Post on 15-Jan-2016
description
Do small systems equilibrate chemically?
Ingrid Kraus
TU Darmstadt
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 2
Outline
• Introduction to the Statistical Model– Ensembles, partition function
• Grand canonical ensemble– Comparison to data
– Extrapolation and predictions for heavy-ion collisions at LHC
– Experimental observables for T and μB determination
– Relevance of resonances
• From Pb+Pb to p+p: system size and energy dependence– Canonical suppression
– Concept of equilibrated clusters
– Comparison to data
• Summary
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 3
• Micro-canonical– closed system
– E, V, N fix
• Canonical– heat bath
– T, V, N fix
• Grand-canonical– open system
– heat bath and particle reservoir
– T, V, fix
Statistical Ensembles
E, V, N
T, V, N
T, Vb, Nb
T, V,
T, Vb, Nb
E
Laplacetransformation
N
canN
NT
cangrand ZeZ )( /
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 4
• Partition function of a grand canonical ensemble
• Energy density Entropy density
• Particle number density Pressure
• Grand-canonical partition function– i: species in the system
– Mesons m < 1.5 GeV, Baryons m < 2 GeV
Partition function and its derivations
),,(ln VTZ
nT
ZT
V
T
)ln(T
ZT
Vs
)ln(1
)ln(1 ZT
Vn
V
ZTP
)ln(
i i VTZVTZ ),,(ln),,(ln
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 5
• Partition function for species i with degenaracy factor gi
• with– (+) for fermions, (-) for bosons
• Model parameters
– T and B S constrained by strangeness neutrality
– V cancels in ratios Q constrained by charge of nuclei
Partition function and model parameters
0
22
1ln2
),,(ln TE
TN
ii
ii
eedppgV
VTZ
QQiSSiBBii NNNN 22iii mpE
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 6
Comparison to Experimental Data
– Accurancy in T, B: few MeV
– Different data selected for fits
A.Andonic, P. Braun-Munzinger, J. Stachel, nucl-th/0511071
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 7
T - B – systematics, extrapolation to LHC
hep-ph/0511094Chemical decoupling conditions
extracted from SIS up to RHIC
Feature common behavior
On the freeze-out curve:
TLHC ≈ TRHIC ≈ 170 MeV
T ≤ TC ≈ 170 MeV
μB from parametrised freeze-out
curve:
μB (√(sNN) = 5.5TeV) = 1 MeV
Nucl. Phys. A 697 (2002) 902
Grand canonical ensemble
for Pb+Pb predictions
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 8
Predictions for Pb+Pb
• Reliable for stable particles
• Benchmark for resonances
• Errors:
T = 170 +/- 5 MeV
μB = 1 + 4 MeV
- 1
All calculations with THERMUS hep-ph/0407174
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 9
Extraction of thermal parameters from data
• determine μB from p/p
• sensitivity on T– increases with mass
difference
– decay contribution affect
lighter particles stronger
– increasing feed-down with
increasing T
– decay dilutes T dependence
• T from and/or
K
_
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 10
Resonance Decays
• Hadron Resonance gas
• no resonance contribution
•
– 50% from feed-down
– both exhibit same T dependence
• K decay exceeds thermal at LHC
• – thermal production ≈ constant
– resonance contribution dominant
• 75% of all from resonances
j
thermaljij
thermalii NNN
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 11
• Grand canonical ensemble– large systems, large number of produced hadrons
• Canonical ensemble– small systems / peripheral collisions, low energies
– suppressed phase-space for particles related to conserved charges
– density of particle i with strangeness S approxiamtely
• S: order of Bessel functions
• x: sum over strange hadrons, related to volume
– Volume enters as additional parameter V
– here: radius R of spherical volume V
Canonical suppression
)(
)(
0 xI
xInn Scanonicalgrandi
canonicali
)()( RnVnn iicanonicali
T, V, N
T, Vb, Nb
T, V,
T, Vb, Nb
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 12
Canonical suppression
– Stronger suppression for
multi-strange hadrons
– Suppression depends on
strangeness content, not
difference
(expected from S)
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 13
