Particle Asymmetry Decay — Pseudoexperiment Analysis

Thomas Griffiths — Data Analysis and Machine Learning, University of Edinburgh

This notebook (AsymmetryAnalysis.ipynb) simulates a measurement of the decay $B \to DX$, of the kind used in real experiments to probe the matter/antimatter asymmetry of the universe. Events are generated from a known decay-time PDF, then an unbinned maximum-likelihood fit (via iminuit) recovers the lifetime ($\tau$), oscillation frequency ($\Delta m$), and asymmetry ($V$) parameters from the simulated data — first in isolation, then in the presence of a background component.

Repeating the generate-and-fit cycle over many pseudoexperiments quantifies how biased and how precise each parameter estimate is, and how that changes once a background is mixed in.

The physics model

Signal PDF (unnormalized):

$$p_{sig}(t;, \tau,, \Delta m,, V) \propto \left(1 + V \sin(\Delta m, t)\right) \exp\left(-\frac{t}{\tau}\right)$$

Background PDF, fixed width $\sigma = 0.5,\mu s$:

$$p_{bkg}(t;,\sigma) \propto \exp\left(-\frac{t^2}{2\sigma^2}\right)$$

Combined model with background fraction $f_{bkg}$ as a free parameter:

$$p_{comb}(t) = (1-f_{bkg}),p_{sig}(t) + f_{bkg},p_{bkg}(t)$$

Both PDFs are re-normalized over the detector's recorded window, $t_{min} = 0.5,\mu s$ to $t_{max} = 10,\mu s$, using trapezoidal integration (a quad-based version was tried first but was too slow for repeated use across hundreds of pseudoexperiments).

Nominal parameter values used to generate pseudo-data throughout: $\tau_g = 1.5,\mu s$, $\Delta m_g = 20,\text{MHz}$, $V_g = 0.1$.

What's in the notebook

1. Step 1 — Signal-only fit
   • Event generation by inverse-CDF sampling (an accept/reject "box method" was also implemented but is ~5x slower and left commented out for reference).
   • A single dataset (10,000 events) generated and fit with an unbinned NLL to sanity-check the pipeline before scaling up.
   • 100 pseudoexperiments to determine the bias and precision on $\tau$, $\Delta m$, $V$.
2. Step 2 — Adding a background
   • Background model and combined signal+background PDF (above).
   • Mixed datasets generated at three background fractions: 1%, 10%, 20%.
   • For each fraction, every pseudoexperiment's mixed dataset is fit twice — once with a signal-only model (which ignores the background) and once with the correct signal+background model — to directly measure the bias introduced by ignoring a real background.

Headline results

Step 1 (signal-only, 100 pseudoexperiments, no background):

Parameter	Outcome
$\tau$	Effectively unbiased — bias ~0.06 standard errors from zero
$\Delta m$	Bias not statistically significant (within 1σ); tightly constrained
$V$	Biased high — mean is ~1.14 standard errors above the true value

Step 2 (effect of an unmodeled background):

Background fraction	Signal-only fit (mis-specified)	Signal+background fit
1%	Negligible deviation from Step 1	Negligible deviation
10%	$\tau$ and $V$ increasingly underestimated; $\Delta m$ comparatively robust	All parameters recovered with negligible bias
20%	Bias grows further on $\tau$ and $V$	$f_{bkg}$ itself recovered correctly (<1σ from truth)

The mechanism: the background peaks near $t_{min}$, which a signal-only fit misreads as a faster exponential decay (biasing $\tau$ low) and as reduced oscillation amplitude (biasing $V$ low). $\Delta m$ is comparatively shielded because the oscillation spacing survives at later times where the background has died away. Modeling the background explicitly, with $f_{bkg}$ free, removes the systematic bias in all three signal parameters across all tested fractions.

Running

No external data files are needed — this is a fully self-contained simulation notebook. Requirements: numpy, pandas, matplotlib, scipy, iminuit.

Run top to bottom: later cells depend on variables defined earlier (events, the f_bkg list, results_sig / results_mix), so cells aren't safe to run out of order.

Known limitations / future work

• In the Step 2 pseudoexperiment loop, both the signal-only and signal+background fits call only migrad() (not hesse()), so their reported parameter errors come from MIGRAD's internal estimate rather than the Hessian-refined errors used in Step 1. Worth revisiting if the Step 2 uncertainties are used quantitatively elsewhere.
• f_bkg is used both as the list of background fractions and as a per-iteration loop variable in the Step 2 histogram cell, which overwrites it; the later pseudoexperiment loop happens to redefine the list before using it, but the two shouldn't share a name.
• Only three background fractions and one background shape are tested; a natural extension would be a finer scan over $f_{bkg}$ or an alternative background model.

Acknowledgements

Project specification: DAML Project 1 — Particle Asymmetry Decay, William Barter, University of Edinburgh, October 2025.