Updated Constraints on the Dark Matter Interpretation of CDMS-II-Si Data
PPrepared for submission to JCAP
Updated Constraints on the DarkMatter Interpretation ofCDMS-II-Si Data
Samuel J. Witte a and Graciela B. Gelmini a a Department of Physics and Astronomy, UCLA, 475 Portola Plaza, Los Angeles, CA 90095(USA)E-mail: [email protected], [email protected]
Abstract.
We present an updated halo-dependent and halo-independent analysis of viablelight WIMP dark matter candidates which could account for the excess observed in CDMS-II-Si. We include recent constraints from LUX, PandaX-II, and PICO-60, as well as pro-jected sensitivities for XENON1T, SuperCDMS SNOLAB, LZ, DARWIN, DarkSide-20k, andPICO-250, on candidates with spin-independent isospin conserving and isospin-violating in-teractions, and either elastic or exothermic scattering. We show that there exist dark mattercandidates which can explain the CDMS-II-Si data and remain very marginally consistentwith the null results of all current experiments, however such models are highly tuned, mak-ing a dark matter interpretation of CDMS-II-Si very unlikely. We find that these models canonly be ruled out in the future by an experiment comparable to LZ or PICO-250.
Keywords: dark matter theory, dark matter experiment a r X i v : . [ h e p - ph ] M a y Introduction
Despite an overwhelming amount of evidence for the existence of dark matter, very little isknown about it beyond what is inferred from its gravitational influence. Motivated largelyby theoretical expectations, weakly interacting massive particles (WIMPs) with mass at the
GeV to (cid:39)
100 TeV -scale remain among the most studied candidates.Direct dark matter experiments search for the energy deposited into nuclei in under-ground detectors by collisions with WIMPs gravitationally bound to the galactic halo. Whileno definitive detections have been made, a number of collaborations have observed potentialdark matter signals [1–7]; however, such observations are typically viewed to be in conflictwith the null results of many other experiments [8–23].The difficulty in making definitive statements regarding the nature of potentially viablesignals arises from the fact that there exists a vast amount of uncertainty in the analysisof direct dark matter detection data. This is because both the particle physics and theastrophysics entering the computation of the expected scattering rates are, at best, poorlyunderstood. In standard analyses of direct detection data, assumptions must be made onthe local dark matter density, the dark matter velocity distribution, the dark matter-nucleiinteraction, and the scattering kinematics. Bounds are then placed as a function of the darkmatter mass and overall scale of the cross section. The obvious problem is that adjustingassumptions, e.g. on the velocity distribution, unevenly alters the predicted rates in differentexperiments. This happens to be particularly true for the region of parameter space wherepotential dark matter signals have arisen, as this region sits near the low-energy threshold ofmany experiments.In recent years, ‘halo-independent’ data comparison methods that avoid making anyassumptions about the local dark matter halo characteristics have been developed, therebyreducing the uncertainty in experimental comparisons (see e.g. [24–52]). The original halo-independent analyses were rather limited in that putative signals often required averagingthe signal over some energy range, potentially removing valuable information and making thecomparison with upper limits ambiguous (see e.g. [24, 26, 27, 32]). Recently, methods were de-veloped which, for putative signals, allow for the construction of halo-independent confidencebands, resulting in a better comparison between upper limits and potential signals [36, 51, 53].These methods, however, rely on the ability to use an extended likelihood [54] for at leastone of the experiments observing a putative signal. At the moment, CDMS-II-Si is the onlyexperiment that has claimed a potential dark matter signal for which such a method can beapplied.Halo-dependent analyses strongly constrain the excess observed by CDMS-II-Si (seee.g. [33, 39, 55]). A halo-independent analysis performed on the CDMS-II-Si data in 2014showed that the only WIMP candidates still consistent with the upper limits of null searcheswere those with spin-independent isospin-violating interactions, and either elastic or exother-mic scattering [53]. Here, we revisit the viability of the CDMS-II-Si excess, using both halo-dependent, assuming the standard halo model (SHM), and halo-independent analyses, incor-porating the latest bounds produced by LUX (using their complete exposure) [21], PandaX-II [22], and PICO-60 [23]. We also assess the projected sensitivity of XENON1T [56, 57],LZ [58, 59], DARWIN [60], DarkSide-20k [61, 62], PICO-250 [63], and the high-voltage ger-manium detectors of SuperCDMS to be