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arxiv: 2605.27340 · v1 · pith:2N7LRR2Unew · submitted 2026-05-26 · ⚛️ physics.ins-det · hep-ex

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

Yiqi Liu , Xuewei Liu , Benda Xu This is my paper

Pith reviewed 2026-06-29 14:44 UTC · model grok-4.3

classification ⚛️ physics.ins-det hep-ex
keywords silicon photomultiplierSiPMcharge responseclosed-formafterpulsingoptical cross-talkcharacteristic functionnegative binomial
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The pith

A characteristic-function model yields a closed-form eight-parameter expression for the full charge response of silicon photomultipliers including recursive cross-talk and afterpulsing.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a unified closed-form analytical expression for the charge response spectrum of silicon photomultipliers. This expression accounts for pedestal noise, single-electron response, optical cross-talk modeled as a Galton-Watson branching process, and afterpulsing modeled as a negative binomial distribution from a maximum-entropy mixture. The characteristic function factors these components independently, leading to a Lambert W function for cross-talk and avoiding numerical convolutions. Such a model matters because it enables direct computation of likelihoods for energy reconstruction in neutrino experiments without simulation overhead. If correct, it unifies treatment of correlated avalanche effects absent in standard photomultiplier tubes.

Core claim

Within the characteristic-function framework the full charge spectrum factorises into pedestal, single-electron-response, and avalanche-count statistics. Prompt internal optical cross-talk is a Galton-Watson branching process with Poisson offspring whose total-progeny probability-generating function admits a Lambert W closed form via Lagrange-Bürmann inversion. Afterpulsing is a per-avalanche geometric chain obtained as the maximum-entropy Poisson-Gamma mixture, whose sum over N avalanches is negative binomial. The resulting eight-parameter expression supplies an explicit per-channel charge-time likelihood for event-level energy reconstruction.

What carries the argument

The characteristic-function factorisation of the charge spectrum together with the Lambert W closed form for the Galton-Watson total-progeny PGF and the negative-binomial representation of recursive afterpulsing.

If this is right

  • The model captures correlated cross-talk and afterpulsing on equal footing with pedestal and SER.
  • It provides analytical closure that encompasses both the Poisson afterpulsing limit and recursive chains.
  • An explicit likelihood for charge-time data is obtained without numerical convolution at inference time.
  • The eight-parameter form can be applied directly to event-level energy reconstruction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the closed-form holds, detector simulations in large neutrino experiments could replace convolution steps with direct evaluation, reducing computational cost.
  • The maximum-entropy derivation of the geometric afterpulse distribution suggests a way to test whether real SiPM data obey the same entropy-maximising statistics.
  • The branching-process treatment of cross-talk may extend to other photodetectors that exhibit similar internal optical coupling.

Load-bearing premise

The afterpulsing process is correctly described by the maximum-entropy Poisson-Gamma mixture that yields a geometric distribution for each avalanche.

What would settle it

Measurement of the charge spectrum from a SiPM illuminated by a pulsed light source at intensities where multiple avalanches and afterpulses occur; systematic deviation from the predicted eight-parameter distribution would falsify the closed-form model.

Figures

Figures reproduced from arXiv: 2605.27340 by Benda Xu, Xuewei Liu, Yiqi Liu.

Figure 1
Figure 1. Figure 1: FIG. 1. Charge spectrum and BIC-selected fit (M1: no AP, no DCR) for PCB6 at 53 V ( [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. BIC-selected fit (M3: DCR only, no AP) for PCB6 at 54 V ( [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. BIC-selected fit (M4: full model) for PCB6 at 60 V ( [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. MLE parameter correlation matrix at 53 V (forced M2 fit, 9 parameters). The dominant [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. MLE parameter correlation matrix at 60 V (M4, 10 parameters). The [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Effective gain [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Cross-talk parameter [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Afterpulse rate [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Mean afterpulse charge fraction [PITH_FULL_IMAGE:figures/full_fig_p030_10.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript claims to derive a unified closed-form analytical expression for the full charge response spectrum of silicon photomultipliers, factorizing the characteristic function into independent pedestal, single-electron-response, and avalanche-count components. Prompt cross-talk is treated as a Galton-Watson branching process with Poisson offspring, yielding a Lambert-W closed form for the total-progeny PGF via Lagrange-Bürmann inversion. Afterpulsing is modeled as a per-avalanche geometric distribution obtained from the maximum-entropy Poisson-Gamma mixture, whose sum over N avalanches is negative binomial; the resulting eight-parameter model supplies an explicit per-channel charge-time likelihood for event-level reconstruction without numerical convolution.

Significance. A validated closed-form likelihood free of truncation or convolution would enable faster and more precise event reconstruction in high-statistics neutrino detectors that rely on SiPM arrays. The factorization into physically motivated components and the explicit handling of recursive correlated avalanches represent a technical advance over existing numerical or truncated models, provided the maximum-entropy afterpulsing assumption and the branching-process inversion are shown to be both mathematically consistent and physically realistic.

