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

## TL;DR

This paper presents analytic methods for pulse shape discrimination in experiments, using Gaussian and Poisson noise models to improve signal-background separation.

## Contribution

The paper provides novel analytic expressions for pulse shape discrimination under Gaussian and Poisson noise models using a Bayesian approach.

## Key findings

- Conditional distributions of pulse integrals are derived for both Gaussian and Poisson noise models.
- A Bayesian method is introduced to calculate posterior mean background acceptance probabilities.
- The method allows for determining ROC curves via numerical integration instead of Monte Carlo simulations.

## Abstract

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 Xp and the temporal integral over the entire pulse Xt.
For both event classes, for both a Gaussian noise model and a Poisson noise model, I present
analytic expressions for the conditional distribution of Xp given knowledge of the
observed value of Xt 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 Xt. My method enables one to determine
receiver-operating-characteristic curves by numerical integration rather than by Monte Carlo
simulation for these two noise models.