Extraction of moments of net-particle event-by-event fluctuations in - PowerPoint PPT Presentation

Extraction of moments of net-particle event-by-event fluctuations in the CBM experiment V. Vovchenko, I. Kisel for the CBM collaboration DPG Spring Meeting Darmstadt, Germany 15 March 2016 HGS-HIRe Helmholtz Graduate School for Hadron and Ion Research

Outline • Higher-order fluctuations on phase diagram • Rate of statistical convergence of different moments • Efficiency corrections • GEANT simulation and reconstruction of fluctuations • Summary 2 / 17

Introduction A future fixed target experiment at FAIR facility. Up to 10 7 Au+Au collisions per second at 4-11 A GeV (SIS100) and 11-35 A GeV (SIS300). Measurement of bulk and rare probes. Physics programme Equation of state at high baryonic densities Phase transitions at high µ B QCD critical point, probed by e-by-e fluctuations Subthreshold production of hadrons Hypernuclei production 3 / 17

Higher-order moments of fluctuations Let N be a random variable and P ( N ) its probability distribution. N k P ( N ) � � N k � = k -th moment: N σ 2 = � (∆ N ) 2 � = � ( N − � N � ) 2 � Variance: σ 2 = σ 2 = κ 2 Scaled variance: width M κ 1 � N � = � (∆ N ) 3 � S σ = κ 3 Skewness: asymmetry σ 2 κ 2 = � (∆ N ) 4 � − 3 � (∆ N ) 2 � 2 κσ 2 = κ 4 Kurtosis: peakedness σ 2 κ 2 and so on... In heavy-ion collisions N can be conserved charge (baryon, electric, strangeness) or some particle number in a specific phase-space region 4 / 17

Fluctuations in thermodynamics Why are fluctuations interesting? In thermodynamics fluctuations are related to susceptibilities χ ( n ) χ ( n ) = ∂ n ( p / T 4 ) ∂ ( µ/ T ) n σ 2 M = χ (2) S σ = χ (3) κσ 2 = χ (4) χ (1) , χ (2) , χ (2) , Fluctuations are very sensitive to QCD equation of state and can be used to study QCD phase transitions 3 2 /M σ Near CP ∼ increasing powers of ξ 1 0.5 0.1 10.00 χ (2) ∼ ξ 2 2 χ (3) ∼ ξ 4 . 5 T/T c 2 1.00 χ (4) ∼ ξ 7 10 1 gas liquid 0.10 mixed phase Infinite system: ξ → ∞ at CP 0.01 0 0 1 2 3 In HIC ξ � 2 − 3 fm n/n c 5 / 17

0.5 1 35 10.00 30 2 25 T (MeV) 10 20 15 1.00 0.1 10 gas liquid 5 0.10 0.01 0 880 890 900 910 920 930 (MeV) S 0 0 -1 1 -1 35 35 40.00 -10 40.00 1 -1 10.00 10.00 30 30 - 1 0 25 25 0 10 T (MeV) T (MeV) 100 -10 100 10 20 20 10 1.00 1.00 0.00 15 0.00 15 -1.00 -1.00 1 10 10 gas liquid gas liquid 5 5 -10.00 -10.00 -40.00 0 -40.00 0 880 890 900 910 920 930 880 890 900 910 920 930 (MeV) (MeV) Fluctuations in T − µ plane: VDW nuclear matter Nuclear matter as van der Waals system of nucleons -3 n (fm ) 35 0.18 30 0.16 0.13 25 T (MeV) 0.11 20 0.09 15 0.07 10 0.04 gas liquid 5 0.02 0 0.00 880 890 900 910 920 930 (MeV) Vovchenko et al., PRC 91, 064314 (2015) and PRC 92, 054901 (2015) 6 / 17

