The correlation between transverse momentum and multiplicity of charged particles in a two-component model

Transcription

1 The correlation between transverse momentum and multiplicity of charged particles in a two-component model E.V. Andronov, V.V. Vechernin St Petersburg State University June 8, 013 The XXI International Workshop High Energy Physics and Quantum Field Theory, Repino

2 Introduction Two-stage scenario of particle production Creation of emitters Decomposition of emitters

3 Introduction Figure 1: The backward and forward pseudorapidity windows η = ln [ tg ( )] θ LRC observables the event multiplicity in the Backward or Forward window n B, n F the event mean transverse momentum in the Backward or Forward window p tb 1 nb n B i=1 p tbi, p tf 1 nf n F i=1 p tfi

4 Introduction LRC studying B F as a function of F Types of correlations: n B n F - the correlation between charged particle multiplicities in B and F windows p tb n F - the correlation between the event mean transverse momentum in the B window and the charged particle multiplicity in the F window p tb p tf - the correlation between the event mean transverse momenta in B and F windows

5 Introduction Correlation coefficients b abs n n n B n F n B n F n F n F ; ( n B nf = f (n F ), b = d n B nf dn F ) b rel n n babs n n n F n B b abs p t n p tb n F p tb n F n F n F ; ( p tb nf = g (n F ), b = d p tb nf dn F ) b rel p t n babs p t n n F p tb For more information - see e.g. I.Altsybeev talk Forward-backward correlations in pp collisions with ALICE detector, QFTHEP 013, June 4

6 Model Figure : Primary and secondary emitters produce particles in the B and F windows

7 Calculation of the n n correlation coefficient Let the probability to have 1 primary emitters and secondary emitters be ω ( 1, ). Let { 1, } C. Then n F = ω (C) n F C = (n F 1 + n F ) ω ( 1, ) P 1 (n F 1 ) P (n F ) ; C n F,n 1 F 1, n B n F = (n F 1 + n F ) (n B 1 + n B ) P 1 (n F 1 ) P 1 (n B 1 ) P (n F ) P (n B ). n B,n 1 B 1, Here n F 1,n F P 1 (B 1) = 1 δ 1 B1 p i1=1 B B1 (B i1 ) ; i1 {Bi1} i1=1 p B1 (n) = 1; n p B1 (n) = µ B1. n n

8 Calculation of the n n correlation coefficient b abs n n n B n F n B n F n F n F ; b rel n n b abs n n n F n B where b abs n n = D 1 µ B1 µ F 1 + cov ( 1, ) (µ B1 µ F + µ B µ F 1 ) + D µ B µ F 1D µf 1 + D µf + D 1 µ F 1 + D µ F + cov ( 1, ) µ F 1 µ F, b rel n n = b abs n n 1µ F 1 + µ F 1µ B1 + µ B, 1, = C ω (C) 1,; D 1, = 1, 1, ; cov (1, ) = 1 1. If we assume that µ F 1 = µ F, µ B1 = µ B, D µf 1 = D µf, D µb1 = D µb, 1 + =,: b rel n n D µ F D µf + D µ F V.V. Vechernin, arxiv: v1 (010); Proc. of Baldin ISHEPP XX, Volume II, 10 (010)

9 Model Figure 3: Interaction scheme P ( ) = C [ ] r ξ (1 r)[ ξ ]

10 n n correlations b rel n n n B n F n B n F n F n F n F n B W µf 1 = 0.05 W µf 1 = 0.5 V = D, W µ F 1 = Dµ F 1 µ F 1

11 p t n correlations p tb 1 nb n B i=1 p tbi p tb = C p tb n F = C ω (C) p tb C = k 1n B 1 + kn B ω ( 1, ) P 1 (n B n n B,n B 1 + n 1 ) P (n B ) B 1 B 1, ω (C) p tb C n F C = k 1n B 1 + kn B ω ( 1, ) n n B,n B 1 + n B 1 B 1, ( 1µ F 1 + µ F ) P 1 (n B 1 ) P (n B ), where P 1 (B 1) = {Bi1} δ B1 1 i1=1 B i1 1 i1=1 p B1 (B i1 ) ; µ F 1 -the mean charged particles multiplicity in the F window from a primary emitter k 1-the mean transverse momentum of particles in the B window from a primary emitter

