Long-Range Rapidity Correlations and Fluctuations in Model with Two Types

of Sources

Evgeny Andronov evgeny.andronov1@gmail.com

SPbSU, Department of High Energy and Elementary Particles Physics,

Laboratory of Ultra-High Energy Physics

St. Petersburg, 04/12/14

International Conference dedicated to the Novozhilov’s 90-th anniversary,

In Search of Fundamental Symmetries

Outline

•  Introduction •  Model with two types of strings •  pT-n correlation analysis •  Results for forward-backward

fluctuations •  Summary

2 Evgeny Andronov

3 Evgeny Andronov

Introduction

[1]A.Capella,    U.P.Sukhatme,    C.I.Tan    and    J.Tran    Thanh    Van,    Phys.    Le@.    B81    (1979)    68;    Phys.    Rep.  236    (1994)    225.           [2]A.B.Kaidalov,    Phys.    Le@.,    116B  (1982)  459.

Two-stage scenario of particles production [1,2]: 1.  Formation of quark-gluon strings 2.  Strings decays General assumptions for central rapidity region: •  Translational invariance in rapidity of the charged particles spectra

from a single string •  Identical decay properties for all rapidity intervals with the same

widths

4

String fusion model

Single string Overlapping strings

M.A.Braun    and    C.Pajares,    Phys.    Le@.    B287    (1992)    154;    Nucl.    Phys.    B390    (1993)    542,    549;   M.A.Braun    and    C.Pajares,    Phys.    Rev.    Le@.    85    (2000)    4864;                   N.S.Amelin,    M.A.Braun    and    C.Pajares,    Phys.    Le@.  B306  (1993)    312;   M.A.Braun,    C.Pajares    and    V.V.Vechernin,    Internal    Note/FMD    ALICE-­‐‑-­‐‑-­‐‑INT-­‐‑-­‐‑-­‐‑2001-­‐‑-­‐‑-­‐‑16

onetstrnewt pNp 22 =

Multiplicity Transverse momentum

Evgeny Andronov

With high enough density in the transverse region strings could overlap and interact, impacting on different observables:

Long-range pseudorapidity correlations

5 Evgeny Andronov

Observables: •  Event multiplicity (extensive) •  Event mean transverse

momentum (intensive)

Correlation strength:

•  pT-n correlations are more sensitive to string fusion effects than n-n correlations (b=0 without string fusion)

•  b<0 for the fixed number of strings case •  Transition from negative to positive values for fluctuating number of strings

M.A.  Braun,  R.S.  Kolevatov,  C.  Pajares,  V.V.  Vechernin,  Eur.Phys.J.  C32,    (2004),  535-­‐‑546    

Fluctuations. Strongly intensive quantities.

6 Evgeny Andronov

Joint fluctuations of two extensive quantities A and B Two families of strongly intensive quantities [1,2] (do not depend on the number of sources and fluctuations in number of sources): •  • 

Proposal to study forward-backward fluctuations:

[1]  Gorenstein,  M.  &  Gazdzicki,  M.,  Phys.Rev.  C84,  (2011),  014904   [2]  Gazdzicki,  M.;  Gorenstein,  M.  &  Mackowiak-­‐‑Pawlowska,  M.,  Phys.Rev.  C88,  (2013),  024907  

in an independent particles model

Model with strings of two types

E.  A.,  V.  Vechernin,  PoS(QFTHEP2013),  054  (2014). E.  A.,  V.  Vechernin,  to  be  published  in  the  proceedings  of  QC  and  HS  XI  2014  

Effectively takes into account string fusion effects: N1 non-fused strings and N2 fused strings in an event with probability

7 Evgeny Andronov

Event multiplicities:

Event mean transverse momentum:

Presence of this fraction leads to the impossibility of explicit analytical calculation of the pT-n correlation parameter b

One fused string – result of interaction of two identical primary strings Number of primary identical strings N=N1+2N2

Model with strings of two types

8 Evgeny Andronov

Fraction shows up in two places:

Average multiplicity in the forward window from a string of two types:

Due to the fusion:

Average transverse momentum of the particles in the backward window from a string of two types:

Due to the fusion:

Model with strings of two types

9 Evgeny Andronov

Proposed approximation:

One type limit holds within this approximation:

Then average values:

- the scaled variance of the number of primary strings

10

Long-range pT-n correlation parameter

Evgeny Andronov

- mean value of the number of primary strings

Blue  –  approximation Red  –  basic  formula Negligible difference

- fixed number of primary strings

Evgeny Andronov 10

N

+Poissonian dist.

- the scaled variance of the number of primary strings

11

Long-range pT-n correlation parameter

N - mean value of the number of primary strings

Evgeny Andronov

+Poissonian dist.

With approximation

2

1

0.1 0.1

1

2

0 0 Evgeny Andronov

Forward-backward fluctuations

12 Evgeny Andronov

•  Sigma is strongly intensive – no dependence on and •  Correct normalization only for Poissonian sources

for strings of one type and symmetrical windows

N

- scaled variance for multiplicity from a single string

Forward-backward fluctuations

13 Evgeny Andronov

In the model with strings of two types:

•  Sigma is strongly intensive only for

•  Correct normalization only for

Forward-backward fluctuations If fusion changes decay features:

Evgeny Andronov 14

•  Predictions for the pT-n correlation strength with and without approximation were compared.