installed at SNOLAB [64]. We show that models withhighly exothermic kinematics and a neutron-to-proton coupling ratio f n /f p set to minimizethe scattering rate in xenon-target experiments are not currently excluded, nor can they be– 2 –ejected by XENON1T.In Sec. 2 we review the halo-independent analysis and the procedure for constructing thetwo-sided pointwise halo-independent confidence band. The analysis for each experiment isexplained in Sec. 3. In Sec. 4, we present our results, specifically focusing on isospin conservingand isospin-violating [65, 66] (with f n /f p = − . and f n /f p = − . ) interactions with elasticand exothermic scattering [67–69]. We conclude in Sec. 5. Here, we briefly review the generalized halo-independent analysis implemented in Sec. 4,concentrating on the extended halo independent (EHI) analysis [53] in the following subsection(the reader is encouraged to consult [27, 30, 32, 39, 70] for additional details).In direct detection experiments, the differential rate per unit detector mass of a target T , induced by collisions with a WIMP of mass m , as a function of nuclear recoil energy E R is given by d R T d E R = ρm C T m T (cid:90) v (cid:62) v min ( E R ) d v f ( v , t ) v d σ T d E R ( E R , v ) , (2.1)where m T is the mass of the target element, ρ is the local dark matter density, C T is the massfraction of a nuclide T in the detector, d σ T / d E R is the dark matter-nuclide differential crosssection in the lab frame, and f ( v , t ) is the dark matter velocity distribution in the lab frame.The temporal dependence of f ( v , t ) arises from Earth’s rotation about the Sun. For thehalo-dependent analyses in Sec. 4, we assume the SHM, i.e. f ( v , t ) is an isotropic Maxwellianvelocity distribution in the Galactic frame, with the astrophysical parameters adopted in [37].The integration in Eq. (2.1) runs over all dark matter particle speeds larger than orequal to v min ( E R ) , the minimum speed necessary to impart an energy E R to the nucleus.Should multiple target nuclides be present in the detector, the total differential scatteringrate is given by d R d E R = (cid:88) T d R T d E R . (2.2)For elastic scattering, the value of v min is given by v min = (cid:115) m T E R µ T , (2.3)where µ T is the WIMP-nuclide reduced mass. It may be possible that the dominant WIMP-nuclei interaction proceeds instead through an inelastic collision, whereby the dark matterparticle χ scatters into a new state χ (cid:48) with mass m (cid:48) = m + δ (with | δ | (cid:28) m ) [67–69]. In thelimit that µ T | δ | /m (cid:28) , v min ( E R ) is instead given by v min ( E R ) = 1 √ m T E R (cid:12)(cid:12)(cid:12)(cid:12) m T E R µ T + δ (cid:12)(cid:12)(cid:12)(cid:12) , (2.4)where δ < ( δ > ) corresponds to an exothermic (endothermic) scattering process. Eq. (2.4)can be inverted to find the possible range of recoil energies which can be imparted by a darkmatter particle with speed v in the lab frame E T, − R ≤ E R ≤ E T, + R , where E T, ± R ( v ) = µ T v m T (cid:32) ± (cid:115) − δµ T v (cid:33) . (2.5)– 3 –t should be clear from Eq. (2.5) that for endothermic scattering, for which δ > , there existsa non-trivial kinematic endpoint for the WIMP speed given by v Tδ = (cid:112) δ/µ T > , such thatdark matter particles traveling at speeds v < v Tδ cannot induce nuclear recoils. In this paperwe will be focusing exclusively on elastic ( δ = 0 ) and exothermic ( δ < ) scattering, for which v Tδ = 0 . Interpreting the CDMS-II-Si data using models with endothermic spin-independentinteractions are clearly experimentally rejected. Notice that Eq. (2.5) implies only a finiterange of recoil energies around the energy E R ( v Tδ ) = µ T | δ | /m T can be probed for inelasticscattering.Experiments do not directly measure the recoil energy of the nucleus, but rather a proxyfor it that we denote E (cid:48) . The differential rate in this new observable energy E (cid:48) is given byd R d E (cid:48) = (cid:88) T (cid:90) ∞ d E R (cid:15) ( E R , E (cid:48) ) G T ( E R , E (cid:48) ) d R T d E R , (2.6)where (cid:15) ( E R , E (cid:48) ) is the detection efficiency and G T ( E R , E (cid:48) ) is the energy resolution; jointly,these two functions give the probability that a detected recoil energy E (cid:48) resulted from a truenuclear recoil energy E R .Changing the order of integration in Eq. (2.6) allows the differential rate to be expressedas d R d E (cid:48) = σ ref ρm (cid:90) v (cid:62) v Tδ d v f ( v , t ) v d H d E (cid:48) ( E (cid:48) , v ) , (2.7)where we have definedd H d E (cid:48) ( E (cid:48) , v ) ≡ (cid:88) T C T m T (cid:90) E T, + R E T, − R d E R (cid:15) ( E R , E (cid:48) ) G T ( E R , E (cid:48) ) v σ ref d σ T d E R ( E R , v ) if v (cid:62) v Tδ , if v < v Tδ .