Bunch collision frequency

Sep 2018
1
0
Hey guys, so I am asked to prove that the luminosity of two colliding beams, where the bunches are described by Gaussians, is:
L = f*Nb*N1*N2/(4*pi*sigma_x*sigma_y)

(see eq. 16 in link: https://cds.cern.ch/record/941318/files/p361.pdf)

Where f=revolution frequency (I imagine this is the time it take for a bunch to go around the synchrotron/cyclotron/device), Nb=number of bunches per beam, N1 and N2=number of particles per bunch in beam 1 and beam 2 respectively, and sigma_x and sigma_y are the variances in x and y for the Gaussian.

The bunches are described by the distribution p(x,y,z)=C*exp(-x²/(2*sigma_x²)-y²/(2*sigma_y²)-z²/(2*sigma_z²)).

I calculate the overlap integral and get that the luminosity in one collision is:
L_collision = N1*N2/(8*pi*sigma_x*sigma_y)

I am almost certain that this is correct. However, to get the total luminosity, they then multiply this by f*Nb*2 !!! Could someone explain where this comes from?
In my opinion, if we were to multiply L_collision by the total number of collisions in the device, we should multiply it by f*2*Nb², or simply by Nb if we are only considering the collisions at a specific point in the device.

Thanks in advance.