Multiplicity Fluctuations Josef Uchytil FNSPE CTU in Prague 27. 9. - PowerPoint PPT Presentation

Multiplicity Fluctuations Josef Uchytil FNSPE CTU in Prague 27. 9. 2017 Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 1 / 25

Outline Statistical moments 1 Multiplicity fluctuations within a simple model 2 Multiplicity fluctuations within a Hadron Gas Model (HRG) with 3 chemical equilibrium Multiplicity fluctuations within a Hadron Gas Model (HRG) without 4 chemical equilibrium Conclusion and outlook 5 Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 2 / 25

Statistical moments ′ : ϕ m ( X ) ′ = E ( X m ) m-th statistical moment ϕ m ( X ) m-th central moment ϕ m ( X ) : ϕ m ( X ) = E ( X − EX ) m first four central moments are of great significance mean: M = ϕ 1 , variance: σ 2 = ϕ 2 skewness: S = ϕ 3 /ϕ 3 / 2 - measure of the assymetry of the probability 2 distribution kurtosis: κ = ϕ 4 /ϕ 2 2 - measure of the ”tailedness”of the probability distribution Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 3 / 25

Skewness (left) and kurtosis (right). Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 4 / 25

Calculation of the multiplicity fluctuations within the statistical model grandcanonical and canonical ensemble assumed, event-by-event distributions of conserved quantities - characterized by the moments (M, σ , S, κ ) introduction of the following products: S σ = ϕ 3 /ϕ 2 , κσ 2 = ϕ 4 /ϕ 2 , M /σ 2 = ϕ 1 /ϕ 2 , S σ 3 / M = ϕ 3 /ϕ 1 -the volume term in the distribution gets obviously cancelled; direct comparison of experimental measurement and theoretical calculation possible large volume limit (V → ∞ ) - all statistical ensembles (MCE, CE, GCE) equivalent Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 5 / 25

Partition functions in statistical ensembles - GC formalism HRG model - all relevant degrees of freedom contained in the partition function confined, strongly interacting matter - interactions that result in resonance formation included nj �� + ∞ � z j ( n j ) λ GC partition function: Z GC ( λ j ) = � j exp j where n j =1 n j � n j m j z j ( n j ) = ( ∓ 1) n j +1 g j V 2 π 2 n j Tm 2 � is the single particle partition j K 2 T function K 2 . . . modified Bessel function, V . . . volume of the hadron gas λ j = exp( µ j T ) . . . fugacity for each particle species j , m j . . . hadron mass µ j . . . chemical potential of a particle species j , g j = 2 J j + 1 . . . spin degeneracy ∓ . . . upper sign for fermions, lower sign for bosons Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 6 / 25

Partition functions in statistical ensembles - canonical formalism constraint - fixed charges → partition function not factorized into one-species expressions let � Q = ( Q 1 , Q 2 , Q 3 ) = ( B , S , Q ) · · · vector of charges let � q j = ( q 1 , j , q 2 , j , q 3 , j ) = ( b j , s j , q j ) · · · vector of charges of the hadron species j Wick-rotated fugacities: λ j = exp[ i � i q i , j φ i ] Canonical partition function: � 2 π �� 3 � 1 d φ i e − iQ i φ i Z � Q = Z GC ( λ j ) i =1 2 π 0 h . . . set of hadron species: λ j → λ h λ j Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 7 / 25

Results of the first four moments Z � ∂ Z � Q − nj � qj 1 � ∞ � N h � = ∂λ h | λ h =1 = � Q n j =1 z j ( n j ) Z � j ∈ h Z � Q Q Z � ∂ Z � � � �� � + ∞ Q − nj � qj N 2 1 ∂ � � = λ h Q | λ h =1 = � n j =1 n j z j ( n j ) + h Z � ∂λ h ∂λ h j ∈ h Z � Q Q Z � � + ∞ � + ∞ Q − nj � qj − nk � qk � n j =1 z j ( n j ) � n k =1 z k ( n k ) j ∈ h k ∈ h Z � Q Z � � � � ∂ Z � ��� � + ∞ Q − nj � qj N 3 1 ∂ λ h ∂ n j =1 n 2 � � Q | λ h =1 = � = λ h j z j ( n j ) + h j ∈ h Z � ∂λ h ∂λ h ∂λ h Z � Q Q �� Z � � � + ∞ � + ∞ Q − nj � qj − nk � qk 3 n j =1 n j z j ( n j ) � n k =1 z k ( n k ) + j ∈ h k ∈ h Z � Q Z � � + ∞ � + ∞ � + ∞ Q − nj � qj − nk � qk − nl � ql � n j =1 z j ( n j ) � n k =1 z k ( n k ) � n l =1 z l ( n l ) j ∈ h k ∈ h l ∈ h Z � Q Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 8 / 25