(2.8)Here, we have explicitly factored out an overall normalization σ ref from the differential crosssection. For spin-independent interactions, the differential WIMP-nucleus cross section isgiven by d σ SIT d E R ( E R , v ) = σ p µ T µ p [ Z T + ( A T − Z T )( f n /f p )] F T ( E R )2 µ T v /m T , (2.9)where F T ( E R ) is the nuclear form factor that accounts for the decoherence of the dark matter-nuclide interaction at large momentum transfer. Here, we take this to be the Helm formfactor [71]. Thus we take σ ref = σ p , the WIMP-proton cross section. Interactions with spin-or nuclear magnetic moment-dependencies produce smaller rates in silicon relative to othertarget elements employed by experiments which have not observed an excess.Let us define the halo function ˜ η ( v min , t ) ≡ ρσ ref m (cid:90) ∞ v min d v F ( v, t ) v , (2.10)where the function F ( v, t ) is the local dark halo speed distribution, given by F ( v, t ) = v (cid:82) d Ω v f ( v , t ) . Using Eq. (2.10), the differential rate in E (cid:48) can be written asd R d E (cid:48) = − (cid:90) ∞ v δ d v ∂ ˜ η ( v, t ) ∂v d H d E (cid:48) ( E (cid:48) , v ) . (2.11)– 4 –pplying integration by parts on Eq. (2.11), and noting that ˜ η ( ∞ , t ) = 0 and d H / d E (cid:48) ( E (cid:48) , v δ ) =0 , the differential rate can be expressed asd R d E (cid:48) = (cid:90) ∞ v δ d v min ˜ η ( v min , t ) d R d E (cid:48) ( E (cid:48) , v min ) , (2.12)where we have defined a WIMP model and experiment dependent “differential response func-tion” d R / d E (cid:48) as d R d E (cid:48) ( E (cid:48) , v min ) ≡ ∂∂v min (cid:20) d H d E (cid:48) ( E (cid:48) , v min ) (cid:21) . (2.13)Approximating the time dependence of the halo function as ˜ η ( v min , t ) (cid:39) ˜ η ( v min ) + ˜ η ( v min ) cos(2 π ( t − t ) / year ) , (2.14)and integrating Eq. (2.12) over E (cid:48) , the unmodulated R and annual modulation amplitude R of the rate, integrated over an observable energy bin [ E (cid:48) , E (cid:48) ] , is given by R α [ E (cid:48) ,E (cid:48) ] ≡ (cid:90) ∞ v δ d v min ˜ η α ( v min ) (cid:90) E (cid:48) E (cid:48) d E (cid:48) d R d E (cid:48) (2.15) = (cid:90) ∞ v δ d v min ˜ η α ( v min ) R [ E (cid:48) ,E (cid:48) ] ( v min ) , (2.16)where α = 0 , and the second line has defined the energy integrated “response function” R .In order to place an upper limit on the function ˜ η ( v min ) (hereby denoted ˜ η ( v min ) ), wenote that at a particular point in the v min − ˜ η plane, the halo function producing the smallestnumber of events in a particular experiment is a downward step-function with the step locatedat the particular ( v min , ˜ η ) point. This is a consequence of the fact that, by definition, ˜ η ( v min ) is a monotonically decreasing function of v min . As first shown in [24], CL limits on ˜ η , ˜ η lim ,are placed by determining the CL limit on the rate, R lim [ E (cid:48) ,E (cid:48) ] , and inverting Eq. (2.16),i.e. ˜ η lim ( v min ) = R lim [ E (cid:48) ,E (cid:48) ] (cid:82) v min v δ d v R [ E (cid:48) ,E (cid:48) ] ( v ) . (2.17) It was shown in [36, 53] that an extended likelihood is maximized by a piece-wise constanthalo function ˜ η BF ( v min ) with a number of steps less than or equal to the number of eventsobserved, and furthermore that a two-sided pointwise halo-independent confidence band canbe constructed around this best-fit halo function, ˜ η BF . A stream of velocity (cid:126)v s with respectto the Galaxy, such that | (cid:126)v s + (cid:126)v ⊕ | = v min (where (cid:126)v ⊕ is Earth’s velocity with respect to theGalaxy) would produce an ˜ η function proportional to Θ( | (cid:126)v s + (cid:126)v ⊕ | − v min ) . Thus a piecewise ˜ η ( v min ) function could be interpreted as corresponding to a series of streams, one for eachof its downward steps. More recently, it was shown that this formalism can be extendedto more generalized likelihood functions that include at least one extended likelihood [51].Here, we briefly summarize the process outlined in [53] for producing a two-sided pointwisehalo-independent confidence band using an extended likelihood function (which we apply inSec. 4 to the CDMS-II-Si data) of the form L = e − N E [˜ η ] N obs (cid:89) a =1 M T d R tot d E (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:48) = E a , (2.18)– 5 –here N E [˜ η ] is the total number of expected events, N obs is the number of observed events,d R tot / d E (cid:48) is the total differential rate, and E (cid:48) a is the detected energy of event a .The confidence band is defined as the region in the v min − ˜ η plane satisfying ∆ L [˜ η ] ≡ L [˜ η ] − L min ≤ ∆ L ∗ , (2.19)where L [˜ η ] is two times the minus log likelihood, L min is the value of L [˜ η ] evaluated with thebest-fit halo function ˜ η BF ( v min ) , and ∆ L ∗ corresponds to the desired confidence level. Thatis to say, we seek the collection of all halo functions that produce changes in the log likelihoodfunction less than or equal to the desired value ∆ L ∗ .While this is a viable definition, in practice finding this complete set of halo functionsis not possible. Instead, we