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```									       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 Ange
Why distributed mass?
coalescence:        convolution

valence mass  hadron mass ( half or third…)

w(m)

Zimányi, Lévai, Bíró, JPG 31:711,2005

Conditions: w ( m ) is not constant
zero probability for zero mass
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)
or
2. Take eos (fit QCD)  find a single w ( m )  rates, spectra

Type1 : w ( m )    m  T  g ( T ,  )        quasiparticle model
   m    m 
 a
 m    b 0 
m 
Type1 : w ( m )  A ( m 0 ) e         0      
with m 0 ( T ,  )

Type 2 : p lattice ( T , 0 )  w ( m / T c )      " the" mass distribution
Consistent quasiparticle
thermodynamics
p(  , T )   w(m) pm (  , T )dm   (  , T )
p                    w         
s      w(m) sm dm      pm dm 
T                    T         T
p                    w         
n      w(m)nm dm       pm dm 
                             
e   n  T s  p   w(m)em dm  

This is still an ideal gas (albeit with an infinite number of components) !
Consistent quasiparticle
thermodynamics
 2    2
Integrability (Maxwell relation):               
 T T 

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

1      m
mass distribution : w(m)     f( )
M     M
t  m / M , g  M / T , gt  m / T
pressurein Boltzmann approximation :
 m 2T 2       m        ( gt ) 2
pm (  , T )   ( )       K2     T 4          K 2 ( gt )
T    2      T            2
pressure – mass distribution


p
 (g)         f (t ) K ( gt ) dt integral transformation
pSB   0

SB limit :  (0)   f (t )dt  1
0

z2
Meijer transformation : K(z)  K 2 ( z )
2
Laplace transformation : K(z)  exp (  z)

our favorite ansatz :         f ( t )  Ae  at  b / t

b K 2 ( 2 ab )
m M
a K 1 ( 2 ab )

integrability  M(  , T) satisfies :

M                          M
                            
  g   3          0
T                   T       
f ( t ) = M w( m )

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

T / M (T, 0)
0.1
M ( 0 , T )  0.28 T                         ,   T c  170 MeV
0.024  ( 3.36 T ) 3
T and µ-dependence of mass
scale M

Boltzmann approximation starts to fail
pressure – mass distribution 2

            2 2
g t
 ( g )   f(t)            K2 ( gt ) dt     Meijer transformation
0
2
c  i
2                1
f (t )       i  ( g ) gt I 2 ( gt ) dg
i c 
inverse transformation


4 d  (i )
f (t )              J 2 (t )
t  2 i
Analytically solvable case

a2
f (t )  2 1  2 t  a 
4a
 ( g )  e  ag   
t    t

 ( g )   F ( x)e dx
 gx
Laplace transform
0
1
4
f (t )      u    1  u F (tu )du inverse Meijer transform
2

   0
Example for inverse Meijer trf.

p
p SB                  (g)

F ( x)          f (t )
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
σ(g) =   exp(-λg) / (1+exp((g-a)/b) ) fit to normalized pressure

1/g=
MASS GAP: fit exp(λg) * data

g=
Fermi eos fit  mass
distribution

(1   ) e   g
 (g)               g
with   e   a ,                  1/ b
1  e

d                
f ( t )  (1   )
4
        J 2 (t )              n   sin  (   n )  
t   0
                n0

                                                                  
 t

4                     2          
n n                     n       2

f ( t )  (1   )              1             (  )                      1                
t   t            t   2
n 1          t                           t2    
                                                                       

mass gap (threshold behavior)
asymptotics:
4             
f (t )                     
t 2         1  

  2.4  10     4
Moments of the mass
distribution
                                            
B  n2 1 , 3   g n 1 ( g ) dg
2   1
t
n
f ( t ) dt              

0
 (n)            2
0

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?

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