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On description of the correlation between multiplicities in windows separated in azimuth and rapidity
V.V. Vechernin St.-Petersburg State University
28 June 2013

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Outline

Connection of the forward-backward (FB) correlation coefficient b with two-particle correlation function C2 Triggered and untriggered di-hadron correlations Model with strings as independent identical emitters Taking into account the string fusion and the FSI (final state interactions) Connection between the ridge and the flows

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Definitions

Connection of the FB correlation coefficient with two-particle correlation function - 1
By definition the two-particle correlation function C2 is defined through the inclusive 1 and double inclusive 2 distributions: C2 (F , F ; B , B ) = 1 ( , ) = d 2N , d d 2 (F , F ; B , B ) -1 1 (F , F )1 (B , B ) d 4N d F d F d B d B (1) (2)

2 (F , F ; B , B ) =

To measure the 1 one has by definition to take a small window around , , then n 1 ( , ) , (3) here n is the mean multiplicity in the acceptance . One has to reduce the acceptance until the ratio (3) becomes constant.
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Definitions

Connection of the FB correlation coefficient with two-particle correlation function - 2
To measure the 2 one has by definition to take TWO small windows: F F around F , F and B B around B , B , then 2 (F , F ; B , B ) nF nB . F F B B (4)

One has to reduce the acceptances of the observation windows until the ratio (4) becomes constant. So by (3) and (4) the definition (1) means the following experimental procedure of the determination of the correlation function C2 : C2 (F , F ; B , B ) nF nB - nF nB , nF nB (5)

where nF and nB are the event multiplicities in TWO small windows: F F around F , F and B B around B , B .
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Definitions

Connection of the FB correlation coefficient with two-particle correlation function - 3
Traditionally one uses the following definition of the FB correlation coefficient: b
abs



nF nB - nF n nF 2 - nF 2

B

or

b

rel



nF b nB

abs

(6)

For small FB windows by (5) we have babs = n
F

nB
F

Dn

C2 (F , F ; B , B ) ,

b

rel

=

nF 2 C2 (F , F ; B , B ) DnF (7)

Note that for small forward window: DnF nF . So by (7) we see that the traditional definition of the FB correlation coefficient in the case of TWO small observation windows coincides with the standard definition of two-particle correlation function C2 upto some common factor nB or nF , which depends on the width of windows.
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Definitions

Connection of the FB correlation coefficient with two-particle correlation function - 4
Note that one can go in C2 to the variables: = F - B , = F - B , C = (F + B )/2 C = (F + B )/2 (8) (9)

and EXPERIMENTALLY check up the dependence of the two-particle correlation function C2 on C for the different configurations and separations between FB observation windows. Summing up, we see that by the standard definition (1) the experimental determination of the two-particle correlation function C2 (F , F ; B , B ) requires (5) the measurements of the event multiplicities nF and nB in TWO SMALL windows: F F around F , F , and B B around B , B , which is performed in our approach.
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Definitions

Triggered and untriggered di-hadron correlations - 1
Untriggered di-hadron correlation function C ( , ) S /B - 1 (10) takes into account all possible pair combinations of particles produced in given event in some ONE LARGE pseudorapidity window (-Y , Y ), where d 2N (11) S= d d and the B is the same but in the case of uncorrelated particle production. Experimentalists obtain the B by the event mixing procedure. Note that (10) has only indirect connection with the standard definition (1) of the two-particle correlation function C2 (F , F ; B , B ). It's easy to show, that C2 (F , F ; B , B ) = C ( , ) only in the case when the pseudorapidity translation invariance (the independence C2 on C ) takes place.
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Definitions

Triggered and untriggered di-hadron correlations - 2
Note also that even in the presence of the translation invariance the details of the event mixing procedure can lead to the loss in C ( , ) the common "pedestal", which takes place in C2 ( , ) (see arXiv:1305.0857 for details). Important that the experimental procedure (5): C2 (F , F ; B , B ) nF n
B

- nF n nF nB

B

=

n n

F F

nB nB

-1 ,

(13)

based on the event-by-event multiplicity observations in TWO small windows F F around F , F , and B B around B , B , which is performed in our approach in correspondence with the standard definition (1) of the two-particle correlation function C2 enables in any case to determine the correlation function C2 without using of the event mixing procedure.
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Definitions

Triggered and untriggered di-hadron correlations - 3
Triggered di-hadron correlation: C ( , ) S /B - 1 where S= d 2N d d (14)

(15)

and the B is the same but in the case of uncorrelated particle production. It takes into account all possible pair combinations of particles produced in given event in some ONE LARGE pseudorapidity window (-Y , Y ), with some additional conditions on the momenta of these particles. Usually they take the momentum of one (trigger) particle belonging to some higher momentum interval than the momentum of its pair.

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Definitions

Triggered and untriggered di-hadron correlations - 4
Clear that this modification can be implemented in our approach based on the event-by-event nF and nB multiplicity observations in TWO small windows F F and B B by taking into account only the particles belonging to corresponding momentum intervals pF and pB .

