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Quark-Gluon Plasma Physics 5. Statistical Model and Strangeness
Prof. Dr. Klaus Reygers Heidelberg University SS 2017
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
Strangeness production in hadronic interactions
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⇤ = (uds), ⌃ = (qqs), ⌅ = (qss), ⌦� = (sss)
Particles with strange quarks:
Creation in collisions of hadrons:
Example 1:
associated production of strangeness
Example 2: p + p ! p + p + ⇤+ ⇤̄, Q = 2m⇤ ⇡ 2230MeV
"hidden strangeness"
p + p ! p + K+ + ⇤, Q = m⇤ +mK+ �mp ⇡ 670MeV
K+ = (us̄), K� = (ūs), K 0 = (ds̄), K̄ 0 = (d̄s), � = (ss̄),
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
Strangeness production in the QGP
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QQGP ⇡ 2ms ⇡ 200MeV
Q value in the QGP significantly lower than in hadronic interactions
This reflects the difference between the current quarks mass (QGP) and the constituent quark mass (chiral symmetry breaking)
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
Strangeness enhancement: One of the earliest proposed QGP signals
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J. Rafelski, B. Müller, PRL 48 (1982) 1066
assumed typical lifetime of the plasma
strangeness per baryon
Strangeness equilibration was expected to be sufficiently fast
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
Quark composition of the ideal QGP
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� = eµ/T
"Boltzmann approximation" (neglect "±1"): first term of the sum
Quarks: fermions ("upper sign"), mu = 2.2 MeV, md = 4.7 MeV, ms = 96 MeV,
In a QGP with μ = 0 and 150 < T < 300 MeV:
[But s-quarks eventually have to materialize as constituent quarks …]
Particle densities for a non-interacting massive gas of fermions (upper sign)/bosons (lower sign):
0.15 0.20 0.25 0.30
0.93
0.94
0.95
0.96
0.97
0.98
T (GeV)
2n s
n u+n d
2(ns + ns̄)
nu + nū + nd + nd̄⇡ 0.92–0.98
upper sign: fermions, lower sign: bosons
ni = gi4⇡
(2⇡)3
Z 1
0
p2 dp
exp
✓pp2+m2�µ
T
◆± 1
=gi2⇡2
m2T1X
k=1
(⌥1)k+1
k�kK2
✓km
T
◆
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
Fraction of strange quarks: A+A vs. e+e–, πp, and pp
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√ s (GeV)
λ S
RHIC
SPSPbPb
AGSAuAu
K+p collisionsπ+p collisionspp collisionspp
– collisions
e+e- collisions
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1 10 102
103
�s =2hss̄i
huūi+ hdd̄i
ratio of newly created valence quark pairs before strong decays (ρ, Δ, …)
arXiv:0907.1031
“Wròblewski factor”, Acta Phys. Pol. B16 (1985) 379
Strangeness indeed enhanced in nucleus-nucleus collisions relative to e+e–, πp, and pp collisions
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
Strangeness Enhancement in Pb-Pb relative to p-Pb at √sNN = 17.3 GeV
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1
10
Strangeness |S|
Enha
ncem
ent E
h- K0S Λ Λ–
Ξ Ξ–
Ω+Ω–
0 1 12 2 3
Figure 4: Strange particle enhancement versus strangeness content.
