While cRT vanilla c58 was tuned using the expected surface-event leakage on calibration data, the performance of the cut can also be independently measured on two datasets — WIMP-search
Figure 7.1: Expected gamma leakage vs. recoil energy threshold for T3Z5 in Runs 125–128. Recall that the ionization yield requirement for WIMP search has a cut excluding events within the−3.5σ
electron recoil band. The effect of the variation of this requirement is also shown. Courtesy: Tobias Bruch.
multiple scatters inside the nuclear-recoil band, and WIMP-search multiple scatters outside the nuclear-recoil band. A combination of all three estimates of surface-event leakage provides the final number. Note that another possible category, WIMP-search single scatters outside the nuclear-recoil band, suffers from low statistics in addition to energy and face distribution systematic differences from WIMP-search nuclear-recoil single scatters. Thus this category is not used to produce an estimate. The following describes the procedure that David Moore and I undertook to compute this leakage estimate.
7.3.1
Premise
The basic premise of measuring surface-event leakage detector is given by the simple formula:
n=X
i
Ni ri (7.1)
wherenis the expected number of WIMP-search nuclear-recoil single scatters passing surface-event rejection, summed over all detectors, just as before; Ni is now the number of nuclear-recoil single
scatters failing surface-event rejection on detector i; ri is an estimator for the cut performance in
terms of a pass-fail ratio on that detector. This formulation statistically decouples the number of nuclear-recoil singles passing or failing surface-event rejection from the total number of WIMP-search
events. Note that using the count of WIMP-search nuclear-recoil single scatters failing surface- event rejection did not violate blinding since WIMP-search nuclear-recoil single scatters passing the surface-event rejection cut were not counted till the analysis was finished. The pass-fail ratiosri, and
thusnwere estimated using three different methods, each using one of the three datasets mentioned earlier. In practice, we estimate n as the sum of the leakage contribution from endcap detectors,
nec, and that from the interior detectors,nint.
7.3.2
Poisson and Bayesian estimates
The leakage estimates for all three methods were made in two stages. The first stage used values of
Ni extrapolated from Runs 123–124, instead of using the true count of WIMP-search nuclear-recoil
single scatters failing surface-event rejection in Runs 125–128 [194]. This enabled a check of the surface-event rejection cut, before applying the cut to WIMP-search data. Also, poisson error bars were propagated for estimates ofriandNi, leading to large uncertainties on the value ofn, because
of low statistics. No systematic errors were estimated. This was deemed sufficient for a check of surface-event rejection performance.
After the first stage, the surface-event rejection cut was applied to determine the number of nuclear-recoil single scatters failing surface-event rejection, providing the true Ni for this analysis
of Runs 125–128 (see Section 7.4). In the interest of reduced uncertainties on the leakage estimate, the second stage used the trueNi, and a Bayesian prescription developed by Jeff Filippini for Runs
123-4, the details of which are presented in Appendix C of [72]. The estimate is made by Monte Carlo simulation of event counts using∼100,000 throws. This has the advantage of marginalizing over nuisance parameters such as pass-fail ratios and reducing complications due to correlated vari- ables. The final estimate of expected leakage was taken to be the median value of the simulated distribution with a 68% credibility interval between [15.87, 84.13] percentile values. Priors for the poisson distribution governing event counts and an appropriate multinomial distribution governing detector energy and face distribution of events are picked from prior “conjugate” families to these distributions. The details of the implementation of second-stage estimates are documented in [195].
7.3.3
Method 1: Using WIMP-search multiples in the NR band
Multiple scatters inside the nuclear-recoil band should have the least systematic differences from nuclear-recoil single scatters, assuming no systematic difference between single and multiple scatter
ri. Thus testing the cut on this population provides an estimate ofrifor nuclear-recoil single scatters. ri=
bi Bi
wherebi andBi are counts of WIMP-search nuclear-recoil multiple scatters passing and failing the
surface-event rejection cut, respectively, on thei-th detector.
The first estimate ofnint yielded 1.312±0.475 events, much greater than, but within error bars
of the 0.5 event target leakage for the surface-event rejection cut. This number also disagreed with the first-stage estimate from Method 2, using WIMP-search singles outside the nuclear-recoil band. A survey of events failing timing cuts showed that these occurred at the lowest recoil energies. This prompted a retuning ofcRT vanilla c58, using revised energy binning, as described under Method 2. After the retuning, the first-stage estimate ofnintwas 0.513±0.272 and agreed with the estimate
from calibration data.
The second-stage estimate gave a final expected leakage summed over interior detectors of
nint= 0.65−+00..4629(stat.)±0.13(syst.) (7.3)
A number using this method was not computed for the endcap detectors (T3Z6 and T4Z6) because the systematic error would be too large in the absence of multiple scatters on their exterior faces. Method 1 provides smallest systematic errors, but greatest statistical errors.
7.3.4
Method 2: Using WIMP-search multiples outside the NR band
WIMP-search multiples outside the nuclear-recoil band avoid any possible systematic differences between single and multiple scatters, but have systematic differences from nuclear-recoil single scat- ters in energy and ionization yield. The systematics reweighting scheme from Equation 6.1 can be employed here to account for these. ri is thus estimated by weighting measurements of the pass-fail
ratio in detector face and energy bins in the following way:
ri= X e,f s(e,fi) m (i) e,f Me,f(i) (7.4)
wherem(e,fi) and Me,f(i) are counts of WIMP-search multiple scattersoutside the nuclear-recoil band, failing and passing the surface-event rejection cut, respectively, on detector i. s(e,fi) are bin-wise expected fractions of total WIMP-search nuclear-recoil single scatter count, estimated from WIMP- search nuclear-recoil multiple scatters for Runs 125-128. Recall, that to tune the surface-event rejection cut, we useds(e,fi) estimated from Run 123–124 WIMP-search nuclear-recoil surface events.