� ∂ ∂ Z � = 1 � ∂ � ∂ � ���� N 4 Q � � λ h λ h λ h | λ h =1 = h ∂λ h ∂λ h ∂λ h ∂λ h Z � Q   + ∞ + ∞ + ∞ Z � Z � Q − n j � q j Q − n j � q j − n k � q k � � n 3 � � n 2 � � j z j ( n j ) +4 j z j ( n j ) z k ( n k )  Z � Z � Q Q n j =1 n j =1 n k =1 j ∈ h j ∈ h k ∈ h   + ∞ + ∞ Z � Q − n j � q j − n k � q k � � � � + 3 n j z j ( n j ) n k z k ( n k )  Z � Q n j =1 n k =1 j ∈ h k ∈ h   + ∞ + ∞ + ∞ Z � Q − n j � q j − n k � q k − n l � q l � � � � � � + 6 n j z j ( n j ) z k ( n k ) z l ( n l )  Z � Q n j =1 n k =1 n l =1 j ∈ h k ∈ h l ∈ h + ∞ + ∞ + ∞ � � � � � � + [ z j ( n j ) z k ( n k ) z l ( n l ) j ∈ h n j =1 k ∈ h n k =1 l ∈ h n l =1 + ∞ Z � Q − n j � q j − n k � q k − n l � q l − n m � q m � � z m ( n m ) ] Z � Q n m =1 m ∈ h Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 9 / 25

Asymptotic fluctuations in the canonical ensemble N j ! � N j � N j e − � N j � 1 Poissonian distribution of fluctuations: P GC = Canonical partition function: � 2 π �� 3 � 1 d φ i e − iQ i φ i Z � Q = Z GC ( λ j ) i =1 2 π 0 integration performed in the complex w plane: w i = exp[ i φ i ] Z � Q = 1 dw Q w − B − 1 w − S − 1 w − Q − 1 j z j (1) w b i B w s i S w q i � � � exp � dw B dw S (2 π i ) 3 B S Q Q obviously: w − ( B , Q , S ) = exp[ − ( B , Q , S ) ln w B , Q , S ] B , Q , S w ) = w b j − 1 w s j − 1 w q j − 1 ; ρ B , S , Q = B , S , Q g ( � B S Q V z k (1) S w q k V w b k B w s k f ( � w ) = − ρ B ln w B − ρ S ln w S − ρ Q ln w Q + � k Q 1 � � � q j = dw Q g ( � w ) exp[ Vf ( � w )] Z � dw B dw S Q − � (2 π i ) 3 method: saddle-point expansion Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 10 / 25

Multiplicity fluctuations for a simple model I. classical pion gas - no b or s quarks → � Q = (0 , 0 , Q ) saddle point: w 0 = λ Q only π + and π − considered ν = V , g ( w ) = 1 / w , f ( w ) = − ρ Q ln w + z π V ( w + 1 w ) �� j = ± 1 z π (1) w q j � 1 dw q w − Q − 1 � Z Q = exp q 2 π i Q s π + = s π − = 0; m π + = m π − = 139 . 57 MeV � p 2 + m 2 V d 3 p exp( − � z j (1) = (2 J j + 1) j ) = (2 π ) 3 � p 2 + m 2 V d 3 p exp( − � j ) (2 π ) 3 Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 11 / 25

Multiplicity fluctuations for a simple model II. � � � � γ ( λ Q ) − α ( λ Q ) � Z π Q = Z GC 1 λ Q + 1 1 1 + O ( V − 2 ) − λ 2 λ Q 2 π f ′′ ( λ Q ) V λ Q λ 3 Q f ′′ ( λ Q ) Q Q � Q − Q i = Z GC ′′ ( λ Q )[1 + 1 1 Z π V [ γ ( λ Q ) λ Q +1 2 π f Q + ( q j − 1) α ( λ Q ) − 1 1 ′′ ( λ Q )] + O ( V − 2 )] 2( q j − 1)( q j − 2) λ 2 λ Q Q f thermodynamical limit V → ∞ : Z π = z π λ ± 1 Q + O ( V − 1 ) = � π ± � GC + O ( V − 1 ) � π ± � = z π Q ∓ 1 Z π Q Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 12 / 25

Fluctuations in a hadron resonance gas model with chemical equilibrium Susceptibilities and cumulants: = ∂ l ( P / T ) 4 χ ( i ) ∂ ( µ i / T ) l | T l χ ( i ) 1 1 1 = VT 3 � N i � c = VT 3 � N i � χ ( i ) 1 (∆ N i ) 2 � 1 (∆ N i ) 2 � � � 2 = c = VT 3 VT 3 χ ( i ) 1 (∆ N i ) 3 � 1 (∆ N i ) 3 � � � 3 = c = VT 3 VT 3 �� (∆ N i ) 2 � 2 � χ ( i ) 1 (∆ N i ) 4 � 1 (∆ N i ) 4 � � � 4 = c = − 3 VT 3 VT 3 Equilibrium pressure: P / T 4 = i ln Z M / B 1 � ( V , T , µ B , µ Q , µ S ) m i VT 3 ln Z M / B = ∓ Vg i d 3 k ln(1 ∓ z i exp( − ǫ i / T )) � m i (2 π ) 3 Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 13 / 25

Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 14 / 25

Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 15 / 25

Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 16 / 25

Inclusion of resonances VT 3 ∂ ( P / T 4 ) � ∂ ( µ h / T ) | T = � N h � + � N R � � n h � R (1) R where � N h � and � N R � are the means of the primordial numbers of hadrons and resonances, respectively. The sum runs over all the resonances in the model. 26 particle species we consider stable: π 0 , π + , π − , K + , K − , K 0 , ¯ K 0 , η and p , d , λ 0 , σ + , σ 0 , σ − , Ξ 0 , Ξ − , Ω − and their respective anti-baryons r b R r n R � n h � R ≡ � h , r b R r - the branching ratio of the decay-channel and n R h , r = 0 , 1 , . . . - number of hadrons h formed in that specific decay-channel. Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 17 / 25

Josef Uchytil (FNSPE CTU in Prague) Multiplicity Fluctuations 27. 9. 2017 18 / 25