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Higgs Mass Reconstruction

Course: PHYS 152 - Machine Learning in Physics Institution: UC Santa Cruz Term: Winter 2024

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Overview

A neural network regression model to reconstruct the invariant mass of the Higgs boson from simulated H→μ⁺μ⁻ decay events. The model learns the non-linear relationship between collider coordinates and particle mass.

Physics Background

Relativistic Kinematics

The four-momentum vector is defined as:

\[ \mathbf{p}^{\mu} = \left(\frac{E}{c}, \mathbf{p} \right) \]

The invariant mass of a parent particle can be reconstructed from its decay products using:

\[ m = \sqrt{2p_{T1}p_{T2}\left[\cosh(\eta_1-\eta_2)-\cos(\phi_1-\phi_2)\right]} \]

Collider Coordinates

  • Transverse momentum (\(p_T\)): Momentum perpendicular to the beam axis
  • Pseudorapidity (\(\eta\)): \(\eta = -\ln\left(\tan\frac{\theta}{2}\right)\)
  • Azimuthal angle (\(\phi\)): Angle in the transverse plane

Model Architecture

A fully-connected neural network with 9 layers:

Input (35 features)
    → Linear(35, 100) → Tanh
    → Linear(100, 75) → Tanh
    → Linear(75, 60)  → Tanh
    → Linear(60, 50)  → Tanh
    → Linear(50, 40)  → Tanh
    → Linear(40, 30)  → Tanh
    → Linear(30, 20)  → Tanh
    → Linear(20, 10)  → Tanh
    → Linear(10, 1)   → Sigmoid
Output (invariant mass)

Training

Parameter Value
Learning Rate 0.01
Momentum 0.2
Epochs 1000
Batch Size 49,999
Test Size 10,000
Loss Function MSE
Optimizer SGD

Input Features

35 features from simulated collider data:

  • Number of jets, leptons, b-tagged jets
  • Missing transverse energy (MET)
  • Lepton kinematics (\(p_T\), \(\eta\), \(\phi\)) for 2 leptons
  • Jet kinematics (\(p_T\), \(\eta\), \(\phi\), b-tag) for up to 6 jets

Results

The model successfully learns to predict the Higgs boson mass (~125 GeV) from low-level collider features, demonstrating that the neural network has effectively learned:

  • Relativistic kinematics
  • Collider coordinate transformations
  • The non-linear relationship between detector observables and invariant mass

Tools

  • PyTorch
  • scikit-learn (preprocessing, train/test split)
  • h5py (HDF5 data loading)
  • pandas, NumPy, Matplotlib