Theoretical description of the forward-backward multiplicity correlations in windows separated in azimuth and rapidity

Transcription

1 Theoretical description of the forward-backward multiplicity correlations in windows separated in azimuth and rapidity Vladimir Vechernin St.-Petersburg State University 16 September 2014 Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 1 / 33

2 Outline Connection of the forward-backward (FB) correlation coefficient b with two-particle correlation function C 2 The b and C 2 in the model with independent identical emitters (strings) Fitting of the model parameters by FB multiplicity correlations between two small windows, separated in azimuth and rapidity Comparison of the FB correlation in large 2π windows with the model results Connection with the di-hadron correlation analysis Conclusions Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 2 / 33

3 Definitions Connection of the FB correlation coefficient with two-particle correlation function - 1 Traditionally one uses the following definition of the FB correlation coefficient: b abs n F n B n F n B D nf where D nf = n 2 F n F 2 (1) To avoid the trivial influence of absolute values of n F and n B on the correlation coefficient we go to the relative or scaled observables: For these observables ν F = n F / n F and ν B = n B / n B (2) b rel = ν F ν B 1 ν 2 F 1 = n F n B b abs. (3) Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 3 / 33

4 Definitions Connection of the FB correlation coefficient with two-particle correlation function - 2 The two-particle correlation function C 2 is defined through the inclusive ρ 1 and double inclusive ρ 2 distributions: C 2 (η F, φ F ; η B, φ B ) = ρ 2(η F, φ F ; η B, φ B ) ρ 1 (η F, φ F )ρ 1 (η B, φ B ) 1 (4) ρ 1 (η, φ) = d 2 N dη dφ, ρ 2(η F, φ F ; η B, φ B ) = For a small window δη δφ around η, φ we have ρ 1 (η, φ) d 4 N dη F dφ F dη B dφ B (5) n δη δφ, (6) here n is the mean multiplicity in the acceptance δη δφ. Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 4 / 33

5 Small windows Connection of the FB correlation coefficient with two-particle correlation function - 3 For two small windows: δη F δφ F around η F, φ F and δη B δφ B around η B, φ B we have n ρ 2 (η F, φ F ; η B, φ B ) F n B. (7) δη F δφ F δη B δφ B The formulae (6) and (7) are the base for the experimental measurement of the one- and two-particle densities of charge particles ρ 1 and ρ 2, and hence of the two-particle correlation function C 2 (4), for which by (6) and (7) we have: C 2 (η F, φ F ; η B, φ B ) n F n B n F n B n F n B. (8) where n F and n B are the event multiplicities in these two small windows. Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 5 / 33

6 Small windows Connection of the FB correlation coefficient with two-particle correlation function - 4 Comparing with traditional definition of the FB correlation coefficient (1), for small FB windows by (8) we have Note that for small forward window: b rel = b rel = n F 2 D nf C 2 (η F, φ F ; η B, φ B ) (9) D nf = n F [1 + n F C 2 (η F, φ F ; η F, φ F )], (10) n F 1 + n F C 2 (η F, φ F ; η F, φ F ) C 2(η F, φ F ; η B, φ B ) (11) So by (11) we see that the traditional definition of the FB correlation coefficient in the case of small observation windows coincides with the standard definition of two-particle correlation function C 2 upto some common factor, which depends on the width of the forward window. Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 6 / 33

7 Small windows Connection of the FB correlation coefficient with two-particle correlation function - 5 One can go in C 2 to the variables: η sep = η F η B, η C = (η F + η B )/2 (12) φ sep = φ F φ B, φ C = (φ F + φ B )/2 (13) and using the connection (8) or (11) check up experimentally the dependence of the two-particle correlation function C 2 on η C for the different configurations and separations between FB observation windows. In the central rapidity region and for small windows we have C 2 (η F, φ F ; η B, φ B ) = C 2 (η sep, φ sep ) (14) b rel = D nf = n F [1 + n F C 2 (0, 0)], (15) n F 1 + n F C 2 (0, 0) C 2(η sep, φ sep ) (16) Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 7 / 33

