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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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