Ratio-Based Pulse Shape Discrimination: Analytic Results for Gaussian and Poisson Noise Models

In experiments in a range of fields including fast neutron spectroscopy and astroparticle physics, one can discriminate events of interest from background events based on the shapes of electronic pulses produced by energy deposits in a detector. Here, I focus on a well-known pulse shape discrimination method based on the ratio of the temporal integral of the pulse over an early interval X_p and the temporal integral over the entire pulse X_t. For both event classes, for both a Gaussian noise model and a Poisson noise model, I present analytic expressions for the conditional distribution of X_p given knowledge of the observed value of X_t and a scaled energy deposit corresponding to the product of the full energy deposit and a relative yield factor. I assume that the energy-dependent theoretical prompt fraction for both classes are known exactly. With a Bayesian approach that accounts for imperfect knowledge of the scaled energy deposit, I determine the posterior mean background acceptance probability given the target signal acceptance probability as a function of the observed value of X_t. My method enables one to determine receiver-operating-characteristic curves by numerical integration rather than by Monte Carlo simulation for these two noise models.