Quantum Treatment of Multiple Scattering and Collective Flow in HBT Cheuk-Yin Wong Oak Ridge National Laboratory & University of Tennessee Kromeriz, Czech Republic August 17, 2005 The pion environment after the QGP phase transition Why quantum treatment? The path integral method to study multiple collisions and collective flow in HBT Conclusions C.Y.Wong, J. Phys. G 29, 2151 (2003) C.Y.Wong, J. Phys. G 30, S1053 (2004) W. N. Zhang et al., Chin. Phys. Lett. 21, 1918 (2004)

Pion environment after the QGP phase transition Pion density ~ /fm 3 Pion matter is dense at high T. Pion energy ~ GeV Pions are energetic. Average pi-pi collision energy ~ GeV. Right after chemical freeze-out, pi-pi collisions are frequent and are predominantly elastic. We are interested in the dynamics of 2 identical pions in such an environment after chemical freeze-out.

Conventional HBT Assumptions As the source expands after chemical freeze-out, the detected pions collide with other pions in the medium when they travel to the freeze-out points. As a result of these random collisions, the initial source evolves into a chaotic source at freeze-out. The source observed in an HBT measurement is the chaotic freeze-out source, and the HBT radii will correspond to those of the freeze-out configuration.

Why we need to re-examine basic HBT assumptions Because the Hanbury-Brown-Twiss intensity interferometry is purely a quantum-mechanical phenomenon, the dynamics of the interfering pions must be investigated within a quantum-mechanical framework. We need to study the interference of waves using probability amplitudes, instead of the conventional description of incoherent hadron cascades in terms of probabilities and cross sections. Pion probability amplitude is best described by Feynman’s path integral method.

Path integral method to study HBT We work in the source C.M. system. Consider a π + produced at x with momentum propagates in the pion medium to the freeze- out point x f, and is detected at the detected point x d with momentum k.

Pions are subject to collective flow. We can describe collective flow in terms of a long-range density-dependent mean field that arises from the pion medium equation of state. Pions are subject to short-range ππ scattering. Pions have initial collective flow from earlier QGP expansion and phase transition.

Lagrangian for a pion in a pion medium

Eqaution of motion of a pion in medium

Pion propagtion amplitude

Property of ππ collisions ←Absorptive process

ππ phase shifts

Role of ρ meson in π + π – scattering The width of ρ about 150 MeV in free space and is broadened substantially in medium. ρ meson mean life time ~ 0.5 fm/c. ρ meson orbiting time ~ 2πr ~ 2 to 3 fm/c ρ meson mean life time << ρ meson orbiting time It is unlikely for π + π – to complete the orbital motion of a ρ meson before the ρ meson breaks up. π + π – scattering is essentially an elastic scattering with an energy dependent amplitude.

Interference of two histories Consider two identical pions which propagate in the pion medium to the freeze-out points, and are detected at x d1 and x d2 with momenta k 1 and k 2.

Two-pion production and propagation Amplitude for the production and propagation of two pions k 1 and k 2 Wave function for the production and propagation of two identical pions Probability for the detection of two pions with momenta k 1 and k 2

Interference of two histories

With collective flow and multiple collisions,

Conclusions 1.After the QGP phase transition and chemical freeze-out, elastic scattering dominates. The pion amplitudes can be described by the Feynman path integral method. 2.The source measured by HBT is the initial source with momenta shifted by the mean field and the initial source distribution modified by phase shift functions. 3.The real parts of the phase shift functions tend to cancel and the imaginary parts of the collision phase shifts lead to an absorption due to the π + π –  π 0 π 0 reaction in the I=0 channel. There is a secondary extended source due to the inverse π 0 π 0  π + π – reaction. 4.If π + π – scattering dominated by the I=1 elastic scattering channel, absorption is relatively small and the HBT source is closer to the chemical freeze-out source with shifted momenta. 5.If π + π – scattering dominated by the I=0 π + π –  π 0 π 0 absorptive channel, absorption is relatively strong and the HBT source is closer to the traditional thermal freeze-out source.