Note that with small momentum intervals pF and pB we simply go to the distributions: 1 ( , , p ) = and 2 (F , F , pF ; B , B , pB ) =
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d 3N n = d d dp p

(16)

nn d 6N = FB d F d F dpB d B d B dpB ...
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Mo dels

Model with strings as independent identical emitters - 1
N ( ) = N 1 ( ) , 1 N (F , B ; ) = N 2 (F , B ; ) + N (N - 1)1 (F )1 (B ) . 2 (18) (19)

Then after averaging over N the one- and two-particle densities of charge particles are given by 1 ( ) = N 1 ( ) , (20) 2 (F , B ; ) = N [2 (F , B ; )-1 (F )1 (B )]+ N 2 1 (F )1 (B ) (21) and 2 (F , B ; ) - 1 (F )1 (B ) = (22) = N [(2 (F , B ; ) - 1 (F )1 (B )] + DN 1 (F )1 (B ) , where DN is the event-by-event variance DN = N of emitters.
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2

-N

2

of the number
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Mo dels

Model with strings as independent identical emitters - 2
Then we find

C2 (F , B ; ) =

(F ,B ;)+N N

,

where N is the event-by-event scaled variance N = DN / N of the number of emitters and (F , B ; F - B ) = 2 (F , B ; F - B ) -1 1 (F )1 (B ) (23)

is the two-particle correlation function for charged particles produced from a decay of a single string.

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Mo dels

Model with strings as 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 1 ( ) = µ0 = const , then 1 ( ) = N µ0 = const , (25) ( , ) + N C2 ( , ) = . (26) N Recall that and are the distances between the centers of forward and backward windows in rapidity and azimuth. So we see that this common "pedestal" in C2 ( , ) is physically important. By (26) we see that from the height of the "pedestal" (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.
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(F , B ; ) = (F - B ; ) ,

(24)


Mo dels

Taking into account the string fusion and the FSI - 1
local fusion (overlaps) M.A. Braun, C. Pajares Eur.Phys.J. C16, 349, (2000) n = µ0 k Sk /0 ,
2 pt

k

k

=p

2 0

k,

k = 1, 2, 3, ...

(27)

global fusion (clusters) M.A. Braun, F. del Moral, C. Pajares, Phys.Rev. C65, 024907, (2002) p
2 t cl

=p

2 0

kcl ,

n

cl

= µ0

kcl Scl /0 ,

kcl = k 0 /Scl (28)

the cellurar version of SFM Vechernin V.V., Kolevatov R.S., hep-ph/0304295; hep-ph/0305136 Braun M.A., Kolevatov R.S., Pajares C., Vechernin V.V. Eur.Phys.J. C32 (2004) 535.
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Mo dels

Taking into account the string fusion and the FSI - 2

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Mo dels

Taking into account the string fusion and the FSI - 3
As discussed, in the central rapidity region di-hadron correlation function: 2 ( , ) C ( , ) = C2 ( , ) -1 (29) 2 1 Consider very simple model, in which we will not take into account the two-particle correlation between particles originating from the decay of a same string (( , ) = 0). Then for a given string configuration i = 1, ..., K (a given event) , we can assume i2 (1 , 2 ) = i1 (1 )i1 (2 ) , (30) where i1 () is an inclusive distribution of charged particles produced by the given string configuration i with taking into account the string fusion and the FSI:
i (an n=1
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i1

()

i0

[1 + 2

cos n + b sin n)] =

i n

i0

[1 + 2
n=1

i i vn cos n( - n )] .

(31)
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Mo dels

Taking into account the string fusion and the FSI - 4
Here i0 =
i an = i bn = i vn =

1 2

i1 ()d , i1 () cos n d , i1 () sin n d ,
i i i tg nn = bn /an .

(32) (33) (34) (35)

1 2 i0 1 2
i 0

i i an2 + bn2 ,

Then the flows are given by vn = 1 K
K i vn = i =1

1 K

K i i an2 + bn2 . i =1

(36)

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Mo dels

Connection between the ridge and the flows - 1
In this approximation we have for the di-hadron correlation function: C () = C2 (1 - 2 ) where 1 = 1 K 1 K 2 (1 - 2 ) -1 , 2 1 (37)

K

i1 ( + i ) ,
i =1 K

(38)

2 (1 - 2 ) =

i1 (1 + i )i1 (2 + i )
i =1

(39)

and i1 () is given by (31). Here i is an additional common RANDOM phase, which arises due to the event-by-event fluctuation of the reaction plane.
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Mo dels

Connection between the ridge and the flows - 2
Averaging over events with this additional common random phase i gives (we add an additional averaging over this phase also for each string configuration, which corresponds to the rotation of a given string configuration): 1 = Recall that i0 = 1 K
K i =1

1 2

2 0

i ( + i ) d i =

1 K

K

i0 i0 .
i =1

(40)

1 2

i1 ()d .

