Particle Physics Guide

A skill for analyzing particle physics data, covering event reconstruction, histogram analysis, statistical methods for discovery, and the standard tools used in high-energy physics (HEP) research. Includes ROOT, uproot, pyhf, and HEPData workflows.

Data Formats and Access

HEP Data Ecosystem

Format	Description	Typical Size	Access Tool
ROOT (.root)	Columnar binary format, HEP standard	GB-TB	ROOT, uproot
NanoAOD	Compact analysis format (CMS)	~1 KB/event	uproot, coffea
DAOD_PHYS	Derived analysis format (ATLAS)	~10 KB/event	ROOT, uproot
HepMC	Monte Carlo event record	Variable	pyhepmc
HEPData	Published results (YAML/JSON)	KB	hepdata_lib

Reading ROOT Files with uproot

import uproot
import awkward as ak
import numpy as np

def load_nanoaod(filepath: str, tree_name: str = "Events",
                  branches: list[str] = None) -> ak.Array:
    """
    Load a NanoAOD ROOT file into an awkward array.
    branches: list of branch names to load (None = all)
    """
    with uproot.open(filepath) as f:
        tree = f[tree_name]
        if branches is None:
            branches = tree.keys()
        events = tree.arrays(branches, library="ak")

    print(f"Loaded {len(events)} events")
    print(f"Branches: {events.fields}")
    return events

# Example: Load muon data
events = load_nanoaod("nano_data.root", branches=[
    "nMuon", "Muon_pt", "Muon_eta", "Muon_phi", "Muon_mass",
    "Muon_charge", "Muon_pfRelIso04_all", "Muon_tightId",
])

Event Selection and Reconstruction

Dimuon Invariant Mass

def compute_invariant_mass(pt1, eta1, phi1, mass1,
                            pt2, eta2, phi2, mass2):
    """
    Compute invariant mass of a particle pair from 4-momentum components.
    Uses the relativistic energy-momentum relation.
    """
    # Convert to Cartesian 4-vectors
    px1 = pt1 * np.cos(phi1)
    py1 = pt1 * np.sin(phi1)
    pz1 = pt1 * np.sinh(eta1)
    e1 = np.sqrt(px1**2 + py1**2 + pz1**2 + mass1**2)

    px2 = pt2 * np.cos(phi2)
    py2 = pt2 * np.sin(phi2)
    pz2 = pt2 * np.sinh(eta2)
    e2 = np.sqrt(px2**2 + py2**2 + pz2**2 + mass2**2)

    # Invariant mass of the pair
    m_inv = np.sqrt(
        (e1 + e2)**2 - (px1 + px2)**2 - (py1 + py2)**2 - (pz1 + pz2)**2
    )
    return m_inv

def select_z_candidates(events):
    """
    Select Z -> mu+mu- candidates from NanoAOD events.
    Requires exactly 2 opposite-sign muons passing quality cuts.
    """
    # Quality cuts
    muon_mask = (
        (events.Muon_pt > 20) &            # pT > 20 GeV
        (abs(events.Muon_eta) < 2.4) &     # |eta| < 2.4
        (events.Muon_tightId == True) &     # tight muon ID
        (events.Muon_pfRelIso04_all < 0.15) # relative isolation
    )

    # Apply mask and require exactly 2 muons
    good_muons = events[muon_mask]
    dimuon_events = good_muons[ak.num(good_muons.Muon_pt) == 2]

    # Opposite sign requirement
    opposite_sign = (
        dimuon_events.Muon_charge[:, 0] * dimuon_events.Muon_charge[:, 1] < 0
    )
    z_candidates = dimuon_events[opposite_sign]

    # Compute invariant mass
    m_inv = compute_invariant_mass(
        z_candidates.Muon_pt[:, 0], z_candidates.Muon_eta[:, 0],
        z_candidates.Muon_phi[:, 0], z_candidates.Muon_mass[:, 0],
        z_candidates.Muon_pt[:, 1], z_candidates.Muon_eta[:, 1],
        z_candidates.Muon_phi[:, 1], z_candidates.Muon_mass[:, 1],
    )

    return m_inv

Statistical Methods for Discovery

Hypothesis Testing with pyhf

import pyhf

def build_counting_model(signal: float, background: float,
                          bkg_uncertainty: float) -> dict:
    """
    Build a simple counting experiment model in pyhf.
    signal: expected signal yield
    background: expected background yield
    bkg_uncertainty: relative uncertainty on background
    """
    model = pyhf.simplemodels.uncorrelated_background(
        signal=[signal],
        bkg=[background],
        bkg_uncertainty=[bkg_uncertainty * background],
    )

