168
M. M. Aggarwal
2 n|n ≡ ≡e
ιn(φ 1 −φ 2 )
≡
1
P M,2
M
i, j=1
(i = j)
e
ιn(φ i −φ j )
(13.16)
4 n,n|n,n ≡ ≡e
ιn(φ 1 +φ 2 −φ 3 −φ 4 )
≡
1
P M,4
M
i, j,k,l=1
(i = j =k =l)
e
ιn(φ i +φ j −φ k −φ l ))
(13.17)
where P M,n = M!/(M − n)!n!, M is the event multiplicity and φ is azimuthal angle
of the particle.
2nd order cumulant is obtained by decomposing |Q n |
2 ,
| Q n |
2
=
M
i, j=1
e
ιn(φ i −φ j )
(13.18)
where φ i (φ j ) is the azimuthal angle of the particle. Indices i and j can be either
the same or different. The decomposition of | Q n |
2 contains contributions from 2particle correlations when indices are different and auto-correlations when indices
are same, as given below:
| Q n |
2
= =2 n,n P M,2 + 1.M
(13.19)
So one gets
2 n,n =
(| Q n |
2
− M)
M(M − 1)
(13.20)
Here Q n is Q-vector:
Q n =
M
i=1
exp(nφ i )
(13.21)
Here summation runs over all the particles in an event. Similarly one can decompose
|Q n |
4 as
| Q n |
4
= =4 n,n|n,n M(M − 1)(M − 2)(M − 3)
+(3 2n|n,n + +3 n,n|2n )M(M − 1)(M − 2)
++2 n|n [M(M − 1)2!(M − 2)2! + M(M − 1)2!2!]
++2 2n|2n M(M − 1) + 1.[M(M − 1)2! + M]
(13.22)
thus for 3 2n|n,n and 3 n,n|2n one has to decompose Q 2n Q
∗
n Q
∗
n and Q n Q n Q
∗
2n .