consider the subset of ˜ η functions which minimize L [˜ η ] subject tothe constraint ˜ η ( v ∗ ) = ˜ η ∗ . (2.20)We define L c min ( v ∗ , ˜ η ∗ ) to be the minimum of L [˜ η ] subject to the constraint in Eq. (2.20), andwe define the function ∆ L c min ( v ∗ , ˜ η ∗ ) as ∆ L c min ( v ∗ , ˜ η ∗ ) ≡ L c min ( v ∗ , ˜ η ∗ ) − L min . (2.21)Should the point ( v ∗ , ˜ η ∗ ) lie within the confidence band, then at least one halo functionpassing through this point should satisfy ∆ L [˜ η ] ≤ ∆ L ∗ . It follows that ∆ L c min ( v ∗ , ˜ η ∗ ) ≤ ∆ L ∗ .On the other hand, should ∆ L c min ( v ∗ , ˜ η ∗ ) > ∆ L ∗ , there should not exist any halo functionscontained within the confidence band passing through ( v ∗ , ˜ η ∗ ) . Thus, a two-sided pointwiseconfidence band can be constructed by finding at each value of v min , the values of ˜ η ∗ around ˜ η BF which satisfy ∆ L c min ( v min , ˜ η ∗ ) ≤ ∆ L ∗ . For the results presented in Sec. 4, we plot thecontours of ∆ L ∗ = 1 . and 2.7, which for a chi-squared distribution with one degree offreedom correspond to and CL confidence bands, respectively [53]. Compatibility ofthese confidence bands with upper limits can then be assessed at a given CL by determiningwhether there exists a non-increasing halo function ˜ η ( v min ) which is entirely contained withina particular band and does not exceed any of the upper limits. A confidence band is said tobe excluded if no such halo function can be constructed. Here, we present current halo-dependent and halo-independent constraints on the CDMS-II-Si and regions for a variety of elastic and exothermic spin-independent interactionmodels. We focus explicitly on isospin conserving ( f n /f p = 1 ), ‘Ge-phobic’ (defined by thechoice of neutron and proton couplings which minimizes scattering in germanium, i.e. f n /f p = − . ), and ‘Xe-phobic’ models (defined by the choice of neutron and proton couplings whichminimizes scattering in xenon, i.e. f n /f p = − . ). Halo-independent constraints are presentedfor three representative choices of m and δ , which had been selected in [53] as parameters inthe halo-dependent analyses which appeared to provide good compatibility of the CDMS-II-Sisignal and the upper bounds from null searches.Upper limits in this section are presented for the following experiments: SuperCDMS [19],CDMSlite (2016 result) [18], XENON100 [20], LUX (2013 result) [72], LUX (2016 result) [21], In the limit that N obs is large, Wilk’s theorem states that the log-likelihood ratio follows a chi-squareddistribution which may not exactly apply with only 3 events. The LUX2013 bound is presented assuming zero observed events. This bound has been shown to be wellrepresentative of the true bound [33]. – 6 – = f n / f p = - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TSuperCDMS Ge ( HV ) LZDARWINPICO - - -
1. 10. 100. - - - - - - - - - - -
39. m [ GeV ] Log ( σ p [ c m ]) Figure 1 . Halo-dependent comparison of CDMS-II-Si (dark red) and (light red) regionswith current
CL upper limits from SuperCDMS (brown), CDMSlite2016 (magenta), XENON100(blue, solid), LUX2013 (purple, dotted), LUX2016 (purple, solid), PandaX-II (grey), and PICO-60(green, solid), for an elastic isospin conserving spin-independent interaction. Also shown are projecteddiscovery limits (dashed) for XENON1T (blue), SuperCDMS SNOLAB Ge HV (black), LZ (purple),DARWIN (orange), DarkSide-20k (yellow), and PICO-250 (green).
PandaX-II [22], and PICO-60 [23]. Also shown are projected bounds for XENON1T [57], Su-perCDMS SNOLAB Ge High-Voltage (which we call SuperCDMS Ge(HV)) [64], LZ [58, 59],DARWIN [60], DarkSide-20k [61, 62], and PICO-250 [63]. The procedure for constructing theLUX2013 bound was previously outlined in [30, 33, 34]. We describe here the process usedbelow to produce the remaining experimental bounds.
The procedure for analyzing the CDMS-II-Si data follows the procedure outlined in [30,33, 34]. Specifically, we consider the three event signal with energies 8.2, 9.5, and 12.3 keV.CDMS-II-Si had an exposure of 140.2 kg-days and an energy window of keV to keV. Usinga profile likelihood ratio test, a preference was found for the WIMP+background hypothesisover the background-only hypothesis with a p − value of 0.19% [7]. We use an E R -dependentefficiency identical to that shown in Fig. 1 of [7] (solid blue line). Since the energy resolutionfor silicon in CDMS-II has not been measured, we use a Gaussian resolution function withthe energy resolution used for CDMS-II’s germanium detectors, taken from in Eq. 1 of [73], σ ( E (cid:48) ) = (cid:112) . + 0 . × E (cid:48) / keV keV. To estimate the differential background rate foreach observed event, we take the differential background rates from [74] and normalize eachcomponent such that 0.41, 0.13, and 0.08 events are expected from surface events, neutrons,and Pb respectively [7]. This procedure reproduces the preferred regions shown in Fig. 4of [7].
