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.

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?