Pulse-shape discrimination against low-energy Ar-39 beta decays in liquid argon with 4.5 tonne-years of DEAP-3600 data

The DEAP-3600 detector searches for the scintillation signal from dark matter particles scattering on a 3.3 tonne liquid argon target. The largest background comes from ${}^{39}$Ar beta decays and is suppressed using pulse-shape discrimination (PSD). We use two types of PSD estimator: the prompt-fraction, which considers the fraction of the scintillation signal in a narrow and a wide time window around the event peak, and the log-likelihood-ratio, which compares the observed photon arrival times to a signal and a background model. We furthermore use two algorithms to determine the number of photons detected at a given time: (1) simply dividing the charge of each PMT pulse by the mean single-photoelectron charge, and (2) a likelihood analysis that considers the probability to detect a certain number of photons at a given time, based on a model for the scintillation pulse shape and for afterpulsing in the light detectors. The prompt-fraction performs approximately as well as the log-likelihood-ratio PSD algorithm if the photon detection times are not biased by detector effects. We explain this result using a model for the information carried by scintillation photons as a function of the time when they are detected.