8 Large windows Connection of the FB correlation coefficient with two-particle correlation function - 6 For windows of arbitrary width in azimuth and rapidity, situated in the central rapidity region, a model-independent way, we obtain: b rel = D nf = n F [1 + n F I FF ], (17) n F 1 + n F I FF I FB (η sep, φ sep ) (18) where 1 I FB (η sep, φ sep ) = dy 1 dϕ 1 dy 2 dϕ 2 C 2 (η 1 η 2 ; φ 1 φ 2 ) δy F δϕ F δy B δϕ B δy F δϕ F δy B δϕ B (19) 1 I FF = (δy F δϕ F ) 2 dy 1 dϕ 1 dy 2 dϕ 2 C 2 (η 1 η 2 ; φ 1 φ 2 ) (20) δy F δϕ F δy F δϕ F A.Capella, A.Krzywicki, Phys.Rev.D18, 4120 (1978). C.Pruneau, S.Gavin, S.Voloshin, Phys.Rev.C66, (2002). Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 8 / 33

9 The model Model with independent identical emitters - 1 ρ N 1 (η) = Nλ 1 (η), (21) ρ N 2 (η F, η B ; φ sep ) = Nλ 2 (η F, η B ; φ sep ) + N(N 1)λ 1 (η F )λ 1 (η B ). (22) Then after averaging over N the one- and two-particle densities of charge particles are given by ρ 1 (η) = N λ 1 (η), (23) ρ 2 (η F, η B ; φ sep ) = N [λ 2 (η F, η B ; φ sep ) λ 1 (η F )λ 1 (η B )]+ N 2 λ 1 (η F )λ 1 (η B ) (24) and ρ 2 (η F, η B ; φ sep ) ρ 1 (η F )ρ 1 (η B ) = (25) = N [(λ 2 (η F, η B ; φ sep ) λ 1 (η F )λ 1 (η B )] + D N λ 1 (η F )λ 1 (η B ), where D N is the event-by-event variance D N = N 2 N 2 of the number of emitters. M.A.Braun, C.Pajares, V.V., Phys.Lett.B493, 54 (2000). Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 9 / 33

10 The model Model with independent identical emitters - 2 Then we find C 2 (η F, η B ; φ F φ B ) = Λ(η F,η B ;φ F φ B )+ω N N, where ω N is the event-by-event scaled variance ω N = D N / N of the number of emitters and Λ(η F, η B ; φ F φ B ) = λ 2(η F, η B ; φ F φ B ) λ 1 (η F )λ 1 (η B ) 1 (26) is the two-particle correlation function for charged particles produced from a decay of a single emitter (string). A.Capella, A.Krzywicki, Phys.Rev.D18, 4120 (1978). Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 10 / 33

11 The model Model with independent identical emitters - 3 In the central rapidity region, where each string contributes to the particle production in the whole rapidity region, one has the translation invariance in rapidity λ 1 (η) = µ 0 = const, Λ(η F, η B ; φ sep ) = Λ(η F η B ; φ sep ), (27) then ρ 1 (η) = N µ 0 = const, (28) C 2 (η sep, φ sep ) = Λ(η sep, φ sep ) + ω N N. (29) So we see that this common pedestal in C 2 (η sep, φ sep ) is physically important. By (29) we see that from the height of the pedestal (ω N / N ) one can obtain the important physical information on the magnitude of the fluctuation of the number of emitters N at different energies and centrality fixation. Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 11 / 33

12 The model FB correlation in the model - 1 FB multiplicity correlation strength b rel in the case of the observation windows of arbitrary width in azimuth and rapidity, which are situated in the central rapidity region, are given by b rel = [ω N + J FB (η sep, φ sep )]µ 0 δ F 1 + [ω N + J FF ]µ 0 δ F, (30) (the relative variables are using, V.V. arxiv: , ), where 1 J FB (η sep, φ sep ) = dη 1 dφ 1 dη 2 dφ 2 Λ(η 1 η 2 ; φ 1 φ 2 ), δη F δφ F δη B δφ B δη F δφ F δη B δφ B (31) 1 J FF = (δη F δφ F ) 2 dη 1 dφ 1 dη 2 dφ 2 Λ(η 1 η 2 ; φ 1 φ 2 ), (32) δη F δφ F δη F δφ F Λ(η; φ) is the pair correlation function for a single string, ω N = N2 N 2 N is the e-by-e scaled variance of the number of strings, δ F δη F δφ F /2π is the acceptance of the forward window, µ 0 is the average rapidity density of the charged particles from one string. Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 12 / 33

