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Dileptons from off-shell transport approach

Dileptons from off-shell transport approach. E lena Bratkovskaya 5.07.2008 , HADES Collaboration Meeting XIX, GSI, Darmstadt. Overview. Study of in-medium effects in heavy-ion collisions require: off-shell transport dynamics in-medium transition rates time-integration methods

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Dileptons from off-shell transport approach

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  1. Dileptons from off-shell transport approach Elena Bratkovskaya 5.07.2008 , HADES Collaboration Meeting XIX, GSI, Darmstadt

  2. Overview • Study of in-medium effects in heavy-ion collisions require: • off-shell transport dynamics • in-medium transition rates • time-integration methods • Bremsstrahlung • HSD results and comparison of transport models • Elementary channels: • h-Dalitz decay • D-Dalitz decay • pp, pn and pd reactions vs. new HADES data

  3. Dileptons from transport models Theory (status: last millenium < 2000) : Implementation of in-medium vector mesons (r,w) scenarios (= ‚dropping‘ mass and ‚collisional broadening‘) in on-shelltransport 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 (r,w,f) scenarios (= ‚dropping‘ mass and ‚collisional broadening‘) in off-shelltransport models: • HSD (>2000) • BRoBUU (Rossendorf) (2006)

  4. Changes of the particle properties in the hot and dense baryonic medium r meson spectral function 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 How to treat in-medium effects in transport approaches?

  5. From Kadanoff-Baym equations to transport equations Generalized transport equations = first order gradient expansion of the Wigner transformed Kadanoff-Baym equations: Operator <> - 4-dimentional generalizaton of the Poisson-bracket drift term Vlasov term backflow term Backflow term incorporates the off-shell behavior in the particle propagation ! vanishes in the quasiparticle limit collision term =‚loss‘ term -‚gain‘ term The imaginary part of the retarded propagator is given by the normalized spectral function: For bosons in first order gradient expansion: GXP – 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 ) W. Cassing et al., NPA 665 (2000) 377; 672 (2000) 417; 677 (2000) 445

  6. 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 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) !

  7. On-shell limit 1) 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 ! 2) 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 !

  8. ‚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 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

  9. 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 the 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)

  10. 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 E.L.B. &W. Cassing, NPA 807 (2008) 214 On-shell model: low mass r and w mesons live forever and shine dileptons! The off-shell spectral function becomes on-shell in the vacuum dynamically by propagation through the medium!

  11. Collision term in off-shell transport models 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!

  12. Spectral function in off-shell transport model Collisional width of the particle in the rest frame (keep only loss term in eq.(1)): with Spectral function: total width: Gtot=Gvac+GColl • Collisional width is defined by all possible interactions in the local cell • Assumptions used in transport model (to speed up calculations): • 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

  13. Modelling of in-medium spectral functions for vector mesons In-medium scenarios: dropping mass collisional broadening dropping mass + coll. broad. • m*=m0(1-a r/r0) G(M,r)=Gvac(M)+GCB(M,r) m* & GCB(M,r) Collisional width GCB(M,r) = g r <u sVNtot> r-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(s,r) E.L.B., NPA 686 (2001), E.L.B. &W. Cassing, NPA 807 (2008) 214

  14. Modelling of in-medium off-shell production cross sections for vector mesons • Low energy BB and mB interactions (s ½ < 2.2 GeV) • High energy BB and mB interactions (s ½ > 2.2 GeV) New in HSD: implementation of the in-medium spectral functions A(M,r) for broad resonances inside FRITIOF Originally in FRITIOF (PYTHIA/JETSET): A(M) with constant width around the pole mass M0 E.L.B. &W. Cassing, NPA 807 (2008) 214

  15. Time integration method for dileptons Cf. G.Q. Li & C.M. Ko, NPA582 (1995) 731 ‚Reality‘: e+ only ONE e+e- pair with probability ~ Br(r->e+e-)=4.5 .10-5 r r w e- ‚Virtual‘ – time integ. method: t0 tabs e+ r time e- tF Calculate probability P(t) to emitan e+e- pair at each time t and integrate P(t) over time! r: t0 < t < tabs w: t0 <t < infinity tF – final time of computation in the code t0 – production time tabs – absorption (or hadronic decay) time

  16. The time integration method for dileptons in HSD Dilepton emission rate: e+ r e- t0=0 time tF Dilepton invariant mass spectra: tF < t < infinity 0 < t < tF The time integration method allows to account for the in-medium dynamics of vector mesons!