The XENON100 bound is produced in the manner outlined in [30], but using the updated477 day exposure [20]. This procedure accurately reproduces the bound shown in Fig. 11 of– 7 – = f n / f p = - - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TSuperCDMS Ge ( HV ) LZDARWINPICO - - -
1. 10. 100. - - - - - - - - - - -
36. m [ GeV ] Log ( σ p [ c m ]) δ = = f n / f p = - - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TSuperCDMS Ge ( HV ) LZDARWINPICO - - - - - - - - - v min [ km / s ] Log ( η ˜ [ d ays - ]) Figure 2 . (Left) Halo-dependent analysis and (Right) halo-independent analysis for m = 9 GeV,assessing the compatibility of the CDMS-II-Si and
CL regions (shown in darker and lighterred) with the
CL upper limits and projected sensitivities of other experiments, for an elasticspin-independent contact interaction with f n /f p = − . . We include SuperCDMS (brown), CDMSlite(magenta), XENON100 (blue, solid), LUX2013 (purple, dotted), LUX2016 (purple, solid), PandaX-II (grey), and PICO-60 (green, solid) upper limits, and the projected sensitivities (dashed lines)of XENON1T (blue), SuperCDMS Ge(HV) (black), LZ (purple), DARWIN (orange), DarkSide-20k(yellow), and PICO-250 (green). Also shown is the best-fit halo function ˜ η BF to the CDMS-II-Si data(dark red step function) and the v min value corresponding to the event with the largest observed recoilenergy, assuming E R = E (cid:48) = 12 . keV (vertical dot dashed dark red line). [20]. The CDMSlite bound (hereby CDMSlite2016) is constructed using results from the recentlyreported 70.1 kg-day exposure. The detector efficiency and quenching factor are taken fromFig. 1 and Eq. 3 of [18], respectively. The energy of detected events is read off the inset inFig. 3 in [18], but only between detected energies of 0.36 and 1.04 keVee, and the maximumgap method is then applied. This procedure reproduces the published bound.
The LUX bound is computed by using the complete LUX exposure (approximately . × kg-days). The efficiency and fractional resolution as functions of E R are extracted from Fig. 2(black solid line) and Fig. 5 of [21], respectively. The bound is obtained by determining thecross section required to produce a total of 3.2 events. As mentioned in [21] this procedurereproduces the CL combined LUX exclusion limit.
The constraint for PandaX-II is based on the . × kg-day run data published in 2016.To reproduce the published bound, the nuclear recoil efficiency function is taken from Fig. 2of [22] (black line), and the recoil energies of the three observed events are read off Figs. 4– 8 –nd 14 of [22]). Applying the maximum gap method [75] yields a bound that reproduces wellthe published bound for m (cid:46) GeV, and is slightly stronger at larger masses by a factor of (cid:46) . . The constraint for PICO-60 is based on the recent kg-day run of C F [23]. Here, werestrict our attention to scattering off fluorine, as this element accounts for (cid:39) of thetarget mass and has a lower threshold than carbon (after considering the bubble nucleationefficiency in Fig. 4 of [15]). PICO-60 is run at a thermodynamic threshold of . keV, howeverthis threshold does not correspond to the threshold recoil energy in fluorine required tonucleate a bubble. We take this threshold to be keV using the efficiency function shownin Fig. 4 of [15] for a 3.2 keV thermodynamic threshold (although this is only determinedfor a GeV WIMP with a spin-independent interaction). Using Poisson statistics with zeroobserved events and zero expected background, we find this threshold perfectly reproducesthe published bound [23].
