Homework 4

Calorimeter properties

You are asked to instrument an electromagnetic calorimeter 12 m downstream of an interaction point at the LHC in the pseudo-rapidity range of 2.5 to 3. A colleague simulated the processes
B⁺ → π⁰π⁺ and D⁰ → K⁺K⁻π⁻π⁺π⁰, with π⁰ → γγ for you and gave you access to csv files containing 4-vectors of the photons (⃗p, E in GeV). Assume that all photons originate from (0,0,0):

• File 1 download link (click on Cancel, then download in the middle of the page)
• File 2 download link

a) Which is the maximum (quadratic) cell size, such that in at least 68.3% of the cases, the photons from the π⁰ decay end up in separate cells? Check both processes and make histograms for the opening angle between photons and their distance on the calorimeter surface.

b) What is the mass resolution of the π⁰ peaks given the reconstructed results of a) and an ECAL resolution of σ_E/E = 4.2%/√E[GeV].
Hint: in order to emulate the reconstruction step, apply a Gaussian smearing to the simulated energy of an accepted candidate. Prepare a histogram that shows the π⁰ mass peak.

c) Comment on the results. Do you think it would be possible to build both calorimeters? Think about, and name at least 4 typical constraints, and technical challenges that you would face when trying to build the calorimeter. Remember to cite your sources.

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Code Implementation and Methodology

Overview

The analysis is performed using Python with the following key libraries:

• NumPy: Numerical computations and array operations
• Pandas: CSV data loading and manipulation
• Matplotlib: Visualization and histogram generation
• SciPy: Gaussian fitting for mass peak analysis

Configuration

The code includes configurable parameters at the top:

USE_RANDOMNESS = True  # Toggle between random and deterministic energy smearing
RANDOM_SEED = 42       # Ensures reproducibility of results

Analysis Workflow

1. Data Loading

The CSV files contain 4-vectors (px, py, pz, E) for both photons from each π⁰ decay. For each decay process:

• B⁺ → π⁰π⁺: 13,876 simulated events
• D⁰ → K⁺K⁻π⁻π⁺π⁰: 13,381 simulated events

2. Opening Angle Calculation

The opening angle θ between two photons is calculated using the scalar product of their momentum vectors:

$$\cos(\theta) = \frac{\vec{p}_0 \cdot \vec{p}_1}{E_0 \cdot E_1}$$

For photons (massless particles), the momentum magnitude equals the energy: |p⃗| = E.

3. Calorimeter Position Projection

Photons originating from (0,0,0) are projected onto the calorimeter surface at z = 12 m using ray tracing:

$$x_{\text{calo}} = \frac{12 \cdot p_x}{p_z}, \quad y_{\text{calo}} = \frac{12 \cdot p_y}{p_z}$$

The distance between two photons on the calorimeter is:

$$d = \sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}$$

4. Cell Size Determination

The maximum cell size is determined by the 31.7% quantile of the distance distribution. This ensures that in at least 68.3% of cases, the two photons hit separate cells (distance > cell size). The 31.7% quantile means that 31.7% of events have distances smaller than the cell size (photons in the same cell), while 68.3% have larger distances (photons in separate cells).

5. Event Selection

Events are accepted if the distance between photons exceeds the cell size (photons in separate cells). This selection is crucial for proper mass reconstruction.

6. Energy Smearing

To simulate realistic detector resolution, Gaussian energy smearing is applied:

$$\sigma_E = \frac{4.2%}{\sqrt{E[\text{GeV}]}} \cdot E = 0.042 \cdot \sqrt{E}$$

For each photon energy, a random error is drawn from N(0, σ_E) and added to the true energy. The fixed random seed ensures reproducibility.

7. Invariant Mass Reconstruction

The invariant mass is calculated using the relativistic formula:

$$m_{\text{inv}} = \sqrt{(E_1 + E_2)^2 - |\vec{p}_1 + \vec{p}_2|^2}$$

The momentum vectors are reconstructed using the measured energies while preserving the original directions.