  17. Summary I Accounting of in-medium effects requires : • off-shell transport models • time integration method

  18. e+ e+ g* N N R g* R N R N N N N e- N e- = Dilepton channels in HSD • All particles decaying to dileptons are first produced in BB, mB or mm collisions • ‚Factorization‘ of diagrams in the transport approach: • The dilepton spectra are calculated perturbatively with the time integration method.

  19. g*->e+e- Phase-space corrected soft-photon cross section: Soft-Photon-Approximation (SPA): NN bremsstrahlung - SPA N N -> N N e+e- ‚quasi- elastic‘ N N -> N N elastic NN ‚off-shell‘ correction factor SPA implementation in HSD: e+e- production in elastic NN collisions with probability:

  20. Bremsstrahlung – a new view on an ‚old‘ story New OBE-model (Kaptari&Kämpfer, NPA 764 (2006) 338): • pn bremstrahlung is larger by a factor of 4 than it has been calculated before (and used in transport calculations before)! • pp bremstrahlung is smaller than pn, however, not zero; consistent with the 1996 calculations from F.de Jong in a T-matrix approach 2007 (HADES): The DLS puzzle is solved by accounting for a larger pn bremsstrahlung !!!

  21. HSD: Dileptons from p+p and p+d - DLS • bremsstrahlung is the dominant contribution in p+d for 0.15 < M < 0.55 GeV at ~1-1.5 A GeV

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

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

  24. Ernst et al, PRC58 (1998) 447 Bremsstrahlung in UrQMD 1.3 (1998) SPA: • SPA implementation in UrQMD (1998): e+e- production in elastic NN collisions (similar to HSD) • Bremsstrahlung-UrQMD’98 smaller than bremsstrahlung from Kaptari’06 • by a factor of 3-6 • „old“bremsstrahlung: missing yield for p+d and A+A at 0.15 < M < 0.55 GeV at 1 A GeV (consistent with HSD employing „old SPA“)

  25. D. Schumacher, S. Vogel, M. Bleicher, Acta Phys.Hung.A27 (2006) 451 Dileptons from A+A - UrQMD 2.2 (2007) NO bremsstrahlung in UrQMD 2.2

  26. C. Fuchs et al., Phys. Rev. C67 025202(2003) HADES - RQMD‘07 Dileptons from A+A - RQMD (Tübingen) DLS - RQMD‘03 1 A GeV • NO bremsstrahlung in RQMD (missing yieldfor p+d at 0.15 < M < 0.55 GeV at ~1-1.5 A GeV) • too strong D-Dalitz contribution (since no time integration?)

  27. M. Thomere, C. Hartnack, G. Wolf, J. Aichelin, PRC75 (2007) 064902 Bremsstrahlung in IQMD (Nantes) HADES: C+C, 2 A GeV SPA implementation in IQMD : e+e- bremsstrahlung production in each NN collision(i.e. elastic and inelastic) ! - differs from HSD and UrQMD’98 (only elastic NN collisions are counted!)

  28. H.W. Barz, B. Kämpfer, Gy. Wolf, M. Zetenyi, nucl-th/0605036 Bremsstrahlung in BRoBUU (Rossendorf) SPA implementation in BRoBUU : e+e- production in each NN collision(i.e. elastic and inelastic) ! - similar to IQMD (Nantes)

  29. Summary II Transport models give similar resultsONLY with the same initial input ! => REQUESTS: „unification“ of the treatment of dilepton production in transport models: • Similar cross sections for elementary channels • Time-integration method for dilepton production • Off-shell treatment of broad resonances + Consistent microscopic calculations for e+e- bremsstrahlung from NN and mN collisions!

