Baseline Statistical Hadronization Model Transport Model Conclusion

Bc Meson enhancement at LHC

Yunpeng Liuin collabration with Prof. Carsten Greiner and Dr. Andriy

Kostyuk

Transport Group Meeting

10-05-2012

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Baseline Statistical Hadronization Model Transport Model Conclusion

Outline

1 Baseline

2 Statistical Hadronization Model

3 Transport Model

4 Conclusion

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Baseline Statistical Hadronization Model Transport Model Conclusion

Motivation

initial productionregeneration

z

t

RHIC

J/

Pro

duct

ion

Prob

abili

tyψ

1

Energy Density

sequential suppression

statistical regeneration

Figure: Satz: arXiv:1101.3937

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Baseline Statistical Hadronization Model Transport Model Conclusion

Motivation

RAA =NAA

NcollNpp∝σQσQ̄

σ(QQ̄)

dσcc̄

dy∼ 10−1 mb

dσbb̄

dy∼ 10−2 mb

dσJ/ψ

dy∼ 10−3 mb

dσΥ

dy∼ 10−5 mb

dσBc

dy∼ 10−5 mb

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Baseline Statistical Hadronization Model Transport Model Conclusion

Bc in Vacuum

V (r) = −α

r+ σr (1)

with

α = π/12 (2)

σ = 0.2 GeV2 (3)

mBc(1S) = 6.36 GeV (4)

mBc(1P ) = 6.72 GeV (5)

mBc(2S) = 6.90 GeV (6)

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Baseline Statistical Hadronization Model Transport Model Conclusion

Bc cross section

Particle M/GeV Γ/keV dσpp/dy

J/ψ 3.096 93 3.4 µb(1.96 TeV)[?]B+c 6.277 0.00000145 15.5 nb (1.96 TeV, pt > 6 GeV)[?]

Υ(1S) 9.460 54 30.36 nb(1.8 TeV, pt < 16 GeV)[?]

Pythia simulation(4× 108 events):

dσB+c(1.96TeV, pt > 0)/dy ∼ 2.7nb(≪ Experiments)

σB+c(pt > 6GeV )

σB+c(pt > 0)

∼2

7.

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Baseline Statistical Hadronization Model Transport Model Conclusion

Estimation for LHC energy:

dσB+c

dy(2.76TeV) ≈ 4×

dσB+c

dy(pt > 6GeV, 1.96TeV) = 62nb

which gives the ratio

dσ/dy(Bc)

dσ/dy(b)=

62nb

20µb= 3.1× 10−3 (7)

consistent with estimation based on CDF measurement[?, ?]

σ(B+c )

σ(b̄)= 2.08+1.06

−0.95 × 10−3

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Baseline Statistical Hadronization Model Transport Model Conclusion

Suppose the distribution is in the form of

dσB+c

dy(pt) ∝ pt

(

1 +p2t

(n− 2)〈p2t 〉

)−n

(8)

From Pythia simulation, we can also estimate:

〈p2t 〉 = 25.1GeV2 n ≈ 4.165 (9)

The heavy quark production cross section is estimated fromexperiments as

dσcc̄dy

= 620µb, (10)

dσbb̄dy

= 20µb, (11)

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Baseline Statistical Hadronization Model Transport Model Conclusion

Statistical Hadronization Model (SHM)

The original Formula of SHM (balance equation)[?]

Ndircc̄ =

1

2gcN

thoc + g2cN

thcc̄ (12)

where

Ndircc̄ : # of directly produced charm pair. (13)

gc : fugacity of charm and anti-charm. (14)

N thoc : # of thermal charmed mesons due to g.c.e. (15)

N thcc̄ : # of thermal hidden charms. (16)

Main Parameters: T , g.

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Baseline Statistical Hadronization Model Transport Model Conclusion

Event by event effect

Ndircc̄ =

1

2gcN

thoc + g2cN

thcc̄ (17)

X ∼ P (λ) (18)

〈X〉 = λ (19)

〈X2〉 = λ(λ+ 1) = λ2(1 +1

λ) (20)

An updated version of the balance equation [?]

Ndircc̄ =

1

2gcN

thoc + g2c (1 +

1

Ncc̄)N th

cc̄ (21)

Main Parameters: T , g, V .

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Baseline Statistical Hadronization Model Transport Model Conclusion

Parameters of SHM

The parameters were fit in the following form[?]

T = Tlim1

1 + e2.6−1

0.45ln

√

sNN

GeV

, (22)

with

Tlim = 164MeV, (23)

at LHC energy, we take T = Tlim.The volume is fit at LHC energy as V∆y=1 ≈ 4160fm3.

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Baseline Statistical Hadronization Model Transport Model Conclusion

SHM with Bc meson

The balance equation

Ndircc̄ =

1

2gc(N

thoc + gbN

thBc) + g2c (1 +

1

Ncc̄)N th

cc̄

Ndirbb̄ =

1

2gb(N

thob + gcN

thBc) + g2b (1 +

1

Nbb̄

)N thbb̄

Rough estimation

Ndircc̄ =

1

2gcN

thoc → gc = 32

NBc

Ndircc̄

=gcN

thBc

N thob

≈ gce−(MBc−MB+ )/T =

gc445

≈ 8.1%

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Baseline Statistical Hadronization Model Transport Model Conclusion

Solution (only B+c (J

P = 0−) considered)

gc = 31.1

gb = 2.39× 108

Nob̄ = 0.59617 = 98.3% ·Ndirbb̄

NB+c

= 0.00669 = 1.08% ·Ndirbb̄

NΥs = 0.00569 = 0.60% ·Ndirbb̄

Ndirbb̄ = 0.60854

NBc

Ndirbb̄

= 1.1% (24)

NdirBc

Ndirbb̄

=σdirBc

σdirbb̄

=62 nb

20 µb= 0.3% (25)

which leads to RAA = 1.1/0.3 = 3.5.13 / 27

Baseline Statistical Hadronization Model Transport Model Conclusion

The Transport Model

For Bc meson:

(∂t + ~v · ∇)fBc= −αfBc

+ β (26)

For the fireball:

∂µTµν = 0 (27)

with EoS of ideal gas of partons and hadrons in Bag model.

