD0 from Eq. 1 is definedas the total cross section in the q → σ SD0 = 322 J + 1 G µ T S T (0) . (31)The differential cross section can then be written asd σ SD d q = 14 µ T v σ SD0 F ( q ) , (32)where µ T = m χ m T / ( m χ + m T ) is the reduced mass ofthe WIMP-nucleus system. Results are presented in the“proton-only” model where a p = 1 and a n = 0, implying a = a = 1, and the “neutron-only” model where a p =0 and a n = 1, implying a = − a = 1. Results arealso normalized to the scattering of a WIMP and a freeproton/neutron as σ SD0 = 4 π J + 1) (cid:18) µ T µ p/n (cid:19) S p/nT (0) σ SD p/n , (33)where σ SD p/n is the free proton/neutron standard cross sec-tion, µ p/n is the proton-/neutron-WIMP reduced mass,and S p/nT (0) is S T (0) evaluated in the proton-/neutron-only models.Limits set on σ SD p/n using the Run 2 data and analysisare presented in Fig. 32. The limits were computed us-ing the same framework as the spin-independent limitsthat is described in Sec. II B, including using the op-timum interval method [31] and sampling the analysisuncertainties. The median and 95 % uncertainty bandfrom the resulting set of limits are shown in the figure foreach model. The low threshold of CDMSlite gives world-leading limits for WIMP masses (cid:46) (cid:46) c forthe neutron-only and proton-only models, respectively.Limits were also computed using the older spin-structuremodel of Ref. [60], which does not include two-body cur-rents. In the neutron-only case, only a mild improve-ment of 8 % is seen using the newer Klos et al. model.However, using the newer model improves the proton-only limit by a factor of ∼
7, a direct consequence of theWIMP-proton-neutron two-body current increasing theproton-only structure function.Limits are also placed jointly on the coupling coeffi-cients a p and a n for four different WIMP masses. Resultsin this plane were computed by converting the coefficientsto polar coordinates, a p = a sin θ and a n = a cos θ , andobserving that for a given θ , S T ( q ) ∝ a . The proton-and neutron-only models are recovered for θ = π/ , θ were scanned, and an upperlimit was placed on a for each angle. Appendix A dis-cusses different methods for computing these limits andincludes justification for the chosen approach. Limits inthe a p vs. a n plane are given in Fig. 33 for m WIMP of 2,5, 10, and 20 GeV/ c . Regions outside of the ellipses areexcluded. The limits were again computed by samplingthe analysis uncertainties with the median and 95 % in-tervals for each WIMP mass given in the figure. VIII. SUMMARY AND OUTLOOK
This paper described in detail the CDMSlite techniquefor extending dark matter direct detection searches toWIMP masses of ∼ c by achieving analysisthresholds as low as 56 eV ee . New analysis techniqueswere presented and applied to the first two CDMSlitedata sets taken with the SuperCDMS Soudan experi-ment, yielding new limits on spin-dependent interactionsand a better understanding of the effects of astrophysicaluncertainties on the limits.6 m WIMP
GeV /c $ − − − − − − − − − σ S D n c m $ − − σ S D n [ pb ] m WIMP
GeV /c $ − − − − − − − − − σ S D p c m $ − − σ S D p [ pb ] Figure 32. Upper limits on the spin-dependent free neutron σ SD n (left) and free proton σ SD p (right) WIMP scattering crosssections in the proton- and neutron-only models, respectively. For both, the median (90 % C.L) (thick black solid curve)upper limit from CDMSlite Run 2 is compared to other selected direct-detection limits from PANDAX-II (thick-green dottedcurve) [61], LUX (thick-green dot-dashed curve) [62], XENON100 (thick-green dashed curve) [63], PICO-60 (magenta upwardtriangles) [64], PICO-2L (magenta downward triangles) [65], PICASSO (purple dot-dashed band) [66], CDEX-0 (thin-reddashed curve) [67, 68], and CDEX-1 (thin-red solid curve) [68]. The orange band surrounding the Run 2 result is the 95 %uncertainty interval on the upper limit. The Run 2 limits are the most sensitive for m WIMP (cid:46) (cid:46) c for theneutron- and proton-only models, respectively. -200 -100 0 100 200 a n -800-600-400-2000200400600800 a p m WIMP = 2 GeV /c -20 -10 0 10 20 a n -80-60-40-20020406080 a p m WIMP = 5 GeV /c -10 -5 0 5 10 a n -50050 a p m WIMP = 10 GeV /c -10 -5 0 5 10 a n -50050 a p m WIMP = 20 GeV /c Figure 33. Median (90 % C.L.) upper limit and associated 95 % uncertainty (thick black solid curve and orange bands) onthe WIMP-nucleon coupling coefficients a p and a n from CDMSlite Run 2 for WIMP masses of 2 (top left), 5 (top right), 10(bottom left), and 20 (bottom right) GeV/ c . Areas outside the ellipses are excluded for each WIMP mass. ee . This data set will be used to de-velop improved CDMSlite analysis techniques, including:a salting scheme to mitigate analyzer bias, further under-standing of the electric-field influence on fiducial volume,and low-energy background modeling to test backgroundsubtraction techniques.The SuperCDMS Collaboration is also designing a newexperiment, SuperCDMS SNOLAB, where the CDMSlitetechnique will be used in detectors designed specificallyfor high-voltage operation. Planned improvements withsuch detectors include [69]: two-sided biasing, which di-minishes the reduced bias region of the detector; increas-ing the surface area coverage of the phonon sensor; oper-ating at higher applied potentials; and fabricating TESswith lower operational temperatures for the phonon read-out. With the latter two improvements, the SuperCDMSCollaboration aims at thresholds (cid:46)
10 eV ee that will cor-respondingly provide sensitivity to WIMP masses as lowas 400 MeV/ c [70].The SuperCDMS Collaboration gratefully acknowl-edges technical assistance from the staff of the SoudanUnderground Laboratory and the Minnesota Departmentof Natural Resources, as well as the many contributionsof David Caldwell, who passed away during the writ-ing of this article. The iZIP detectors were fabricated inthe Stanford Nanofabrication Facility, which is a memberof the National Nanofabrication Infrastructure Network,sponsored and supported by the NSF. Part of the re-search described in this article was conducted under theUltra Sensitive Nuclear Measurements Initiative and un-der Contract No. DE-AC05-76RL01830 at Pacific North-west National Laboratory, which is operated by Battellefor the U.S. Department of Energy. Funding and sup-port were received from the National Science Founda-tion, the Department of Energy, Fermilab URA VisitingScholar Award No. 13-S-04, NSERC Canada, and Mul-tiDark (Spanish MINECO). Fermilab is operated by theFermi Research Alliance, LLC, under Contract No. De-AC02-07CH11359. SLAC is operated under Contract No.DEAC02-76SF00515 with the United States Departmentof Energy. Appendix A: Setting limits on spin-dependentcoupling coefficients with two-body currents
A model-independent method for setting joint limitson the spin-dependent coupling constants a p and a n wasderived by Tovey et al. in Ref. [71]. In that work, theauthors derive a simple expression relating the allowedvalues of the coupling constants, for a given WIMP mass,as π G µ p ≥ a p (cid:113) σ Lp ± a n (cid:113) σ Ln , (A1)where G F is Fermi’s constant, σ Lp/n are the limits onthe free-proton/-neutron cross sections for the givenWIMP mass (assuming a proton-/neutron-only interac-tion), the small difference between the WIMP-proton µ p and WIMP-neutron µ n reduced masses is ignored,and the sign in the brackets is the same as the ratio ofnuclear spin-group expectation values (cid:104) S n (cid:105) / (cid:104) S p (cid:105) . Thisexpression is derived from the observation that the al-lowed total-nucleus cross section σ SD0 must be smallerthan the limit set upon it by a given analysis σ L . Equa-tion A1 is then found by using the expression for thezero-momentum spin structure function S T (0) withouttwo-body currents, found by taking δa (0) → S T (0)from Klos et al. [59] changes this derivation and result.Starting with σ SD0 /σ L ≤ σ SD0 andEq. 30 for S T (0) gives1 ≥ J + 1) G µ T Jπ × (cid:34) | ( a + a (cid:48) ) (cid:104) S p (cid:105)| (cid:112) σ L ± | ( a − a (cid:48) ) (cid:104) S n (cid:105)| (cid:112) σ L (cid:35) , (A2)where the sign of the ± is determined by the sign of( a − a (cid:48) ) (cid:104) S n (cid:105) / ( a + a (cid:48) ) (cid:104) S p (cid:105) . The limits on the totalcross section are not factored out as they are next rewrit-ten in terms of the limits on the free-proton/-neutroncross sections σ Lp/n in the proton-/neutron-only models,as given by Eq. 33. In the denominator of the left term,the proton-only model form is used, while the neutron-only form is used under the right term. The resultinginequality after changing coupling bases to that of theproton and neutron couplings is8 π G µ p ≥ | a p + ( a p − a n ) δa (0) | (cid:113) σ Lp |(cid:104) S p (cid:105)|| [2 + δa (0)] (cid:104) S p (cid:105) − δa (0) (cid:104) S n (cid:105)|± | a n − ( a p − a n ) δa (0) | (cid:113) σ Ln |(cid:104) S n (cid:105)||− δa (0) (cid:104) S p (cid:105) + [2 + δa (0)] (cid:104) S n (cid:105)| . (A3)The simpler Eq. A1 is recovered by taking the limit of notwo-body currents ( δa (0) → a p , a n ) to the polar ( a, θ ) as a p = a sin θ (A4) a n = a cos θ. 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