Momentum Resolution of a Tracking Detector

Monte Carlo simulation and reconstruction of charged-particle trajectories in a silicon-style tracking detector with a magnetic field, including a from-scratch weighted least-squares fitter, bending-angle-based reconstruction of the transverse momentum pT, and a full residual/pull analysis of the estimator and its uncertainties.

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Overview

A charged particle enters the detector at (z = 0, x = x0) with angle s0 to the beam axis, crosses five detection planes, is bent by a homogeneous magnetic field, and is measured again by three further planes. From the eight discrete hit positions alone, the task is to reconstruct the trajectory and the transverse momentum — and, more importantly, to reconstruct honest uncertainties on them.

The project is structured in two parts:

Part	Question	Method
A — Tracking resolution	How well can x0 and s0 be recovered from five discretised hits, with no field?	Weighted linear regression on the hit positions; residual and pull analysis over 1000 trajectories
B — Momentum resolution	How well can pT be recovered from the bending angle across the magnet?	Circular-arc propagation through the field, straight-line fits before and after, pT from the angular difference, with two alternative uncertainty models

Everything is written in NumPy/SciPy; no tracking or fitting framework is used.

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Physics setup

• Detector — 5 planes before and 3 planes after the magnet, spaced Δz = 2 cm, measuring the x coordinate with cells of width 500 µm and no sub-cell resolution. The hit is recorded at the cell centre with uncertainty σ_x = cell width / √12.
• Magnet — homogeneous field of length L = 10 cm between planes 5 and 6, oriented along y; B ∈ {0.5, 1.0, 1.5, 2.0} T.
• Particles — x0 ~ N(0, 1 mm), s0 ~ N(0, 0.1 rad), charge q = ±1 assigned at random, pT ∈ {0.1, 0.3, 1, 2, 5, 10, 20} GeV/c.

Inside the field the particle follows a circular arc of radius ρ = pT / (qB), so the bending angle θ ≈ L·q·B / pT gives the reconstruction relation pT ≈ L·q·B / θ.

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Method notes

Three implementation decisions carry most of the work and are worth reading the code for:

1. A hand-rolled weighted least-squares fitter instead of curve_fit. Fitting x = x0 + tan(s0)·z directly with scipy.optimize.curve_fit badly underestimated the uncertainty on s0, because the nonlinearity of tan is not captured by the linearised covariance estimate — the failure showed up as pull distributions with σ well above 1. Reparametrising in terms of the slope reduced but did not eliminate the effect. The final version implements weighted linear regression under a least-squares criterion directly (manual_least_squares), which recovers correctly calibrated uncertainties at the cost of runtime. The pull distributions were used as the diagnostic that drove this iteration — they are the reason the first two approaches were rejected.

2. Exact arc termination via root finding. The end of the circular arc was initially placed using the small-angle approximation θ ≈ L·q·B / pT. Over many trajectories this truncated the most strongly deflected arcs too early. The final version solves for the exit angle of each arc individually with scipy.optimize.root_scalar, which makes the geometry correct for arbitrary numbers of trajectories.

3. Two uncertainty models for pT, switchable at call time. Naive Gaussian error propagation from the two fitted angles ignores their correlation and underestimates σ_pT. A Monte Carlo alternative resamples both angles and takes the spread of the reconstructed pT as the uncertainty; this improves the pull means but overestimates σ_pT at high momentum. Both are implemented and selected via the pt_calculation_way parameter, with the analytic version as default, so the trade-off stays visible rather than being hidden behind one choice.

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Results

Part A — tracking resolution (1000 trajectories, no field)

Pull distributions are essentially ideal: σ = 0.996 ± 0.022 for both x0 and s0, with means consistent with zero. The estimator is unbiased and its uncertainties are correctly calibrated.

Part B — momentum resolution (pT = 0.3 GeV/c, B = 0.5 T)

Residual σ = 0.010 GeV/c, i.e. a relative resolution of roughly 3.3 %. The pull distribution has μ = -0.32, σ = 1.07.

Scaling behaviour. Resolution improves with larger B and with smaller pT, both of which shrink the radius of curvature ρ = pT/(qB) and therefore increase the measurable deflection.

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Known limitations

Stated explicitly because they are the interesting part of the analysis:

1. Negative pull bias in pT. The reconstructed bending angle is systematically overestimated, so pT is systematically underestimated. With only three closely spaced planes after the magnet, the fitted slope is very sensitive to small x errors; because the slope enters through a division by the z lever arm, upward fluctuations produce larger deviations than downward ones, and the resulting asymmetry biases the angle. The effect grows with B and with extreme pT.
2. Underestimated σ_pT at large pT and B. The angles before and after the magnet are correlated, which the analytic propagation does not account for; the step-by-step geometric construction (centre → arc points → exit normal) compounds this.
3. Discretisation artefacts. At high pT the residual histograms break into distinct bands rather than approaching a Gaussian, an artefact of the finite cell width. Smaller cells would mitigate it.

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Repository contents

.
├── momentum_resolution.ipynb   # drives the simulation and presents the results
├── src/
│   └── tracking.py             # simulation, fitting and reconstruction functions
├── figures/                    # key figures, exported from the notebook
├── report/
│   ├── momentum_resolution_report.pdf   # full write-up with all results
│   └── midterm_presentation.pptx        # mid-course status presentation
├── requirements.txt
└── README.md

The simulation and reconstruction functions live in src/tracking.py; the notebook imports them and drives the experiments.

The notebook is committed with outputs, so all figures and numbers render directly on GitHub without re-running it.

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Running it

git clone https://github.com/florian-hellwig/particle_tracking.git
cd particle_tracking
python -m venv .venv && source .venv/bin/activate
pip install -r requirements.txt
jupyter lab momentum_resolution.ipynb

Python 3.10+. Then Restart Kernel and Run All Cells. The 1000-trajectory runs and the per-arc root finding are the slow steps; reduce nr_trajectories in the setup cell for a quick pass.

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Authors and contributions

Four-person course project, PHY 241 Data Analysis II, University of Zurich, spring 2025. The report is submitted under all four names — Florian Hellwig, Felix Huber, Mariano Fasano and Manuel Tuor.

Within the project:

• Part A, tracking resolution: Florian Hellwig, sole author — including the weighted least-squares fitter and the residual/pull analysis.
• Part B, momentum resolution: started by Mariano Fasano, developed further by Florian Hellwig, and completed jointly.
• Report: written by Florian Hellwig, with an early draft of one subsection contributed by Felix Huber.