Chemical Equilibration at the Hagedorn Temperature Jaki Noronha-Hostler Collaborators: C. Greiner...
-
date post
19-Dec-2015 -
Category
Documents
-
view
214 -
download
0
Transcript of Chemical Equilibration at the Hagedorn Temperature Jaki Noronha-Hostler Collaborators: C. Greiner...
Chemical Equilibration at the Hagedorn Temperature
Jaki Noronha-HostlerCollaborators: C. Greiner and I. Shovkovy
Outline
• Motivation: understanding chemical freeze-out in heavy ion collisions– Hagedorn Resonances
• Master Equations for the decay
– Parameters• Estimates of Equilibration times
– Baryon anti-baryon decay widths• Conclusions and Outlook
BBnHS
Motivation• Standard hadron gas:
• 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…
BBBchem nv - chemical eq. time- chemical eq. time
RHIC
3
-3
40mb30
1
560 ,fm 040 ,mb30
fm
cvnn eqB
eqB
.
..
Can’t explain Can’t explain
apparent equilibriumapparent equilibrium
c
fm 10
Production of anti-baryons
• production
• production
pnNp
Y
KnN
KnN
KnN
3
2
KnnNY Y
detailed balancedetailed balance
• annihilation rateannihilation rate
c
fm 3-1
11
BNYKnnYN
YYv
Y
chemicalchemical equilibration timeequilibration time
mb 50 pNYN
•Rapp and Shuryak, PRL 86, 2980 (2001)•Greiner and Leupold, J. Phys. G 27, L95 (2001)
• Baryon anti-baryon production lower by a factor of 3-4
• Can be produced through where HS are mesonic Hagedorn resonances with time scales of =1-3 fm/c.– Greiner, Koch-Steinheimer, Liu, Shovkovy, and Stoecker
Motivation
Huovinen and KapustaHuovinen and Kapusta
BBnHS
Hagedorn Resonances• In the 1960’s Hagedorn found a fit for an exponentially growing mass
spectrum
• Provides extra degrees of freedom near the critical temperature to “push” hadrons into equilibrium
dme0
M
m
T
m
HmF )( 4
520
2
)(
mm
AmF
dNR (i )
dt= ¡ ¡ toti NR (i ) +
X
n
¡ toti ;¼<i ;n(T)(N¼)nB i ! n¼
+ ¡ toti ;B ¹B<<n>¼B ¹Bi ;<n> (T)(N¼)<n>N 2
B ¹B
dN¼
dt=
X
i
X
n
¡ toti ;¼nB i ! n¼¡NR (i ) ¡ <(T)(N¼)n
¢
+X
i
¡ toti ;B ¹B < n >³NR (i ) ¡ <<n>¼B ¹B
i ;<n> (T)(N¼)<n>N 2B ¹B
´
dNB ¹B
dt= ¡
X
i
¡ toti ;B ¹B
¡N 2B ¹BN
<n>¼ <i ;<n>(T) ¡ NR (i )
¢
(1)
dNR (i )
dt= ¡ ¡ toti NR (i ) +
X
n
¡ toti ;¼<i ;n(T)(N¼)nB i ! n¼
+ ¡ toti ;B ¹B<<n>¼B ¹Bi ;<n> (T)(N¼)<n>N 2
B ¹B
dN¼
dt=
X
i
X
n
¡ toti ;¼nB i ! n¼¡NR (i ) ¡ <(T)(N¼)n
¢
+X
i
¡ toti ;B ¹B < n >³NR (i ) ¡ <<n>¼B ¹B
i ;<n> (T)(N¼)<n>N 2B ¹B
´
dNB ¹B
dt= ¡
X
i
¡ toti ;B ¹B
¡N 2B ¹BN
<n>¼ <i ;<n>(T) ¡ NR (i )
¢
(1)
12 1
2
12 a
b
f (x;y) =
(x + 1 if y = 0xy if y 6= 0
GeV 7MM
0.5A
MeV 180T
MeV 500
H
0
m
Master Equations for the decay
iiRBB
n
eqeqBB
eqiR
BBiBB
iBB
n
eqeqBB
eqiR
iRBBii n
n
eqeqiRiRnii
BB
n
eqeqBB
eqiR
BBin
ni
n
eqeqiRiiRi
iR
NNN
N
N
Nn
dt
dN
NN
N
N
NNn
N
NNNnB
dt
dN
NN
N
N
NB
N
NNN
dt
dN
2 2
2 2
2 2
,
,,
,,
BBnHS
• master equationmaster equation
"""" gainlossdt
dn
Parameters
• Hagedorn States (mesonic, non-strange) M=2-7 GeV
• Branching Ratios– Gaussian distribution:
• Decay Widths
2
2
2
2
1
nn
ni eB
165
3.06.0
m
mn i
8.11
26.0
m
mi
102881680 ..][ ii mMeV
Hammer ‘72Hammer ‘72
Future: microcanonical Future: microcanonical modelmodel
Ranges from Ranges from ii=250-1090 MeV=250-1090 MeV
Estimates of Equilibration times: HS $ nπ
• Case 1: Pions are held in equilibrium
• Case 2: Hagedorn States are held in equilibrium
c
fm 47301810 ,
1..
c
fm26900170 ,
N
eq
..
nN
ieq
iRtoti
eff
Estimates of Equilibration times: HS$ nπ
• Case 3: Both are out of equilibrium
– Quasi-equilibrium- when the right hand side goes to zero before full equilibrium is reached.
i n
n
iRiRnitoti
niRni
n
iRtoti
iR
N
NNNnB
dt
dN
NBN
NN
dt
dN
2eq
eq
2eq
eq
= 0= 0 (Quasi-equilibrium) (Quasi-equilibrium)
Estimates of Equilibration times: HS $ nπ
• Quasi-equlibrium is reached on the time scales of Case 1 and Case 2
• Because resonances decay into many pions a small deviation of the pions from equilibrium makes it more difficult for the resonances to reach equilibrium
0eq
eq
n
iRiR N
NNN
n
iR
iR
N
N
N
N
eqeq
Baryon anti-Baryon decay widths
MeV 43545
4020
toti
toti
totBBi
B
,
,,
..
(Fuming Liu)(Fuming Liu)
n
GeV 4HSM
Estimates of Equilibration times:
• Case 1: Pions are held in equilibrium
c
fm 06211690 , ..,,
ieqBB
eqiRtot
BBieffBBi N
N
Reminder: HS appear only Reminder: HS appear only near Tnear Tc!c!
BBnHS
Estimates of Equilibration times:
• Case 2: Hagedorn States are held in equilibrium
• Case 3: Pions and Hagedorn States are held in equilibrium
BBnHS
Estimates of Equilibration times:
• Case 4: All are out of equilibrium
BBnHS
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