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

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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

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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

,

,,

,,

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• 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

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Estimates of Equilibration times:

• Case 4: All are out of equilibrium

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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