Off-shell relativistic transport theory

1. Off-shell relativistic transport theory Lecture II: Elena Bratkovskaya 27-31 Aug. 2008 , 20th Indian-Summer School of Physics& 4th HADES Summer School ‚Hadrons in the Nuclear Medium‘

2. From classical dynamics to quantum-field dynamics

3. Outlook of the Lecture 1: Quantum mechanical description of the many-body system = = Correlation dynamics ! BBGKY-Hierarchy (Bogolyubov, Born, Green, Kirkwood and Yvon) !!! truncation scheme !!! 2-body density matrix: 2PI= 2-particle-irreducible approach 1PI = 1-particle-irreducible approach = TDHF approximation 2-body correlations Vlasov + Uehling-Uhlenbeck equation

4. Quantum-field dynamics: EoM • Lagrangian density can be constructed from field f and its derivatives: • The functional action can be constructed by integrating over space-time • Then by enforcing the stationary action principle (variational principle), the Euler-Lagrange equationsemerge: But the BBGKY hierarchy problem appears again! For solution employ a consistent 2PI truncation scheme

5. A simple example: Ф4 - theory The interacting field theory for spinless massive scalar bosons provides a ‚theoretical laboratory‘ for testing aproximation schemes Lagrangian density: Use a closed-time-path (CTP) method: Path-ordered Green functions are defined along the contour by: Heisenberg picture

6. Two-particle irreducible approaches Two-particle irreducible (2PI) effective action G (general form!): (.)p means convolution integral over the closed time-path; G0 - free propagator • Ф(G) is the sum of all closed 2PI diagrams built up by the full G(x,y) E.g.:Ф4 – theory: Ф(G)up to 3-loop order; ~ 2nd orded in l x x y d: dimension of space • Self-energies are defined by the variation of Ф w.r.t G(y,x): cut a line in Ф and stretch:  e.g. Ф4 – theory

7. Equations of motion for G(x,y) are obtained from the variation of the effective action: where the full path-ordered Green function G is defined bythe Dyson-Schwinger equation (DSE): Cut Green function into different pieces: Note: only two Green functions are independent! Tc / Ta denote time ordering on the upper/lower branch of the real-time contour Do the same separation for the self-energy:  DSE becomes 2x2 matrix equation

8. 2PI self-energies in Ф4 - theory Local in space and time part: tadpole Nonlocal part: sunset ‚potential‘ term (~λ) leads to the generation of an effective mass for the field quanta interaction term (~λ2)

9. Kadanoff-Baym equations of motion for Ф4 - theory obtained from the variation of the effective action 1) potential term interaction term d: dimension of space 2) Thus, the Kadanoff-Baym equations include the influance of mean-field on the particle propagation (generated by the tadpole diagram) as well as scattering processes as inherent in the sunset diagram. Go ahead and solve KBE with some initial condition !

10. Retarded and advanced Green functions < < • retarded/advanced Green functionscontain only spectral information (no information on particle density)! From Dyson-Schwinger equations => • retarded/advanced Green functions only depend on retarded/advanced quantities

11. Kadanoff-Baym equation for heavy-ion dynamics Let‘s go now from Ф4 – theory to the real dynamics of heavy-ion collisions Dynamics of heavy-ion collisions is a many-body problem!  Correct way to solve the many-body problem including all quantum mechanical features  Kadanoff-Baym equations for Green functions (KBE: 1962)

12. e.g. for bosons: Kadanoff-Baym equation Kadanoff-Baym equations for Green functions S< (S<=>G) Greens functions S / self-energies S : Sxyret - retarded (ret) Green function Sxyadv - advanced (adv) Sxya (Sxyc) - (anti-)causal (a,c ) Ta(Tc) – (anti-)time-ordering operators h=+1 for bosons, h=-1 for fermions

13. Wigner transformation of the Kadanoff-Baym equation • do Wigner transformation of the Kadanoff-Baym equation X=(x+y)/2 – space-time coordinate, P – 4-momentum Convolution integrals convert under Wigner transformation as Operator M is a 4-dimentional generalizaton of the Poisson-bracket: an infinite series in the differential operator M • consider only contribution up tofirst order in the gradients  = astandard approximationof kinetic theory which is justified if the gradients in the mean spacial coordinate X are small

14. From Kadanoff-Baym equations to transport equations I • separate all retarded and advanced quantities – Geen functions and self energies – into real and imaginary parts: The imaginary part of the selfenergy corresponds to the width GXP ; then from Dyson-Schwinger equation: The imaginary part of the retarded propagator is given by the normalized spectral function AXP: algebraic solution The spectral function AXP in first order gradient expansion (for bosons) : The real part of the retarded propagator in first order gradient expansion : AXP and ReSXPret in first order gradient expansion depend ONLY on SXPret !

15. drift term Vlasov term backflow term collision term =‚loss‘ term - ‚gain‘ term From Kadanoff-Baym equations to transport equations II After the first order gradient expansion of the Wigner transformed Kadanoff-Baymequations and separation into the real and imaginary parts one gets: 1. Generalized transport equations: Backflow term incorporates the off-shell behavior in the particle propagation ! vanishes in the quasiparticle limit 2. Generalized mass-shell equations: !Eqs. (1) and (2) are not fully consistent =>differ by higher order gradienttermssince (1) contains ReSXPret in the backflow term

16. From Kadanoff-Baym equations to transport equations III • The Botermans-Malfliet solution (1990): • separate spectral information from occupation density: • N -number distribution • A - spectral function • - width of spectral function = reaction rate of particle (at phase-space position XP) Greens function S< characterizes the number of particles (N) and their properties (A – spectral function ) • rewrite with =>‚correction term‘ = collision term /AXP andof higher gradient order => has to be omitted for consistency ! => as a consequence: NS -> N , S< -> S< .G/A => Generalized transport equations can be written: (3) ! now consistent in gradient order with the mass-shell equation !

