Distortion of the HBT image by mean field interaction

1. Distortion of the HBT image by mean field interaction Koichi Hattori ・KH, T.Matsui, Prog. Theor. Phys. 122 (2009) 1301 ・KH, Prog.Theor.Phys.124,(2010) 869-909

2. First part: qualitative study ＊ RHIC data on the HBT interferometry a.k.a.“HBT puzzle” ＊ Effects of the mean field interaction in the freeze-out stage Modification of the propagation after the emission ⇒　How does it result in the distortion of HBT image ? Second part: quantitative study ＊ To what extent are the HBT radii modified at RHIC energy ? ＊ Time-dependent phenomenological mean field interaction

3. Motivation In the RHIC experiment, hydrodynamical simulations reproduce the measured V2 of the hadron spices . Using the same parameter set, there appeared deviations in the HBT radii . Sideward Outward In the conventional hydrodynamical modeling, we assume the free streaming of pions after the thermal freeze-out, and the several approx. common to other models. (beyond this talk).

4. Final state interactions * Two-body interaction Mutual Coulomb force between a pair ⇒ Corrected by an elaborate experimental analysis * One-body interaction Interaction of each pion and the rest of the system ・Coulombic interaction ・Strong interaction - meson mean field G.Baym, P.Braun-Munzinger(1996) G.Cramer, G.Miller, J.Wu, J-H Yoon(2005) S.Pratt(2006) KH, T.Matsui(2009) KH (2010)

5. Imaginary part: Real part: Coherent scattering with medium particles Attempts for consistent description of hadronic phase by kinetic theories An attenuation by π+πρ is implemented in UrQMD, Bass and Dumitru (2000) A Vlasov term is highlighted in Matsuo and Matsui (2008) Mean field interaction in the hadronic phase Coherent interactions with evaporating particles described as a one-body mean-field interaction. Big picture: Need a consistent description with both effects taken into account. We examine the effects within a simplified framework. ρ meson

6. x Aclose look at interference term Chu, Gardner, Matsui, Seki (1994) Simple formula by the Fourier transform Back to the free case: Mean field interaction distorts the kernel of the integral transform.

7. Unit vectors oriented in the outward and sideward Mean field interaction Induces a coordinate shift: Distorted Fourier transform: Phase shift by the classical action

8. Plane wave Compare to the free case !! Effective distribution Original distribution Jacobian of the coordinate transform How the phase shift results in the distorted image Distorted phase due to the mean field

9. Gaussians with different ranges Central potential: Real part: Surrounding halo Dense region Imaginary part: Dense region A toy model of mean field interaction Classical action in the transverse plane: Imaginary part ⇒ Attenuation: Integral along the classical trajectory Real part of the mean filed potential ⇒ phase shift

10. Effects of real and imaginary parts Effect of real part Repulsion Repulsion + attenuation Attraction + attenuation Attraction Distorted profile of the apparent distribution Original distribution: Gaussian with R = 5 fm Outward Sideward

11. Second part: quantitative study ＊ Time-dependent phenomenological mean field interaction ＊ To what extent the HBT radii are modified at RHIC energy ?

12. q Medium pion p p Escaping pion Two-body ππ forward scattering amplitude in the vacuum A time-dependent model of mean field interaction Phenomenological pion self-energy Thermal pion Distribution after the“freeze-out” We assume: * free streaming after the“freeze-out” * cylindrical geometry with Bjorken flow Imaginary part: Real part: Coherent scattering with medium particles ΠΠ scattering amplitude * Elastic scattering in s-wave & p-wave below 1 GeV ρ meson

13. Medium pion: π± ,π0 Strong attenuation near rho resonance(770) P-wave: attractive with rho resonance s-wave + p-wave: attractive Forward ππ scattering amplitude B.Ananthanarayan and the authors below(2001) G.Colangelo, J.Gasser, H.Leutwyler(2001) Isospin average in the isospin symmetric limit:

14. t(fm/c) Au+Au collision z(fm) Bjorken flow: Radial flow: (Simple model with linear profile) Involved parameters: T=130 MeV, μ=30 MeV, ζ=0.06 Profile of the freeze-out time: Distribution after the “freeze-out” Free streaming after the“freeze-out” Thermal distribution specified by T, μ and the flow velocity uμ

15. Time evolution at k=150 MeV: Profiles of the effective mass

16. Absorption by the imaginary part of the potential Modification of the transverse spectrum at Y=0 In linear scale In log scale Deceleration by the real part of the potential

17. Effect of the real part Effect of the real & imaginary parts Modification of the Gaussian parameters 15% 15% Upgrades of hydro-modeling, Pratt (2009)

18. Summary We examined effects of the final state mean field interaction outside the freeze-out surface on the basis of ππ scatterings . We found modifications of the HBT radii without remarkable effect on the single-particle spectrum . Prospects • Consistent treatments thought out the hadronic phase • would result in a sizable effect in pion HBT interferometry . • Effects of the mean field interaction originated in interactions • with other species should be examined. • How about the interaction with nucleon in lower-energy collision • with a larger baryon chemical potential ? • N + π  ⊿