Calibration, event reconstruction, data analysis, and limit calculation for the LUX dark matter experiment

The LUX experiment has performed searches for dark-matter particles scattering elastically on xenon nuclei, leading to stringent upper limits on the nuclear scattering cross sections for dark matter. Here, for results derived from $1.4\times {10}^4\mathrm{kg}$ days of target exposure in 2013, details of the calibration, event-reconstruction, modeling, and statistical tests that underlie the results are presented. Detector performance is characterized, including measured efficiencies, stability of response, position resolution, and discrimination between electron- and nuclear-recoil populations. Models are developed for the drift field, optical properties, background populations, the electron- and nuclear-recoil responses, and the absolute rate of low-energy background events. Innovations in the analysis include in situ measurement of the photomultipliers’ response to xenon scintillation photons, verification of fiducial mass with a low-energy internal calibration source, and new empirical models for low-energy signal yield based on large-sample, in situ calibrations.

Figure 1

Schematic of the LUX detector. LUX is a two-phase xenon time projection chamber containing an active LXe mass of 250 kg that is viewed from above and below by PMT arrays.

Figure 2

Photograph of the LUX detector, with its inner vessel removed.

Figure 3

Photograph of the inside of the LUX detector, with its lower PMT array removed.

Figure 4

Photograph of LUX installed in the Davis Cavern water shield at SURF.

Figure 5

A schematic diagram of the various thresholds applied to raw waveforms.

Figure 6

Measured maximum sphe pulse height for PMT 31. The 1.5 mV POD threshold is indicated by the vertical dashed line. Based on a fit of the single-photoelectron pulse height distribution, it is estimated that 95.6% of the sphe in PMT 31 produce a pulse that exceeds the POD threshold.

Figure 7

Single photoelectron acceptance distribution for all 122 photomultiplier tubes.

Figure 8

Schematic view of the LUX Data Processing Framework.

Figure 9

An example of a double-scatter event waveform. The search regions are labeled A and B, with the first, A, prioritized. In addition to the main signal, this event contains three SDPs, with one shown as an inset to indicate the noise threshold of the pulse finder. The bottom panel shows the location of pulses within the entire event record, with the $x$ axis as the number of 10 ns data samples either side of the trigger pulse, located at zero.

Figure 10

Pulse classification decision tree.

Figure 11

Map of reconstructed positions in the $xy$-plane from data obtained after an injection of ${}^{83\mathrm{m}}\mathrm{Kr}$ in the detector. The observed striped pattern is the result of a focusing effect of the electrons near the gate wires, created by the difference between the fields in the drift and the extraction regions. There are significant differences in the intensity of these strips. These variations are most likely to be caused by small variations in the pitch of the wires producing fluctuations of the bulk field which are not observed for the S2 right below the gate wires [34].

Figure 12

From top to bottom: Profile of the absolute number of background events as a function of $\mathrm{S}1$ size, $\mathrm{S}2$ size, depth, and radius, combining all known sources of background. S1 is cut off at 50 phd (photons detected), the WIMP search cut, causing a dropoff in $\mathrm{S}2$ spectrum. The black points and crosses are the data, while the smooth gray center lines come from a simulation based on a data-informed full background model. The green bands represent the 1-sigma uncertainties in the simulation; they are $\sim 30\%$ of the value of the background model, aggregating uncertainties across different background components [40]. The disagreement at high $\mathrm{S}2$ comes from the NEST overestimating the high-energy charge yield for electron recoils. Because this was at high energy, the impact on the WIMP search sensitivity was negligible. (This was corrected in the following run, 332 live days, with new calibrations.)

