Chemical Equilibration at the
Hagedorn Temperature
Jaki Noronha-Hostler
Collaborators: C. Greiner and I. Shovkovy
Outline
• Motivation: understanding chemical freeze-out in
heavy ion collisions
– Hagedorn Resonances
• Master Equations for the decay
HS  n  BB
– Parameters
• Estimates of Equilibration times
– Baryon anti-baryon decay widths
• Conclusions and Outlook
Motivation
• Standard hadron gas:
chem   B vB nB
- chemical eq. time
  30mb , nBeq  nBeq  0.04 fm -3 , v  0.56c
 
1
RHIC


3
30mb  0.4 fm 
 fm 
10  
 c 
Can’t explain
apparent equilibrium
• Kapusta and Shovkovy, Phys. Rev. C 68, 014901-1 (2003)
• Greiner and Leupold, J. Phys. G 27, L95 (2001)
• Huovinen and Kapusta, Phys. Rev. C 69, 014901 (2004)
• Some suggest long time scales imply that the hadrons are “born in
equilibrium”
– Heinz ,Stock, Becattini…
Production of anti-baryons
• p production
• annihilation rate
p  N  n
• Y production
 N Y   N p  50 mb
 Y  Y  
1
  N  n  K
  N  n  2 K
  N  n  3K
Y  N  n  nY K
detailed balance
1
 NY n  n K vYN
Y

fm
 1- 3
c
B
chemical equilibration time
•Rapp and Shuryak, PRL 86, 2980 (2001)
•Greiner and Leupold, J. Phys. G 27, L95 (2001)
Motivation
• Baryon anti-baryon production lower by a factor of 3-4
Huovinen and Kapusta
• Can be produced through HS  n  BB where HS are mesonic
Hagedorn resonances with time scales of =1-3 fm/c.
– Greiner, Koch-Steinheimer, Liu, Shovkovy, and Stoecker
Hagedorn Resonances
•
M
In the 1960’s Hagedorn found a fit for an exponentially growing mass
spectrum
m
TH
F ( m) 
F
(
m
)
e
dm

m0
•
m0  500 MeV
A
m
2
 m02

5
4
TH  180 MeV
A  0.5
MM  7 GeV
Provides extra degrees of freedom near the critical temperature to “push”
hadrons into equilibrium
Master Equations for the decay
HS  n  BB
• master equation
dn
 " loss "" gain"
dt
dN R i 
dt
n
N Reqi   N 
eq  N  
 eq 
 i N R i    i , N R i   eq  Bi n  i , BB
eq 2 
N BB   N 
n
 N 
n
N BB  2
n
n
eq




N




dN
N
N
R i 
2
eq





  i , nBi n N R i   N R i   eq    i , BB n N R i   eq 2  eq  N BB 



dt
N BB   N 
i
n
 N   i



 N eq  N  n

dN BB
R i 
2


 eq  N BB   N R i  
  i , BB n
2 
eq
 N   N 

dt
i
 BB

Parameters
• Hagedorn States (mesonic, non-strange)
M=2-7 GeV
m
n  0.6  0.3
• Branching Ratios
m
i

– Gaussian distribution:
Bin
1

e
 2
 n  n
2
 2 2
• Decay Widths
i [ MeV ]  0.168mi  88.102
Hammer ‘72
 5  16
  0.26
mi
m
Future: microcanonical
model
 1  1.8
Ranges from i=250-1090 MeV
Estimates of Equilibration times: HS $ nπ
• Case 1: Pions are held in
equilibrium
1
fm
  ,   0.181  0.473

c
• Case 2: Hagedorn States
are held in equilibrium
eff   itot
i
N Reqi 
Neq
n ,   0.017  0.269
fm
c
Estimates of Equilibration times: HS$ nπ
• Case 3: Both are out of equilibrium
dN R i 
dt
n




N

 Bi n  N R i  
 itot   N Reqi   eq
 n2

 N 


n
 




dN
N
tot
eq




  i nBi n N R i   N R i   eq  

dt
i n2
 N  

= 0 (Quasi-equilibrium)
– Quasi-equilibrium- when the right hand side
goes to zero before full equilibrium is reached.
Estimates of Equilibration times:
HS $ nπ
•
Quasi-equlibrium is reached on the time scales of Case 1 and Case 2
n
N R i   N
•
eq
R i 
 N 
 eq   0
 N 
N R i 
N Reqi 
 N 

  eq
 N 
n
Because resonances decay into many pions a small deviation of the pions from
equilibrium makes it more difficult for the resonances to reach equilibrium
Baryon anti-Baryon decay widths
tot
itot

B

i ,
, BB
n
 0 .2  0.4  
tot
i ,
 45  435 MeV
M HS  4 GeV
(Fuming Liu)
Estimates of Equilibration times:
HS  n  BB
• Case 1: Pions are held in equilibrium
ieff
, BB
eq

N
fm
R i  
tot


  i , BB  eq ,   0.169  1.062
c
i
 N BB 
Reminder: HS appear only
near Tc!
Estimates of Equilibration times:
HS  n  BB
• Case 2: Hagedorn States are
held in equilibrium
• Case 3: Pions and Hagedorn
States are held in equilibrium
Estimates of Equilibration times:
HS  n  BB
• Case 4: All are out of equilibrium
Conclusions and Outlook
• Our preliminary results and time scale estimates indicate
that baryon anti-baryon pairs can be born out of
equilibrium.
• Fully understand time scales when all particles are out of
equilibrium
• Include a Bjorken expansion to observe the fireball cooling
over time (already done)
• Improve branching ratios by using a microcanonical model
• Include non-zero strangeness… in the baryon anti-baryon
part