12 p t n correlations cov (p tb, n F ) = p tb n F p tb n F = ω ( 1, (( ) ) 1 1 µf 1 + 1, + ( ) ) k 1 n B 1 + k n B µf P 1 (n B n n B,n B 1 + n 1 ) P (n B ). B 1 B One should make an approximation: n B 1 + n B n B n B = 1 µ B1 + µ B Then cov (p tb, n F ) = ω ( 1, (( ) ) 1 1 µf 1 + 1, + ( ) µf ) k 1 1 µ B1 + k µ B 1 µ B1 + µ B. ote: if k 1 = k, then cov (p tb, n F ) = 0, if k 1 k, then cov (p tb, n F ) 0

13 p t n correlations: numerical results b rel p t n p tb n F p tb n F n F n F n F p tb W µf 1 = 0.05 W µf 1 = 0.5

14 Modification Possible functions r = 1 e const r = e const1 ( const) Monte-Carlo simulation n F n B b abs n n n B n F n B n F n F n F

15 Conclusions Expressions for the n n and p t n correlation coefficients were obtained in the simple toy model with interaction n n correlation coefficient may increase with the inclusion of the interaction as well as decrease Only negative p t n correlations take place in this model Application of Monte-Carlo simulations allows us to express correlation coefficients as a function of average number of emitters

16 Plans calculation of n n and p t n correlation coefficients via Monte-Carlo simulation comparison with the experimental results ( independent seeds [1], negative pt n correlations [], etc.) addition of other types of emitters 1 Eva Sicking on behalf of the ALICE Collaboration. CER LHC Seminar Multiple Parton Interactions in ALICE A49 Collaboration, G.A. Feofilov, R.S. Kolevatov, V.P. Kondratiev, P.A. aumenko, V.V. Vechernin Proc. of Baldin ISHEPP XVII, Volume I, (005)

17 Thank you!

18 Backup

19 Introduction Comment Multiplicity distribution depends on probability distribution of emitter configuration and multiplicity distribution from one emitter. Two-stage scenario A. Capella, U.P. Sukhatme, C. I. Tan, J. Tran Thanh Van Phys. Lett. B81, 68 (1979); Phys. Rep. 36, 5 (1994). A.B. Kaidalov, Phys. Lett. B116, 459 (198). A.B. Kaidalov, K.A. Ter-Martirosyan, Phys. Lett. B117, 47 (198). V.A. Abramovsky, O.V. Cancelli, JETP Letters, 31, 566 (1980).

20 Introduction Shortcoming of the n n correlations The n n correlations, arised from the interaction between emitters, are suppressed by the correlations, arised from the fluctuations in number of primary emitters[1]. Therefore in [1,, 3] it is supposed to use the event mean transverse momentum in the backward and forward window as a dynamical variable. One will consider only n n and p t n correlations. Referencies [1] M.A. Braun, C. Pajares, Eur. Phys. J. C16, 349 (000) [] M.A. Braun, C. Pajares, Phys. Rev. Lett. 85, 4864 (000) [3] E.V. Shuryak, ucl. Phys. A661, 119 (1999)

21 n n correlations Model The amount of] emitters in collision should be even [1, ]. So if ξ =, then = ξ. [ ξ Primary and secondary parameters are connected [3, 4, 5, 6]: µ F = µ F 1, µ B = µ B1 [1] A. Capella, U.P. Sukhatme, C. I. Tan, J. Tran Thanh Van Phys. Lett. B81, 68 (1979); Phys. Rep. 36, 5 (1994). [] J. Dias de Deus, R. Ugoccioni, A. Rodrigues, Eur. Phys. J. C16, 537 (000). [3] T.S. Biro, H.B. ielsen, J. Knoll, ucl. Phys. B45, 449 (1984). [4] A. Bialas, W. Czyz, ucl. Phys. B67, 4 (1986). [5] M.A. Braun, C. Pajares, Eur. Phys. J. C16, 349 (000). [6] M.A. Braun, C. Pajares, Phys. Rev. Lett. 85, 4864 (000).

22 n n correlations Correlation coefficient ( ) ( ) r 1 V 1 b rel n n = ( ) ( ) r 1 V 1 + r ( 1 ) ( 1 V ) + V ( ) ( ) + r 1 1 V Wµ F 1 + V + W µf 1 V = D, Wµ = Dµ F 1 F 1 µ F 1

23 p t n correlations b abs p t n p tb n F p tb n F n F n F ; b rel p t n babs p t n n F p tb b rel p t n = ( ) 1 r + r 1 r + r ( 1 r + r r ( 1 ) ( V 1 ( ) 1 r + r, P () C r (1 r) ) ( ) ( ) + r 1 1 V Wµ F 1 + V + W µf 1 + )

