Distributed mass partons in quark matter Consistent e os with mass distribution

1. Equationof state for distributed mass quark matterT.S.Bíró, P.Lévai, P.Ván, J.ZimányiKFKI 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? 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:

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 • 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): • w independent of T and µ  Φ constant • single mass scale M  Φ(M) and ∂ p / ∂ M = 0.

7. pressure – mass distribution

8. Adjust M(µ,T) to pressure

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

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

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

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

22. MASS GAP: fit exp(λg) * data g =

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

24. asymptotics:

25. zoom

26. 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

27. Conclusions • Lattice eos data demand finite width T-independent mass distribution, this is unique • Adjusted < m >(T) behaves like the fixed mass in the quasiparticle model • 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

28. 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?