This paper presents a search for standard model Higgs boson production in association with a $W$ boson using events recorded by the CDF experiment in a dataset corresponding to an integrated luminosity of 5.6 fb-1. The search is performed using a matrix element technique in which the signal and background hypotheses are used to create a powerful discriminator. The discriminant output distributions for signal and background are fit to the observed events using a binned likelihood approach to search for the Higgs boson signal. We find no evidence for a Higgs boson, and 95% confidence level (C.L.) upper limits are set on the Higgs boson production rate. The observed limits range from 3.5 to 37.6 relative to the standard model expectation for Higgs boson masses between 100 and 150 GeV. The 95% C.L. expected limit is estimated from the median of an ensemble of simulated experiments and varies between 2.9 and 32.7 relative to the production rate predicted by the standard model over the Higgs boson mass range studied.
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