•  Negative values of the pT-n correlation strength were obtained for the fixed number of strings case.

•  Transition from the negative values of the pT-n correlation strength to the positive ones.

•  Forward-backward correlations were studied within model of independent strings of one and two types

•  Strongly intensive quantity Sigma could lose its main property, if string fusion produces non-poissonian sources

15 Evgeny Andronov

Summary

Thank you for your attention!

16 Evgeny Andronov

Back-up

17 Evgeny Andronov

Long-range correlations. General remarks.

•  LRC are governed by the fluctuations in number of strings and by the string fusion effects

•  n-n correlation coefficient is zero without these fluctuations and fusion effects

•  pT-n correlation coefficient is zero without fusion effects.

Evgeny Andronov 18

Model with strings of two types

E.  A.,  V.  Vechernin,  PoS(QFTHEP2013),  054  (2014). E.  A.,  V.  Vechernin,  to  be  published  in  the  proceedings  of  QC  and  HS  XI  2014  

Effectively takes into account string fusion effects: N1 non-fused strings and N2 fused strings in an event with probability

19 Evgeny Andronov

String interaction function:

One fused string – result of interaction of two identical primary strings Number of primary identical strings N=N1+2N2

•  No interaction for small number of strings (small density)

•  Total fusion for large number of strings (high density)

Forward-backward fluctuations

Evgeny Andronov 20

Parametrization:

One- and two-particle densities

Two-particle correlation function:

Forward-backward fluctuations

Evgeny Andronov 21

Correlations from a single string!

Experimental studies on LRC

p+(anti-)p, 900 GeV (1988) Linear regression

Pb+Pb, 158 AGeV/c (2005)

pT-­‐‑n

Negative pT-n correlations [1]  R.E.  Ansorge  et  al.  (UA5  Collaboration),  Z.  Phys.,  C37-­‐‑191,  (1988).

[2]  G.  Feofilov  et  al.  (NA49  Collaboration),  Proc.  of  Baldin  ISHEPP  XVII,  Volume  I,  222  (2005)

22 Evgeny Andronov

Long-range pseudorapidity correlations

1) Linear regression 2) Correlator formula

Two definitions of the correlation parameter for two observables B and F

23 Evgeny Andronov

•  n-n •  pT-n •  pT-pT

n – charged particles multiplicity

Для того чтобы упростить вычисления, ранее было предложено использоваь следую-щее приближение:

B1 +B2 → N1 µB1 +N2 µB2

Далее будут приведены результаты как с использованим приближения, так и без него.Также снова будет рассмотрен случай с нефлуктуирующим числом первичных источ-ников.

ω [X] =�X2� − �X�2

�X�Нет зависимости от N и ω [N ].

B(1) +B(2) ≈ N1µB1 +N2µB2

η = −ln

�tg

�θ

2

��

dN

d η

δηF

δηB

∆η

�B�F = a+ b · F

b =�B · F � − �B� · �F �

�F 2� − �F �2

pt =1

n

n�

i=1

p(i)t

5

- event mean value of transverse momentum

Observable types

Correlation types

24

Long-range pseudorapidity correlations

Evgeny Andronov

Mechanism of particle production in the model

with independent strings

A.Capella,    U.P.Sukhatme,    C.I.Tan    and    J.Tran    Thanh    Van,    Phys.    Le@.    B81    (1979)    68;    Phys.    Rep.  236    (1994)    225.           A.B.Kaidalov,    Phys.    Le@.,    116B(1982)459

25 Evgeny Andronov

26

nF = F !P F( )F" = nF one

nB = B !P B( )B" = nB one

Single string case

Evgeny Andronov

27

In case of sufficiently large gap between windows one string produces particles in both windows independently!

( ) FBBF

FB nnBFPBFnn ⋅=⋅⋅=∑,

,

P F,B( ) = P B( ) !P F( )( )

Single string case

Evgeny Andronov

bn!n = 0

nF = F !P F( )F" = nF one

nB = B !P B( )B" = nB one

28

In case of sufficiently large gap between windows one string produces particles in both windows independently!

Single string case

nF

Phys.Rev.D  vol.34,  num.11(1986)

e+ e- at 29 GeV

bn!n = 0

b = 0.002± 0.006Experiment:

Evgeny Andronov

nF = F !P F( )F" = nF one

nB = B !P B( )B" = nB one

P F,B( ) = P B( ) !P F( )( )

( ) FBBF

FB nnBFPBFnn ⋅=⋅⋅=∑,

,

Two-stage scenario of particles production.

29

p-p, low energies

I stage: strings creation.