24 Formalism otations and definitions Let us consider the set of 1 emitters of the first type, emitters of the second type. Let the probability of a configuration be ω ( 1, ). ω (C)... ω ( 1, )... ; ω (C) = 1; 1, C 1, 1ω ( 1, ) = 1; 1, ω ( 1, ) = ; 1 1 n F = F 1 + F = F i1 + F i ; n B = B 1 + B = B i1 + B i ; i1=1 i=1 {B i1, B i } B 11,..., B 1 1; B 1,..., B, i1 = 1,..., 1, i = 1,..., ; {F i1, F i } F 11,..., F 1 1; F 1,..., F, i1 = 1,..., 1, i = 1,..., C i1=1 i=1

25 Formalism Basic formulae Let k (j1) i1 = k (j1) i1 (j1 = 1,..., B i1) be transverse momenta of particles, produced by the i1-th emitter in the backward window, k (j) i = k (j) i (j = 1,..., B i) be transverse momenta of particles, produced by the i-th emitter in the backward = q (j1) i1 (j1 = 1,..., F i1) - transverse momenta of particles, produced by the i1-th emitter in the forward window, q (j) i (j = 1,..., F i ) - by the i-the in the forward window. window. Similarly, q (j1) i1 P C {B i1,b i } (p tb ) δ p tb 1 B i1 ( ρ 1 i1=1 j1=1 k (j1) i1 ) [ 1 B i1 B i k (j1) i1 + i1=1 j1=1 i=1 j=1 dk (j1) i1 B i ( ρ i=1 j=1 k (j) i ) k (j) i dk (j) i = q (j) i ] [ 1 B i1 + i1=1 i=1 ] 1 B i

26 Formalism Basic formulae ρ 1 (k) dk = ρ (k) k dk = k, ρ (k) dk = 1, ρ 1 (k) k dk = k 1, P C {B i1,b i } (p tb ) dp tb = 1, P (n B, n F ) = C ω (C) P C (n B, n F ), P (p tb, n B, n F ) = C ω (C) P C (p tb, n B, n F ), P (n B, n F ) = P (p tb, n B, n F ) dp tb, P C (n B, n F ) = P C (p tb, n B, n F ) dp tb, P (p tb, n B, n F ) dp tb = 1, P C (p tb, n B, n F ) dp tb = 1 n B,n F n B,n F

27 Formalism Basic formulae For independent emitters: P C (n B, n F ) = {B i1,b i } p (B i, F i ), i=1 δ nb 1 i1=1 B i1+ i=1 B i {F i1,f i } δ nf 1 i1=1 F i1+ i=1 F i 1 i1=1 p 1 (B i1, F i1 ) In case of the long-range correlations: p 1 (B i1, F i1 ) = p B1 (B i1 ) p F1 (F i1 ), p (B i, F i ) = p B (B i ) p F (F i ). Correlations arise only due to fluctuations of the number and types of emitters from event to event

28 Formalism Basic formulae Factorization: P C (n B, n F ) = P C (n B ) P C (n F ), P C (n B ) = P C (n F ) = {B i1,b i } {F i1,f i } δ nb δ nf 1 i1=1 B i1+ i=1 B i 1 i1=1 F i1+ i=1 F i 1 i1=1 1 i1=1 p F1, (n) = p B1, (n) = 1, n n n p F1, (n) = µ F1,, n p B1, (n) = µ B1,. n n p B1 (B i1 ) p B (B i ), i=1 p F1 (F i1 ) p F (F i ), i=1

29 Formalism Basic formulae Similarly P C (p tb, n B, n F ) = P C (p tb, n B ) P C (n F ), P C (p tb, n B ) = P C (p tb ) = n B {B i1,b i } P C (p tb ) dp tb = 1. δ nb P C (p tb, n B ) = 1 i1=1 B i1+ i=1 B i {B i1,b i } 1 P C {B i1,b i } (p tb ) p B1 (B i1 ) p B (B i ), i1=1 i=1 1 P C {B i1,b i } (p tb ) p B1 (B i1 ) p B (B i ), i1=1 i=1

30 Formalism Basic formulae otations: P 1 (B 1) = {Bi1} δ B1 1 i1=1 B i1 P (B ) = {Bi} δ B i=1 B i P 1 (F 1) = {Fi1} δ F1 1 i1=1 F i1 P (F ) = {Fi} δ F i=1 F i 1 i1=1 i=1 1 i1=1 i=1 p B1 (B i1 ) ; p B (B i ) ; p F1 (F i1 ) ; p F (F i ).