Figure 13

The averaged $\mathrm{S}2$ shapes from NR ($\mathrm{D}\text{−}\mathrm{D}$ neutron calibration data, in blue) and ER (${\mathrm{CH}}_3\mathrm{T}$, in red). Slow simulation (yellow) refers to a G4/NEST prediction from first principles based on the electrical and geometric properties of the detector. Fast simulation (green) refers to the parametrized PMT hit-map library mode of LUXSim that is forced to agree with data as closely as possible. A drift time cut of $100–120\mu \mathrm{s}$ is applied in all cases so that $\mathrm{D}\text{−}\mathrm{D}$ and tritium can be compared directly. In contrast to $\mathrm{S}1$ signals, no significant differences are expected for different interaction types.

Figure 14

Comparison of simulation results (blue) to data (black, with error bars representing the statistical uncertainties in each bin) for the ${}^{131\mathrm{m}}\mathrm{Xe}$ cosmogenic-neutron activation peak, which is the only monoenergetic ER in the background that has sufficient statistics to allow such a comparison to be made. Presented here is the reconstructed “combined” energy scale, calculated from both $\mathrm{S}1$ and $\mathrm{S}2$ and which therefore has improved resolution as compared to the use of a single channel alone, for the data, while for the simulation the $x$ axis represents the Monte Carlo truth energy. The width of the peak is determined by the binomial statistics of $\mathrm{S}1$ light collection, sphe resolution, electron lifetime, extraction efficiency, $\mathrm{S}2$ collection, single-electron pulse size variance, and recombination fluctuations. To reproduce the mean precisely, the values of $\mathrm{S}1$ and $\mathrm{S}2$ collection efficiencies had to be varied by 5% (upward), well within their uncertainties at the 1–2 sigma level, and the recombination fluctuations are increased by 1-sigma compared to a best fit at low energies. The default NEST mean yields from V0.98, 2013, are assumed without modification. The source of the slight asymmetry in the data was likely a background component.

Figure 15

Top: the relative light collection of $\mathrm{S}1$ signals as a function of the depth in the detector. Bottom: The asymmetry in the $\mathrm{S}1$ light collection efficiencies between the top and bottom PMT arrays, defined as $(\mathrm{T}-\mathrm{B})/(\mathrm{T}+\mathrm{B})$, where T and B are the values of $\mathrm{S}1$ recorded in the top and bottom PMT arrays, respectively (simultaneous fit to both quantities). In both plots, data are presented in black with statistics-only error bars, while the results of the simulation are shown in blue.

Figure 16

${\chi }^2$ map for Teflon reflectivity and photon absorption length in liquid xenon, the two free parameters that have the greatest impact on both relative and absolute light collection. The degeneracy between the reflectivity of the Teflon and the absorption length for photons in the liquid can be seen, with each having similar effects on the reduction in the light collected. As the former asymptotes to $\sim 96\%$, a reasonable value [57], the latter approaches infinity.

Figure 17

The pulse-area spectrum for one of the LUX PMTs under a range of conditions. The black histogram shows single photoelectrons stimulated by exposure to a pulsed blue LED, the red histogram shows the signals recorded as a consequence of VUV photons emitted from gaseous xenon, and the blue histogram shows the signals recorded as a consequence of VUV photons emitted from liquid xenon. Evidence of two phe emission from the VUV photon exposures is clearly evident.

Figure 18

The spectrum of ionization signals from single electrons, shown as the blue data points. The spectrum, fitted with a skew-normal distribution, gives a mean of $24.66\pm 0.02(\mathrm{stat})$ phd and a $1\text{−}\sigma$ width of $5.95\pm 0.02(\mathrm{stat})$ phd. The black line shows a simulation of this spectrum produced from first principles, based on a $5.8\mathrm{kV}/\mathrm{cm}$ field in a 1.6 bar, 5 mm “gas gap.” with an $\mathrm{S}2$ light collection efficiency of 10%. The falloff in the data for large phd values is caused by the switch from single-electron to $\mathrm{S}2$ pulse definition.

Figure 19

Variation of the mean of the single-electron distribution in the plane of the liquid surface.

Figure 20

The gain as a function of time for ten representative PMTs.

