= Photon-recoil lag correlation
{title2=$\ell=t_\gamma-t_e$}
A <photon-recoil lag correlation> asks how often a <photon> record occurs at a specified time displacement from an <Electron> record. Let $r_e,r_\gamma$ be observed separate-channel rates, $S$ the rate of detected genuine pairs before a timing cut, and $f(\ell)$ their normalized timing <probability density function>. For an original dilute, stationary model with observation time $T$, neglecting end effects and <detector dead time>, the expected pair density is
$$
\frac{d\mathbb E[N]}{d\ell}
=T\bigl[r_er_\gamma+Sf(\ell)\bigr],\qquad
\int_{-\infty}^{\infty} f(\ell)\,d\ell=1.
$$
This counts all pairs of records, rather than at most one <detector trigger> per <Electron>. It uses independent unrelated detections for the flat term and an added same-event contribution. A <normal distribution> for timing spread is a chosen approximation, not a claim about the historical instrument. Intrinsic emission delay, counter response and readout uncertainty can all affect the observed <probability density function>.
Define the accepted lag interval by $|\ell|<w/2$: \b[$w$ is its full width]. Integrating the flat baseline gives the existing <accidental coincidence> expectation $Tr_er_\gamma w$. The additional area is $TS\int_{-w/2}^{w/2}f(\ell)\,d\ell$. Moving a window of the same full width away from the peak tests the baseline under the model's stationarity assumption. It does not remove correlated instrumental backgrounds or changing rates automatically.
\Image[/photon-recoil-joint-lag-correlation.png]
{title=Lag-dependent pairs and a <joint probability distribution> carry information absent from singles totals}
{id=image-photon-recoil-joint-lag-correlation}
{description=Original chosen models, not historical counts or fitted data. Left: a normalized <normal distribution> of genuine-pair lags with <standard deviation> $1\,\mathrm{ms}$ is added to an independent-pair baseline. Shaded intervals have full width $2\,\mathrm{ms}$, centered at zero and $5\,\mathrm{ms}$. Right: a separate exact ideal-bin model ignores timing spread; the two populations have identical singles expectations but different joint expectations.}
For the lag model, choose $T=1000\,\mathrm s$, $r_e=2\,\mathrm{s^{-1}}$, $r_\gamma=0.5\,\mathrm{s^{-1}}$, $S=0.1\,\mathrm{s^{-1}}$ and timing <standard deviation> $\sigma=1\,\mathrm{ms}$. A full width $w=2\,\mathrm{ms}$ gives $2$ expected <accidental coincidences> and $100\operatorname{erf}(1/\sqrt2)\simeq68.27$ accepted genuine pairs, using the <error function>. The total is approximately $70.27$, whereas the displaced interval from $4$ to $6\,\mathrm{ms}$ has approximately $2.003$ expected pairs. A finite sample fluctuates around these expectations; these values are not measured counts or a significance calculation.
The right panel isolates the information lost by retaining singles totals alone. In a separate idealized model, stipulate $M=10^6$ independent $1\,\mathrm{ms}$ bins, with <Bernoulli distribution> indicators $X,Y$ for <Electron> and <photon> detection in a bin. Set $p_e=\Pr(X=1)=0.002$ and $p_\gamma=\Pr(Y=1)=0.0005$. <Independence> predicts $\Pr(X=1,Y=1)=p_ep_\gamma=10^{-6}$, hence one expected double detection. Alternatively choose
$$
\Pr(X=1,Y=1)=0.000101,\quad
\Pr(X=1,Y=0)=0.001899,\quad
\Pr(X=0,Y=1)=0.000399,\quad
\Pr(X=0,Y=0)=0.997601.
$$
These nonnegative <probabilities> sum to one and preserve both marginal <probabilities>, yet predict $101$ double detections in expectation. Their <covariance> is $0.000101-p_ep_\gamma=0.000100$. This is an exact <joint probability distribution> illustrating pairing; it does not model the timing spread in the left panel or reconstruct either historical hypothesis quantitatively. Its lesson is why two singles totals cannot decide whether the observed events share a cause.
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