8. Mass Resolution Analysis

A Gaussian fit is applied to the mass distribution histogram to extract:

• μ: Peak position (reconstructed π⁰ mass)
• σ: Peak width (absolute mass resolution)
• σ/μ: Relative mass resolution

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Answer to Task a): Maximum Cell Size

B⁺ → π⁰π⁺ Decay

Results:

• Maximum cell size: 11.77 cm × 11.77 cm (31.7% quantile)
• Acceptance rate: 68.3% of events (9,477 out of 13,876)

Opening Angle Statistics:

• Mean: 1.04°
• Median: 0.78°
• Range: [0.05°, 15.97°]

Distance on Calorimeter Statistics:

• Mean: 22.13 cm
• Median: 16.52 cm
• Range: [0.98 cm, 336.60 cm]

See histogram: histograms_opening_angle_distance_B+.png

D⁰ → K⁺K⁻π⁻π⁺π⁰ Decay

Results:

• Maximum cell size: 85.60 cm × 85.60 cm (31.7% quantile)
• Acceptance rate: 68.3% of events (9,138 out of 13,379)

Opening Angle Statistics:

• Mean: 7.00°
• Median: 6.08°
• Range: [0.17°, 18.70°]

Distance on Calorimeter Statistics:

• Mean: 147.86 cm
• Median: 128.35 cm
• Range: [3.65 cm, 395.16 cm]

See histogram: histograms_opening_angle_distance_D0.png

Interpretation

The cell size is chosen as the 31.7% quantile of the distance distribution, which ensures that 68.3% of photon pairs have separations larger than the cell size and thus hit separate cells. This threshold corresponds roughly to one standard deviation (±1σ) in a Gaussian distribution.

The calorimeter design accepts that approximately 31.7% of events will have photons landing in the same cell (distance < cell size), making them unresolvable. These events must be rejected for mass reconstruction. The 68.3% threshold balances detector granularity (cost, complexity) with event acceptance rate.

Comparison of decay processes:

• B⁺ decay: Small opening angles (median 0.78°) result in small separations (median 16.52 cm), requiring fine granularity (11.77 cm cells)
• D⁰ decay: Larger opening angles (median 6.08°) result in larger separations (median 128.35 cm), allowing coarser granularity (85.60 cm cells)

The D⁰ decay has approximately 7.3 times larger cell size than B⁺, making it significantly easier and cheaper to instrument.

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Answer to Task b): Mass Resolution

B⁺ → π⁰π⁺ Decay

Mass Reconstruction Results:

• Reconstructed π⁰ mass (μ): 135.00 MeV/c²
• Theoretical π⁰ mass: 135.0 MeV/c²
• Deviation: -0.00 MeV/c²

Mass Resolution:

• Absolute resolution (σ): 1.69 MeV/c²
• Relative resolution (σ/μ): 1.25%

See histogram: histogram_pion_mass_B+.png

D⁰ → K⁺K⁻π⁻π⁺π⁰ Decay

Mass Reconstruction Results:

• Reconstructed π⁰ mass (μ): 135.02 MeV/c²
• Theoretical π⁰ mass: 135.0 MeV/c²
• Deviation: +0.02 MeV/c²

Mass Resolution:

• Absolute resolution (σ): 4.48 MeV/c²
• Relative resolution (σ/μ): 3.32%

See histogram: histogram_pion_mass_D0.png

Interpretation

The mass resolution is determined by:

1. Detector energy resolution: σ_E/E = 4.2%/√E[GeV], which dominates for high-energy photons
2. Geometric effects: Uncertainty in photon direction affects momentum reconstruction
3. Event selection: Only events with well-separated photons (distance > cell size) are used

The Gaussian fit to the invariant mass distribution provides a precise measurement of both the peak position and width. Both decay processes reconstruct the π⁰ mass extremely accurately, with deviations < 0.02 MeV/c² from the theoretical value of 135.0 MeV/c².

Comparison of mass resolution:

• B⁺ decay: σ = 1.69 MeV/c², σ/μ = 1.25% (better resolution due to higher photon energies, mean ~10 GeV)
• D⁰ decay: σ = 4.48 MeV/c², σ/μ = 3.32% (worse resolution due to lower photon energies, mean ~1.3 GeV)

The relative resolution follows the expected 1/√E dependence: higher energy photons in B⁺ decay lead to approximately 2.7× better resolution than in D⁰ decay. Both values are within the typical range of 1-5% for electromagnetic calorimeters in this energy regime.

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Answer to Task c): Feasibility and Technical Challenges

Discusion on the results and the typical constraints and technical challenges that have to be faced when constructing a calorimeter are found in Task c).pdf.

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Use of Generative AI

Generative AI was used while working on this homework task.

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Authors

• Colin Beckmann [colin.beckmann@edu.ruhr-uni-bochum.de]
• Fabian Steube [fabian.steube@edu.ruhr-uni-bochum.de]

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License

This project was created as an homework assignment.

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