The bins were at first the same: Low energy (10–30 keV) charge-side, low energy (10–30 keV) phonon- side, high energy (30–100 keV) charge-side, high energy (30–100 keV) phonon-side. Note that in the absence of tagged charge-side events for endcaps1, m(endcap)
e,charge and M (endcap)
e,charge were conservatively
estimated using all singles on that endcap. s(e,fendcap) were corrected by reassigning a fraction of
untagged events as phonon- and charge-side events. The reassigned fraction was determined using the average singles-to-multiples ratio for internal detectors. The resulting count was split between the two faces using the average phonon-to-charge count ratio.
The first-stage estimate yieldednint= 0.298±0.117 andnec= 0.283±0.298. Based on disagree-
ment with the Method 1 estimate, and the observed surface-event energy distribution for multiple scatters inside the nuclear-recoil band, the energy binning in the systematics reweighting scheme was revised to be 10–20 keV, 20–30 keV and 30–100 keV [196]. This made for a total of six bins (charge-side and phonon-side for each energy bin.) The cut, cRT vanilla c58 was therefore re- optimized for target leakage of 0.5 events on calibration data using the new bins. After the retuning, the first-stage estimate of nint was 0.287±0.165 and for nec was 0.529±0.563 for endcaps. The
estimate did not change very much, but was in better agreement with Method 1. The final Method 2 leakage estimate, using the Bayesian technique, was:
nint= 0.72−+00..5625(stat.)±0.18(syst.) (7.5) nec= 0.29−+00..3314(stat.)±0.05(syst.) (7.6)
Method 2 provides larger systematic errors than Method 1, but smaller statistical errors.
7.3.5
Method 3: Using calibration data
Method 3 uses exactly the same formulation as Method 2, except that m(e,fi) and Me,f(i) are counted for calibration data inside and outside the nuclear-recoil band. Using the revised six-bin systematic reweighting scheme, the first estimate of leakage wasnint= 0.934±0.115 andnec= 0.205±0.098,
higher than the original 0.5 target. Following retuning of the surface-event rejection cut, these estimates werenint= 0.448±0.094 andnec= 0.057±0.098. The final Method 3 leakage estimate,
using the Bayesian technique, was:
nint= 0.69−+00..1210(stat.)±0.19(syst.) (7.7) nec= 0.14−+00..0604(stat.)±0.06(syst.) (7.8)
7.3.6
Combined estimate
The correlated estimates from all three methods were combined in the Bayesian Monte Carlo simu- lation to provide the following combined leakage estimate:
nint= 0.66+0−0..1109(stat.) +0.16 −0.15(syst.) (7.9) nec= 0.16+0−0..0604(stat.) +0.05 −0.05(syst.) (7.10) ntot= 0.82+0−0..1210(stat.) +0.20 −0.19(syst.) (7.11)
The distributions obtained from the simulation fornint,nec, andntot are plotted in Figure 7.2.
Figure 7.2: Surface-event leakage posterior distributions obtained from a Bayesian Monte Carlo simulation for Runs 125–128. Courtesy: David Moore.
7.3.7
Systematic errors
Systematic errors quoted above were computed separately from statistical ones, using the Bayesian technique. The following sources of systematic errors were accounted for:
1. Single scatters vs. multiple scatters: The computations for surface-event leakage in Method 1 assume no difference in the pass-fail ratios of the two. In the absence of suffi- cient statistics in the nuclear-recoil band to quantify a possible systematic difference, we use all surface events. In Runs 123–124, the ratio of pass-fail ratios between single and multiple scatters was measured to be∼0.8, whereas in Runs 125–128, it was measured to be∼1.24. Thus a 20% systematic error is assigned.
2. Choice of prior: The values ofni turn out to be sensitive to the choice of prior in some cases.
Using 20,000 mock datasets, the prior parameters are picked to approximately minimize bias and ensure good coverage around the “true” values ofniin these mock sets. A 3–6% systematic
error is assigned to the chosen prior based on the spread in the optimal parameters for different “true” leakages.
3. Endcap systematics: The reassignment of untagged events to charge-side and phonon-side to get corrects(e,chargeendcap)is based on the average singles-to-multiples ratio for internal detectors.
The systematic error assigned to this is the standard deviation in computing this average and is ∼5%. Also, the counts formendcape,f andM
endcap
e,f included all available untagged events. If
the same correction used fors(e,chargeendcap)is used here too, the systematic error is under a percent.
4. Errors in reweighting for low energy events: After unblinding, we discovered a possible residual systematic error due to lack of events with very low charge energies in the calibration sample. This is better explained after introducing the problem in Section 7.5.3. For now, I just quote the additional expected leakage (already accounted for in the combined estimate above): 0.19 and 0.04 events for interior and endcap detectors, respectively, for both Methods 1 and 2, and 0.23 and 0.06 events for interior and endcap detectors, respectively, for Method 3. A 50% systematic error is assigned to this component of leakage to account for possible errors in its estimation.