1. Study of in-medium effects within transport approaches Elena Bratkovskaya 25.06.2008, Workshop ‚Hadrons@FAIR‘ FIAS, Uni. Frankfurt, Frankfurt am Main

2. 2 Changes of the particle properties in the hot and dense baryonic medium How to treat in-medium effects in transport approaches? In-medium models:  chiral perturbation theory  chiral SU(3) model  coupled-channel G-matrix approach  chiral coupled-channel effective field theory predict changes of the particle properties in the hot and dense medium, e.g. broadening of the spectral function   meson spectral function L. Tolos et al., NPA 690 (2001) 547

3. 3 Dynamics of heavy-ion collisions –> complicated many-body problem! Correct way to solve the many-body problem including all quantum mechanical features  Kadanoff-Baym equations for Green functions S < (from 1962) Greens functions S / self-energies  : e.g. for bosons  do Wigner transformation retarded (ret), advanced (adv) (anti-)causal (a,c ) consider only contribution up to first order in the gradients consider only contribution up to first order in the gradients = a standard approximation of kinetic theory which is justified if the gradients in = a standard approximation of kinetic theory which is justified if the gradients in the mean spacial coordinate X are small the mean spacial coordinate X are small

4. 4 From Kadanoff-Baym equations to transport equations drift term Vlasov term collision term = ‚loss‘ term -‚gain‘ term Operator <> - 4-dimentional generalizaton of the Poisson-bracket backflow term Generalized transport equations = first order gradient expansion of the Wigner transformed Kadanoff-Baym equations: The imaginary part of the retarded propagator is given by the normalized spectral function: For bosons in first order gradient expansion:  XP – width of spectral function = reaction rate of particle (at phase- space position XP) W. Cassing et al., NPA 665 (2000) 377; 672 (2000) 417; 677 (2000) 445 Backflow term incorporates the off-shell behavior in the particle propagation ! vanishes in the quasiparticle limit ! vanishes in the quasiparticle limit Greens function S < characterizes the number of particles (N) and their properties (A – spectral function )

5. 5 General testparticle off-shell equations of motion Employ testparticle Ansatz for the real valued quantity i S < XP - insert in generalized transport equations and determine equations of motion ! General testparticle off-shell equations of motion: with Note: the common factor 1/(1-C (i) ) can be absorbed in an ‚eigentime‘ of particle (i) ! W. Cassing, S. Juchem, NPA 665 (2000) 377; 672 (2000) 417; 677 (2000) 445

6. 6 On-shell limit 2) Γ(X,P) such that C X  =0 and C P  =0 E.g.: Γ = const  =Γ vacuum (M)  ‚Vacuum‘ spectral function with constant or mass dependent width  : (backflow term vanishes also!) spectral function A XP does NOT change the shape (and pole position) during propagation through the medium (backflow term vanishes also!) 1) Γ(X,P)  0 quasiparticle approximation : A(X,P) = 2   (P 2 -M 2 ) A(X,P) = 2   (P 2 -M 2 )|| Hamiltons equation of motion - independent on Γ ! Backflow term - which incorporates the off-shell behavior in the particle propagation - vanishes in the quasiparticle limit !  Hamiltons equation of motion - independent on Γ !