Figure 21

The gain of each PMT was measured each week throughout the run with an LED calibration to stimulate sphe emission. Shown is a histogram of the standard deviations of these gains, each normalized to the PMT’s mean gain. Most gains are stable to better than 2%.

Figure 22

The variation in the size of single-electron $\mathrm{S}2$ pulses over the duration of the WIMP search.

Figure 23

The measured electron lifetime over the duration of the experiment. Sporadic dips in electron lifetime are a result of temporary stoppages in the LXe recirculation.

Figure 24

The scintillation (squares) and ionization (circles) response of the LUX detector over time to the decays of ${}^{83\mathrm{m}}\mathrm{Kr}$.

Figure 25

Histogram of the number of the observed tritium events as a function of the corrected radius and for a drift time between 38 and $305\mu \mathrm{s}$ (solid black line). The red line represents a simulation of a perfectly uniform distribution of events. The blue line represents an estimate of the background observed during the lifetime of the tritium calibration, multiplied by a factor of 50.

Figure 26

Xenon mass calculated directly from tritium counting as a function of the radius of the fiducial volume. The $\pm 1$ sigma band is represented by the gray band. The drift time considered is between 38 and $305\mu \mathrm{s}$, and the mass between the gate and anode is $250.9\pm 2.1\mathrm{kg}$.

Figure 27

The difference between the xenon mass calculated from the counting of tritium events and the xenon mass obtained from the geometry of the fiducial volume as function of the radius of the fiducial volume. The $\pm 1\sigma$ region is represented by the gray band.

Figure 28

Comparison of centripetal uncertainties in Mercury in $\mathrm{D}\text{−}\mathrm{D}$ data (vertical black crosses), tritium data (diagonal black crosses), single electrons (black squares), and the root mean square between the reconstructed radius and the original radius, obtained in $\mathrm{D}\text{−}\mathrm{D}$ simulations (triangles).

Figure 29

Comparison of the centripetal and polar uncertainties in $\mathrm{D}\text{−}\mathrm{D}$ simulations to the root mean square of the difference between the reconstructed position and the true position.

Figure 30

Top: The crosses show the average reconstructed $x$ position as a function of reconstructed $y$ from $\mathrm{D}\text{−}\mathrm{D}$ calibration data. In both cases, $x$ and $y$ are corrected for the effects of the nonuniform electric field. The dashed line shows a linear fit to the $\mathrm{D}\text{−}\mathrm{D}$ beam direction. Bottom: Deviation from the linear fit in the $x$ direction.

Figure 31

Top: uncorrected $xy$ positions for ${}^{83\mathrm{m}}\mathrm{Kr}$ data with drift times between 200 and $300\mu \mathrm{s}$. Bottom: The same data, after the correction to $xy$ position has been applied.

Figure 32

$\mathrm{S}{1}_c$ vs $\mathrm{S}{2}_c$ for a compilation of LUX line source data. ${}^{83\mathrm{m}}\mathrm{Kr}$ and ${}^{137}\mathrm{Cs}$ data were collected during dedicated calibration runs. All other lines were present in the low background WIMP search data. ${}^{127}\mathrm{Xe}$ [66], ${}^{129}\mathrm{Xe}$, and ${}^{131}\mathrm{Xe}$ were only present early in Run 3 due to their cosmogenic origin. The anticorrelation between $\mathrm{S}1$ and $\mathrm{S}2$ is due to recombination.

Figure 33

Extraction of the gain factors $g_1$ and $g_2$, adhering to the method described in the text. Each point represents a single line source listed in Table 2.

Figure 34

The mean and width of the ER band as determined from Gaussian fits to the tritium population in $\mathrm{S}1$ slices are depicted in blue, with a tuned MC based on the fundamental yields extracted using the known beta energy spectrum, at the same discrete points in $\mathrm{S}1$, in red. The latter is based on a fit to the yields, but not to the band, and it is thus in a sense a prediction.

