cases/case24/task.md

# When does a weak-field hierarchy faithfully describe assisted pair production?

## Background and central research question

The momentum spectrum of pairs created from the vacuum by a slowly varying
strong field may be modified by a much weaker oscillating field.
A total spectrum alone does not establish whether a finite expansion in the
weak field is quantitatively reliable or what physical processes its terms
represent.

Investigate this problem with a probability-level hierarchical quantum-
kinetic treatment. Determine whether the hierarchy gives a faithful and
physically interpretable account of dynamically assisted pair production in
the two field regimes specified below. Use your calculated asymptotic spectra
to establish how the strong and weak fields combine, whether interference is
material, and what truncation and physical scope your evidence supports. Do
not assume the outcome in advance.

Design the task-specific derivation, computation, evidence and scientific
argument needed to answer this one research question. The organization of the
argument and the number and form of the numerical outputs are yours.

## Physical system and indispensable inputs

Use natural units, $\hbar=c=m=1$, with $E_{\rm cr}=m^2/e$. Work in temporal
gauge with a spatially homogeneous, linearly polarized vector potential

$$
A_{\rm tot}(t)={\cal A}(t)+A^{\gamma}(t),
$$

where

$$
{\cal A}(t)={\cal E}E_{\rm cr}\,t\exp(-t^2/T^2),
$$

and

$$
A^{\gamma}(t)=\frac{\epsilon E_{\rm cr}}{\Omega}
\exp(-t^2/\tau^2)\sin(\Omega t).
$$

The electric field is $E(t)=-\dot A_{\rm tot}(t)$. Neglect spatial gradients
and backreaction, and begin in the free vacuum before either pulse. Evaluate
the asymptotic field-induced particle-number spectrum as a function of
$p_\parallel$ at $p_\perp=0$.

Investigate both physical instances:

$$
\begin{array}{c|ccccc}
 & {\cal E} & \epsilon & \Omega/m & mT & m\tau \\
\hline
\mathrm{I} & 0.30 & 0.05 & 1.0 & 20 & 20 \\
\mathrm{II} & 0.15 & 0.02 & 0.70 & 40 & 40
\end{array}
$$

Develop the hierarchy as an expansion in the weak-field nonlinearity

$$
\xi=\frac{e\epsilon E_{\rm cr}}{m\Omega},
\qquad \gamma=\xi^{-1}>1,
$$

while retaining the response to ${\cal A}(t)$ nonperturbatively. The expansion
is at the level of the Wigner/particle distribution. Use a standard spatially homogeneous
equal-time Wigner or algebraically equivalent quantum-kinetic formulation.
Derivation of the task-specific hierarchy, its truncation and the evidentiary
strategy needed to support your claim remain part of the research work.

State the continuum or phase-space normalization used for the calculated
particle-number spectrum and apply it consistently to the total and every
order-resolved contribution. Preserve the underlying unscaled numerical data
whenever a display rescaling is used.

## Numerical freedom and scope

The time and momentum domains, hierarchy depth, solver, grid, step size,
tolerances, reliability strategy, diagnostics and evidence representation are
student choices. Define enough conventions to make the calculation
reconstructible. No fixed plot, momentum range, panel sequence, display floor
or author-specific numerical setting is prescribed.

Conclusions must remain within the spatially homogeneous, prescribed-field,
vacuum-initialized and weak-oscillating-field hierarchy stated here. Do not
infer backreaction, spatial focusing, experimental observability or behavior
outside the regime supported by your calculation.

## Submission and resources

Submit one self-contained scientific report and an auditable offline
calculation package containing producing code, reconstructible numerical data
for material claims, a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge, and concise run and environment
information. Evidence must be sufficiently reliable for the claims made, but
no particular validation procedure is required.

Resource limits are supplied exclusively by the accompanying
`resource_limits.json`. Its fixed CPU allocation, candidate-active
wall time, network boundary and temporary-storage allowance are binding.
No separate aggregate CPU-hour limit applies.

## Reference and integrity policy

This examination is conducted without network access or live literature
retrieval. You may rely on your existing general scientific knowledge and, if any are
explicitly supplied with the examination, on those offline materials or
software documentation. Record every such source actually consulted. For formulas,
methods, constants, or conventions used from general knowledge rather than a
consulted source, state enough of the method basis, definition, value, units,
and convention to make the calculation auditable; do not invent a citation.
You are not required to identify or prove a publication date or to retrieve
literature during the examination.

The hidden target article and its identifying metadata, figures or captions,
supplementary material, author code or data, target-specific reproduction
code, solution or scoring material, online solutions, and worked
reproductions of this case must not be used. All submitted scientific figures
and numerical data must be generated by your own implementation of the stated
problem.

See [submission requirements](../../submission_requirements.md) for the common submission entry points.