Prompt Decay: Mechanisms and Signatures

• Prompt decay is defined as a rapid transition from an excited, active state to a quiescent or absorbed state, observable in systems such as GRBs, particle decays, and stochastic networks.
• Scaling relations—like energy-dependent pulse widths and lag–luminosity correlations—provide quantitative insights that constrain physical models and diagnostic interpretations.
• Analytical methods, including high latitude emission models and eigenvalue analyses in queueing networks, offer refined, practical tools for understanding underlying decay mechanisms.

Prompt decay describes rapid diminution or transformation processes associated with a variety of physical, probabilistic, or quantum systems, typically denoting the swift transition from an initial active or excited state to a quiescent or absorbed state, observable across gamma-ray bursts (GRBs), particle physics (especially heavy-flavor meson dynamics), atmospheric neutrino production, and stochastic networks. Its characterization hinges on temporal profiles, correlation properties, invariant or quasi-stationary distributions, and underlying physical or stochastic mechanisms driving the fast decline.

1. Temporal and Spectral Signatures of Prompt Decay in Astrophysical Transients

In gamma-ray bursts, prompt decay refers to the rapid decline of flux following the main emission episode. Detailed analysis of X-ray flares in GRBs reveals that high-energy flare profiles exhibit faster rise and decay, peaking before their low-energy counterparts; the observed pulse width ($w$) and its temporal derivatives scale with energy as $w \propto E^{-0.32\pm0.01}$, $t_r \propto E^{-0.37\pm0.01}$, and $t_d \propto E^{-0.29\pm0.01}$ (Margutti et al., 2010). These relations mirror the properties of prompt gamma-ray pulses but with a key distinction: flare parameters evolve with time, becoming wider and exhibiting larger peak lag as peak luminosity and spectral hardness decline.

Central to the prompt decay phenomenology is the flare’s spectral peak energy ($E_{p,i}$), which exponentially decays in synchrony with flux, characterized by e-folding times on the order of 20–29 seconds after the peak (Margutti et al., 2010). This decay reflects an internal engine-driven mechanism and is quantitatively tracked via parameterizations of the pulse model: $$I(t) = A \lambda \exp\left[-\frac{\tau_1}{t-t_s} - \frac{t-t_s}{\tau_2}\right], \quad t > t_s$$ where $t_s$ is the pulse start time, $\tau_1$ and $\tau_2$ are rise/decay parameters, and the derived pulse properties become tightly correlated with spectral behaviors.

2. Lag–Luminosity Relations and Statistical Constraints

The lag–luminosity relation, a haLLMark scaling law, reveals that the peak isotropic luminosity of flares ($L_{p,\mathrm{iso}}^{0.3-10\,\text{keV}}$) scales inversely with their peak lag as $$L_{p,\mathrm{iso}} \propto t_{\mathrm{lag}}^{-0.95\pm0.23}$$, persisting across five decades in time and energy (Margutti et al., 2010). This relation consistently binds both prompt pulses and flares, and is expressed with normalization as: $L_{p,\mathrm{iso}}^{(0.3-10\,\text{keV})} = 10^{50.82\pm0.20} \cdot (t_{\mathrm{lag}})\,^{(-0.95\pm0.23)}$ Time-resolved analyses within individual flares yield $E_{p,i}(t) \propto L_{\mathrm{iso}}(t)^{1.0\pm0.2}$, further reducing the number of independent observables and providing stringent constraints on physical models for prompt decay.

3. Mechanistic Interpretations: GRB Central Engine, High Latitude Emission, and Model Diagnostics

Prompt decay in the context of GRB afterglows is often explained via high latitude emission (HLE), where the observed steep decline results from off-axis photons arriving after cessation of on-axis dissipation. The model-specific HLE timescale is: $\tau_{\mathrm{HLE}} \simeq R_\gamma/(2\Gamma^2 c)$ where $R_\gamma$ is the radius at which prompt emission ends, and $\Gamma$ the bulk Lorentz factor. Correct matching to the burst duration ($t_{\mathrm{burst}}$) dictates $$R_\gamma \simeq 6 \times 10^{15} (t_{\mathrm{burst}}/10\,\text{s})(\Gamma/100)^2$$ cm (Hascoët et al., 2012). This requirement is naturally satisfied only for internal shock models, wherein the last internal shocks occur at radii proportional to the longest observed variability timescale, aligning $\tau_{\mathrm{HLE}}$ with $t_{\mathrm{burst}}$ and reproducing observed steep decay slopes ($\alpha\approx3$). Contrastingly, in photospheric models ($R_{\text{ph}} \lesssim 10^{13}$ cm), $\tau_{\mathrm{HLE}}$ is too short, requiring instead a universal central engine shut off. Magnetic reconnection models require additional assumptions about dissipation scales for compatibility (Hascoët et al., 2012). Thus, prompt decay becomes a diagnostic of emission site physics and engine behavior.

