Coulomb interacting Bose-Einstein correlations in Fourier space

TL;DR

The paper addresses the challenge of incorporating Coulomb final-state interactions into Bose-Einstein correlation analyses in femtoscopy for Lévy-stable source shapes encountered in heavy-ion collisions. It introduces a Fourier-transform-based method that first transforms the interacting two-particle wave function and then couples it to the source function in Fourier space, using a regulator to swap integrals and derive a tractable expression. The key result is a fast, exact-like formula for the spherically symmetric case: $C_s(k) = |N|^2 [ f_s(0) + f_s(2k) + \frac{\eta}{\pi} (\mathcal{A}_{1s} + \mathcal{A}_{2s}) ]$, with $\mathcal{A}_{1s}$ and $\mathcal{A}_{2s}$ defined by integrals involving $f_s(q)$ and the hypergeometric function $_2F_1$, enabling dramatic speed-ups over prior numerical schemes. The method supports practical experimental analyses, reduces computation from days to minutes on desktop hardware, and is complemented by a publicly available software package, though its current formulation assumes spherical symmetry with plans for generalization.

Abstract

In high-energy heavy-ion physics experiments, a state of matter is created that existed in the early Universe: the quark-gluon plasma. This strongly interacting matter exists in today's experiments only within a range of a few femtometers and for a duration of a few femtometers per speed of light, making its resolution with optical tools impossible. However, there is a method that allows for a closer look into the structure of the quark-gluon plasma: femtoscopy. Initially used in astronomy, femtoscopy is based on the quantum mechanical indistinguishability of identical particles, which causes them to arrive at detectors in a correlated manner. The measurable correlation is related to the spacetime structure of the particle-emitting source, which in heavy-ion physics is the quark-gluon plasma created in collisions. For free particles, a relatively simple relationship exists between the source and the correlation (essentially a Fourier transform). However, this relationship becomes complex when accurately accounting for the repulsive Coulomb interaction between final-state electrically charged particles. Our paper presents a new method that is more precise than previously used ones, yet less computationally demanding, especially for the exotic source function shapes. Mathematically, the method is interesting because it exactly handles many emerging integrals and limits. Practically, it is ready for use in experimental analyses.

Figures (3)

• Figure 1: $\mathcal{A}_{1s}$ and $\mathcal{A}_{2s}$ integrands plotted for $\alpha = 1.3$ and $R = 12$. The graphs show that the functions are smooth, making them easily integrable. The only critical point is at $x = 1$ where the function becomes logarithmically oscillatory, but it remains bounded, so this does not affect integrability.
• Figure 2: Example correlation functions for pions (left) and kaons (right), plotted for four different $R$ and two $\alpha$ values. $Q = 2 k$, and the $K$ argument is dropped for simplicity. At a given $R$ value, the shape of the correlation function with increasing $\alpha$ values from $\alpha\,{=}\,0.6$ to $\alpha\,{=}\,2$ goes smoothly through the shaded region.
• Figure 3: Difference between the correlation function calculated with a numerical integral method described in Ref. Kincses:2019rug ($^{\textnormal{table}}C_2(Q)$) and the correlation function calculated with the wave-function Fourier method described in the current paper ($^{\textnormal{WFF}}C_2(Q)$). $\Delta C_2(Q)$ is plotted for 6 different $\alpha$ values and two $R$ values, for pions (left) and kaons (right) separately. At a given $\alpha$ value, $\Delta C_2(Q)$ goes smoothly through the shaded region when increasing $R$ values from $R = 3\textnormal{ fm}$ to $R = 12\textnormal{ fm}$.