13 The model FB correlation in the model - 2 η sep, φ sep are the separations between the centers of FB windows in rapidity and azimuth. b rel = b LR + b SR, (33) b LR ω N µ = 0 δ F 1 + [ω N + J FF ]µ 0 δ F, (34) b SR = µ 0 δ F 1 + [ω N + J FF ]µ 0 δ F J FB (η sep, φ sep ) (35) The Long-Range (LR) contribution arising due to e-by-e fluctuation in the number of emitters (strings). The Short-Range (SR) contribution originating from the pair correlation function Λ(η; φ) of a single string. Note that at Λ(η, φ) = 0 we have J FB = J FF = 0 and bλ=0 SR = 0, but b Λ=0 rel = b LR Λ=0 = ω Nµ 0 δ F 1 + ω N µ 0 δ F 0. M.A.Braun, R.S.Kolevatov, C.Pajares, V.V., Eur.Phys.J.C32, 535 (2004). Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 13 / 33

14 The model FB correlation in the model - 3 Note that by (31) and (32) for windows of small acceptances in rapidity and azimuth, when δη η cor and δφ φ cor (where η cor and φ cor are the characteristic correlation lengthes, defined by the behavior of the pair correlation function Λ(η, φ) (39) of a single string), we have J FB (η sep, φ sep ) Λ(η sep, φ sep ), (36) and the formula (30) takes the simple form J FF Λ(0, 0) (37) b rel [ω N + Λ(η sep, φ sep )]µ 0 δ F 1 + [ω N + Λ(0, 0)]µ 0 δ F, (38) which enables to fit the model parameters by experimental observation of the FB correlations between two small windows, varying the separation between these windows in azimuth and rapidity. Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 14 / 33

15 The model The pair correlation function of a single string The parametrization for the pair correlation function Λ(η, φ) of a single string (basing on the Schwinger mechanism of a string decay, V.V. arxiv: ): ( ) Λ(η, φ) = Λ 1 e η η 1 e φ2 φ Λ 2 e η η 0 η 2 + e η+η 0 η 2 e ( φ π) 2 φ 2 2. (39) This formula has the nearside peak, characterizing by parameters Λ 1, η 1 and φ 1, and the awayside ridge-like structure, characterizing by parameters Λ 2, η 2, η 0 and φ 2 (two wide overlapping hills shifted by ±η 0 in rapidity, η 0 - the mean length of a string decay segment). We imply that in formula (39) φ π. (40) If φ > π, then we use the replacement φ φ + 2πk, so that (40) was fulfilled. With such completions the Λ(η, φ) meets the following properties Λ( η, φ) = Λ(η, φ), Λ(η; φ) = Λ(η, φ), Λ(η, φ + 2πk) = Λ(η, φ) (41) Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 15 / 33

16 The model Calculation of the integrals The integrals (31) and (32) in the case of symmetric arbitrary windows δη F = δη F δη and δφ F = δφ F δφ reduce to δη J FB (η sep, φ sep ) = (δηδφ) 2 dη δη δφ δφ δη J FF = I FB (0, 0) = (δηδφ) 2 dη δη = 4(δηδφ) 2 dη 0 δη δφ 0 dφ Λ(η+η sep, φ+φ sep ) t δη (η) t δφ (φ), δφ δφ dφ Λ(η, φ) t δη (η) t δφ (φ) = (42) dφ Λ(η; φ) (δη η) (δφ φ). (43) where t δy (y) is a triangular weight function arising at integration due to phase: t δy (y) = [θ( y)(δy + y) + θ(y)(δy y)] θ(δy y ). (44) Then the η φ factorization for near and away side contributions in the fit (39) for Λ(η; φ) enables to reduce the (42) and (43) to single integrals. Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 16 / 33

17 Fitting of the model parameters Fitting of model parameters by FBC in small windows δη=0.2, δφ=π/4 windows φ sep =0 φ sep =90 φ sep =180 LRC φ sep =0 φ sep =90 φ sep = δη=0.2, δφ=π/4 windows φ sep =0 φ sep =45 φ sep =135 LRC φ sep =0 φ sep =45 φ sep =135 correlation coefficient - b corr correlation coefficient - b corr pp 0.9 TeV distance between windows - η sep pp 0.9 TeV distance between windows - η sep Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 17 / 33

18 Fitting of the model parameters Fitting of model parameters by FBC in small windows δη=0.2, δφ=π/4 windows φ sep =0 φ sep =90 φ sep =180 LRC φ sep =0 φ sep =90 φ sep = δη=0.2, δφ=π/4 windows φ sep =0 φ sep =45 φ sep =135 LRC φ sep =0 φ sep =45 φ sep =135 correlation coefficient - b corr correlation coefficient - b corr pp 7 TeV 0.02 pp 7 TeV distance between windows - η sep distance between windows - η sep Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 18 / 33

