Closed-Form Analytical Charge Response Model for Silicon Photomultipliers with Recursive Correlated Avalanches

Silicon photomultipliers (SiPMs) have become the preferred photodetectors in next-generation neutrino experiments, yet no unified closed-form analytical expression free of truncation and numerical convolution has been established for their full charge response spectrum, which must simultaneously capture correlated cross-talk and afterpulsing effects absent in conventional photomultiplier tubes (PMTs). We present a unified closed-form model for the SiPM charge response within the characteristic-function framework, treating pedestal noise, single-electron-response (SER) charge, internal optical cross-talk, and afterpulsing on equal footing. The characteristic-function representation factorises the full charge spectrum into three independent physical components: pedestal, single-electron response (SER), and avalanche count statistics. Prompt internal optical cross-talk is modelled as a Galton-Watson branching process with Poisson offspring; building on the Generalised Poisson count statistics identified by Vinogradov, we derive a Lambert $W$ closed form for the total-progeny PGF via Lagrange-B\"{u}rmann inversion, providing the analytical handle needed for efficient event-level reconstruction. Afterpulsing is modelled as a per-avalanche geometric chain, derived as the maximum-entropy Poisson-Gamma mixture: the exponential prior-maximum-entropy for a positive continuous yield with fixed mean-marginalised over a Poisson count yields the geometric per-avalanche distribution, whose $N$-avalanche total is Negative Binomial. This naturally encompasses the Poisson afterpulsing limit and recursive afterpulse chains while preserving analytical closure. The resulting eight-parameter expression is further applied to derive an explicit per-channel charge-time likelihood for event-level energy reconstruction without numerical convolution at inference time.