Modeling Falling Backgrounds with Exponential Mixtures

Technical Summary: Modeling Falling Backgrounds with Exponential Mixtures

Problem Statement
Searches for new physics at the Large Hadron Collider (LHC), particularly "bump hunts" for resonant signals, rely on distinguishing localized excesses from smoothly falling Standard Model (SM) background distributions. Current background modeling strategies face a dichotomy:

1. Flexible Parametric Models: Universal approximators like polynomials are attractive but often require many parameters, are difficult to optimize, and risk overfitting signals or generating non-physical behaviors in distribution tails.
2. Rigid Analysis-Specific Models: Functions like the "dijet function" are motivated by leading-order Quantum Chromodynamics (QCD) and parton distribution functions. However, they rely on specific approximations and ad hoc corrections that may require revision as datasets grow in size and complexity. Furthermore, these rigid forms struggle in distribution extrema (tails) where sideband constraints are unavailable.

There is a significant need for generic, reusable fitting solutions that balance flexibility with physical constraints to avoid extensive, analysis-specific development.

Methodology
The authors propose the finite exponential mixture as a semi-parametric class of functions for modeling falling backgrounds. The approach is grounded in Extreme Value Theory (EVT), specifically the Pickands–Balkema–de Haan theorem, which states that tail distributions converge to a Generalized Pareto Distribution (GPD).

• Theoretical Basis: The GPD with a positive shape parameter ($\xi > 0$) is mathematically equivalent to an exponential-gamma mixture (Lomax distribution). The authors generalize this to a finite mixture of exponential distributions to accommodate pre-asymptotic behaviors and mixed regimes where standard GPD assumptions break down.
• Mathematical Formulation: The density is defined as a linear combination of $k$ exponential components:
  $$f_k(x|\vec{\beta}, \vec{w}) = \sum_{i=1}^k w_i \beta_i e^{-\beta_i x}$$
  where $\vec{\beta}$ are rate parameters and $\vec{w}$ are weights constrained to sum to 1 (via a softmax transformation).
• Properties: This class can approximate any completely monotone function arbitrarily closely. While not all physical distributions are strictly completely monotone (e.g., near threshold turn-ons or finite energy cutoffs), the class is sufficiently flexible to approximate falling tails while remaining constrained enough to prevent spurious peaks or non-monotone structures.
• Implementation:
  • Models are fitted using maximum likelihood estimation (MLE) with the MINUIT/MIGRAD algorithm.
  • Model selection (determining the number of components $k$) is performed using the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC).
  • For binned data (ATLAS), the integral of the exponential mixture is solved analytically to ensure numerical stability.
  • For small datasets (CMS), unbinned likelihoods are used.

Key Contributions and Results
The paper validates the exponential mixture against two distinct published datasets and through simulation studies:

1. ATLAS Dijet Spectrum ($n \approx 28.6$ million):

   • The exponential mixture was compared to the standard dijet function.
   • Result: The exponential mixture (selected with 4 components via AIC) achieved a slightly better $\chi^2$ (90 vs. 96) and comparable residuals across the mass range, despite using more parameters. The AIC and BIC criteria agreed on the optimal complexity.

2. CMS High-Mass Diphoton Spectrum ($n = 5,036$):

   • The mixture was compared against four existing background functions ($f_1$–$f_4$) used in the official analysis.
   • Result: The 2-component exponential mixture achieved a $\chi^2$ (50.6) comparable to or better than the original functions. While the BIC preferred simpler models, the AIC-selected mixture provided superior bias characteristics in subsequent tests.

3. Simulation Studies (Pseudo-datasets):

   • Pseudo-datasets were generated from the four CMS background functions to test bias, coverage, and spurious signal yields.
   • Bias: The AIC-selected exponential mixture exhibited small bias, frequently crossing the true density value and remaining within the $\pm 1$ standard deviation band of the true model. It outperformed the BIC-selected mixture.
   • Coverage: The AIC-selected model maintained nominal coverage for masses $m_{\gamma\gamma} < 2000$ GeV. In the tails, it showed slight under-coverage but remained among the best-performing models.
   • Spurious Signal: When fitting signal-plus-background models, the exponential mixture produced small spurious signal yields (consistent with zero within uncertainty), indicating it does not artificially generate "bumps."

Significance and Claims
The authors claim that the finite exponential mixture offers a flexible, asymptotically motivated, and reusable alternative to traditional background modeling.

• Generality: It performs comparably to existing methods across datasets varying by orders of magnitude in size and background complexity.
• Constraint: Unlike polynomials, the exponential mixture is sufficiently rigid to avoid overfitting and generating non-physical structures, yet flexible enough to approximate a wide variety of falling spectra.
• Limitations: The paper modestly acknowledges that the method assumes completely monotone behavior. It cannot inherently model threshold turn-on effects or finite-energy cutoffs (which break complete monotonicity) without modification or separate modeling. Additionally, the performance relies on heuristics for model selection (AIC vs. BIC) and optimization (random restarts), and the choice of fit range (upper threshold) impacts normalization.

The paper concludes that while the exponential mixture is a promising general-purpose basis for resonant searches, further work is needed to optimize complexity selection, improve global optimization methods, and integrate specific physics-motivated structures for non-monotone effects.