The projected bound for XENON1T [57] is computed assuming a 2 ton-year exposure, a flatefficiency of . , and an effective light yield, a low-energy threshold, and an energy resolutionequal to those used in the XENON100 analysis of [30]. This procedure produces a sensitivitylimit consistent with the ± σ confidence intervals of the 2 ton-year sensitivity limit shown inFig. 8 of [57]. SuperCDMS plans to operate the next generation of their experiment at SNOLAB beginningin 2020; the discovery limits produced here are based on the recent projected sensitivityfor their high-voltage germanium, Ge(HV), detectors. Specifically, we assume 8 Ge(HV)detectors, each with an exposure of 44 kg-days. We also assume perfect efficiency in theenergy range . keV (taken from Table VIII of [64]) to keV (taken to be consistent withthe energy range suggested in the caption of Table V of [64]), perfect energy resolution,an ionization yield given by Lindhard theory (with parameters taken from [76]), and zeroobserved events. Using the maximum gap method (which coincides with using a Poissonlikelihood in this case) we obtained a CL limit very similar to the Ge(HV) limit shownin Fig. 8 of [64]. SuperCDMS also plans to run a high-voltage silicon detector which is notincluded here because its projected sensitivity is inferior across most of the parameter space.Also note that if the energy ranges of the HV detectors could be extended to energies beyond keV, these experiments could gain sensitivity to the exothermic models considered here. The projected sensitivity for LZ is produced using the same energy resolution and efficiencyfunction used in the LUX2016 analysis, and assuming a total exposure of . ton-years(i.e. a . ton fiducial volume with 1000 live-days) [58, 59]. We then apply the maximum gapmethod, under the assumption of zero observed events, with which we reproduce a sensitivitylimit comparable to that shown in Fig.4 of [59].– 9 – = -
50 keV f n / f p = - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TSuperCDMS Ge ( HV ) LZDARWINPICO - - -
1. 10. - - - - - - - - - - -
39. m [ GeV ] Log ( σ p [ c m ]) δ = -
50 keVm = f n / f p = - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TLZPICO - - - - - - - - v min [ km / s ] Log ( η ˜ [ d ays - ]) δ = -
50 keV f n / f p = - - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TSuperCDMS Ge ( HV ) LZDARWINPICO - - -
1. 10. - - - - - - - - - - -
38. m [ GeV ] Log ( σ p [ c m ]) δ = -
50 keVm = f n / f p = - - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TLZPICO - - - - - - - - - v min [ km / s ] Log ( η ˜ [ d ays - ]) δ = -
50 keV f n / f p = - - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TSuperCDMS Ge ( HV ) LZDARWINPICO - - -
1. 10. - - - - - - - - - - -
36. m [ GeV ] Log ( σ p [ c m ]) δ = -
50 keVm = f n / f p = - - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TLZPICO - - - - - - - - - v min [ km / s ] Log ( η ˜ [ d ays - ]) Figure 3 . (Left) Halo-dependent analysis and (Right) halo-independent analysis for m = 3 . GeV,assessing the compatibility of the CDMS-II-Si and
CL regions (shown in darker and lighterred) with the
CL upper limits and projected sensitivities of other experiments, for an exothermicspin-independent contact interaction with δ = − keV. Results are shown for isospin conservingcouplings (top), ‘Ge-phobic’ couplings (middle), and ‘Xe-phobic’ couplings (bottom). Experimentsincluded are identical to those shown in Fig. 2. – 10 – .10 DARWIN The projected sensitivity limit for DARWIN is based on the design presented in [60], for aliquid xenon experiment with a 200 ton-year exposure. Following [60], we consider an energyrange of keV to . keV and a constant detection efficiency of . We approximate theenergy resolution as a Gaussian with σ = E R × . , which is roughly consistent with Fig. 1 of[60]. Assuming zero observed events, the bound is obtained using the maximum gap method.This procedure is found to produce a sensitivity limit in strong agreement with that shownin Fig. 7 of [60]. The projected sensitivity for DarkSide-20k is produced assuming a flat nuclear recoil efficiencyof . between energies keV and keV (and zero elsewhere), a 60 ton-year exposure (i.e. a20 ton fiducial volume run for 3 years), and by applying the maximum gap method with zeroobserved events [61, 62]. DarkSide-20k is not sensitive for the nuclear recoils imparted toargon nuclei by the particular candidates in our halo-independent analyses (we show v min ≤ km/s), thus no DarkSide-20k bounds appear in the halo-independent plots. The projected sensitivity for PICO-250 is produced assuming perfect detection efficiency forenergies above keV (see Sec. 3.6), a kg fiducial volume, a year runtime (or alternatively,a kg fiducial volume run for one year), and by using Poisson statistics with zero observedevents and zero expected background [23, 63]. As in Sec. 3.6, we only consider scattering offfluorine. For the purpose of providing context, we begin by plotting in Fig. 1 a comparison of the and
CDMS-II-Si regions with the current and projected
CL limits of otherexperiments, assuming the conventional elastic spin-independent contact interaction withisospin conserving couplings. Null results from LUX2013 and SuperCDMS have excludedthis model at the
CL in both halo-dependent and halo-independent analyses (there existsmall discrepancies in the preferred CDMS-II-Si regions of [53] and those presented below, amistake that arose because the factor of 2 in the definition of L [˜ η ] was missing in [53]) [33, 53].We present in Fig. 2 a halo-dependent (left) and halo-independent (right) analysis of anelastic spin-independent contact interaction with ‘Xe-phobic’ couplings (i.e. f n /f p = − . ).In the halo-dependent analysis, the CL CDMS-II-Si region is excluded by the
