1. Equation of state for distributed mass quark matter T.S.Bíró, P.Lévai, P.Ván, J.Zimányi KFKI RMKI, Budapest, Hungary Distributed mass partons in quark matter Consistent eos with mass distribution Fit to lattice eos data Arguments for a mass gap Strange Quark Matter 2006, 27.03.2006. Los Angeles

2. 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)

3. Previous progress (state of the art…) valence mass + spin-dependent splitting : too large perturbations (e.g. pentaquarks) Hagedorn spectrum (resonances): no quark matter, forefactor uncertain QCD on the lattice: pion mass is low resonances survive Tc quasiparticle mass m ~ gT leads to p / p_SB < 1

4. Strategies 1.guess w ( m )  hadronization rates  eos (check lattice QCD) 2. Take eos (fit QCD)  find a single w ( m )  rates, spectra o r

5. Consistent quasiparticle thermodynamics This is still an ideal gas (albeit with an infinite number of components) !

6. Consistent quasiparticle thermodynamics Integrability (Maxwell relation): 1.w independent of T and µ  Φ constant 2.single mass scale M  Φ(M) and ∂ p / ∂ M = 0.

7. pressure – mass distribution

8. Adjust M(µ,T) to pressure

9. t = m / M f ( t ) = M w( m )

10. T / M (T, 0) All lattice QCD data from: Aoki, Fodor, Katz, Szabó hep-lat/0510084

12. Adjusted M(T) for lattice eos

16. T and µ-dependence of mass scale M Boltzmann approximation starts to fail

17. pressure – mass distribution 2

18. Analytically solvable case

19. Example for inverse Meijer trf.

20. eos fits to obtain σ(g)  f(t) ● sigma values are in (0,1) ● monotonic falling ● try exponential of odd powers ● try exponential of sinh ● study - log derivative numerically ● fit exponential times Wood-Saxon (Fermi) form All lattice QCD data from: Aoki, Fodor, Katz, Szabó hep-lat/0510084

23. exp(-λg) / (1+exp((g-a)/b) ) fit to normalized pressure 1 / g = σ(g) =

24. Maximum Entropy Method The original problem: find normalized positive f(t) to data σ(g) in k=1…m points

25. Maximum Entropy Method The dual problem: find k=1…m Lagrange multipliers λ to each k guaranteed: f(t) is a probability distribution not guaranteed: the re-fit of data is acceptable

27. MASS GAP argument: log derivative

28. MASS GAP: fit exp(λg) * data g =

29. Fermi eos fit  mass distribution mass gap (threshold behavior)

30. asymptotics:

31. zoom

32. Moments of the mass distribution n = 0 limiting case: 1 = 0 ·  n < 0 all positive mass moments diverge due to 1/m² asymptotics n > 0 inverse mass moments are finite due to MASS GAP

33. Here σ(g) was approximated linearly

34. Conclusions 1) Lattice eos data demand finite width T- independent mass distribution, this is unique 2) Adjusted (T) behaves like the fixed mass in the quasiparticle model 3) Strong indication of a mass gap: best fit to lattice eos: exp · Fermi SB pressure achieved for large T all inverse mass moments are finite - d/dg ln σ(g) has a finite limit at g=0

35. Interpretation  Does the quark matter interact?  Mass scale vs mean field: * M(T) if and only if Φ(T) * w(m) T-indep.  Φ const.  What about quantum statistics and color confinement?  From what do (strange) hadrons form?  How may the Hagedorn spectrum be reflected in our analysis?