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

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

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)

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

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 )

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

On-shell limit 2) Γ(X,P) such that 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 !

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)

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!

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!

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

Spectral function in off-shell transport model Assumptions used in transport model (to speed up calculations): Collisional width  in low density approximation:  Coll (M,p,  ) =  VN tot  Collisional width  in low density approximation:  Coll (M,p,  ) =  VN tot  replace  VN tot  by averaged value G=const:  Coll (M,p,  ) =  G replace  VN tot  by averaged value G=const:  Coll (M,p,  ) =  G Collisional width of the particle in the rest frame (keep only loss term in eq.(1)): Spectral function:  Collisional width is defined by all possible interactions in the local cell total width:  tot =  vac +  Coll with

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

Modelling of in-medium off-shell production cross sections for vector mesons Low energy BB and mB interactions Low energy BB and mB interactions (s ½ < 2.2 GeV) High energy BB and mB interactions High energy BB and mB interactions (s ½ > 2.2 GeV) New in HSD: implementation of the in-medium spectral functions A(M,  ) for broad resonances inside FRITIOF Originally in FRITIOF (PYTHIA/JETSET): A(M) with constant width around the pole mass M 0 E.L.B. &W. Cassing, NPA 807 (2008) 214

Time integration method for dileptons   t0t0t0t0 t abs time FFFF  F – final time of computation in the code t 0 – production time t abs – absorption (or hadronic decay) time  e-e-e-e- e+  e-e-e-e- e+ ‚Reality‘: ‚Virtual‘ – time integ. method: only ONE e+e- pair with probability ~ Br(  ->e+e-)= Calculate probability P(t) to emit an e+e- pair at each time t and integrate P(t) over time!  : t 0 < t < t abs  : t 0 <t < infinity Cf. G.Q. Li & C.M. Ko, NPA582 (1995) 731

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

Summary I Accounting of in-medium effects requires : 1)off-shell transport models 2)time integration method

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

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

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

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 bremsstrahlung is the dominant contribution in p+d for 0.15 < M < 0.55 GeV at ~1-1.5 A GeV

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 !

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

Bremsstrahlung in UrQMD 1.3 (1998) Ernst et al, PRC58 (1998) 447  Bremsstrahlung-UrQMD’98 smaller than bremsstrahlung from Kaptari’06 by a factor of 3-6 SPA: SPA implementation in UrQMD (1998): e + e - production in elastic NN collisions (similar to HSD) SPA implementation in UrQMD (1998): e + e - production in elastic NN collisions (similar to HSD) „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“) „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“)

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

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

Bremsstrahlung in IQMD (Nantes) M. Thomere, C. Hartnack, G. Wolf, J. Aichelin, PRC75 (2007) 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!) HADES: C+C, 2 A GeV

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

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

Part II Elementary channels: Elementary channels:   -Dalitz decay   -Dalitz decay pp, pn and pd reactions vs. new HADES data pp, pn and pd reactions vs. new HADES data

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

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

 -Dalitz decay similar results for the  -Dalitz electromagnetic decay from different models ! similar results for the  -Dalitz electromagnetic decay from different models ! starting point: the same Lagrangian for the  N-vertex starting point: the same Lagrangian for the  N-vertex small differences are related to a different treatment of the 3/2 spin states small differences are related to a different treatment of the 3/2 spin states

 -spectral function The main differences in the dilepton yield from the  -Dalitz decay are related not to the electromagnetic decay but to the treatment of  - dynamics in the transport models !

   – dynamics vs. TAPS data Constraints on ,  by TAPS data: Constraints on ,  by TAPS data: HSD: good description of TAPS data on ,  multiplicities and m T -spectra HSD: good description of TAPS data on ,  multiplicities and m T -spectra => ,  dynamics under control ! E.L.B. &W. Cassing, NPA 807 (2008) 214

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

Quasi-free pn (pd) reaction: HADES 1.25 GeV "quasi-free" p+n 1.25 GeV  e+e- >9 0 PLUTO   HSD predictions: underestimates the HADES p+n (quasi-free) data at 1.25 GeV: 1)0.2<M<0.55 GeV:  -Dalitz decay by a factor of ~10 is larger in PLUTO than in HSD since the c hannels d + p  p spec + d +  (‘quasi-free’  -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 

Quasi-free pn 1.25 GeV:  -channel 1) p + n  d +  2) d + p  p spec + d +  Add the following channels: Now HSD agrees  with PLUTO on the  - Dalitz decay!

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

Ratio 1.25 GeV HSD shows a qualitative agreement with the HADES data on the ratio: accounting for the subthreshold  production via N(1520) decay should improve the agreement !

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

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

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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

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

Test in HSD: bremsstrahlung production in NN collisions (only elastic vs. all) In HSD assume: e + e - production from „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) !

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 E.B., W. Cassing and U. Mosel, NPA686 (2001) 568 Total deuteron energy:  Dispersion relation I. (used): (fulfill the binding energy constraint)  Dispersion relation II.: I II