10.2 Statistical methods in particle physics

Statistical methods are the backbone of particle physics analysis. They help scientists make sense of complex data from particle colliders and detectors. Without these tools, we couldn't separate signal from noise or draw meaningful conclusions about the fundamental nature of matter.

These methods cover everything from basic probability to advanced techniques like Monte Carlo simulations. They're essential for discovering new particles, measuring their properties, and testing theories about how the universe works. Understanding these methods is key to interpreting experimental results in particle physics.

Statistical Concepts for Particle Physics

Probability Distributions and Likelihood Functions

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• Statistical analysis in particle physics applies probability theory and statistical inference to interpret experimental data and draw conclusions about fundamental particles and interactions
• Probability distributions model various physical processes and experimental outcomes
  • Gaussian (normal) distribution models continuous variables with symmetric spread around a mean
  • Poisson distribution models discrete count data (particle decays)
  • Binomial distribution models binary outcome processes (particle detection efficiency)
• Likelihood functions represent the probability of observing experimental data given a particular theoretical model or set of parameters
  • Crucial for parameter estimation and hypothesis testing
  • Mathematically expressed as $L(\theta|x) = P(x|\theta)$, where $\theta$ represents model parameters and $x$ represents observed data
• Bayesian and frequentist approaches offer different perspectives on probability and parameter estimation
  • Bayesian approach incorporates prior knowledge and calculates posterior probabilities
  • Frequentist approach focuses on the sampling distribution of estimators and hypothesis tests

Monte Carlo Methods and Maximum Likelihood Estimation

• Monte Carlo methods simulate complex physical processes, estimate uncertainties, and generate probability distributions
  • Used to model detector response, particle interactions, and background processes
  • Enable comparison between simulated and experimental data
  • Example: Simulating particle showers in a calorimeter to estimate energy resolution
• Maximum likelihood estimation determines best-fit values of physical parameters from experimental data
  • Maximizes the likelihood function with respect to model parameters
  • Mathematically expressed as $\hat{\theta} = \arg\max_{\theta} L(\theta|x)$, where $\hat{\theta}$ is the maximum likelihood estimate
  • Provides a powerful method for parameter estimation in complex models
• Background estimation and signal extraction techniques separate events of interest from background processes
  • Sideband methods use control regions to estimate background in signal region
  • Template fitting combines signal and background shapes to fit observed data
  • Example: Estimating background under a particle mass peak using polynomial fit to sidebands

Hypothesis Testing in Particle Physics

Formulating and Testing Hypotheses

• Hypothesis testing involves formulating null and alternative hypotheses about particle existence or properties
  • Null hypothesis (H0) typically assumes no new physics or standard model prediction
  • Alternative hypothesis (H1) proposes new particle or deviation from standard model
• Likelihood ratio test compares competing hypotheses to determine best model for observed data
  • Test statistic defined as $\lambda = -2 \ln \frac{L(H_0|x)}{L(H_1|x)}$
  • Large values of $\lambda$ favor the alternative hypothesis
• Goodness-of-fit tests assess how well theoretical models match experimental observations
  • Chi-square test compares observed and expected frequencies
  • Kolmogorov-Smirnov test evaluates the difference between cumulative distribution functions

Advanced Analysis Techniques

• Profile likelihood methods handle nuisance parameters and incorporate systematic uncertainties
  • Nuisance parameters are "profiled out" by maximizing likelihood for each value of parameter of interest
  • Reduces impact of systematic uncertainties on parameter estimation and hypothesis testing
• Multivariate analysis techniques extract maximum information from complex datasets
  • Neural networks learn complex non-linear relationships in data
  • Boosted decision trees combine multiple weak classifiers into strong classifier
  • Example: Distinguishing Higgs boson signal from background using multiple kinematic variables
• Unfolding techniques correct for detector effects and reconstruct true distribution of physical observables
  • Inverse problem of removing detector smearing and inefficiencies
  • Methods include matrix inversion, iterative Bayesian unfolding, and regularized unfolding
  • Example: Reconstructing true jet energy spectrum from measured calorimeter energy deposits

Statistical Significance and Confidence Intervals

Quantifying Statistical Significance

• Statistical significance in particle physics expressed in standard deviations (sigma) from background-only hypothesis
  • 5-sigma threshold (p-value < 2.87 × 10^-7) often used to claim discovery
  • Corresponds to probability of false positive less than one in 3.5 million
• P-values quantify probability of obtaining results as extreme as observed, assuming null hypothesis is true
  • Small p-values indicate strong evidence against null hypothesis
  • Calculated by integrating tail of test statistic distribution
• Look-elsewhere effect accounts for increased probability of finding apparent signals when searching over wide range of possibilities
  • Adjusts p-value for multiple testing
  • Example: Searching for new particle resonance over wide mass range increases chance of statistical fluctuation

Confidence Intervals and Limit Setting

• Confidence intervals express uncertainty in parameter estimates and provide range of plausible values
  • Frequentist confidence intervals contain true parameter value with specified probability (confidence level)
  • Bayesian credible intervals represent posterior probability distribution of parameter
• Feldman-Cousins method constructs confidence intervals with proper coverage, especially near physical boundaries
  • Unified approach for upper limits and two-sided intervals
  • Avoids unphysical results (empty confidence intervals)
• CLs method sets exclusion limits in searches for new particles
  • Conservative approach avoids excluding signals in cases of low sensitivity
  • Defined as ratio of p-values: $CL_s = \frac{p_{s+b}}{1-p_b}$
  • Exclusion claimed when CLs < 0.05 (95% confidence level)

Evaluating Systematic Uncertainties

Sources and Propagation of Systematic Uncertainties

• Systematic uncertainties arise from various sources in particle physics experiments
  • Detector calibration (energy scales, efficiencies)
  • Background modeling (theoretical cross-sections, shape uncertainties)
  • Theoretical predictions (parton distribution functions, higher-order corrections)
  • Data processing methods (reconstruction algorithms, selection criteria)
• Propagation of systematic uncertainties through complex analyses requires sophisticated techniques
  • Covariance matrices capture correlations between uncertainties
  • Monte Carlo methods generate ensembles of pseudo-experiments with varied systematic effects
• Nuisance parameter techniques incorporate systematic uncertainties into statistical models
  • Auxiliary measurements or control regions constrain nuisance parameters
  • Profile likelihood incorporates nuisance parameters in hypothesis tests and confidence intervals

Uncertainty Reduction and Result Combination

• Sensitivity studies assess impact of individual systematic uncertainties on final results
  • Identify dominant sources of uncertainty
  • Guide efforts to reduce most impactful uncertainties
• Techniques for reducing systematic uncertainties improve precision of particle physics measurements
  • Data-driven background estimation reduces reliance on simulation
  • In-situ calibration methods constrain detector uncertainties using known physics processes
  • Example: Using Z → ee events to calibrate electron energy scale in Higgs boson measurements
• Combination of results from multiple experiments or analysis channels requires proper treatment of uncertainties
  • Correlations between systematic uncertainties must be accounted for
  • BLUE (Best Linear Unbiased Estimator) method combines measurements with correlated uncertainties
  • Example: Combining Higgs boson coupling measurements from ATLAS and CMS experiments at the LHC