[Paper Review] R_{out}/R_{sid} and Opacity at RHIC

Larry McLerran, Sandra S. Padula|ArXiv.org|May 8, 2002

High-Energy Particle Collisions Research2 references3 citations

TL;DR

This paper proposes that the experimentally observed $ R_{\text{out}}/R_{\text{sid}} < 1 $ at RHIC arises from surface emission in an opaque, hot source with early decoupling. Using a covariant current ensemble formalism and a simplified model of 1+1D longitudinal expansion with blackbody-like surface emission, the authors show that opacity and early-time emission naturally suppress $ R_{\text{out}} $, reproducing the data trend without requiring transverse flow or complex hydrodynamics.

ABSTRACT

One of the most dramatic results from the first RHIC run are the STAR results for pi+- pi+- interferometry. They showed that the ratio of the so-called R_{out} and R_{sid} radii seem to decrease below unity for increasing transverse momentum of the pair (K_T). This was subsequently confirmed by PHENIX, which also extended the K_T range of the measurements. We consider here the effects of opacity of the nuclei on this ratio, and find that such a small value is consistent with surface emission from an opaque source.

Motivation & Objective

• To explain the experimentally observed $ R_{\text{out}}/R_{\text{sid}} < 1 $ at RHIC, which contradicts standard HBT models predicting $ R_{\text{out}}/R_{\text{sid}} \sim 2 $.
• To investigate whether source opacity and surface emission can account for the suppression of $ R_{\text{out}} $ relative to $ R_{\text{sid}} $.
• To assess the role of early decoupling and time-dependent temperature in shaping the $ K_T $-dependence of the HBT radii.
• To test whether a simplified model with surface emission and 1+1D expansion can quantitatively describe the RHIC data without full hydrodynamic simulations.

Proposed method

• Uses the Covariant Current Ensemble formalism to compute two-particle HBT correlation functions, enabling a relativistic treatment of emission sources.
• Introduces a simplified model of a cylindrical source undergoing 1+1D longitudinal expansion with a fixed emission time $ \Delta t \sim R_T $.
• Considers two limits: fully transparent source (emission from entire volume) and fully opaque source (emission only from surface), with surface emission weighted by $ \cos\phi $, where $ \phi $ is the angle between the emission direction and surface normal.
• Assumes blackbody-like emission from the surface at high temperature, with energy conservation linking parton/gluon flux to pion emission.
• Computes $ R_{\text{out}} $ and $ R_{\text{sid}} $ as functions of $ K_T $, incorporating time delays due to emission at different times.
• Compares results to preliminary STAR and PHENIX data for $ \pi^+\pi^+ $ and $ \pi^-\pi^- $ pairs, using emissivity parameter $ \kappa $ to tune agreement.

Experimental results

Research questions

• RQ1 Can source opacity alone explain the $ R_{\text{out}}/R_{\text{sid}} < 1 $ observed in RHIC data?
• RQ2 How does surface emission with $ \cos\phi $-dependent flux affect the $ K_T $-dependence of $ R_{\text{out}} $ and $ R_{\text{sid}} $?
• RQ3 What role does early decoupling play in suppressing $ R_{\text{out}} $, and how does it compare to hydrodynamic models?
• RQ4 Why does the ratio $ R_{\text{out}}/R_{\text{sid}} $ decrease with increasing $ K_T $, and can this be explained without transverse flow?
• RQ5 To what extent does early, high-temperature surface emission dominate over later, cooler emission in shaping the HBT radii?

Key findings

• In the opaque source limit, $ R_{\text{out}} \sim \sqrt{\Delta t^2 (K_T/E_K)^2 + (0.2R_T)^2} $ and $ R_{\text{sid}} \sim R_T $, leading to $ R_{\text{out}}/R_{\text{sid}} < 1 $ when $ \Delta t \sim R_T $, which matches the RHIC data trend.
• The transparent source model always yields $ R_{\text{out}}/R_{\text{sid}} > 1 $, confirming that opacity is essential to reproduce the data.
• The model reproduces the $ K_T $-dependence of $ R_{\text{out}} $ and $ R_{\text{sid}} $, with $ R_{\text{sid}} $ nearly flat and $ R_{\text{out}} $ decreasing with $ K_T $, consistent with STAR and PHENIX data for $ \kappa = 1 $.
• About 80% of the emission originates from the surface, primarily at early times, due to high emissivity and blackbody-like radiation from the hot surface.
• The longitudinal decoupling time is significantly shorter than in standard hydrodynamic models, which helps explain the weak $ K_T $-dependence of the source size at RHIC.
• The model suggests that the $ R_{\text{out}}/R_{\text{sid}} $ ratio is robust and dimensionless, making it less sensitive to model details, while the absolute radii depend strongly on time and size scales, highlighting the need for a full dynamical model with early surface emission.