[Paper Review] Model independent analysis of nearly Lévy correlations

T. Novák, T. Csörgő+2 more|arXiv (Cornell University)|Apr 19, 2016

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper presents a model-independent method for analyzing two-particle short-range correlations with nearly Lévy or stretched exponential shapes, using orthogonal polynomials (Lévy polynomials) tailored to the Lévy weight function exp(−Q^αR^α). It generalizes earlier Laguerre (α=1) and Gaussian (α=2) expansions, enabling systematic fitting of HBT and ridge correlation data across diverse experimental conditions without assuming a specific source model.

ABSTRACT

A model-independent method for the analysis of the two-particle short-range correlations is presented, that can be utilized to describe e.g. Bose-Einstein (HBT), dynamical (ridge) or other correlation functions, that have a nearly Lévy or streched exponential shape. For the special case of Lévy exponent alpha = 1, the earlier Laguerre expansions are recovered, for the alpha = 2 special case, a new expansion method is obtained for nearly Gaussian correlation functions. Multi-dimensional Lévy expansions are also introduced and their potential application to analyze rigde correlation data is discussed.

Motivation & Objective

• To develop a model-independent framework for analyzing two-particle short-range correlations with nearly Lévy or stretched exponential shapes.
• To generalize existing Laguerre and Gaussian expansions to arbitrary Lévy index α ∈ [0,2] via orthogonal polynomial expansions.
• To enable systematic fitting of experimental correlation functions without assuming a Fourier-transformed source model.
• To extend the method to multidimensional correlations, such as those in ridge or angular correlation measurements.
• To provide a unified analytical tool for HBT and dynamical correlation data with non-Gaussian, non-exponential shapes.

Proposed method

• The method employs a series expansion of the correlation function C₂(Q) in terms of orthonormal Lévy polynomials {Lₙ(t|α)} with respect to the weight function w(t|α) = exp(−t^α), where t = QR.
• The expansion takes the form C₂(t) = N{1 + λ exp(−t^α) Σ cₙ Lₙ(t|α)}, allowing direct fitting to experimental data.
• Lévy polynomials are constructed via a Gram-Schmidt process using moments μₙ,ₐ = ∫₀^∞ tⁿ exp(−t^α) dt = (1/α)Γ((n+1)/α).
• The method recovers Laguerre polynomials for α=1 and introduces a new orthogonal basis for α=2 (Gaussian-like shape) on the positive real line.
• For multidimensional correlations, a dimensionless scaling variable t is defined using covariance matrices (e.g., Rᵢⱼ for HBT or σᵢⱼ for angular correlations), reducing multivariate data to a one-dimensional effective problem.
• The approach is applicable to Bose-Einstein, ridge, and other correlation functions that exhibit nearly Lévy behavior in their short-range structure.

Experimental results

Research questions

• RQ1 How can two-particle correlation functions with nearly Lévy or stretched exponential shapes be systematically analyzed without assuming a specific source model?
• RQ2 What is the mathematical structure of orthogonal polynomials for arbitrary Lévy index α, and how do they generalize known expansions for α=1 and α=2?
• RQ3 Can the method be extended to multidimensional correlation data, such as those from ridge or angular correlation measurements?
• RQ4 How do the resulting Lévy polynomials behave for different values of α, and what are their explicit forms?
• RQ5 What is the connection between the expansion coefficients cₙ and the deviation of the correlation shape from the ideal Lévy form?

Key findings

• For α=1, the Lévy polynomials reduce to the standard Laguerre polynomials, recovering the expansion previously used in model-independent HBT analysis.
• For α=2, a new orthogonal polynomial basis is derived for the Gaussian-shaped correlation function, valid only on the positive real line t ≥ 0.
• The first three Lévy polynomials are explicitly computed: L₀(t|α) = 1, L₁(t|α) = (1/α){Γ(1/α)t − Γ(2/α)}, and L₂(t|α) involves combinations of gamma functions of (n+1)/α.
• The method enables a systematic, model-independent fit of correlation functions by decomposing deviations from the ideal Lévy shape into orthogonal components via coefficients cₙ.
• The multidimensional extension uses a scaling variable t = (∑ᵢⱼ Rᵢⱼ qᵢ qⱼ)^½ or t = (∑ᵢⱼ σᵢⱼ Δᵢ Δⱼ)^½, allowing the same functional form to describe HBT and ridge correlations.
• The approach is robust for experimental data where the standard source-Fourier transform assumption may fail, providing a flexible alternative to model-dependent analyses.