Figure 35

The measured events from the D-D exposure defining the nuclear recoil. There are 19249 events remaining after all cuts. The black data points are the Gaussian fit mean values for each $\mathrm{S}1$ spike bin. The red data points are the corresponding Gaussian fit mean values for the simulated nuclear-recoil band. The black triangles and red dashed line indicate the 90% one-sided limits from data and simulation, respectively. The lower magenta dot-dashed line indicates the $\mathrm{S}2$ threshold at 165 phd, and the upper magenta dot-dashed line shows the applied upper cut for $\mathrm{S}2$ ($\mathrm{S}2<5000\mathrm{phd}$). The lower density of points in the center reflect the differential cross section behavior for 2.45 MeV neutrons incident on a xenon target. Error bars on the black and red data points are statistical only and are generally too small to see.

Figure 36

The measured events defining the nuclear-recoil band are shown in the scatter plot as a function of energy in ${\mathrm{keV}}_{\mathrm{ee}}$ and $\mathrm{keV}\mathrm{nr}$. They are the same 19249 events, after analysis cuts, shown in Fig. 35. The density scale is different from the previous nuclear-recoil band plot due to normalization by bin width.

Figure 37

Calibrations of the detector response in the fiducial volume and leakage of ER events below the NR mean. Panel (a) shows the ER (tritium) calibration, while panel (b) shows the NR calibration (obtained with monoenergetic neutrons from a $\mathrm{D}\text{−}\mathrm{D}$ generator). Solid lines show band means, while dashed lines indicate the $\pm 1.28\sigma$ contours (blue for ER and red for NR). Also shown is the $\mathrm{S}2$ threshold applied in the analysis (dot-dashed magenta line). Panel (c) shows ${\log }_{10}(\mathrm{S}2/\mathrm{S}1)$ normalized to the mean of the ER band as a function of the $\mathrm{S}1$ for the tritium data. The 248 events that fall below the NR mean are plotted using a different marker (vertical cross) for clarity.

Figure 38

Leakage (left) and discrimination (1-leakage, right) vs $\mathrm{S}1$. Values obtained from a Gaussian extrapolation (blue and green triangles) are shown alongside those obtained from counting tritium events. The dashed line corresponds to the average discrimination of 99.80% in the data count. To achieve good agreement on observed leakage (data count), the $g_1$ and $g_2$ used in simulations had to be adjusted by 5%, well within uncertainty. The Monte Carlo neither underestimates nor overestimates the leakage fraction systematically, alternating between the two within error, so, for both total leakage and Gaussian only, MC (first principles) and data agree fairly well.

Figure 39

A cleanliness cut applied to golden events, shown by the dashed lines, which removes periods of high-rate single-electron background, is defined as a function of extra nonsignal area contained within the 1 ms window of each event. The 165 phd $\mathrm{S}2$ threshold used in the analysis is also shown (vertical dot-dashed line).

Figure 40

Top: Cumulative number of 1 ms time window slices from a randomly picked WIMP search data set as a function of the event area contained within, showing 99% acceptance at 80 phd. Bottom: Cleanliness cuts (dashed lines) applied to a tritium calibration data set, confirming an acceptance of $\gtrsim 99\%$ for good areas above 400 phd. Yellow triangles indicate the 99% percentile in each good area bin; the vertical dot-dashed line shows the 165 phd $\mathrm{S}2$ threshold.

Figure 41

Comparison of the single-scatter efficiency curve obtained from the simulation of NR events (black dots; the line is added just to guide the eye) with the efficiency obtained from analysis of $\mathrm{D}\text{−}\mathrm{D}$ data (blue empty squares), in the relevant region of small $\mathrm{S}1$ pulses. For larger signals ($\mathrm{S}1>5\mathrm{phd}$), the efficiency stays flat at 100% up to the $S150\mathrm{phd}$ upper limit.

Figure 42

Distribution in ${\log }_{10}(\mathrm{S}2/\mathrm{S}1)$ vs $\mathrm{S}1$ of events caused by random coincidence of $\mathrm{S}1$-only and $\mathrm{S}2$-only events. The blue and red lines have the same meaning as in Fig. 37.