(41)

The 1 is the mean multiplicity density.


2 (1 - 2 ) = (i0 )2 + 2
n=1
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i (i0 vn )2 cos(n) .

(42)

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Mo dels

Connection between the ridge and the flows - 3
Then C () = where C= 2 i0
2 i (i0 vn )2 cos(n)+C = 2 n=1 n=1

(

i0 i 2 v ) cos(n)+C i0 n (43) (44)

(i0 )2 - i0 i0 2

2

.

Usually experimentalists can't measure the constant C (see, V.V. arxiv:1305.0857). Recall that
i i i (i0 vn )2 = (i0 )2 [(an )2 + (bn )2 ] =

(45)
2

=

1 2

i () cos n d

2

+

1 2

i () sin n d

,
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Connection between the ridge and the flows - 4
Further rude evaluation of (43) is possible only if we will consider that i0 weakly depends on i : i0 i0 = Then C () = 2
n=1

1 K

K

i0 = const .
i =1

(46)

i (vn )2 cos(n) ,

(47)

at that C = 0. We see that even in this simple model (without initial internal correlations) the final state interactions (FSI) lead through the direct flow (n = 1) to the formation of the ridge phenomenon in resulting correlation function.
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Mo dels

Connection between the ridge and the flows - 5
Note that even in the last very rude approximation (47) the C () is expressed not through the flows:
i vn = vn =

1 K

K i vn = i =1

1 K

K i an2 + b i =1 i2 n

,

(48)

ms but rather through the "mean squared flows"(vn ):

ms vn

i (vn )2 =

1 K

K i (vn )2 = i =1

1 K

K i i (an2 + bn2 ) . i =1

(49)

Using this notation we have in the rude approximation


C () = 2
n=1
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ms (vn )2 cos(n) .

(50)
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Backup slides

Backup slides

Backup slides

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Backup slides

Connection between two-particle and di-hadron correlations1
The di-hadron correlation function C (y , ) S /B - 1 (51)

takes into account all possible pair combinations of particles produced in given event in some ONE LARGE pseudorapidity window y (-Y , Y ), where d 2N S= (52) d y d 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 (51) through the two-particle correlation function:
Y /2

S (y , ) =
-Y /2
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dy1 dy2 2 (y1 , y2 ; ) (y1 - y2 - y )
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Backup slides

Connection between two-particle and di-hadron correlations2
In the central rapidity region, when the translation invariance takes place within the whole rapidity interval (-Y /2, Y /2), we have 2 (y1 , y2 ; ) = 2 (y1 - y2 ; ) and one can fulfill the integration in (53): S (y , ) = 2 (y ; ) tY(y ) where the tY (y ) is a "triangular"weight function ty (y ) = [(-y )( y + y ) + (y )( y - y )] ( y - |y |) . (55) (54)

.: The "triangular"weight function arising due to phase space .
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Backup slides

Connection between two-particle and di-hadron correlations3
In the denominator of (51) we should replace the 2 (y1 , y2 ; ) by the product 1 (y1 )1 (y2 ), which due to the translation invariance in rapidity reduces simply to 2 . Then 0 B (y , ) = 2 tY(y ) . 0 Substituting into (51) we get C (y , ) = 2 (y ; ) - 1 = C2 (y , ) , 2 0 (57) (56)

We see that if the translation invariance in rapidity takes place within the whole interval (-Y /2, Y /2), then the definition (51) for the di-hadron correlation function C leads to the standard two-particle correlation function C2 (1) (see meanwhile the remark below).
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Backup slides

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 (51): S (y , ) = 2 (y ; ) tY(y ) == N (y ; ) tY(y ) = 2 = [ N (y , ) + N 2 ]µ2 tY(y ) , 0
Y /2

(58)

B (y , ) =
-Y /2 Y /2

dy1 dy2 1 (y1 )1 (y2 ) (y1 - y2 - y ) =

=
-Y /2

dy1 dy2 N (y1 ) N (y2 ) (y1 - y2 - y ) = 1 1 = 2 tY(y ) = N 2 µ2 tY(y ) , 0 0 (59)

we have noted that 1 (y ) = µ0 . Then by C = S /B - 1 we get C (y , ) =
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N + (y , ) = C2 (y , ) , N
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Backup slides

Comments on the event mixing - 2
But if instead of (59) one has
Y /2

B (y , ) =
-Y /2

dy1 dy2 N (y1 )N (y2 ) (y1 -y2 -y ) = N 2 µ2 tY(y ) 1 1 0

as it sometimes takes place in a di-hadron data analysis (or if some other artificial normalization conditions for the B (y , ) are being used), then instead of (60) by C = S /B - 1 we get C (y , ) = N (y , ) , N2 (61)

which does not correspond to the standard two-particle correlation function C2 (y , ), defined by (1). Compare (61) with (60) we see that in this case the resulting C (y , ) 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 (y , ) and, therefore, is equal to zero in the absence of the pair correlation from one string.
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