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Distortion of HBT Images by Mesonic Mean Field and Their Correlation Functions in Collider Experiments

This study focuses on the distortion of images in Heavy-Boson-Two particle interferometry due to mesonic mean field interactions. It explores the impact on source distribution functions and pion pairs in ultra-relativistic heavy ion collisions, considering mutual Coulomb forces and one-body interactions. The analysis evaluates amplitude distortion using semi-classical approximation and examines phase shifts caused by mean field interactions. The apparent distortion on source distribution is discussed, along with effects like repulsive and attractive mean fields. The work aims to understand the dynamics of expanding sources and the strength of mean fields, with implications for absorption effects in experimental results.

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Distortion of HBT Images by Mesonic Mean Field and Their Correlation Functions in Collider Experiments

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  1. Distortion of HBT images by mesonic mean field Koichi Hattori in collaboration with Professor Matsui nucl-th/0905.3210, to be published on Prog. Theor. Phys.

  2. HBT interferometry in collider experiments x1 x2 detector１ Correlation function is related to the Fourier transform of source distribution function. detector２ ρ: distribution function Conventional analysis assumes the free streaming of pions after the thermal equilibrium has broken, random phase, factorization of tow-body distribution function and the Gaussian profile of source(beyond this talk). Detecting pion pair in the products of ultra-relativistic heavy ion collisions (limited to central collisions in this talk) Symmetrization of the amplitude to detect pion pair carrying momentum k1 and k2 k: Averaged momentum q: Relative momentum

  3. Distortion of the amplitude in meson clouds after thermal freeze-out Thermal distribution of pion spectrum on the freeze-out surface is possibly shifted by subsequent final state interactions. * Two-body interaction mutual Coulomb force between the pair ⇒ Corrected with Gamow factor M.Gyulassy, S.K.Kauffmann, L.W.Wilson(1979) * One-body interaction Interaction of each pion and the rest of the system ・Coulombic interaction ・Strong interaction - mesonic 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) We examine distortion of the amplitude in the frame work of semi-classical approximation.

  4. X Back to the free case Simple formula by the Fourier transform Plane wave Aclose look at interference term Chu, Gardner, Matsui, Seki (1994) One-body amplitude Analysis on the one-body amplitude provides how the images are distorted due to mean field interaction. We evaluate phase shift given by the classical action of the trajectory initiated at X and terminated with momentum k1 or k2.

  5. Plane wave Phase shift Reflecting the observation of two pions, interference arises as the difference of the phases. Semi-classical analysis of the phase shift Assuming boost invariance in the incident beam direction, we examine distortion of 2 dimensional images on the transverse plane.

  6. Unit vector Outward // K Sideward ⊥ K Apparent shift of emission points results in a simple formula. New coordinate The difference of phase shifts Expand the action with respect to relative momentum q. - Correlation arises on the small q region.

  7. Apparent distortion on source distribution Distortion due to mean field Using the coordinate transform , the effect of mean field on the amplitude is transferred to the distribution function. Plane wave Apparent distribution Distribution of the initial momentum Jacobian of the coordinate transform

  8. 20MeV 15MeV 30fm at most in the dense region Repulsive: Attractive: at least Strength of mean field: Classical trajectories refracted by mean field potential - initiated at X and terminated with momentum k S.Pratt(2006) Repulsive Attractive A schematic model of mean field -Non-relativistic analysis Classical action on the transverse plane: Dense region Surrounding halo Central potential: Radial coordinate

  9. outward(X’) Free (Gaussian) outward(X’) sideward(Y’) sideward(Y’) Repulsive Focusing the images? Good analogy on geometrical optics?? Attractive Stretching the images? Results: Profile of the apparent distribution

  10. Straight-line approximation “Glauber approximation” Phase shift along classical trajectory X Repulsive Attractive “Glauber approx.” qualitatively reproduces the distortion of images. The shifts in the momentum space is more essential than the refraction in the coordinate space for HBT interferometry.

  11. Repulsive Attractive Momentum dependence Averaged final momentum K=100(MeV) Mean field is effective for small K. Averaged final momentum K=150(MeV) • Effects of absorption • imaginary part of • mean field potential K=100(MeV) with absorption The effect of absorption is similar to that of attractive potential.

  12. Short summary of this talk • Mean field generated in the vicinity of source provides possible distortion of the HBT images. The difference of two phase shifts plays an important role. With schematic static mean field model • The effect of absorption is also significant for the apparent distribution. Work now in progress: more realistic model ・How does the dynamics of expanding source reflect on the mean field? ・How strong is the mean field? Repulsive or attractive ? What is the origin of strong absorption, if it exit?

  13. q Medium pion p p Detected pion Two-body ππ forward scattering amplitudein the vacuum A model of dynamical mean field based on elementary ππ scattering Modification of pion mass in medium expansion inhomogeneous system Pion self-energy A.Schenk(1991) ΠΠ scattering amplitude * s-wave & p-wave * I=0,1,2 isospin channels Phase space distribution after freeze-out We assume: * free streaming after the freeze-out * cylindrical source as initial distribution (Bjorken flow)

  14. t(fm/c) z(fm) A simple model of expanding pion gas ＊ Free streaming after the freeze-out ＊Initial distribution on the freeze-out surface Thermal distribution with Bjorken flow Time evolution of the sorce profile (integrated out by momentum p) Au+Au collisions (fm^-3) (fm^-3) Longitudinal coordinate z(fm) Radial coordinate r(fm) Along the axis of cylinder On transverse plane

  15. I=0 I=2 I=1 Total Scattering amplitude of ππ collisions Mandelstam variable： squared center of mass energyof ππ B.Ananthanarayan and the authors below(2001) G.Colangelo, J.Gasser, H.Leutwyler(2001) Low energy scattering blow 1 GeV s-wave: I=0,2 p-wave: I=1 Strong absorption near rho resonance(770) Weak attraction owing to the contribution from I=1 Real part of Tππ(s) Imaginary part of Tππ(s)

  16. Dependence on isospin channel - Comparison of s-wave and s-wave + p-wave(rho meson) at p=100(MeV) S-wave + p-wave: attravtive meff (MeV) meff (MeV) S-wave: repulsive Radial coordinate r(fm) Radial coordinate r(fm)

  17. Momentum dependence of self-energy s-waveand s-wave + p-wave(rho meson) Meff (MeV) Im(Π) (MeV^2) p(MeV) Red line: I=0,2 139.5 Blue line: I=0,1,2 Stronger absorption due to rho meson resonance Momentum p(MeV) Repulsive at higher momentum Effective mass given by real part of self-energy Imaginary part of self-energy

  18. Applying for HBT interferometry, how is the image distorted ? Numerical result is coming soon.

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