CLupper limits of LUX2016, PandaX-II, and PICO-60. This is consistent with the results of [55].In the halo-independent analysis, the upper limit of PandaX-II does not entirely exclude the
CL CDMS-II-Si region, the LUX2016 limit only marginally excludes the
CL CDMS-II-Si region, and only the very recent PICO-60
CL bound definitively excludes
CLCDMS-II-Si region. This is shown for m = 9 GeV, but other choices of masses lead to similarresults. In the halo-independent analysis, we also show the v min value corresponding to theenergy of the event with the largest observed energy, assuming E R = E (cid:48) = 12 . keV (shownwith vertical dot dashed dark red line). Had our analysis of the CDMS-II-Si data assumed aperfect energy resolution, the location of the highest step of the best-fit ˜ η would identicallycorrespond to this value of v min ; with finite energy resolution, the locations of the steps of– 11 – = -
200 keV f n / f p = - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TLZDARWINPICO - - -
1. 10. - - - - - - - - - - -
40. m [ GeV ] Log ( σ p [ c m ]) δ = -
200 keVm = f n / f p = - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TLZPICO - - - - - - - - - - v min [ km / s ] Log ( η ˜ [ d ays - ]) δ = -
200 keV f n / f p = - - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TLZDARWINPICO - - -
1. 10. - - - - - - - - - - -
38. m [ GeV ] Log ( σ p [ c m ]) δ = -
200 keVm = f n / f p = - - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TLZPICO - - - - - - - - - - v min [ km / s ] Log ( η ˜ [ d ays - ]) δ = -
200 keV f n / f p = - - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TLZDARWINPICO - - -
1. 10. - - - - - - - - - - -
38. m [ GeV ] Log ( σ p [ c m ]) δ = -
200 keVm = f n / f p = - - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TLZPICO - - - - - - - - - - v min [ km / s ] Log ( η ˜ [ d ays - ]) Figure 4 . Same as Fig. 3, but for δ = − keV and m = 1 . GeV (halo-independent analysesonly). The SuperCDMS Ge(HV) discovery limit is not shown as it cannot probe the WIMP candidatesshown. the best-fit ˜ η function occur at slightly larger values of v min . For highly exothermic models,it becomes important to verify that the dark matter speeds capable of producing such recoilsare physical, i.e. they do not exceed the galactic escape velocity, which for the Standard HaloModel is v esc (cid:39) km/s (in the lab frame).– 12 – = -
225 keV f n / f p = - - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TLZDARWINPICO - - -
1. 10. - - - - - - - - - - -
38. m [ GeV ] Log ( σ p [ c m ]) δ = -
225 keVm = f n / f p = - - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) PandaX - IIXENON1TLZPICO - - - - - - - - - - v min [ km / s ] Log ( η ˜ [ d ays - ]) Figure 5 . Same as Fig. 2 but for δ = − keV. The halo-independent analysis is shown for m = 1 . GeV. The dotted black line in the halo-independent analysis shows the SHM ˜ η ( v min ) function for σ p = 2 × − cm , a value included in the 68% CL region in the halo-dependent analysis. In Figs. 3, 4 and 5 we plot halo-dependent (left) and halo-independent (right) analysesof exothermic spin-independent contact interactions with δ = − keV, δ = − keV, and δ = − keV respectively. In Figs. 3 and 4 results are shown for isospin conserving (top),‘Ge-phobic’ (middle), and ‘Xe-phobic’ (bottom) models. The halo-dependent analyses inFig. 3 show that the present CL limits reject the and
CL CDMS-II-Si regions.The Fig. 3 halo-independent analyses, shown for m = 3 . GeV, illustrate that the CDMS-II-Si
CL region for a ‘Xe-phobic’ interaction with δ = − keV is only excluded bythe recent PICO-60, and not by the PandaX-II or LUX limits. Note that the keV uppercutoff imposed on the recoil energy in the SuperCDMS Ge(HV) data analysis implies that thisexperiment only tests very light exothermic candidates, and does not probe the CDMS-II-Siregions. Similarly, DARWIN’s relatively large low energy threshold prevents this experimentfrom probing the WIMP candidate presented in the halo-independent analysis. This is aconsequence of only showing WIMP speeds less than km/s.The results shown in Fig. 4 are similar to those in Fig. 3, except that in the halo-independent analyses (shown for m = 1 . GeV), the
CL CDMS-II-Si region for the‘Xe-phobic’ interaction with δ = − keV is no longer ruled out for a small set of halo func-tions which deviate considerably from the SHM. It would seem that increasingly exothermicscattering kinematics (i.e. more negative values of δ ) may alleviate the tension between thedark matter interpretation of CDMS-II-Si and the null results of other experiments. This isnot the case, however, as increasingly negative values of δ decrease the range of recoil energiesthat can be imparted by WIMPs (see Eq. (2.5)). This implies that highly exothermic candi-dates traveling at speeds less than the galactic escape velocity may not be able to account forall three events observed by CDMS-II-Si (as illustrated in Fig. 1 of [37]). While the largeststep in the best-fit ˜ η function in Fig. 4 does lie above what is conventionally taken to be thegalactic escape velocity, ∼ km/s in the lab frame, the v min value corresponding to the . keV event is clearly below this value (additionally, there are non-negligible astrophysical– 13 – = -