S 0 0 -1 1 -1 35 35 40.00 -10 40.00 1 -1 10.00 10.00 30 30 - 1 0 25 25 0 10 T (MeV) T (MeV) 100 -10 100 10 20 20 10 1.00 1.00 0.00 15 0.00 15 -1.00 -1.00 1 10 10 gas liquid gas liquid 5 5 -10.00 -10.00 -40.00 0 -40.00 0 880 890 900 910 920 930 880 890 900 910 920 930 (MeV) (MeV) Fluctuations in T − µ plane: VDW nuclear matter Nuclear matter as van der Waals system of nucleons -3 n (fm ) 0.5 1 35 0.18 35 10.00 30 0.16 30 2 0.13 25 25 T (MeV) T (MeV) 0.11 10 20 20 0.09 15 15 1.00 0.1 0.07 10 10 0.04 gas liquid gas liquid 5 5 0.10 0.02 0.01 0 0.00 0 880 890 900 910 920 930 880 890 900 910 920 930 (MeV) (MeV) Vovchenko et al., PRC 91, 064314 (2015) and PRC 92, 054901 (2015) 6 / 17

0 1 -1 35 -10 40.00 10.00 30 - 1 0 25 0 10 T (MeV) 100 100 10 20 1.00 0.00 15 -1.00 1 10 gas liquid 5 -10.00 -40.00 0 880 890 900 910 920 930 (MeV) Fluctuations in T − µ plane: VDW nuclear matter Nuclear matter as van der Waals system of nucleons -3 n (fm ) 0.5 1 35 0.18 35 10.00 30 0.16 30 2 0.13 25 25 T (MeV) T (MeV) 0.11 10 20 20 0.09 15 15 1.00 0.1 0.07 10 10 0.04 gas liquid gas liquid 5 5 0.10 0.02 0.01 0 0.00 0 880 890 900 910 920 930 880 890 900 910 920 930 (MeV) (MeV) S 0 -1 35 40.00 1 -1 10.00 30 25 T (MeV) -10 20 10 1.00 15 0.00 -1.00 10 gas liquid 5 -10.00 0 -40.00 880 890 900 910 920 930 (MeV) Vovchenko et al., PRC 91, 064314 (2015) and PRC 92, 054901 (2015) 6 / 17

Fluctuations in T − µ plane: VDW nuclear matter Nuclear matter as van der Waals system of nucleons -3 n (fm ) 0.5 1 35 0.18 35 10.00 30 0.16 30 2 0.13 25 25 T (MeV) T (MeV) 0.11 10 20 20 0.09 15 15 1.00 0.1 0.07 10 10 0.04 gas liquid gas liquid 5 5 0.10 0.02 0.01 0 0.00 0 880 890 900 910 920 930 880 890 900 910 920 930 (MeV) (MeV) S 0 0 -1 1 -1 35 35 40.00 -10 40.00 1 -1 10.00 10.00 30 30 - 1 0 25 25 0 10 T (MeV) T (MeV) 100 -10 100 10 20 20 10 1.00 1.00 0.00 15 0.00 15 -1.00 -1.00 1 10 10 gas gas liquid liquid 5 5 -10.00 -10.00 -40.00 0 -40.00 0 880 890 900 910 920 930 880 890 900 910 920 930 (MeV) (MeV) Vovchenko et al., PRC 91, 064314 (2015) and PRC 92, 054901 (2015) 6 / 17

Beam energy dependence Can be measured in different acceptance windows at different energies For small window fluctuations approach ideal gas For large window global charge conservation plays role Measurements should be performed in different windows Peculiarities in energy dependence may signal criticality 7 / 17

Needed statistics to measure higher moments How much statistics are needed for accurate estimation of higher moments? For a large sample of Gaussian distributed variables � � � 6 σ 2 24 σ 4 720 σ 8 ∆( κσ 2 ) = ∆( S σ ) = n , , ∆( κ 6 /κ 2 ) = . n n More rigorously: Delta theorem, X. Luo, JPG 39, 025008 (2012) Simulation result 4 10 Dashed: σ Poisson 2 = 5 Solid: σ 2 = 15 ∆ (S σ ) 3 10 Dash-dotted: σ 2 = 40 ∆ ( κσ 2 ) 2 10 ∆ ( κ 6 / κ 2 ) 1 10 0 10 10% -1 10 1% -2 10 -3 10 -4 10 1 2 3 4 5 6 7 8 9 10 11 12 10 10 10 10 10 10 10 10 10 10 10 10 Events 8 / 17

