content_image/374531.jpg content_image/374532.jpg content_image/374534.jpg content_image/374537.jpg content_image/374538.jpg content_image/374540.jpg
The averages are defined as 

with 

<img src='content_image/374531.jpg'>

<img src='content_image/374532.jpg'>

<img src='content_image/374534.jpg'>

where d Γ$_{j}$ = d $^{3}$p$_{j}$ / (2 π ) $^{3}$, j = 1 , 2 , 3, and the local particle distribution function f ( x, p ) is the output from the parton cascade, f ( x, p ) = ∑ i δ $^{(3)}$[ ⃗ x − ⃗ x$_{i}$ ( t )] δ $^{(3)}$[ ⃗ p − ⃗ p$_{i}$ ]. The sum runs over all particles with individual position ⃗ x$_{i}$ and momentum ⃗ p$_{i}$ at time t . 

For a gg → gg or a gg → ggg process involving two incoming particles with ( x, p$_{1}$ ) and ( x, p$_{2}$ ) the relative velocity of the two particles is given by v$_{rel}$ = s/ (2 E$_{1}$E$_{2}$ ), where s = ( p$_{1}$ + p$_{2}$ ) 2 is the invariant mass. The total cross sections read 

<img src='content_image/374537.jpg'>

<img src='content_image/374538.jpg'>

where d Γ ′ j = d $^{3}$p ′ $_{j}$/ (2 π ) $^{3}$/ (2 E ′ $_{j}$), j = 1 , 2 , 3. For a ggg → gg process we define 

<img src='content_image/374540.jpg'>

which is, similar as a cross section, an integral over the final states. The factor of 1 / 2! in Eqs. (13) and (15) is due to the fact that the two outgoing gluons are identical particles. There is no such a factor (1 / 3!) in gg → ggg processes [28], because the three outgoing gluons are distinguished as the radiated gluon ( p ′ $_{3}$), the radiating gluon ( p ′ $_{2}$) and the gluon ( p ′ $_{1}$), which is only deflected and does not radiate. Thus, they are kinematically “distinguishable” particles. 

We note that n in Eq. (12) is the local particle density in the center of mass frame of the Au+Au collision (collision frame) and only equals the Lorentz invariant particle density if the local particle system is at rest, e.g. at the collision center. Also the local collision rates (7)-(9) are calculated in the collision frame and are smaller than those in the rest frame by a Lorentz factor. 

The left panel of Fig. 2 shows the time evolution of the total collision rate per particle 〈 R$_{gg}$$_{→}$$_{gg}$ + R$_{gg}$$_{→}$$_{ggg}$ + R$_{ggg}$$_{→}$$_{gg}$ 〉 . The rate is obtained as the average over all particles in 