  30. Part II • Elementary channels: • h-Dalitz decay • D-Dalitz decay • pp, pn and pd reactions vs. new HADES data

  31. h-production cross section in pp and pn • HSD:good description of the experimental data (Celsius/WASA) on inclusive h production cross section in pp and pn collisions • => h-Dalitzdecay contribution is under control ! E.L.B. &W. Cassing, NPA 807 (2008) 214

  32. D-Dalitz decay Original paper: H.F. Jones, M.D. Scadron, Ann. Phys. 81 (1973) 1

  33. D-Dalitz decay • similar results for the D-Dalitz electromagnetic decay from different models ! • starting point: the same Lagrangian for the gDN-vertex • small differences are related to a different treatment of the 3/2 spin states

  34. D-spectral function Themain differences in the dilepton yield from the D-Dalitz decay are related not to the electromagnetic decay but to the treatment of D-dynamics in the transport models !

  35. p0, h, D – dynamics vs. TAPS data • Constraints on p, h by TAPS data: • HSD: good description of TAPS data on p, h multiplicities and mT-spectra • => p (D), h dynamics under control ! E.L.B. &W. Cassing, NPA 807 (2008) 214

  36. pp @ 1.25GeV : new HADES data PLUTO • D-Dalitz decay is the dominant channel (HSD consistent with PLUTO) • HSD predictions:good description of new HADES data p+p data! E.L.B. &W. Cassing, NPA 807 (2008) 214

  37. "quasi-free" p+n 1.25 GeV e+e->90 PLUTO   Quasi-free pn (pd) reaction: HADES data @ 1.25 GeV  HSD predictions: underestimates the HADES p+n (quasi-free) data at 1.25 GeV: 0.2<M<0.55 GeV: h-Dalitz decay by a factor of ~10 is larger in PLUTO than in HSD since the channels d + p pspec + d + (‘quasi-free’ h-production -dominant at 1.25GeV!) and p + n d +  were NOT taken into account ! Note: these channels have NO impact for heavy-ion reactions and even for p+d results at higher energies! *In HSD:p+d = p + (p&n)-with Fermi motion according to the Paris deuteron wave function

  38. Quasi-free pn (pd) @ 1.25 GeV: h-channel Add the following channels: 1) p + n d +  2) d + p pspec + d +  Now HSD agreeswith PLUTO on the h- Dalitz decay!

  39. Quasi-free pn (pd) @ 1.25 GeV: N(1520) ?! N(1520) 2)M > 0.45 GeV: HSD preliminary result for p+d @1.25 GeV shows that the missing yield might be attributed to subthreshold r-production via N(1520) excitation and decay ?! N(1520) Similar to our NPA686 (2001) 568 Model for N(1520): according to Peters et al., NPA632 (1998) 109

  40. Ratio pd/pp @ 1.25 GeV HSD shows a qualitative agreement with the HADES data on the ratio: accounting for the subthreshold r-production via N(1520) decay should improve the agreement !

  41. Outlook HADES succeeded: the DLS puzzle is solved ! Outlook-1: need new pp,pd andpN data from HADES for a final check! Outlook-2: study in-medium effects with HADES

  42. Thanks to HADES collegues: Yvonne, Gosia, Romain, Piotr, Joachim, Tatyana, Volker … + Wolfgang +

  43. - More slides -

  44. 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) e.g. for bosons Greens functions S / self-energies S : retarded (ret), advanced (adv) (anti-)causal (a,c ) • do Wigner transformation 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

  45. Kaptari&Kämpfer, NPA 764 (2006) 338 OBE-model: N N -> N N e+e- NN bremsstrahlung: OBE-model ‚pre‘ ‚post‘ ‚pre‘ ‚post‘ + gauge terms The strategy to restore gauge invariance is model dependent! charged meson exchange contact terms (from formfactors)

  46. Test in HSD:bremsstrahlung production in NN collisions (only elastic vs. all) In HSD assume: e+e- productionfrom „old“ SPA bremsstrahlung in each NN collision (i.e. elastic and inelastic reactions) => can reproduce the results by Gy. Wolf et al., i.e. IQMD (Nantes) and BRoBUU (Rossendorf) !

  47. Deuteron in HSD In HSD:p+d = p + (p&n) -with Fermi motion according to the momentum distribution f(p) with Paris deuteron wave function • Dispersion relation I. (used): • (fulfill the binding energy constraint) Total deuteron energy: • Dispersion relation II.: I II E.B., W. Cassing and U. Mosel, NPA686 (2001) 568

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