1. Enhancement of hadronic resonances in RHI collisions We explain the relatively high yield of charged Σ ± (1385) reported by STAR We show if we have initial hadrons multiplicity above equilibrium the fractional yield of resonances A*/A (A * →Aπ) can be considerably higher than expected in SHM model of QGP hadronization. We study how non-equilibrium initial conditions after QGP hadronization influence the yield of resonances. Inga Kuznetsova and Johann Rafelski Department of Physics, University of Arizona Work supported by a grant from: the U.S. Department of Energy DE-FG02-04ER4131

2. Time evolution equation for N Δ, N Σ Δ(1232) ↔ Nπ, width Γ≈120 MeV (from PDG); Σ(1385)↔Λπ,width Γ ≈ 35 MeV (from PDG). Reactions are relatively fast. We assume that others reactions don’t have influence on Δ (Σ) multiplicity. and are Lorentz invariant rates

3. Phases of RHI collision QGP phase; Chemical freeze-out (QGP hadronization); We consider hadronic gas phase between chemical freeze- out (QGP hadronization) and kinetic freeze-out; the hadrons yields can be changed because of their interactions; M. Bleicher and J.Aichelin, Phys. Lett. B, 530 (2002) 81 M. Bleicher and H.Stoecker,J.Phys.G, 30, S111 (2004) Kinetic freeze-out : reactions between hadrons stop; Hadrons expand freely (without interactions).

4. Motivations How resonance yield depends on the difference between chemical freeze-out temperature (QGP hadronization temperature) and kinetic freeze- out temperature? How this yield depends on degree of initial non- equilibrium? Explain yields ratios observed in experiment.

5. Distribution functions forin the rest frame of heat bath where x i =m i /T; K 2 (x) is Bessel function; g i is particle i degeneracy; Υ i is particle fugacity, i = N, Δ, Σ, Λ; Multiplicity of resonance:

6. Equations for Lorentz invariant rates

7. Bose enhancement factor: Fermi blocking factor: using energy conservation and time reversal symmetry: we obtained:

8. We obtained: I. Kuznetsova, T. Kodama and J. Rafelski, ``Chemical Equilibration Involving Decaying Particles at Finite Temperature '' in preparation. Equilibrium condition: is global chemical equilibrium. If in initial state then Δ production is dominant. If in initial state then Δ decay is dominant.

9. For Boltzmann distributions : We can write time evolution equation as whereis decay time in medium We assume that no medium effects, τ Δ ≈τ Δ vac We don’t know decay rate, we know decay width or decay time in vacuum τ Δ vac =1/Γ.

10. Model assumptions Δ(1232) ↔ N π is fast. Other reactions do not influence Δ yield or The same for Σ(1385)↔Λπ Large multiplicity of pions does not change in reactions Most entropy is in pions and entropy is conserved during expansion of hadrons as

11. Equation for Υ Δ(Σ) τ is time in fluid element comoving frame.

12. Expansion of hadronic phase Growth of transverse dimension: Taking we obtain: is expansion velocity At hadronization time τ h:

13. Solution for Υ Δ where Using particles number conservation: Δ + N = N 0 tot we obtain equation : The solution of equation is:

14. Non-equilibrium QGP hadronization γ q is light quark fugacity after hadronization Entropy conservation fixes γ q (≠1). Strangeness conservation fixes γ s (≠1). γ q is between 1.6 for T=140 MeV and 1 for T=180 MeV; Initially and Δ (Σ) production is dominant

15. Temperature as a function of time τ

16. The ratios N Δ /N Δ 0, N N /N N 0 as a function of T N Δ increases during expansion after hadronization when γ q >1 (Υ Δ < Υ N Υ π ) until it reaches equilibrium. After that it decreases (delta decays) because of expansion. Opposite situation is with N N. If γ q =1, there is no Δ enhancement, Δ only decays with expansion.

17. N Δ /N tot ratio as a function of T. N tot (observable) is total multiplicity of resonances which decay to N. Dot-dashed line is if we have only QGP freeze-out. Doted line (SHARE) is similar to dot-dashed line with more precise decays consideration. There is strong dependence of resulting ratio on hadronization temperature.

18. N Σ /Λ tot ratio as a function of T Experiment: J.Adams et al. Phys.Rev. Lett. 97, 132301 (2006) S.Salur, J.Phys.G, 32, S469 (2006) Observable: Effect is smaller than for Δ because of smaller decay width

19. Future study Γ≈ 150 MeV; Γ≈50 MeV Γ≈170 MeV

20. Conclusions If we have initial hadrons multiplicity above equilibrium the fractional yield of resonances A*/A can be considerably higher than expected in SHM model of QGP hadronization. Because of relatively strong temperature dependence, Δ/N tot can be used as a tool to distinguish the different hadronization conditions as chemical non-equilibrium vs chemical equilibrium; We have shown that the relatively high yield of charged Σ ± (1385) reported by STAR is well explained by our considerations and hadronization at T=140 MeV is favored.