Monte Carlo simulation: Poisson statistics 10 12 Poisson-distributed numbers with ¯ N = 5 Expected values: σ 2 / M = S σ = κσ 2 = κ 6 /κ 2 = 1 1.4 σ 2 = 5 1.2 1.0 0.8 σ 2 /M 0.6 0.4 0.2 0.0 10 10 11 10 1 2 3 4 5 6 7 8 9 12 10 10 10 10 10 10 10 10 10 10 Events 9 / 17

Monte Carlo simulation: Poisson statistics 10 12 Poisson-distributed numbers with ¯ N = 5 Expected values: σ 2 / M = S σ = κσ 2 = κ 6 /κ 2 = 1 1.4 σ 2 = 5 1.2 1.0 0.8 σ 2 /M 0.6 S σ 0.4 0.2 0.0 10 10 11 10 1 2 3 4 5 6 7 8 9 12 10 10 10 10 10 10 10 10 10 10 Events 9 / 17

Monte Carlo simulation: Poisson statistics 10 12 Poisson-distributed numbers with ¯ N = 5 Expected values: σ 2 / M = S σ = κσ 2 = κ 6 /κ 2 = 1 σ 1.4 2 = 5 1.2 1.0 0.8 σ 2 /M 0.6 S σ 0.4 κσ 2 0.2 0.0 10 10 11 10 1 2 3 4 5 6 7 8 9 12 10 10 10 10 10 10 10 10 10 10 Events 9 / 17

Monte Carlo simulation: Poisson statistics 10 12 Poisson-distributed numbers with ¯ N = 5 Expected values: σ 2 / M = S σ = κσ 2 = κ 6 /κ 2 = 1 1.4 σ 2 = 5 1.2 1.0 0.8 σ 2 /M 0.6 S σ 0.4 κσ 2 κ 6 / κ 2 0.2 0.0 10 10 11 10 1 2 3 4 5 6 7 8 9 12 10 10 10 10 10 10 10 10 10 10 Events Higher fluctuation moments require higher statistics. 9 / 17

Monte Carlo simulation: Poisson statistics 10 12 Poisson-distributed numbers with ¯ N = 15 Expected values: σ 2 / M = S σ = κσ 2 = κ 6 /κ 2 = 1 σ 1.4 2 = 15 1.2 1.0 0.8 0.6 σ 2 /M S σ 0.4 κσ 2 κ 6 / κ 2 0.2 0.0 10 10 11 10 1 2 3 4 5 6 7 8 9 12 10 10 10 10 10 10 10 10 10 10 Events Statistical error grows with ¯ N . Convergence rate will depend on kinematic window. 10 / 17

Monte Carlo simulation: Poisson statistics 10 12 Poisson-distributed numbers with ¯ N = 40 Expected values: σ 2 / M = S σ = κσ 2 = κ 6 /κ 2 = 1 1.4 σ 2 = 40 1.2 1.0 0.8 0.6 σ 2 /M S σ 0.4 κσ 2 κ 6 / κ 2 0.2 0.0 10 10 11 10 1 2 3 4 5 6 7 8 9 12 10 10 10 10 10 10 10 10 10 10 Events Statistical error grows with ¯ N . Convergence rate will depend on kinematic window. 10 / 17

Efficiency corrections Since not all particles are reconstructed and identified, the efficiency corrections are needed The simplest one is the binomial correction Binomial correction assumptions Detection of all particles is independent of each other Probability to register particle is binomial Only a single efficiency parameter ε is needed Original cumulants K i reconstructed from measured k i K 1 = k 1 ε K 2 = k 2 + ( ε − 1) k 1 ε 2 K 3 = k 3 + 3( ε − 1) k 2 + ( ε − 1)( ε − 2) k 1 ε 3 · · · For net-particle numbers ( N = N + − N − ) more complicated correction involving factorial moments exists. 11 / 17