8
0-40% Pb-Pb at √sNN = 17.3 GeV
Strangeness enhancement increases with s quark contents (up to factor 17 for the Ω baryon)
WA97, PLB 449 (1999) 401, CERN-EP/99-29
E =
✓Y
Npart
◆
Pb�Pb/
✓Y
Npart
◆
p�Pb
p-Be reference instead of p-Pb: similar behavior (NA57)
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
Ξ/π and Ω/π enhancement in Pb-Pb at √sNN = 2.76 TeV
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-π++π-+K+K
-π++πpp+
0KΛ2
-π++π+Ξ+-Ξ
-π++π+Ω+-Ω d
dHe 3
-π++π HΛH+ Λ
-+K+Kφ
-+K+KK*+
0
0.2
0.4
1× 3× 0.5× 30× 250× 50× 100× 5 4 10× 2× 1×
ALICE Preliminary = 7 TeVspp = 5.02 TeVNNsp-Pb
V0A Multiplicity (Pb-Side) 0-5% = 2.76 TeV, 0-10%NNsPb-Pb
Extrapolated (Pb-Pb 0-10%)
Extrapolated (p-Pb 0-5%)
K/ p/ /K0 / / d/p He/d H/ /K K*/K3
Interestingly, φ/π very similar in pp, p-Pb, and Pb-Pb
3
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
Particle yields from the hadron resonance gas■ Idea: Freeze-out of the QGP creates an equilibrated hadron resonance gas ■ The HRG then freezes out with a characteristic temperature Tch close to Tc
which determines the yields of different particle species ■ What is the appropriate statistical ensemble for the theoretical treatment?
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system S T, V, N
heat bath T
canonical ensemble: N and V fixed, energy E of the system fluctuates (Es + Eb = E, T is given)
grand-canonical ensemble: V fixed, energy E and particle number N fluctuate (T, μ given)
system S T, V, μ
heat bath T, μ
pp collisions, strangeness locally conserved
central A-A collisions, local strangeness fluctuations possible,“there is a medium”
Braun-Munzinger, Redlich, Stachel, nucl-th/0304013v1
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
Grand canonical ensemble: Large volume limit of the canonical treatment
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Canonical suppression factor Fs:
nK : Density of particles with strangeness K = |S|, S =-1, -2, -3
Modified Bessel function of the first kind
nCK = nGCK · FS
FS =IK (2nGCK V )
I0(2nGCK V )
In :
A. Tounsi, K. Redlich, hep-ph/0111159
Already at moderately central Pb-Pb collisions the grand canonical ansatz is justified
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
Statistical model (hadron gas, grand canonical ensemble)Partition function (particle species i):
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lnZi =Vgi2⇡2
Z 1
0±p2dp ln(1± exp(�(Ei � µi )/T ))
“-” for bosons, “+” for fermions
Particle densities: ni = N/V = �T
V
@ lnZi@µ
=gi2⇡2
Z 1
0
p2 dp
exp((Ei � µi )/T )± 1
For every conserved quantum number there is a chemical potential:
µi = µBBi + µSSi + µI3 I3,i
Use conservation laws to constrain V ,µs ,µI3strangeness:
charge:
baryon number:
X
i
niSi = 0 ! µs
VX
i
niBi = Z + N ! µB
VX
i
ni I3,i =Z � N
2! µI3
Only two parameters left (T, μB) Example: → determine (T, μB) for different √sNN from fits to data
gi = (2 Ji +1) spin degeneracy factorEi2 = pi2 + mi2
n(p̄)/n(p) = exp(�2µB/T )
Boltzmann approximation
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
χ2 fit of the statistical models to LHC data
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LHC, Pb–Pb, 0-10%9 [email protected]
y/d
NYi
eld
d
-410
-310
-210
-110
1
10
210
310
Data, ALICE, 0-10%=33.0/14)dfN/2χStatistical model fit (
3=5330 fmV=0 MeV, b
µ=156.0 MeV, T
=2.76 TeVNNsPb-Pb
+π -π +K -K s0K 0K* φ p p Λ-Ξ
+Ξ -Ω
+Ω d He3 HΛ
3 HΛ3
Dat
a/Fi
t
0.4
0.6
0.8
1
1.2
1.4
1.6=2.76 TeVNNsPb-Pb
Data, ALICE, 0-10%
σ(D
ata-
Fit)/
-3-2-1
012
33=5330 fmV=0 MeV,
bµ=156.0 MeV, TFit:
+π -π +K -K s0K 0K* φ p p Λ -Ξ +Ξ -Ω +Ω d He3 HΛ
3 HΛ
3
T = 156.0± 1.5 MeV, µB = 0± 2 MeV, V = 5330± 410 fm3
π, K±, K0 from charm included (0.7%, 2.6%, 2.9% for the best fit)
