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

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)

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

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

Consistent quasiparticle thermodynamics

∫

∫ ∫

∫ ∫

∫

Φ+=−+=

∂Φ∂

−∂∂

+=∂∂

=

∂Φ∂

−∂∂

+=∂∂

=

Φ−=

dmemwpsTne

dmpw

dmnmwp

n

Tdmp

T

wdmsmw

T

ps

TdmTpmwTp

m

mm

mm

m

)(

)(

)(

),(),()(),(

μ

μμμ

μμμ

This is still an ideal gas (albeit with an infinite number of components) !

Consistent quasiparticle thermodynamics

μμ ∂∂Φ∂

=∂∂Φ∂

TT

22

Integrability (Maxwell relation):

1. w independent of T and µ Φ constant

2. single mass scale M Φ(M) and ∂ p / ∂ M = 0.

pressure – mass distribution

z)((z)

zKz

(z)

dttf

dtgttfp

pg

SB

−=

=

==

=Φ+

=

∫

∫∞

∞

expK :ation transformLaplace

)(2

K :nnsformatio Meijer tra

1)()0( :limit SB

ation transformintegral )( K )()(

2

2

0

0

σ

σ

Adjust M(µ,T) to pressure

t = m / M

f (

t )

= M

w(

m )

T / M (T, 0)

All lattice QCD data from: Aoki, Fodor, Katz, Szabó hep-lat/0510084

Adjusted M(T) for lattice eos

MeVTT

TTM c 170,)36.3(024.0

1.028.0),0(

3=

++=

T and µ-dependence of mass scale M

Boltzmann approximation starts to fail

pressure – mass distribution 2

Analytically solvable case

Example for inverse Meijer trf.

SBp

p

)( gσ

)( xF )(tf

eos fits to obtain eos fits to obtain σσ(g) (g) f(t) 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

exp(-λg) / (1+exp((g-a)/b) ) fit to normalized pressure

1 / g =

σ(g)

=

MASS GAP: fit exp(λg) * data

g =

Fermi eos fit mass distribution

mass gap (threshold behavior)

⎟⎠

⎞⎜⎝

⎛+

+→ε

εβλπ 1

4)(

2ttf

asymptotics:

4104.2 −⋅=ε

zoom

Moments of the mass distribution

( )∫ ∫∞ ∞

−+−

Γ=

0 0

123

21 )(,

)(

12)( dgggB

ndttft nnn σ

π

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

Conclusions

1) Lattice eos data demand finite width T-independent mass distribution, this is unique

2) Adjusted < m >(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

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?