19 Fitting of the model parameters Fitting of model parameters by FBC in small windows - 3 Results of the fitting of the model parameters by FB correlations between two small windows, separated in azimuth and rapidity: s, TeV LRC µ 0 ω N µ 0 Λ η φ SRC µ 0 Λ η φ η ω N = N2 N 2 N is the e-by-e scaled variance of the number of strings, µ 0 is the average rapidity density of the charged particles from one string, i =1 corresponds to the nearside and i =2 to the awayside contributions, η 0 is the mean length of a string decay segment. I.Altsybeev, PhD Thesis, SPbSU, G.Feofilov et al. (for ALICE Collaboration), PoS (Baldin ISHEPP XXI) 075, Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 19 / 33

20 Comparison with experiment Comparison of FBC in large 2π windows with SM - 1 correlation coefficient - b corr π azimuth windows SM δη=0.2 SM δη=0.4 SM δη=0.6 SM δη=0.8 δη=0.2 δη=0.4 δη=0.6 δη=0.8 correlation coefficient - b corr π azimuth windows SM δη=0.2 SM δη=0.4 SM δη=0.6 SM δη=0.8 δη=0.2 δη=0.4 δη=0.6 δη= pp 0.9 TeV 0.1 pp 7 TeV gap between windows - η gap gap between windows - η gap Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 20 / 33

21 Comparison with experiment Comparison of FBC in large 2π windows with SM π azimuth windows, η gap = π azimuth windows, η gap =0 0.6 pp 0.9 TeV 0.6 pp 7 TeV correlation coefficient - b corr correlation coefficient - b corr LRC+SRC LRC SRC 0.9 TeV width of windows - δη 0.1 LRC+SRC LRC SRC 7 TeV width of windows - δη Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 21 / 33

22 Comparison with experiment Comparison of FBC in large 2π windows with SM π azimuth windows, η gap =0 pp 0.9 TeV correlation coefficient - b corr SM 0.9 TeV 7 TeV width of windows - δη Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 22 / 33

23 Comparison with experiment Connection with the di-hadron correlation analysis - 1 The di-hadron correlation function C( y, φ) S/B 1 (45) takes into account all possible pair combinations of particles produced in given event in some ONE LARGE pseudorapidity window y ( Y, Y ), where d 2 N S = (46) d y d φ and the B is the same but in the case of uncorrelated particle production. Experimentalists obtain the B by the event mixing procedure. We can express the enumerator of (45) through the two-particle correlation function: S( y, φ) = Y /2 Y /2 dy 1 dy 2 ρ 2 (y 1, y 2 ; φ) δ(y 1 y 2 y) (47) Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 23 / 33

24 Comparison with experiment Connection with the di-hadron correlation analysis - 2 In the central rapidity region, when the translation invariance takes place within the whole rapidity interval ( Y /2, Y /2), we have ρ 2 (y 1, y 2 ; φ) = ρ 2 (y 1 y 2 ; φ) and one can fulfill the integration in (47): S( y, φ) = ρ 2 ( y; φ) t Y ( y) (48) where the t Y ( y) is a triangular weight function t δη (y) = [θ( y)(δη + y) + θ(y)(δη y)] θ(δη y ). (49) Figure: The triangular weight function arising due to phase space. Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 24 / 33

25 Comparison with experiment Connection with the di-hadron correlation analysis - 3 In the denominator of (45) we should replace the ρ 2 (y 1, y 2 ; φ) by the product ρ 1 (y 1 )ρ 1 (y 2 ), which due to the translation invariance in rapidity reduces simply to ρ 2 0. Then Substituting into (45) we get B( y, φ) = ρ 2 0 t Y ( y). (50) C( y, φ) = ρ 2 ( y; φ) ρ 2 1 = C 2 ( y, φ), (51) 0 We see that if the translation invariance in rapidity takes place within the whole interval ( Y /2, Y /2), then the definition (45) for the di-hadron correlation function C leads to the standard two-particle correlation function C 2 (4) (see meanwhile the remarks below). Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 25 / 33

