Likelihood approach to the first dark matter results from XENON100

Many experiments that aim at the direct detection of dark matter are able to distinguish a dominant background from the expected feeble signals, based on some measured discrimination parameter. We develop a statistical model for such experiments using the profile likelihood ratio as a test statistic in a frequentist approach. We take data from calibrations as control measurements for signal and background, and the method allows the inclusion of data from Monte Carlo simulations. Systematic detector uncertainties, such as uncertainties in the energy scale, as well as astrophysical uncertainties, are included in the model. The statistical model can be used to either set an exclusion limit or to quantify a discovery claim, and the results are derived with the proper treatment of statistical and systematic uncertainties. We apply the model to the first data release of the XENON100 experiment, which allows one to extract additional information from the data, and place stronger limits on the spin-independent elastic weakly interacting massive particles nucleon scattering cross section. In particular, we derive a single limit, including all relevant systematic uncertainties, with a minimum of $2.4\times {10}^{-44}{\mathrm{cm}}^2$ for weakly interacting massive particles with a mass of $50\mathrm{GeV}/c^2$.

Figure 1

All direct measurements of $L_{\mathrm{eff}}$ [24, 25, 26, 27, 28, 29, 30] together with a Gaussian fit (solid blue line) and the uncertainty band indicating the $\pm 1$ and $\pm 2\sigma$ regions (shaded dark and light, respectively).

Figure 2

Expected number of signal events $N_s$ as a function of the $L_{\mathrm{eff}}$ parametrization $t$, for various WIMP masses $m_{\chi }$ as indicated in the figure, and for a cross section $\sigma ={10}^{-39}{\mathrm{cm}}^2$. The lower the WIMP mass $m_{\chi }$, the stronger the systematic uncertainty from $L_{\mathrm{eff}}$.

Figure 3

The normalized WIMP recoil spectrum $f_s(S1)$ for $m_{\chi }=5\mathrm{GeV}/c^2$ and three possible parametrizations of $L_{\mathrm{eff}}$ through the parameter $t$ [Eq. (11)]. Even for low WIMP masses, the spectral shape hardly changes.

Figure 4

Scatter plot of electronic recoils from a ${}^{60}\mathrm{Co}$ source (blue circles) and nuclear recoils from an ${}^{241}\mathrm{AmBe}$ source (red dots). Superimposed are the border lines of the bands $j$ (thin lines) as well as the median of the electronic (upper thick line) and nuclear (lower thick line) recoil bands in the relevant $S1$ parameter range (dashed vertical lines).

Figure 5

The probability density functions $f(q_{\sigma }|H_{\sigma })$ (thick red histogram) and $f(q_{\sigma }|H_0)$ (thin blue histogram) for a WIMP with $m_{\chi }=5\mathrm{GeV}/c^2$ and $\sigma ={\sigma }^{\mathrm{up}}$. One can clearly see the distribution is well approximated by a chi-square distribution (dashed red line). The median value of $f(q_{\sigma }|H_0)$ is indicated together with the test statistic $q_{\sigma }^{\mathrm{obs}}$ observed in data.

Figure 6

Parameter space of spin-independent elastic WIMP-nucleon cross section $\sigma$ as a function of WIMP mass $m_{\chi }$. The sensitivity for the data set analyzed here is shown as light and dark (blue) shaded areas at $1\sigma$ and $2\sigma$ C.L., respectively. The actual limit at 90% C.L., taking into account all relevant systematic uncertainties as derived with the profile likelihood method, is shown as the thick (black) line. Two limits from the same data set, derived for two assumptions of the behavior of $L_{\mathrm{eff}}$, are shown as dotted lines [8]. A limit from CDMS [4] is shown as a thin (orange) line, recalculated assuming an escape velocity of $544\mathrm{km}/\mathrm{s}$ and $v_0=220\mathrm{km}/\mathrm{s}$. Expectations from a theoretical model [37] are shown, as well as the areas (at 90% C.L.) favored by CoGeNT (green) [38] and DAMA (red, without channeling) [39].