    # Observed data (background-only for expected limit)
    data = [background] + model.config.auxdata

    return {"model": model, "data": data}

def compute_cls(model, data, poi_values=None):
    """
    Compute CLs exclusion limits (frequentist hypothesis test).
    Uses the CLs method standard in HEP.
    """
    if poi_values is None:
        poi_values = np.linspace(0, 5, 50)

    obs_cls = []
    exp_cls = []

    for mu in poi_values:
        result = pyhf.infer.hypotest(
            mu, data, model["model"],
            test_stat="qtilde",
            return_expected_set=True,
        )
        obs_cls.append(float(result[0]))
        exp_cls.append([float(v) for v in result[1]])

    return {
        "poi_values": poi_values.tolist(),
        "observed_cls": obs_cls,
        "expected_cls": exp_cls,
    }

Discovery Significance

def discovery_significance(n_observed: float, n_background: float,
                            sigma_b: float = 0) -> dict:
    """
    Compute discovery significance for a counting experiment.
    n_observed: number of observed events
    n_background: expected background
    sigma_b: uncertainty on background
    """
    from scipy.stats import norm

    if sigma_b == 0:
        # Simple Poisson significance
        # Z = sqrt(2 * (n * ln(n/b) - (n - b)))
        if n_observed <= n_background:
            z = 0
        else:
            z = np.sqrt(2 * (
                n_observed * np.log(n_observed / n_background)
                - (n_observed - n_background)
            ))
    else:
        # With systematic uncertainty (profile likelihood approximation)
        tau = n_background / sigma_b**2
        n = n_observed
        b = n_background
        z = np.sqrt(2 * (
            n * np.log((n * (b + tau)) / (b**2 + n * tau))
            - (b**2 / tau) * np.log(1 + tau * (n - b) / (b * (b + tau)))
        ))

    p_value = 1 - norm.cdf(z)

    return {
        "z_significance": round(z, 4),
        "p_value": p_value,
        "is_evidence": z >= 3.0,       # 3 sigma = evidence
        "is_discovery": z >= 5.0,      # 5 sigma = discovery
    }

Histogram Analysis

Binned Fitting

from scipy.optimize import curve_fit

def fit_breit_wigner_plus_bg(bin_centers: np.ndarray,
                               bin_contents: np.ndarray,
                               mass_range: tuple = (80, 100)) -> dict:
    """
    Fit a Breit-Wigner (resonance) + polynomial background to a mass histogram.
    Standard approach for Z boson mass measurement.
    """
    def model(m, N_sig, M_Z, Gamma_Z, a0, a1):
        # Breit-Wigner
        bw = N_sig * Gamma_Z / (2 * np.pi) / (
            (m - M_Z)**2 + (Gamma_Z / 2)**2
        )
        # Linear background
        bg = a0 + a1 * (m - 91.0)
        return bw + bg

    mask = (bin_centers >= mass_range[0]) & (bin_centers <= mass_range[1])
    x = bin_centers[mask]
    y = bin_contents[mask]

    p0 = [1000, 91.2, 2.5, 10, 0]  # initial guess
    popt, pcov = curve_fit(model, x, y, p0=p0, sigma=np.sqrt(y + 1))
    perr = np.sqrt(np.diag(pcov))

    return {
        "M_Z": f"{popt[1]:.3f} +/- {perr[1]:.3f} GeV",
        "Gamma_Z": f"{popt[2]:.3f} +/- {perr[2]:.3f} GeV",
        "N_signal": f"{popt[0]:.0f} +/- {perr[0]:.0f}",
        "chi2_ndf": round(np.sum(((y - model(x, *popt))**2 / (y + 1))) / (len(x) - 5), 2),
    }

Monte Carlo Simulation

Event Generation Pipeline

1. Matrix element calculation (MadGraph, Sherpa, POWHEG)
   --> Hard scattering process (e.g., pp -> Z -> mu+mu-)

2. Parton shower (Pythia, Herwig)
   --> QCD radiation, initial/final state radiation

3. Hadronization (Pythia string model, Herwig cluster model)
   --> Quarks/gluons -> hadrons

4. Detector simulation (Geant4 via CMSSW/Athena, or Delphes for fast sim)
   --> Particle interactions with detector material

5. Reconstruction
   --> Raw hits -> tracks, clusters, physics objects

Tools and Software

• ROOT: C++ data analysis framework (CERN), ubiquitous in HEP
• uproot: Pure Python ROOT file reader (no ROOT dependency)
• awkward-array: Columnar data with variable-length nested structure
• coffea: Analysis framework built on uproot + awkward + dask
• pyhf: Pure Python HistFactory for statistical models
• MadGraph5_aMC@NLO: Automated matrix element generation
• Pythia 8: Monte Carlo event generator (parton shower + hadronization)
• Delphes: Fast detector simulation framework
• HEPData: Repository for published HEP measurements