biro_tamas

Document Sample
biro_tamas Powered By Docstoc
					       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) w(had-m)
              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)
     Adjust M(µ,T) to pressure

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)
Adjusted M(T) for lattice eos
                                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?

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:0
posted:2/15/2012
language:
pages:28