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?