Do small systems equilibrate chemically?

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Do small systems equilibrate chemically?

• Ingrid Kraus
• TU Darmstadt

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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 PbPb to pp system size and energy
  dependence
• Canonical suppression
• Concept of equilibrated clusters
• Comparison to data
• Summary

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Statistical Ensembles

• 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, m fix

E, V, N
Laplace transformation
SE
SN
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Partition function and its derivations

• 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 lt 1.5 GeV, Baryons m lt 2 GeV

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Partition function and model parameters

• Partition function for species i with degenaracy
  factor gi
• with
• () for fermions, (-) for bosons
• Model parameters
• T and mB mS constrained by strangeness
  neutrality
• V cancels in ratios mQ constrained by charge of
  nuclei

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Comparison to Experimental Data
A.Andonic, P. Braun-Munzinger, J. Stachel,
nucl-th/0511071

• Accurancy in T, mB few MeV
• Different data selected for fits

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T - mB systematics, extrapolation to LHC
Chemical 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 (v(sNN) 5.5TeV) 1 MeV Nucl. Phys. A 697
(2002) 902 Grand canonical ensemble for PbPb
predictions
hep-ph/0511094
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Predictions for PbPb

• Reliable for stable particles
• Benchmark for resonances
• Errors
• T 170 /- 5 MeV
• µB 1 4 MeV

- 1
All calculations with THERMUS hep-ph/0407174
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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 W / p and/or W / K

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Resonance Decays

• Hadron Resonance gas
• W no resonance contribution
• X
• 50 from feed-down
• both exhibit same T dependence
• K decay exceeds thermal at LHC
• p
• thermal production constant
• resonance contribution dominant
• 75 of all p from resonances

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Canonical suppression

• 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

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Canonical suppression

• Stronger suppression for multi-strange hadrons
• Suppression depends on strangeness content, not
  difference
• (expected from gS)

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Suppression by undersatured phase-space

• Stronger suppression for multi-strange hadrons
• Suppression depends on difference of strangeness
  content
• (power of gS)

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Suppression in small systems

• Suppressed strangeness production beyond
  canonical suppression
• addressed by canonical treatment and
  undersaturation factor gS
• new equilibrated clusters

SPS v(sNN) 17 AGeV
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Modification of the model

• Statistical Model approach T and µB
• Volume for yields ? radius R used here
• Deviations strangeness undersaturation factor gS
• Fit parameter
• Alternative small clusters (RC) in fireball (R)
  RC R
• Chemical equilibrium in subvolumes canonical
  suppression
• RC free parameter

R
RC
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Fit Example

• All Fits were performed with THERMUS hep-ph/040
  7174
• Fits with gS / RC give better description of data

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System size and energy dependence of T and mB

• T independent of
• System size
• Data selection
• Energy

• µB smaller at RHIC

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System size and energy dependence of the cluster
size

• Small clusters in all systems
• Small system size dependence
• pp
• energy dependence?
• PbPb
• depends on data selection (multistrange hadrons
  needed)

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System size and energy dependence of the cluster
size

• AA clusters smaller than fireball
• RC not well defined for RC 2 fm because
  suppression vanishes

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Canonical Suppression

• Particle ratios saturate at RC 2 - 3 fm
• no precise determination for small strangeness
  suppression

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Summary

• Canonical ensemble
• volume dependend suppression
• stronger suppression modeled with smaller,
  thermally equilibrated clusters
• successful description of pp, CC, SiSi data
• strangeness production in small systems
  reproduced with equilibrated subvolumes
• Outlook
• strangeness production in pp at LHC

• Grand canonical ensemble
• successful description of AuAu, PbPb data
• extrapolations allow for predictions
• determination of thermal parameters with few
  particle ratios
• proper treatment of resonances is mandatory

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Going into formulas

• performing the momentum integration
• () for bosons, (-) for fermions
• mi mass of hadron i
• Particle number density

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Density and Ratios

• Approx. modified Bessel function
• Particle ratio
• Antiparticle/Particle ratio

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System size dependence of T and mB

• µB decreases at mid-rapidity in small systems .
• . as expected from increasing antibaryon /
  baryon ratio

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System size dependence of the cluster size

• Same trend as K / p

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More SPS and RHIC 200 GeV Data
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Model setting with gS

• gS
• sensitive on data sample
• increase with size
• increase with energy

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Extrapolation to LHC

• does strangeness in pp at LHC behave grand
  canonical ?
• multiplicity increases with v(sNN)
• canonical and grand canon. event classes?
• plot from PPR Vol I

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Prediction for pp

• significant increase of ratios at RC 1.5 fm
• K / p and W / X behave differently
• multistrange hadrons suffer stronger suppression
• RC will be determined with ALICE data