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Pion and kaon spectra from distributed mass quark matter

Pion and kaon spectra from distributed mass quark matter. Hadronization by coalescence Quasiparticle mass and QCD eos Mass gap estimates due to Markov inequality Pion p spectra directly and from rho decay. Károly Ürmössy and Tamás S. Bíró KFKI Res.Inst.Part.Nucl.Phys. Budapest. .

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Pion and kaon spectra from distributed mass quark matter

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  1. Pion and kaon spectra from distributed mass quark matter • Hadronization by coalescence • Quasiparticle mass and QCD eos • Mass gap estimates due to Markov inequality • Pion p spectra directly and from rho decay Károly Ürmössy and Tamás S. Bíró KFKI Res.Inst.Part.Nucl.Phys. Budapest 

  2. Further collaborators • József Zimányi • Péter Lévai • Péter Ván • Gábor Purcsel hep-ph / 0607079, 0606076, 0605274, 0612085

  3. Hadronization bycoalescence Entropy vs lattice eos (PLB 650, 193, 2007)

  4. Lattice QCD eos: normalized pressure vs. temperature Aoki, Fodor, Katz, Szabo JHEP 0601:089, 2006

  5. Boltzmann mixtures

  6. Boyle-Mariotte law Perfect fluid expands so that locally Seff is constant. Can Neff and T be reduced by that?

  7. N / S = pV / TS effective number / entropy

  8. Cooling vs expansion (S = const.)

  9. Number reduction (coalescence)

  10. What do we conclude? • Adiabatic cooling with number reduction to its 1 / 2 . . . 1 / 3 • Most of the reduction and cooling happens relatively short, the volume grows with a factor of 3 . . . 30 • N / S is constant for an ideal gas eos of type p ~ Tª • Lower pressure can be achieved by higher mass ideal gas

  11. Is high-T quark matterperturbative?

  12. Thermal probability of Q² values for massless partons < x >

  13. On the average yes, but watch for IR unsafequantities!

  14. 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

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

  16. Quasiparticle mass and QCD eos

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

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

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

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

  21. 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

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

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

  24. High-T behavior of lattice eos

  25. High-T behavior of lattice eos

  26. High-T behavior of lattice eos + Gribov-Zwanziger dispersion constant m ideal + 1/m² ideal

  27. Mass dependence of the relativistic pressure

  28. Boltzmann vs. Bose and Fermi

  29. Fodor et.al.

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

  31. Quasiparticle mass distributionby inverting the Boltzmann integral Inverse of a Meijer trf.: inverse imaging problem!

  32. Mass gap estimates dueto Markov inequality

  33. 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

  34. Particular inequalities Markov Chebyshev

  35. Particular inequalities Chernoff Minimize the log of this upper bound in λ,  get the best estimate!

  36. General Markov inequality Extreme value probability estimate (upper bound) with variable substitution Original Markov: g=id, f=id

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

  38. Markov inequality and mass gap

  39. Markov inequality and mass gap Upper bound for the low-mass part of the mass distribution. I M D F

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

  41. Boltzmann scaling functions  

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

  43. SU(3) LGT upper bounds

  44. 2+1 QCD upper bounds

  45. Hadron spectra from quarks

  46. Tsallis fit to hadron spectra

  47. Fit parameters at large p

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