Analyzing correlation functions with tesseral and Cartesian spherical harmonics

Abstract

The dependence of inter-particle correlations on the orientation of particle relative-momentum can yield unique information on the space-time features of emission in reactions with multiparticle final states. In the present paper, the benefits of a representation and analysis of the three-dimensional correlation information in terms of surface spherical harmonics is presented. The harmonics include the standard complex tesseral harmonics and the real cartesian harmonics. Mathematical properties of the lesser-known cartesian harmonics are illuminated. The physical content of different angular harmonic components in a correlation is described. The resolving power of different final-state effects with regarding to determining angular features of emission regions is investigated. The considered final-state effects include identity interference and strong and Coulomb interactions. The correlation analysis in terms of spherical harmonics is illustrated with the cases of gaussian and blast-wave sources for proton-charged meson and baryon-baryon pairs.

Freeze-out dynamics via charged kaon femtoscopy insNN=200GeV central Au + Au collisions

“…This is consistent with the absence of long source lifetimes, which would indicate a first-order phase transition [6]. Moreover, analysis of three-dimensional (3D) two-pion correlation functions, exploiting the novel technique of Cartesian surface-spherical harmonic decomposition of Danielewicz and Pratt [7,8], revealed significant non-Gaussian features in the pion source function [9]. Furthermore, the extraction of the shape of the pion source function in conjunction with model comparisons has permitted the decoupling of the spatiotemporal observable into its spatial and temporal aspects and the latter into source lifetime and emission duration.…”

“…To obtain the moments, the 3D correlation function C(q), is expanded in a Cartesian harmonic basis [7,8],…”

“…where the relative wave function φ(q, r) serves as a sixdimensional kernel, which in our case incorporates Coulomb interactions and Bose-Einstein symmetrization only [8].…”

Use of cumulants to quantify uncertainties in the HBT measurements of the homogeneity regions

“…Many papers go even further and use sources which are, or can be decomposed into sources, point-like in space-time [4], [10]. A general derivation of the relation between correlation functions and emission functions, which is still [6] considered to be the most detailed, has been given by Anchishkin, Heinz and Renk [11]. Analyzing it one easily finds that the proof breaks down if the sources are not, or at least for the HBT analysis cannot be replaced by, instant sources.…”

“…Homogeneity region K is the region where hadrons with momentum K are produced. For a homogeneity region, much information with little model dependence can be obtained using the imaging method (see [6] and references quoted there). The imaging method, however, yields the distribution of distances between the points where identical hadrons with a given momentum are produced.…”