Figure 43

Distribution in $\mathrm{S}1$ vs $\log 10(\mathrm{S}{2}_b/\mathrm{S}1)$ of background events from 86.3 live days and a sample of ${}^{83\mathrm{m}}\mathrm{Kr}$ events from calibration data. Top: Events within the liquid xenon bulk defined by a radius of less than 15 cm and a drift time between 10 to $320\mu \mathrm{s}$. Bottom: events near the PTFE wall, defined by the distribution of ${}^{210}\mathrm{Pb}$ daughters. More details on the wall reconstruction can be found in Ref. [78]. The spread of events below the main peaks illustrates the incomplete charge collection for events near the wall.

Figure 44

Kernel-density estimated distribution, in $\mathrm{S}1$ and $\mathrm{S}2$, of wall background events reconstructed within the fiducial volume. The predicted count is 24 events within the 145 kg fiducial volume over 95.0 live days.

Figure 45

Electron-equivalent-energy histogram of the LUX Run 3 search events with $\mathrm{S}1<50\mathrm{phd}$ (points) together with the best-fit model (stacked histogram), which has zero signal. The $x$ axis is the linear, combined light-and-charge energy estimator, $E_{\mathrm{ee}}=W·(\mathrm{S}1/g_1+\mathrm{S}2/g_2)$, with $W=13.7\mathrm{eV}$. The roll-off in efficiency either side of 2–9 keV is primarily due to cuts on $\mathrm{S}1$; search events are required to have at least two PMTs in coincidence and a maximum size of 50 phd.

Figure 46

Top: SI WIMP-nucleon interaction differential rate functions for WIMP $\mathrm{mass}=10\mathrm{GeV}$ (blue), 33 GeV (black), and 1 TeV (red). Bottom: SD WIMP-neutron (solid) and WIMP-proton (dashed) interaction rate for WIMP $\mathrm{mass}=10\mathrm{GeV}$ (blue), 33 GeV (black), and 1 TeV (red).

Figure 47

Top: Mean of the ${\log }_{10}(\mathrm{S}2)$ band as a function of $\mathrm{S}1$. Bottom: Gaussian width of the ${\log }_{10}(\mathrm{S}2)$ band as a function of $\mathrm{S}1$. (Black) Nominal value of all nuisance parameters, (red) TIB varied by $+1\sigma$, (blue) $g_2$ varied by $+1\sigma$, and (magenta) ${\kappa }_L$ varied by $+1\sigma$, $g_1$, $N_{ex}/N_i$, $\eta$, and $P$ variations are not plotted but they fall within the $g_2$ and TIB variations.

Figure 48

Top: Number of SI WIMP-nucleon interaction events passing the analysis cuts scaled to the Run 3 exposure for a WIMP mass of 5 GeV. The black line corresponds to nominal values of all nuisance parameters, blue shows $g_2$ varied by $\pm 1\sigma$, and red shows ${\kappa }_L$ varied by $\pm 1\sigma$. Bottom: The number of SD WIMP-neutron (upper) and SD WIMP-proton (lower) interaction events passing the analysis cuts scaled to the Run 3 exposure for a WIMP mass of 50 GeV. In these plots, black corresponds to nominal values of all nuisance parameters; blue shows the structure factors $S_n(u)129$ and $S_p(u)129$, as defined in Sec. C of Ref. [80], varied by $+1\sigma$; and red shows $S_n(u)131$ and $S_p(u)131$ varied by $+1\sigma$.

Figure 49

Negative logarithm of the profile likelihood ratio vs spin-independent WIMP-nucleon cross section at a benchmark mass of $50\mathrm{GeV}/c^2$. The critical value to exclude a cross section at 90% C.L. is in red. The raw observed test statistic is the dashed line. Given the best-fit BG-only model, the median, and the central 68% and 95% ranges are shown in black, green, and yellow, respectively.