4. Prompt Decay in Particle and Atmospheric Physics

In high-energy atmospheric lepton physics, prompt decay refers to the rapid transformation of charm hadrons into muons and neutrinos immediately after their production in cosmic-ray interactions (Gaisser, 2013). Unlike conventional mesons (pion/kaon), which may re-interact before decaying—leading to a softer, anisotropic neutrino spectrum—charm hadrons decay before reinteraction, producing a “harder” (less steep) and nearly isotropic prompt component: $$\phi_\nu(E_\nu) = \phi_N(E_\nu)\left\{ \frac{A_{\pi\nu}}{1+B_{\pi\nu}\cos\theta (E_\nu/\epsilon_\pi)} + \frac{A_{K\nu}}{1+B_{K\nu}\cos\theta (E_\nu/\epsilon_K)} + \frac{A_{\text{charm},\nu}}{1+B_{\text{charm},\nu}\cos\theta (E_\nu/\epsilon_{\text{charm}})} \right\}$$ Critical energy for charm ($\epsilon_{\text{charm}} \sim 3\times10^7$ GeV) ensures prompt decay dominates the background at high energies, crucially affecting neutrino observatories and searches for extra-galactic sources. The charm-to-pion production ratio ($R_c$) parameterizes the relative yield (typically $R_c \lesssim 10^{-3}$).

5. Prompt Decay in Heavy-Flavor Hadron Physics

Prompt decay is also pivotal in decoupling direct meson production from secondary sources. In proton-proton collisions, “prompt” refers to mesons directly produced in hadronization, while “non-prompt” denotes decay products from longer-lived parent hadrons. For instance, in D*⁺ vector meson studies (Collaboration, 2022), prompt D*⁺ mesons exhibit no net spin alignment ($\rho_{00} \simeq 1/3$), whereas non-prompt D*⁺ from beauty hadron decays display enhanced alignment ($$\rho_{00} = 0.455 \pm 0.022\,\mathrm{(stat.)} \pm 0.035\,\mathrm{(syst.)}$$), consistent with helicity conservation in weak decays. The decay angular distribution is modeled as: $$\frac{dN}{d\cos\theta^*} \propto 1 - \rho_{00} + (3\rho_{00} - 1)\cos^2\theta^*$$ where $\theta^*$ is the decay angle with respect to the helicity axis. Disentangling prompt and non-prompt components, particularly via decay vertex displacement and multivariate techniques (e.g., Boosted Decision Trees), underpins the physical interpretation of observed angular momentum transfer and polarization phenomena.

In the case of J/ψ meson production (Prasad et al., 2023), prompt decays are distinguished from non-prompt ones by measuring the pseudoproper decay length,

$$c\tau = \frac{c\,m_{\rm J/\psi}\,(\vec{L}\cdot\vec{p_T})}{|\vec{p_T}|^2}$$

where $L$ connects primary and decay vertices. Machine learning models (XGBoost, LightGBM) achieve up to 99% separation accuracy on simulated pp collision data, facilitating fine-grained studies of QCD production mechanisms.

6. Stochastic Networks and the Analytical Framework for Prompt Decay

Prompt decay in queueing and stochastic networks is analytically characterized by decay parameters governing the exponential rate of transition probability loss in the transient regime (Li et al., 28 Apr 2024). In stopped $M^X/M/1$-queueing networks, the decay parameter $\lambda_\mathcal{C}$ measures the asymptotic exponential decline: $$\frac{1}{t} \log p_{ij}(t) \to -\lambda_\mathcal{C}, \quad t\to\infty$$ Computed via a generating-function approach, the parameter $\lambda_\mathcal{C}$ is associated with the maximal eigenvalue $\rho(q)$ of an operator derived from the network’s node functions, so that $\lambda = -\rho(q)$ with $q$ solving $B(x) + \lambda x = 0$. Invariant and quasi-stationary distributions, defined through

$\sum_j m_j q_{ij} = -\lambda_\mathcal{C} m_i$

capture the stationary behavior before absorption. This analytical structure is generalizable to epidemic models, ecological systems, or other Markovian dynamics where prompt decay is a salient feature.

7. Broader Implications and Cross-Disciplinary Significance

Prompt decay serves as a diagnostic tool for physical mechanisms, whether assessing GRB central engine behavior, determining emission site models, probing QCD subprocesses in particle physics, or analyzing spectral gaps and stability in stochastic systems. Its manifestations—temporal steep falls, scaling relations, and polarization signatures—yield critical constraints on model validity, background estimation in detector physics, and understanding of non-equilibrium processes. The analytical techniques developed for queueing networks, especially those leveraging generating functions and eigenvalue analysis, provide templates for addressing prompt decay in a broad spectrum of systems, enhancing both interpretive power and predictive accuracy across domains.

In sum, prompt decay unifies a set of phenomena characterized by rapid, mechanism-dependent transitions, captured by tight scaling relations, statistical dependencies, and analytically tractable parameters, and remains central to contemporary investigations in astrophysics, high-energy particle physics, atmospheric science, and applied stochastic modeling.