Suppression by undersatured phase-space
– Stronger suppression for
multi-strange hadrons
– Suppression depends on
difference of strangeness
content
(power of S)
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 14
Suppression in small systems
• Suppressed strangeness
production beyond canonical
suppression– addressed by canonical treatment
and undersaturation factor S
– new: equilibrated clusters
SPS √(sNN) = 17 AGeV
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 15
Modification of the model
• Statistical Model approach: T and μB
– Volume for yields → radius R used here
• Deviations: strangeness undersaturation factor S
– Fit parameter
• Alternative: small clusters (RC) in fireball (R): RC ≤ R
– Chemical equilibrium in subvolumes: canonical suppression
– RC free parameter
R
RC
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 16
Fit Example
• All Fits were performed with
THERMUS
hep-ph/0407174
• Fits with S / RC give better
description of data
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 17
System size and energy dependence of T and B
• T independent of– System size
– Data selection
– Energy • μB smaller at RHIC
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 18
System size and energy dependence of the cluster size
• Small clusters in all systems
• Small system size dependence
• p+p– energy dependence?
• Pb+Pb– depends on data selection
(multistrange hadrons needed)
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 19
System size and energy dependence of the cluster size
• A+A: clusters smaller than fireball
• RC not well defined for RC ≥ 2 fm because suppression vanishes
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 20
Canonical Suppression
• Particle ratios saturate
at RC ≈ 2 - 3 fm
– no precise determination
for small strangeness
suppression
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 21
• Grand canonical ensemble– successful description of Au+Au,
Pb+Pb data
– extrapolations allow for predictions
– determination of thermal
parameters with few particle ratios
– proper treatment of resonances is
mandatory
Summary
• Canonical ensemble– volume dependend suppression
– stronger suppression modeled with
smaller, thermally equilibrated clusters
– successful description of p+p, C+C,
Si+Si data
– strangeness production in small
systems reproduced with equilibrated
subvolumes
• Outlook– strangeness production in p+p at LHC
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 22
Going into formulas
• performing the momentum integration
– (+) for bosons, (-) for fermions
– mi: mass of hadron i
• Particle number density
T
kmKe
k
mgTVVTZ i
k
k
TNk
iii
i
122
1
2
2 )1(
2),,(ln
0
22
1ln2
),,(ln TE
TN
ii
ii
eedppgV
VTZ
T
kmKe
k
mgTZT
VTn i
k
TN
k
kiii
i
i
21
1
2
2 )1(
2
)ln(1),(
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 23
• Approx. modified Bessel function
• Particle ratio
• Antiparticle/Particle ratio
Density and Ratios
Tm
TNii
i
ii
eemTg
n
2
2/3
2
)(2
Tmm
TNN
eem
m
g
g
n
n 2121 )(2/3
2
1
2
1
2
1
T
NN
T
N SSBB
een
n 1,1,1
222
1
1
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 24
System size dependence of T and B
• μB decreases at mid-rapidity in small systems ….
• …. as expected from increasing antibaryon / baryon ratio
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 25
System size dependence of the cluster size
Same trend as K /
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 26
More SPS and RHIC 200 GeV Data
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 27
Model setting with S
• S
– sensitive on data sample
– increase with size
– increase with energy
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 28
Extrapolation to LHC
• does strangeness in
p+p at LHC behave
grand canonical ?
• multiplicity increases
with √(sNN)
– canonical and grand
canon. event classes?
plot from PPR Vol I
Ingrid Kraus, TU Darmstadt Hot Quarks 2006, Sardinia, May 16, 2006 29
Prediction for p+p
• significant increase of
ratios at RC ≈ 1.5 fm
• K / and
behave differently– multistrange hadrons
suffer stronger
suppression
• RC will be determined
with ALICE data