Exact time-correlated coincidence modeling with reset boundaries

Abstract

Delayed-coincidence searches identify a rare prompt-delayed signal, but their accidental background is not a simple product of marginal rates: muon vetoes, event dead time, and delayed events created before the current window condition which event sequences can be recorded. We derive exact ordered coincidence rates, within a stated stochastic model, for a recorded stream of uncorrelated prompt-like singles and correlated prompt-delayed sources subject to Poisson reset boundaries and event dead time. The calculation separates two tasks: a Markov history chain carries the delayed events still pending at reset boundaries across windows, and a current-window propagator evaluates the ordered within-window integrals in closed form with block-matrix exponentials, the matrix-analytic toolkit of applied probability. The construction extends to any prescribed finite multiplicity by increasing the block-chain depth. Here we report every ordered one-, two-, and three-fold rate formed from uncorrelated singles, correlated prompts, and recorded delayed events, together with the genuine/accidental split of prompt-delayed pairs, multiplicity efficiencies, and the aggregate rate for multiplicity four or more. For the default window-close convention, we derive a three-term a posteriori error bound for the finite pending-population truncation. An independent streaming toy Monte Carlo validates the ordered rates, two-fold time densities, matched dead-time conventions, and aggregate high multiplicity. Within the stated assumptions, the construction is exact on the retained finite state spaces and provides explicit, computable truncation-error bounds.