26 Comparison with experiment Comment on the normalization By the formulae: we see that calculating C 2 (η F, φ F ; η B, φ B ) = ρ 2(η F, φ F ; η B, φ B ) ρ 1 (η F, φ F )ρ 1 (η B, φ B ) 1, ρ 1 (η, φ) n δη δφ, n ρ 2 (η F, φ F ; η B, φ B ) F n B δη F δφ F δη B δφ B C( η, φ) S B 1 we must to normalize S and B dividing by the number of events (N ev ), not by the number of triggers (N tr ), pairs (N par ) or B(0, 0). Otherwise the C will not coincide with C 2. Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 26 / 33

27 Conclusion Conclusions The observation of multiplicity-multiplicity correlation with two small (in azimuth and rapidity) windows, enables to measure the two-particle correlation function C 2 in accordance with the standard definition even in the case of nonhomogeneous distributions (e.g. as in pa) without using the event mixing procedure. The model with strings as independent identical emitters well describes the FB multiplicity correlation in large 2π windows, when its parameters are fitted by the correlation in small windows. The relative contribution of the Long-Range Correlation (LRC), originating from e-by-e fluctuation in the number of emitters (strings), considerably increases with energy while the contribution of the Short-Range Correlations (SRC), originating from the pair correlation function of a single string remains practically the same. We must to apply the proper normalization to obtain C 2 by the di-hadron analysis. Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 27 / 33

28 Backup - 1 Backup - 1 Backup slides - 1 Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 28 / 33

29 Backup - 1 Fitting the model parameters at 2.76 TeV At present we have not the experimental value of FB correlation coefficient b rel with small windows (δη = 0.2, δφ = π/4) at the 2.76 TeV energy for fitting of the model parameters at this energy. So for a rude evaluation we take the mean value of the parameters at 0.9 and 7 TeV: s, TeV LRC µ 0 ω N Then with these parameters we have: µ 0 Λ η φ SRC µ 0 Λ η φ η Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 29 / 33

30 Backup - 1 Comparison of FBC in large 2π windows with SM correlation coefficient - b corr π azimuth windows SM δη=0.2 SM δη=0.4 SM δη=0.6 SM δη=0.8 δη=0.2 δη=0.4 δη=0.6 δη=0.8 correlation coefficient - b corr π azimuth windows, η gap =0 0.1 pp 2.76 TeV gap between windows - η gap 0.1 SM 0.9 TeV 2.76 TeV 7 TeV width of windows - δη Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 30 / 33

31 Backup slides - 2 Backup - 2 Backup slides - 2 Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 31 / 33

32 Backup - 2 Comments on the event mixing - 1 In the framework of the model with strings as independent identical emitters we have for the enumerator and the denominator of (45): S( y, φ) = ρ 2 ( y; φ) t Y ( y) = ρ N 2 ( y; φ) t Y ( y) = (52) B( y, φ) = = Y /2 Y /2 = [ N Λ( y, φ) + N 2 ]µ 2 0 t Y ( y), Y /2 Y /2 dy 1 dy 2 ρ 1 (y 1 )ρ 1 (y 2 ) δ(y 1 y 2 y) = dy 1 dy 2 ρ N 1 (y 1 ) ρ N 1 (y 2 ) δ(y 1 y 2 y) = = ρ 2 0 t Y ( y) = N 2 µ 2 0 t Y ( y), (53) we have noted that λ 1 (y) = µ 0. Then by C = S/B 1 we get C( y, φ) = ω N + Λ( y, φ) N = C 2 ( y, φ), (54) Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 32 / 33

33 Backup - 2 Comments on the event mixing - 2 But if instead of (53) one has B( y, φ) = Y /2 Y /2 dy 1 dy 2 ρ N 1 (y 1 )ρ N 1 (y 2 ) δ(y 1 y 2 y) = N 2 µ 2 0 t Y ( y) as it sometimes takes place in a di-hadron data analysis (or if some other artificial normalization conditions for the B( y, φ) are being used), then instead of (54) by C = S/B 1 we get C( y, φ) = N N 2 Λ( y, φ), (55) which does not correspond to the standard two-particle correlation function C 2 ( y, φ), defined by (4). Compare (55) with (54) we see that in this case the resulting C( y, φ) does not have an additional contribution reflecting the event-by-event fluctuation in the number of emitters. It depends only on the pair correlation function of a single string Λ( y, φ) and, therefore, is equal to zero in the absence of the pair correlation from one string. Dubna (15-20 September 2014) ISHEPP XXII V. Vechernin 33 / 33