7. 7 for each particle species i (i = N, R, Y, , , K, …) the phase-space density f i follows the transport equations for each particle species i (i = N, R, Y, , , K, …) the phase-space density f i follows the transport equations  with collision terms I coll describing elastic and inelastic hadronic reactions: baryon-baryon, meson-baryon, meson-meson, formation and decay of baryonic and mesonic resonances, string formation and decay (for inclusive particle production: baryon-baryon, meson-baryon, meson-meson, formation and decay of baryonic and mesonic resonances, string formation and decay (for inclusive particle production: BB  X, mB  X, X =many particles) BB  X, mB  X, X =many particles)  with propagation of particles in self-generated mean-field potential U(p,  )~Re(  ret )/2p 0 Numerical realization – solution of classical equations of motion + Monte-Carlo simulations for test-particle interactions Numerical realization – solution of classical equations of motion + Monte-Carlo simulations for test-particle interactions ‚On-shell‘ transport models Basic concept of the ‚on-shell‘ transport models (VUU, BUU, QMD etc. ): 1)Transport equations = first order gradient expansion of the Wigner transformed Kadanoff-Baym equations 2) Quasiparticle approximation or/and vacuum spectral functions : A(X,P) = 2   (p 2 -M 2 ) A vacuum (M) A(X,P) = 2   (p 2 -M 2 ) A vacuum (M)

8. 8 Short-lived resonances in semi-classical transport models In-medium   Vacuum (  narrow states In-medium: production of broad states BUU: M. Effenberger et al, PRC60 (1999) width  Im  ret  width  Im  ret  Spectral function: Example :  -meson propagation through the medium within the on-shell BUU model  broad in-medium spectral function does not become on-shell in vacuum in ‚on-shell‘ transport models!

9. 9 Off-shell vs. on-shell transport dynamics The off-shell spectral function becomes on-shell in the vacuum dynamically by propagation through the medium! Time evolution of the mass distribution of  and  mesons for central C+C collisions (b=1 fm) at 2 A GeV for dropping mass + collisional broadening scenario E.L.B. &W. Cassing, NPA 807 (2008) 214 On-shell model: low mass  and  mesons live forever and shine dileptons!

10. 10 Collision term in off-shell transport models Collision term for reaction 1+2->3+4: Collision term for reaction 1+2->3+4: with The trace over particles 2,3,4 reads explicitly for fermions for bosons The transport approach and the particle spectral functions are fully determined once the in-medium transition amplitudes G are known in their off-shell dependence! additional integration

11. 11 Study of in-medium K + and K - properties in heavy-ion collisions within dynamical transport models Theory (status: last millenium < 2000) : Implementation of in-medium K + and K - scenarios (= ‚dropping‘ of K - mass and ‚enhancement‘of K + mass) in on-shell transport models: BUU (Texas) (1994), HSD (or RBUU) (1997), QMD (Tübingen) (1998), IQMD (Nantes) (2000),... Strangeness production at low energy Baryon-Baryon collisions: meson-Baryon collisions:meson-Baryon collisions: meson-meson collisions: Dominant channel for low energy K - production!

12. 12 Need to know in-medium transition amplitudes G and their off-shell dependence Coupled channel G-matrix approach Transition probability : with G(p, ,T) - G-matrix from the solution of coupled-channel equations  with pion dressing, i.e. including the pion self-energy in the intermediate  states  without pion dressing    W. Cassing, L. Tolos, E.L.B., A. Ramos, NPA 727 (2003) 59 L. Tolos et al., NPA 690 (2001) 547 Off-shell transport => In-medium transition rates: G-matrix approach

13. 13 K - production in A+A within the off-shell transport approach Au+Au (and Ni+Ni) data at 1.5 (1.93) A GeV are consistent with the ‚pion dressing‘ scenario, whereas C+C at 1.8 A GeV are underestimated even without pion dressing Au+Au (and Ni+Ni) data at 1.5 (1.93) A GeV are consistent with the ‚pion dressing‘ scenario, whereas C+C at 1.8 A GeV are underestimated even without pion dressing W. Cassing, L. Tolos, E.L.B., A. Ramos, NPA 727 (2003) 59 Updated plot‘07

14. 14 K - /K + ratio in Au+Au at 1.5 A GeV within the off-shell transport approach Data are consistent with the ‚pion dressing‘ scenario W. Cassing, L. Tolos, E.L.B., A. Ramos, NPA 727 (2003) 59