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Baseline Statistical Hadronization Model Transport Model Conclusion

Dissociation rate

α =1

E

∫

d~k

(2π)3Ekkµpµσg+Bc→b+c̄(s)fg(k

µ, uµ, T )

β =1

2E

∫

d3~k

(2π)32Eg

d3~qb(2π)32Eb

d3~qc̄(2π)32Ec̄

Wb+c̄→Bc+g(s)

fb(~qb, ~x, t)fc̄(~qc̄, ~x, t)(2π)4δ(4)(p+ k − qb − qc̄)(1 + fg)

where

σg+Bc(1S)→b+c̄ = A0 ·(ω/ǫψ − 1)3/2

(ω/ǫψ)5(28)

A0 = (211π)/(27√

µ3ǫ) (29)

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Baseline Statistical Hadronization Model Transport Model Conclusion

Lifetime of Bc due to gluon dissociation

0.2 0.4 0.60

2

4

6

T (GeV)

(fm

)τ

(1S)cB

(1P)cB

(2S)cB

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Baseline Statistical Hadronization Model Transport Model Conclusion

Cornell Potential at Finite Temperature

[

1

2µ∇2 + V (r, T )

]

ψ(~r, T ) = Eψ(~r, T )

−1

−0.5

0

0.5

1

1.5

2

2.5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

R σ1/2

F(R,T)/σ1/2

0.760.810.870.900.961.001.021.071.111.161.231.361.501.651.811.98

Figure: Free energy of heavyquarks by LQCD.

V = F Free energy;

V = U Internal energy.

U = F + TS

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Baseline Statistical Hadronization Model Transport Model Conclusion

Radius of Bc at finite temperature

1.0 1.5 2.0 2.5 3.00.0

1.0

2.0

3.0

cT/T

/ fm

⟩ r

⟨

(1S)cB(1P)cB(2S)cBV=U

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Baseline Statistical Hadronization Model Transport Model Conclusion

Centrality dependence of RAA

0 200 400

0

5

10

15

partN

AA

R

Initial production

Regeneration

Total

V=U

0 200 400

0.0

0.5

1.0

1.5

2.0

2.5

partN

Initial production

Regeneration

Total

V=F

RAA =NAA

NcollNpp∝σQσQ̄

σ(QQ̄)

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Baseline Statistical Hadronization Model Transport Model Conclusion

0 100 200 300 4000.00

0.01

0.02

0.03

0.04

partN

S

V=U Au+Au 200 A GeV regen.V=U Pb+Pb 2.76 A TeVV=F Au+Au 200 A GeV regen.V=F Pb+Pb 2.76 A TeV

S ≡dNBc/dy

dN b/dy · dN c/dy

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Baseline Statistical Hadronization Model Transport Model Conclusion

Mean p2T

0 200 400 0

5

10

15

20

25

partN

)2 (

GeV

⟩ 2 T

p⟨

Initial

Regenerated

Total

V=U

0 200 400

partN

Initial

Regenerated

Total

V=F

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Baseline Statistical Hadronization Model Transport Model Conclusion

Momentum dependence of RAA

0 5 100

10

20

30

(GeV)T

p

AA

R V=U

InitialRegeneratedTotal

0 5 100

2

4

6

(GeV)t

p

V=F

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Baseline Statistical Hadronization Model Transport Model Conclusion

Momentum spectrum

6 8 10 12

-20

-15

-10

(GeV)tE

]3 (

GeV

/c)

pln

[dn/

d

V=U

InitialRegenerated

Total

6 8 10 12

-20

-15

-10

(GeV)t

V=F

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Baseline Statistical Hadronization Model Transport Model Conclusion

Spectra of different states b = 0 fm

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Baseline Statistical Hadronization Model Transport Model Conclusion

Effective temperature

6 8 10 12

-20

-15

-10

(GeV)tE

]3 (

GeV

/c)

pln

[dn/

d

V=U

InitialRegenerated

Total

6 8 10 12

-20

-15

-10

(GeV)t

V=F

dn

d~p= Ae−E/Teff (30)

V=U V=FRegeneration 567 533

Total 569 534

Table: Effective temperature Teff/MeV of Bc.

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Baseline Statistical Hadronization Model Transport Model Conclusion

Elliptic flow at b = 8.4 fm

0 5 10

0.0

0.2

0.4

0.6

(GeV)T

p

2v

Initial

Regenerated Total

V=U

0 5 10

(GeV)t

p

Initial

RegeneratedTotal

V=F

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Baseline Statistical Hadronization Model Transport Model Conclusion

Conclusion

Enhancement of Bc production is expected at LHC energy,which can verify the regeneration mechanism directly.

Contribution from the excited states are sensitive to the heavyquark potential.

Less regeneration at low pT relative to thermal productionmay suggest dissociation of the regenerated quarkonia.

Bc meson carries the information from the fireball liketemperature and flow.

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