Download - Using Higher Moments of Fluctuations and their Ratios in the Search for the QCD Critical Point
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Using Higher Moments of Fluctuations and their Ratios in the Search for the QCD Critical Point
Christiana Athanasiou, MIT4
work with: Krishna Rajagopal (MIT) Misha Stephanov (University of Illinois)
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Outline
• Introduction• Critical Contribution to Particle Multiplicity Fluctuations• Ratios of Fluctuation Observables• Summary
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QCD Phase Diagram
vacuum
Quark-Gluon Plasma
Critical point
μB / MeV
T / MeV
~ 170
~ 940
nuclearmatter
0
Hadron gas Color Superconductor
crossover
Models
Lattice simulations
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Heavy-Ion Collision Experiments
• Locating the critical point from first-principles – hard Heavy-Ion Collision Experiments
• RHIC: Au-Au collisions at • Momentum asymmetry collective flow strongly-coupled QGP
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~ 940
vacuum
Quark-Gluon Plasma
Critical point
μB / MeV
T / MeV
~ 170
nuclearmatter
0
Hadron gas Color Superconductor
crossover
QCD Phase Diagram
RHIC energy scan
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Heavy-Ion Collision Experiments - continued
• As QGP expands and cools, it follows trajectories with approx.
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Quark-Gluon Plasma
μB / MeV
T / MeV
~ 170
~ 940
nuclearmatter
0
Color Superconductor
QCD Phase Diagram
RHIC energy scan
Critical point
Hadron gas
crossover
vacuum
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Heavy-Ion Collision Experiments 2
• As QGP expands and cools, it follows trajectories with approx.
• Chemical freeze-out: system dilute enough that particle numbers freeze
• To maximize critical point (CP) effects vary to get freeze-out point near CP
Event-by-Event fluctuations
• Detector “sees” particle multiplicities from freeze-out conditions
• Find observables that are sensitive to proximity to the CP
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Outline
• Introduction• Critical Contribution to Particle Multiplicity Fluctuations• Ratios of Fluctuation Observables• Summary
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• Critical mode - σ : order parameter of the chiral phase transition
• Correlation length diverges at the CP
• Develops long wavelength correlations at the CP• Effective action
Critical Mode FluctuationsCritical Mode
• Near the CP: with dimensionless and known in the
Ising universality class
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Critical Mode Fluctuations
• at CP in the thermodynamic limit• Finite system lifetime compared to away from the CP (Berdnikov, Rajagopal 00)
• Critical mode fluctuations affect Particle multiplicity fluctuations Momentum distributions Ratios, etc…
of these particles.
• σ couples to pions and protons:
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Measuring fluctuations in particle multiplicities
measure the mean, variance, skewness, etc…
• Can repeat these calculations for pions, net protons, etc• Want to obtain the critical contribution to these quantities• We will use cumulants, e.g.:
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Critical contribution to pion/proton correlators
(Rajagopal, Shuryak, Stephanov 99, Stephanov 08)
ξ2
ξ7
ξ9/2
+ …
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Net protons and mixed correlators
• Note: correlators depend on 5 parameters:
which have large uncertainties
• Net protons: Adapt previous expressions by replacing:
• Can also calculate mixed correlators, e.g. 2 pion – 2 proton:
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Calculating multiplicity cumulants
• Second cumulant – variance:
Poisson - Bose-Einstein effects- Other interactions- Etc..
ignore
• Normalizing:
• For mixed cumulants with i protons and j pions:
• Non-critical contribution to ωipjπ = δi,i+j + δj,i+j + (few %)
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Multiplicity cumulants – critical point signature
• Higher cumulants depend stronger on ξ:
• As we approach the CP ξ increases and then decreases as we move away from it
• CP signature: Non-monotonic behavior, as a function of collision energy, of multiplicity cumulants
• E.g. toy example
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Multiplicity cumulants – example plots
Parametrization (Cleymans et al 05):
and using
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Data on net proton cumulants
where
(STAR Collaboration 2010)
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Critical contribution to proton ω4
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Multiplicity cumulants – movie
Changing the critical μB – the location of the CP:
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Outline
• Introduction• Critical Contribution to Particle Multiplicity Fluctuations• Ratios of Fluctuation Observables• Summary
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Uncertainties of parameters
• Cumulants depend on 5 non-universal parameters:
• have large uncertainties hard to predict the critical contribution to cumulants• By taking ratios of cumulants can cancel some
parameter dependence minimize observable uncertainties
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Ratios of multiplicity cumulants
No parameter dependence
Ratios taken after subtracting Poisson and defined
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Parameter independent ratios
• Parameter and energy independent ratios:
where
• All equal to 1 if CP contribution dominates
• How to use these ratios:• If one sees peaks in the measured cumulants at some μB
• Calculate these ratios around the peak• If equal to 1 Parameter independent way of verifying
that the fluctuations you see are due to the CP
• Poisson contribution:
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Constraining parameters
• If CP found, can constrain parameters by measuring cumulant ratios near the CP
• Parameters appear in certain combinations in the cumulants can only constraint 4 independent (but not unique) combinations
• For example, some choices are:1. using or 2. using or
3. using4. using
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Outline
• Introduction• Critical Contribution to Particle Multiplicity Fluctuations• Ratios of Fluctuation Observables• Summary
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Summary
• We used particle multiplicity fluctuations as a probe to the location of the CP
• Higher cumulants of event-by-event distributions are more sensitive to critical fluctuations
• Constructed cumulant ratios to identify the CP location with reduced parameter uncertainties
• CP signature: Non-monotonic behavior, as a function of collision energy, of multiplicity cumulants
• If CP is found, showed how to use cumulant ratios to constraint the values of the non-universal parameters
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Thank you!