major comments (3)
  1. [Afterpulsing model] Afterpulsing section: the manuscript states that the geometric per-avalanche distribution follows directly from the maximum-entropy Poisson-Gamma mixture with fixed mean, yet provides no derivation or comparison showing that the physical trap-release mechanism produces exactly this distribution (rather than a time-dependent or non-geometric form). Because this assumption supplies the negative-binomial closure and the independent factorization of the characteristic function, its justification is load-bearing for the central claim of an eight-parameter closed-form likelihood.
  2. [Cross-talk model] Cross-talk derivation, Lagrange-Bürmann step: the claim of a Lambert-W closed form for the total-progeny PGF rests on the application of the inversion theorem to the Galton-Watson process with Poisson offspring; the manuscript must exhibit the explicit functional equation and the resulting expression to confirm that no hidden truncation or auxiliary approximation is introduced beyond Vinogradov’s generalised Poisson statistics.
  3. [Overall model] Parameter count and identifiability: the abstract asserts an eight-parameter expression, but the manuscript does not list the eight parameters explicitly or demonstrate that they remain independent once the maximum-entropy and branching-process constraints are imposed; this affects whether the model is truly parameter-efficient for likelihood-based reconstruction.
minor comments (2)
  1. [Introduction] Notation for the characteristic function should be introduced once with a clear definition of its argument (charge or charge-time) to avoid ambiguity when the time component is later adjoined.
  2. [Abstract vs. body] The abstract refers to “recursive correlated avalanches” but the body should explicitly map this phrase to the combination of the branching-process cross-talk and the geometric afterpulsing chains.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and for highlighting the load-bearing aspects of the afterpulsing and cross-talk derivations as well as the parameter enumeration. We address each major comment below with clarifications and commit to targeted revisions that strengthen the manuscript without altering its core claims.

read point-by-point responses

  1. Referee: [Afterpulsing model] Afterpulsing section: the manuscript states that the geometric per-avalanche distribution follows directly from the maximum-entropy Poisson-Gamma mixture with fixed mean, yet provides no derivation or comparison showing that the physical trap-release mechanism produces exactly this distribution (rather than a time-dependent or non-geometric form). Because this assumption supplies the negative-binomial closure and the independent factorization of the characteristic function, its justification is load-bearing for the central claim of an eight-parameter closed-form likelihood.

    Authors: We will insert a dedicated subsection deriving the geometric distribution step-by-step from the maximum-entropy principle applied to a Poisson-Gamma mixture with fixed mean (exponential prior on yield, marginalisation over Poisson count). This supplies the required mathematical closure. On the physical side, the maximum-entropy choice is presented as a principled approximation consistent with the observed afterpulsing statistics in the literature; we will add a short paragraph noting that a full microscopic trap-release simulation lies outside the present scope while the resulting negative-binomial form reproduces the measured spectra shown in the manuscript. revision: yes

  2. Referee: [Cross-talk model] Cross-talk derivation, Lagrange-Bürmann step: the claim of a Lambert-W closed form for the total-progeny PGF rests on the application of the inversion theorem to the Galton-Watson process with Poisson offspring; the manuscript must exhibit the explicit functional equation and the resulting expression to confirm that no hidden truncation or auxiliary approximation is introduced beyond Vinogradov’s generalised Poisson statistics.

    Authors: We will expand the cross-talk section to state the probability-generating-function functional equation for the Galton-Watson branching process with Poisson offspring, followed by the explicit application of the Lagrange-Bürmann inversion formula that yields the Lambert-W expression. This will make clear that the closed form is exact under the stated Poisson-offspring assumption and introduces no truncation beyond the Vinogradov generalised-Poisson framework already cited. revision: yes

  3. Referee: [Overall model] Parameter count and identifiability: the abstract asserts an eight-parameter expression, but the manuscript does not list the eight parameters explicitly or demonstrate that they remain independent once the maximum-entropy and branching-process constraints are imposed; this affects whether the model is truly parameter-efficient for likelihood-based reconstruction.

    Authors: We will add an explicit table or enumerated list of the eight parameters together with their physical interpretations (pedestal offset and width, SER gain and width, prompt cross-talk probability, afterpulsing probability per avalanche, negative-binomial dispersion parameter, and overall scaling). A short paragraph will argue that the characteristic-function factorisation keeps the parameters independent because each controls a distinct physical component; the constraints from maximum-entropy and branching-process derivations do not introduce additional functional relations among them. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation follows from explicit modeling assumptions without reduction to fits or self-citations

full rationale

The paper constructs its eight-parameter closed-form charge response via characteristic-function factorization into pedestal, SER, and avalanche components. Cross-talk uses a Galton-Watson branching process with Poisson offspring, yielding a Lambert-W PGF via Lagrange-Bürmann inversion (building on Vinogradov's external generalized Poisson). Afterpulsing is explicitly chosen as the maximum-entropy Poisson-Gamma mixture producing geometric per-avalanche and negative-binomial totals. These are presented as modeling choices with free parameters, not as outputs of the derivation itself. No equation reduces a claimed prediction back to a fitted input by construction, and no load-bearing step relies on self-citation chains. The central claim of an explicit likelihood without convolution holds under the stated statistical assumptions; any deviation is a correctness issue, not circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the Galton-Watson branching model for prompt cross-talk, the maximum-entropy derivation of the geometric afterpulsing distribution, and the assumption that the three physical components (pedestal, SER, avalanche statistics) factorize exactly in the characteristic-function domain. Eight free parameters are introduced to match data but are not enumerated.

free parameters (1)
  • eight unspecified parameters
    The final expression is described as eight-parameter; these must be fitted to data and are not derived from first principles.
axioms (2)
  • domain assumption Prompt internal optical cross-talk follows a Galton-Watson branching process with Poisson offspring distribution.
    Invoked to obtain the total-progeny PGF via Lagrange-Bürmann inversion.
  • domain assumption Afterpulsing follows the maximum-entropy Poisson-Gamma mixture yielding a geometric per-avalanche distribution.
    Used to obtain the negative-binomial total for N avalanches.

pith-pipeline@v0.9.1-grok · 5842 in / 1547 out tokens · 32752 ms · 2026-06-29T14:44:23.738541+00:00 · methodology

discussion (0)

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