[1]A.Capella,    U.P.Sukhatme,    C.I.Tan    and    J.Tran    Thanh    Van,    Phys.    Le@.    B81    (1979)    68;    Phys.    Rep.  236    (1994)    225.           [2]A.B.Kaidalov,    Phys.    Le@.,    116B(1982)459

Evgeny Andronov

Two-stage scenario of particles production.

30

p-p, low energies

[1]A.Capella,    U.P.Sukhatme,    C.I.Tan    and    J.Tran    Thanh    Van,    Phys.    Le@.    B81    (1979)    68;    Phys.    Rep.  236    (1994)    225.           [2]A.B.Kaidalov,    Phys.    Le@.,    116B(1982)459

qseap-p, high energies

I stage: strings creation.

Evgeny Andronov

Long-range n-n correlation parameter. Monte-Carlo simulations.

 -­‐‑  scaled variance of the number of primary strings

0 0,1

1

2

µ = Dµ = 0.5  -­‐‑  parameters of string decay

31 Evgeny Andronov

N  -­‐‑  mean value of the number of primary strings

n!nrelb =

nFnB

"nBnF ! nB nF

F2n !

2

nF

Long-range n-n correlation parameter. Monte-Carlo simulations.

0 0,1

1

2

Single type of the string limits (r=0 or r=1) Correspondence to the analytical results – V.V.  Vechernin,  Proc.  Of  XX  Baldin  ISHEPP  (2011)  

32 Evgeny Andronov

Long-range n-n correlation parameter. Connection with experiment.

By varying the width and the position of the centrality class one can scan our plot in two directions and search for the predicted effects

33 Evgeny Andronov

N

Width of CC

CC=centrality class

Center of CC

0 0,1 1 2

34

Long-range pT-n correlation parameter. Monte-Carlo simulations.

Absence of correlations without fusion effects!

Evgeny Andronov

0 0,1 1 2

35

Long-range pT-n correlation parameter. Monte-Carlo simulations.

Transition from the negative values to positive.

G.  Feofilov  et  al.  (NA49  Collaboration),  Proc.  of  Baldin  ISHEPP  XVII,  Volume  I,  222  (2005)

Evgeny Andronov

bn!n =DN1

µ + 2DN2µ + 2 2 cov N1,N2( )

N1! µ[ ]+ N2! µ[ ]+DN1µ + 2DN2

µ + 2 2 cov N1,N2( )

bn!n =DNµ

N! µ[ ]+DNµ

Transition to one-type case

36

Model with strings of two types

Evgeny Andronov

Comparison of the n-n correlation coefficients

without fluctuations in the number of primary

strings.

туирующее число померонов. В этом случае не приходится вводить корректирующийфактор reduc, т.к. выбираются только события с постоянным ненулевым числом по-меронов.

Оказалось, что для нефлуктуирующего числа померонов, а значит для нефлук-туирующего числа первичных струн N (приведенная дисперсия числа первичных ис-точников ω [N ] = D(N)

N= 0) удается вычислить коэффициент nB − nF корреляции

(1.26) аналитически:

babsn−n =µB · µF ·N · r (N) · (1− r (N)) ·

�3− 2

√2�

µF2 ·N · r (N) · (1− r (N)) ·

�3− 2

√2�+ µF ·N ·

�1− r (N) +

√2

2r (N)

� ; (1.43)

breln−n = babsn−n ·µF

µB

, (1.44)

или для используемых значений µF = µB = 0.5:

breln−n =0.25 ·N · r (N) · (1− r (N)) ·

�3− 2

√2�

0.25 ·N · r (N) · (1− r (N)) ·�3− 2

√2�+ 0.5 ·N ·

�1− r (N) +

√2

2r (N)

� . (1.45)

Таким образом, сравнивая результаты работы генератора событий с аналитическимвыражением, можно проверять, насколько правильно работает код. В свою очередьподтверждается и справедливость полученных аналитических выражений. На Рис. 1.3представлено подобное сравнение для параметра shift = 15, а на Рис. 1.4 для shift =100. Коэффициент корреляции вычисляется в относительных переменных. Красныелинии соответствуют slope = 20, синие - slope = 4 и черные - slope = 0.2. Для slope =0.2 коэффициент корреляции оказывается отличным от 0 только в точке N = shift(такое возможно только для четного shift). В самом деле,

r (N) =

1− 5 · 10−5 : N ≡ shift+ 25 · 10−5 : N ≡ shift− 20.5 : N ≡ shift

т.е. с хорошей точностью можно считать, что при N �= shift имеется только один типисточников, а значит корреляции отсутствуют. Для нечетного shift:

r (N) =

�1− 7 · 10−3 : N ≡ shift+ 17 · 10−3 : N ≡ shift− 1

что ведет к тому, что во всех точках breln−n = 0. Можно отметить, что случай такогорезкого перехода от отсутствия слияния к полному слиянию рассматривался скореедля изучения математического предела поведения модели для ступенчатой функцииr (N), чем для получения физических результатов.

Как видно из Рис. 1.3-1.4, генератор событий полностью воспроизводит поведе-ние, описываемое аналитическим выражением.

15

Slope=20 Slope=4 Slope=0.2

37 Evgeny Andronov