31 n n correlations Summation n F ω (C) n F C = n F ω ( 1, ) P 1 (F 1) P (F ) = ω ( 1, ) C n F 1, 1, (F i1 ) + (F i ) + F i1 F j1 + F i F j + F i1 F i {F i1,f i } i1=1 1 p F 1 (F i1 ) + i1=1 i=1 i=1 i1 j1,i1,j1=1 i j,i,j=1 i1=1 i=1 p F (F i ) = ) ω ( 1, ) ( 1µ F 1 + µf ( µ F 1 + 1, ) ) ) ) ( µ F + 1µ F 1 µ F = 1 µ F 1 + µf ( µ F 1 ( + µ F + 1 µ F 1 µ F = 1 µ F 1 + µ F + 1 µ F 1 µ F + 1Dµ F 1 + Dµ F

32 n n correlations Generating functions For any distribution p (k) (p (k) 0, k=0 p (k) = 1) GF is introduced: h (z) p (k) z k k=0 Let GF of distributions from one emitter: ϕ 1 (z F, z B ), ϕ (z F, z B ) Then resulting GF: H (z F, z B ) = ω ( 1, ) (ϕ 1 (z F, z B )) 1 (ϕ (z F, z B )) 1,

33 n n correlations Generating functions Differentiating GF, one can find moments of distribution: ϕ 1 zf =z z B =1 = µ F 1, F ϕ zf =z z B =1 = µ B, B ϕ zf =z z B =1 = µ F, F ϕ 1 zf ϕ 1 z B zf =z B =1 = µ B1, zf =z B =1 = µ F 1 µ F 1, ϕ zf zf =z B =1 = µ F µ F ϕ 1 zb zf =z B =1 = µ B1 µ B1, ϕ zb zf =z B =1 = µ B µ B Correlation coefficient. Coincidence of two methods b abs n n = n B n F n B n F n F n F = ( H z B z F ( H z + H F z F H H z B ) z F zb =z F =1 ( ) ) H z F zb =z F =1.

34 p t n correlations p tb = C p tb n F = C ω (C) p tb C = B 1,B k 1B 1 + k B B 1 + B ω (C) p tb C n F C = B 1,B k 1B 1 + k B B 1 + B ( 1µ F 1 + µ F ) P 1 (B 1) P (B ) ; 1, ω ( 1, ) P 1 (B 1) P (B ) 1, ω ( 1, ) cov (p tb, n F ) = ω ( 1, ) ( 1µ F 1 + µ F ) k 1B 1 + k B B 1 + B 1, B 1,B P 1 (B 1) P (B ) 1, ω ( 1, ) P (B1) P 1 (B) = ω ( 1, ) 1, ( 1µ F 1 + µ F ) B 1,B k 1B 1 + k B B 1 + B [P 1 (B 1) P (B ) P (B 1, B )].

35 p t n correlations Possible approximation k 1B 1 + k B P 1 (B 1) P (B ) = B 1 + B B 1,B = k1 B1 1 + k B B B ; k 1B 1 + k B P (B 1, B ) = B 1 + B B 1,B = k1 B1 + k B. B 1 + B ( ) B 1,B k1b 1 + k B P1 (B 1) P (B ) = B 1,B (B 1 + B ) P 1 (B 1) P (B ) ( ) B 1,B k1b 1 + k B P (B1, B ) = B 1,B (B 1 + B ) P (B 1, B ) If k 1 = k, there is a reduction, that is there are no correlations

36 p t n correlations B B = n B C = 1µ B1 + µ B ; B 1 + B = n B = 1 µ B1 + µ B, µ B µ B1 = µ F µ F 1, 1µ F 1 + µ F B B = 1µ F 1 + µ F 1µ B1 + µ B = µ F 1 µ B1. It can be seen that under this assumption correlations vanish even with k 1 k, that is this approximation is too rough

37 p t n correlations In order to avoid numerical summation, it is proposed to make such an approximation: B 1 p tb = k 1 B 1 + B + k B B 1 + B B 1 n p tb n F = k F 1 B 1 + B + k B n F B 1 + B

38 p t n correlations Correlation coefficient b rel p t n = b abs p t n ( µ F 1 1 r + r ) = 1 r + r k 1 ( 1 r + r r (V ) r 4 ( r 3 (V ) ) 1 4 ) 1 r + r 1 4 (V ( ) ) + V ( ) ( ). + r 1 1 V Wµ F 1 + V + W µf 1

39 p t n correlations Correlation coefficient b rel p t n = b abs p t n ( µ F 1 1 r + r ) = 1 r + r k 1 ( 1 r + r 1 4 ) 1 r + r ( ) r (V ) r (V ) V ( r 3 (V ) ) ( ) ( ). + r 1 1 V + V + W µf 1 1 4

40 n n correlations r = e 0.5 ( 150), P() is B with and D = + 10

41 Comparison with Monte-Carlo simulations