1. A L C O R History of the idea Extreme relativistic kinematics Hadrons from quasiparticles Spectral coalescence T.S. Bíró, J. Zimányi †, P. Lévai, T. Csörgő, K. Ürmössy MTA KFKI RMKI Budapest, Hungary From quark combinatorics to spectral coalescence

2. A L C O R: the history Algebraic combinatoric rehadronization Nonlinear vs linear coalescence Transchemistry Recombination vs fragmentation Spectral coalescence

3. Quark recombination : combinatoric rehadronization 1981

4. Quark recombination : combinatoric rehadronization

5. Robust ratios for competing channels PLB 472 p. 243 2000

6. Collision energy dependence in ALCOR

7. 100 10 0 2468 leading rapidity Stopped per cent of baryons AGS SPS RHIC LHC

8. Collision energy dependence in ALCOR 200 100 0 246810 leading rapidity Newly produced light dN/dy AGS SPS RHIC LHC

9. Collision energy dependence in ALCOR 0.2 0.1 0 246810 leading rapidity K+ / pi+ ratio AGS SPS RHIC LHC

10. A L C O R: kinematics 2-particle Hamiltonian massless limit virial theorem coalescence cross section

11. A L C O R: kinematics Non-relativistic quantum mechanics problem

12. Virial theorem for Coulomb Deformed energy addition rule

13. Test particle simulation x y h(x,y) = const. E E E E 1 3 4 2 uniform random: Y(E ) = (  h/  y) dx ∫ 0 E 3 3 E E h=const

14. Massless kinematics Tsallis rule

15. A special pair-energy: E = E + E + E E / E 121 2 12c (1 + x / a) * (1 + y / a ) = 1 + ( x + y + xy / a ) / a Stationary distribution: f ( E ) = A ( 1 + E / E ) c - v

16. Color balanced pair interaction E = E + E + D 121 color state 2 Singlet channel: hadronization color state D + 8 D = 0 singletoctet Octet channel: parton distribution E = E + E - D 121 2 singlet E = E + E + D / 8 121 2 octet

17. Semiclassical binding: E = E + E - D = E + E - D 121 2 Zero mass kinematics (for small  angle): Octet channel: Tsallis distribution singlettotrel kin rel kin E = 4 sin (  / 2) E E E + E 1 1 2 2 2 constant? 4 / E c Singlet channel: convolution of Tsallis distributions - D / 2 virial Coulomb for

18. Coalescence cross section a: Bohr radius in Coulomb potential Pick-up reaction in non-relativistic potential

19. Limiting temperature with Tsallis distribution N = E – j T TE T = E / d  ; c c j=1 d c H Massless particles, d-dim. momenta, N-fold For N  2: Tsallis partons  Hagedorn hadrons ( with A. Peshier, Giessen ) hep-ph/0506132

20. Temperature vs. energy

21. Hadron mass spectrum from X(E)- folding of Tsallis N = 2 N = 3

22. A L C O R: quasiparticles continous mass spectrum limiting temperature QCD eos  quasiparticle masses Markov type inequalities

23. High-T behavior of ideal gases Pressure and energy density

24. High-T behavior of a continous mass spectrum of ideal gases „interaction measure” Boltzmann: f = exp(-  / T)   (x) =  x K1(x)

25. High-T behavior of a single mass ideal gas „interaction measure” for a single mass M: Boltzmann: f = exp(-  / T)   (0) = 

26. High-T behavior of a particular mass spectrum of ideal gases Example: 1/m² tailed mass distribution

27. High-T behavior of a continous mass spectrum of ideal gases High-T limit ( µ = 0 ) Boltzmann: c =  /2, Bose factor  (5), Fermi factor  (5) Zwanziger PRL, Miller hep-ph/0608234 claim: (e-3p) ~ T

28. High-T behavior of lattice eos SU(3)

29. High-T behavior of lattice eos hep-ph/0608234 Fig.2 8 × 32 ³

30. High-T behavior of lattice eos

32. From pressure to mass Pressure of relativistic ideal gas of massive particles: Lattice QCD data from Budapest-Wuppertal: JHEP 0601, 089, 2006. Bielefeld: NPB 469, 419, 1996.

33. Lattice QCD eos + fit Peshier et.al. Biro et.al.

34. Quasiparticle mass distribution by inverting the Boltzmann integral Inverse of a Meijer trf.: inverse imaging problem!

35. Bounds on integrated mdf Markov, Tshebysheff, Tshernoff, generalized Applied to w(m): bounds from p Applied to w(m;µ,T): bounds from e+p –Boltzmann: mass gap at T=0 –Bose: mass gap at T=0 –Fermi: no mass gap at T=0 Lattice data

36. Markov inequality and mass gap T and µ dependent w(m) requires mean field term, but this is cancelled in (e+p) eos data!

37. Boltzmann scaling functions  

38. General Markov inequality Relies on the following property of the function g(t): i.e.: g() is a positive, montonic growing function.

39. Markov inequality and mass gap There is an upper bound on the integrated probability P( M ) directly from (e+p) eos data!

40. SU(3) LGT upper bounds

41. 2+1 QCD upper bounds

42. A L C O R: spectral coalescence p-relative << p-common convolution of thermal distributions convolution of Tsallis distributions convolution with mass distributions

43. Idea: Continous mass distribution Quasiparticle picture has one definite mass, which is temperature dependent: M(T) We look for a distribution w(m), which may be temperature dependent

44. Why distributed mass? valence mass  hadron mass ( half or third…) c o a l e s c e n c e : c o n v o l u t i o n Conditions: w ( m ) is not constant zero probability for zero mass Zimányi, Lévai, Bíró, JPG 31:711,2005 w(m) w(m) w(had-m)

45. Coalescence from Tsallis distributed quark matter

50. Kaons

51. Recombination of Tsallis spectra at high-pT

52. (q-1) is a quark coalescence parameter

53. Properties of quark matter from fitting quark-recombined hadron spectra T (quark) = 140 … 180 MeV q (quark) = 1.22 power = 4.5 (same as for e+e- spectra) v (quark) = 0 … 0.5 Pion: near coalescence (q-1) value

54. SQM 1996 Budapest

56. July 22, 2006, Budapest