225 keVm = f n / f p = - - II - SiSuperCDMSCDMSlite ( ) XENON100LUX ( ) LUX ( ) PandaX - IIXENON1TLZPICO - - - - - - - - - - v min [ km / s ] Log ( η ˜ [ d ays - ]) Figure 6 . Same as right panel of Fig. 5 but for m = 1 . GeV. The dotted black line in the halo-independent analysis shows the SHM ˜ η ( v min ) function for σ p = 3 × − cm , a value included in the90% CL region in the halo-dependent analysis. uncertainties in the value of the galactic escape velocity.) To further illustrate this point, weshow in Fig. 5 an analysis of a ‘Xe-phobic’ dark matter candidate with δ = − keV. It canbe clearly seen in the halo-independent analysis (shown in the right panel for m = 1 . GeV)that the third event of CDMS-II-Si can only be attributed to WIMPs traveling at speeds v (cid:39) km/s (in the lab frame), far above the galactic escape velocity. Notice that wehave not included in our halo-independent analyses a term in the likelihood penalizing largeunphysical halo speeds (as was done e.g. in [41]), which in this case would allow only two ofthe events observed by CDMS-II-Si to be attributed to dark matter. Also shown in Fig. 5 isthe SHM ˜ η function with a normalization set to σ p = 2 × − cm , a value which is allowedthe CL region in the halo-dependent analysis. The halo-independent analysis clearlyrejects this function at the
CL, showing that, for this particular dark matter particlecandidate, the SHM does not fit the CDMS-II-Si data well.For strongly exothermic candidates, a small change in the particle mass leads to aconsiderable change in the range of recoil energies probed by acceptable values of v min . InFig. 6, we show the halo-independent analysis for the same interaction (i.e. spin-independentwith f n /f p = − . and δ = − keV) and a WIMP mass m = 1 . GeV instead of m =1 . GeV. This small change in the mass eliminates the problem of requiring unacceptablylarge WIMP speeds. However, in this case the new PICO-60 90% CL limit rejects the halo-independent CDMS-II-Si 90% CL region, which would otherwise be allowed by all otherbounds. Again, the SHM ˜ η function with values of σ p allowed in the 90% CL region in thehalo-dependent analysis, lies outside the halo-independent 90% CL confidence band. Theseexamples clearly illustrate the point that one cannot continue to lower delta below − keVwith the hope of increasing the viability of a dark matter interpretation of the CDMS-II-Sievents.For completeness, we show in Fig. 7 the viable parameter space in the δ − m plane for‘Xe-phobic’ models. The pink regions are where either one (light pink), two (pink), or allthree (redder pink) events observed by CDMS can be induced by WIMPs traveling at speeds– 14 – n / f p = - v esc =
800 km / s AB - - - - - - - δ [ keV ] m [ G e V ] Figure 7 . Values of m and δ in the ‘Xe-phobic’ model for which one (light pink), two (pink), and allthree (redder pink) of the CDMS-II-Si events are below the galactic escape velocity. The two purpleregions indicate the part of the redder pink region not excluded by current experiments. While stillviable, the light purple, region ‘B’, region provides a worse fit to the CDMS-II-Si data than the darkerpurple region, region ‘A’ (see text for details). v ≤ km/s in the lab frame (a conservative choice for the galactic escape velocity [77]),namely where their recoils are kinematically allowed. The purple regions highlight the subsetof the dark red region that cannot be ruled out by current direct detection experiments (i.e. theviable parameter space where all three observed events can be due to WIMPs bound to thegalactic halo). While the light purple region (region ‘B’) is still viable, the minus log likelihoodevaluated in the light purple region is significantly larger than that of the dark purple region(region ‘A’), indicating a worse fit to the data. This is because two of the observed events arerelatively close in energy and thus, given that the halo function is monotonically decreasing,the data prefers models in which the v min values associated with the two lowest observedrecoils are lower than the v min value of the highest energy event.Fig. 4 and Fig. 5 show that no viable parameter space for ‘Xe-phobic’ interactionswill remain if an experiment like LZ or PICO-250 does not find any dark matter signal(i.e. the purple regions, both ‘A’ and ’B’, shown in Fig. 7 will be rejected). Notice that,even though the exposure of PICO-250 is much smaller than the exposure of LZ, PICOis highly sensitive to light exothermic WIMPs because fluorine is much lighter than xenon(in general exothermic scattering favors lighter target nuclei) and both PICO and LZ havecomparable energy thresholds. In this regard, although argon is much lighter than xenon, thehigher energy threshold of DarkSide-20k makes this experiment insensitive to light exothermicWIMPs. We have presented here updated halo-dependent and halo-independent constraints on darkmatter particle candidates that could explain the CDMS-II-Si data. We have studied candi-– 15 –ates with isospin conserving and isospin-violating spin-independent interactions, with eitherelastic or exothermic scattering. We included constraints from PandaX-II, LUX (completeexposure), and PICO-60, as well as projected sensitivities for XENON1T, SuperCDMS SNO-LAB Ge(HV), LZ, DARWIN, DarkSide-20k, and PICO-250.The results presented show that both spin-independent isospin conserving and ‘Ge-phobic’ ( f n /f p = − . ) interpretations of CDMS-II-Si are excluded at the CL. ‘Xe-phobic’ ( f n /f p = − . ) interpretations, however, are still marginally viable after the recentPICO-60 result if the dark matter particle scatters exothermically with nuclei (with δ (cid:46) − keV), and can only be ruled out in the future by an experiment comparable to LZ or PICO-250. Although still marginally viable, the highly tuned nature of these models make a darkmatter interpretation of the CDMS-II-Si very unlikely. Acknowledgments.
G.G. was supported in part by the US Department of EnergyGrant DE-SC0009937. S.W. was supported by a UCLA Dissertation Year Fellowship.
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