Calculating exclusion limits for weakly interacting massive particle direct detection experiments without background subtraction

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Abstract
Competitive limits on the weakly interacting massive particle (WIMP) spin-independent scattering cross section are currently being produced by Ge76 detectors originally designed to search for neutrinoless double beta decay, such as the Heidelberg-Moscow and IGEX experiments. In the absence of background subtraction, limits on the WIMP interaction cross section are set by calculating the upper confidence limit on the theoretical event rate, given the observed event rate. The standard analysis technique involves calculating the 90% upper confidence limit on the number of events in each bin, and excluding any set of parameters (WIMP mass and cross-section) which produces a theoretical event rate for any bin which exceeds the 90% upper confidence limit on the event rate for that bin. We show that, if there is more than one energy bin, this produces exclusion limits that are actually at a lower degree of confidence than 90%, and are hence erroneously tight. We formulate criteria which produce true 90% confidence exclusion limits in these circumstances, including calculating the individual bin confidence limit for which the overall probability that no bins exceeds this confidence limit is 90% and calculating the 90% minimum confidence limit on the number of bins which exceed their individual bin 90% confidence limits. We then compare the limits on the WIMP cross-section produced by these criteria with those found using the standard technique, using data from the Heidelberg-Moscow and IGEX experiments