[ no p,p̄ in fit: T = 158+1−2 MeV, V = 5170+450−210 fm
3, χ2/Ndf=11.5/12 ]
■ Overall good agreement with data ■ T = 156 ± 1.5 MeV, μB = 0 ± 2 MeV, V = 5330 ± 400 fm3
Andronic, Braun-Munzinger, Stachel arXiv:1106.632, arXiv:1210.7724, arXiv:1311.4662, talk A. Andronic Trento
3σ deviation for protons and anti-protons
Statistical yields for primaries + feed-down from strong decay, e.g., ρ → π+π−, η → π+π−π0, φ → K+ K−
http://www.ectstar.eu/sites/www.ectstar.eu/files/talks/andronic_trento14.pdf
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
√sNN dependence of T and μB
■ Smooth evolution of T and μB with √sNN ■ T reaches limiting value of Tlim = 159 ± 2 MeV
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Energy dependence of T , µB (central collisions)4 [email protected]
40
60
80
100
120
140
160
180
1 10 102
103
√sNN (GeV)
T (M
eV)
fits of yields
dN/dy
parametrization4π
0
100
200
300
400
500
600
700
800
900
1 10 102
103
√sNN (GeV)
µB
(MeV
)
thermal fits exhibit a limiting temperature: Tlim = 159± 2 MeV
T = Tlim1
1+exp(2.60−ln(√sNN (GeV))/0.45)
, µB[MeV] =1307.5
1+0.288√sNN (GeV)
PLB 673 (2009) 142 ...with updates (Tlim was 164± 4 MeV)
update of PLB 673 (2009) 142
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
K/π ratio vs. √sNN
■ Maximum in K+/π+ (“the horn”) was discussed as a signal for the onset of deconfinement at √sNN ≈ a few GeV
■ However, in the GC statistical model the structure can be reproduced with T, μB that vary smoothly with √sNN
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(GeV)NNs10 210 310
dN/d
y ra
tio
0
0.05
0.1
0.15
0.2
0.25
0.3
+π/+K-π/-K
symbols: data, lines: thermal model
A. Andronic, arXiv:1407.5003v1
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
Freeze-out points for √sNN ≳ 10 GeV from thermal model fits coincide with Tc from lattice calculations
■ What is the origin of equilibrium particle yields? ‣ General property of the
QCD hadronization process (“particle born into equilibrium”)
‣ Or does the hadron gas thermalizes via particle scattering after the transition?
■ Possible mechanism for fast thermalization after the transition: multi-hadron scattering resulting from high particle densities
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Braun-Munzinger, Stachel, Wetterich, PLB 596 (2004) 61
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
Strangeness enhancement already in small systems: Multiplicity dependence of Ω/π in pp, p-Pb, and Pb-Pb
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Significant increase in Ω/π with dNch/dη already in pp and p-Pb
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
Even yields in e+e– are not so far from chemical equilibrium
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Becattini, Castorina, Manninen, Satz, 0805.0964Statistical model + phenomenological factor γs < 1, reducing hadron yields by γsN where N is the number of strange quarks (or antiquarks)
T not so different from the one in central A+A
QGP physics SS2017 | K. Reygers | 5. Statistical Model and Strangeness
Summary/questions strangeness
■ Strangeness is enhanced in A-A collisions relative to e+e– and pp ■ LHC: Strangeness enhancement in high-multiplicity pp collisions approaches
the enhancement in Pb-Pb ■ Origin of the strangeness enhancement? ‣ Collisional equilibration? ‣ Or "born into equilibrium"? ‣ Strange quark coalescence ("recombination")? ‣ Or something else?
■ Strangeness provides important information and probably points to QGP formation ‣ But why does the statistical approach also work to some degree in e+e– where
no QGP is expected? ‣ Better understanding of the mechanisms of strangeness enhancement is
needed
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