17. General testparticle off-shell equations of motion W. Cassing , S. Juchem, NPA 665 (2000) 377; 672 (2000) 417; 677 (2000) 445 Employ testparticle Ansatz for the real valued quantity iS<XP - insert in (3) 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) !

18. Limiting cases 1) No explicit energy dependence of Σret(X,P)  C(i) = 0 2) Γ(X,P) = Γ(X) - width depends only on space-time X: P = use M2 as an independent variable => and fix P0 by follows: i.e. the deviation of Mi2 from the pole mass (squared) M02 scales with Γi !

19. On-shell limit 3) Γ(X,P)  0 quasiparticle approximation : A(X,P) = 2 pd(P2-M2) || Hamiltons equation of motion - independent on Γ ! Backflow term - which incorporates the off-shell behavior in the particle propagation - vanishes in the quasiparticle limit ! 4) Γ(X,P) such that E.g.: Γ = const G=Γvacuum(M) ‚Vacuum‘ spectral function with constant or mass dependent width G: spectral function AXP does NOT change the shape (and pole position) during propagation through the medium(backflow term vanishes also!) <=> • Hamiltons equation of motion - independent on Γ !

20. Model cases stable particle unstable particle energy energy momentum momentum mass mass Imag. Imag. Real Real

21. ‚On-shell‘ transport models Basic concept of the ‚on-shell‘ transport models (VUU, BUU, QMD etc. ): 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 pd(p2-M2) Avacuum(M) • for each particle species i (i = N, R, Y, p, r, K, …) the phase-space density fi followsthe BUU transport equations • with collision termsIcoll describing elastic and inelastic hadronic reactions: baryon-baryon, meson-baryon, meson-meson, formation and decay of baryonic and mesonicresonances, string formation and decay (for inclusive particle production: BB -> X , mB ->X, X =many particles) • with propagation of particles in self-generated mean-field potential U(p,r)~Re(Sret)/2p0 • Numerical realization – solution of classical equations of motion + Monte-Carlo simulations for test-particle interactions

22. Short-lived resonances in semi-classical transport models Spectral function: width G ~ -Im Sret /M Vacuum (r =0)narrow states In-medium: production of broad states In-medium r >> r0 • Example : • r-meson propagation through the medium within on-shell BUU model • broad in-medium spectral function does not become on-shell in vacuum in ‚on-shell‘ transport models! BUU: M. Effenberger et al, PRC60 (1999)

23. Problems in the treatment of short-lived resonances in the on-shell semi-classical transport models • Problem: • dynamical changes of spectral function by propagation through the medium are NOT included in the ‚on-shell‘ semi-classical transport equations ! • the resonance spectral function can be changed only due to explicit collisions with other particles in ‚on-shell‘ semi-classical transport models ! Reason for the problem: backflow term*is missing in the explicit ‚on-shell‘ dynamical equations since this backflow term vanishes in the on-shell limit, however, does NOT vanish in the off-shell limit (i.e. becomes very important for the dynamics of broad resonances)! * Generalized transport equations Operator <> - 4-dimentional generalizaton of the Poisson-bracket drift term Vlasov term backflow term ‚loss‘ term‚gain‘ term W. Cassing et al., NPA 665 (2000) 377

24. Off-shell vs. on-shell transport dynamics Time evolution of the mass distribution of r and w mesons for central C+C collisions (b=1 fm) at 2 A GeV for dropping mass + collisional broadening scenario The off-shell spectral function becomes on-shell in the vacuum dynamically by propagation through the medium!

25. Collision term in off-shell transport model Collision term for reaction 1+2->3+4: with The trace over particles 2,3,4 reads explicitly for fermions for bosons additional integration 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!

26. Collisional width of the particle in the rest frame (keep only loss term in eq.(1)): with Spectral function in off-shell transport model Spectral function: total width: Gtot=Gvac+GColl • Collisional width is defined by all possible interactions in the local cell  Assumptions used in transport models (to speed up calculations), e.g.: • Collisional widthin low density approximation: GColl(M,p,r) = g r <u sVNtot> • replace <u sVNtot> by averaged value G=const: GColl(M,p,r) = g r G

27. p p p p p In-medium transition rates: G-matrix approach Need to know in-medium transition amplitudes G and their off-shell dependence Coupled channel G-matrix approach Transition probability : E.g.: in-medium K- in G-matrix approach (L. Tolos et al.): • G(p,r,T) - G-matrix from the solution of coupled-channel equations • with pion dressing, i.e. including the pion self-energy in the intermediate pL, pS states • without pion dressing W. Cassing, L. Tolos, E.L.B., A. Ramos, NPA 727 (2003) 59

28. Transition probability for pY<-> K-p (Y = L,S) • With pion dressing: • L(1405) and S(1385) melt away with baryon density • K- absorption/productionfrom pY collisions are strongly suppressed in the nuclear medium • However, pY is still the dominant channel for K- production in heavy-ion collisions ! S(1385) L(1405) W. Cassing, L. Tolos, E.L.B., A. Ramos, NPA 727 (2003) 59

29. In-medium effects  off-shell transport • 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 the in-medium effects