15. 15 Dileptons from transport models Theory (status: last millenium < 2000) : Implementation of in-medium vector mesons (  ) scenarios (= ‚dropping‘ mass and ‚collisional broadening‘) in on-shell transport models:  BUU/AMPT (Texas) ( > 1995)  HSD ( > 1995)  UrQMD v. 1.3 (1998)  RQMD (Tübingen) (2003), but NO explicit propagation of vector mesons  IQMD (Nantes) (2007), but NO explicit propagation of vector mesons Theory (status: this millenium > 2000) : Implementation of in-medium vector mesons (  ) scenarios (= ‚dropping‘ mass and ‚collisional broadening‘) in off-shell transport models:  HSD (>2000)  BRoBUU (Rossendorf) (2006)

16. 16 Modelling of in-medium spectral functions for vector mesons In-medium scenarios: In-medium scenarios: dropping mass collisional broadening dropping mass + coll. broad. dropping mass collisional broadening dropping mass + coll. broad. m*=m 0 (1-   )  (M,  )=  vac (M)+  CB (M,  ) m* &  CB (M,  m*=m 0 (1-   )  (M,  )=  vac (M)+  CB (M,  ) m* &  CB (M,  Collisional width  CB (M,  ) =  VN tot   meson spectral function: Note: for a consistent off-shell transport one needs not only in-medium spectral functions but also in-medium transition rates for all channels with vector mesons, i.e. the full knowledge of the in-medium off-shell cross sections  (s,  ) Note: for a consistent off-shell transport one needs not only in-medium spectral functions but also in-medium transition rates for all channels with vector mesons, i.e. the full knowledge of the in-medium off-shell cross sections  (s,  ) E.L.B., NPA 686 (2001), E.L.B. &W. Cassing, NPA 807 (2008) 214 E.L.B., NPA 686 (2001), E.L.B. &W. Cassing, NPA 807 (2008) 214

17. HSD: Dileptons from C+C at 1 and 2 A GeV - HADES HADES data show exponentially decreasing mass spectra HADES data show exponentially decreasing mass spectra Data are better described by in-medium scenarios with collisional broadening Data are better described by in-medium scenarios with collisional broadening In-medium effects are more pronounced for heavy systems such as Au+Au In-medium effects are more pronounced for heavy systems such as Au+Au

18. HSD: Dileptons from A+A at 1 A GeV - DLS bremsstrahlung and  -Dalitz are the dominant contributions in A+A for 0.15 < M < 0.55 GeV at 1 A GeV ! bremsstrahlung and  -Dalitz are the dominant contributions in A+A for 0.15 < M < 0.55 GeV at 1 A GeV !

19. 19 Dileptons at SPS NA60CERES NA60 and CERS data are better described by in-medium scenario with collisional broadening NA60 and CERS data are better described by in-medium scenario with collisional broadening

20. 20 Dileptons at RHIC PHENIX: Au+Au PHENIX: pp HSD provides a good description of pp data HSD provides a good description of pp data Standard in-medium effects of vector mesons -- compatible with the NA60 and CERES data at SPS – do not explain the large enhancement observed by PHENIX in the invariant mass from 0.2 to 0.5 GeV in Au+Au collisions at s 1/2 =200 GeV (relative to pp collisions) Standard in-medium effects of vector mesons -- compatible with the NA60 and CERES data at SPS – do not explain the large enhancement observed by PHENIX in the invariant mass from 0.2 to 0.5 GeV in Au+Au collisions at s 1/2 =200 GeV (relative to pp collisions)

21. 21 Summary Accounting of in-medium effects with medium dependent spectral functions requires off-shell transport models ! Accounting of in-medium effects with medium dependent spectral functions requires off-shell transport models ! Coupled-channel G-matrix + off-shell transport approach is the most consistent way to investigate in-medium effects Coupled-channel G-matrix + off-shell transport approach is the most consistent way to investigate in-medium effects

22. 22 Thanks to Laura Tolos Olena Linnyk Wolfgang Cassing