David Griffiths: "Introduction to Elementary Particles", Second Revised Edition, formula (6.47) reads dσdΩ=(ℏc8π)2S|M|2(E1+E2)2|pf||pi| and expresses the differential cross section of a process 1+2→3+4.
According to Griffiths, S is a statistical factor that corrects for double-counting when there are identical particles in the final state. In particlar, in Formula (6.47), S equals 1 if particles 3 and 4 are different and equals 1/2 if the particles are identical. Griffiths says no more about S.
I have tried to Google for more information about S and exactly how to use it, but all I have found is that some web pages like https://www.nikhef.nl/~i93/Master/PP1/2017/Lectures/Lecture2017.pdf omit S and others like http://www.physics.utah.edu/~belz/phys5110/lecture23.pdf include it.
My question concerns the following thought experiment: Suppose A, B and C are types of elementary particles and that A+B→C+C is a possible process. Suppose we let a beam of A-particles intersect with a beam of B-particles. Suppose the luminosity of the collision (the thing to multiply σ by to get an event rate) is L. Suppose we have a detector which detects C-particles which are created by the process and travel in a particular direction within a small, spherical angle dΩ.
Now the question is: Is the expected event rate dσdΩLdΩ or 2dσdΩLdΩ. The argument for the latter would be that the detector could detect either of the C's in the final state. If the latter is correct, then we first have to multiply by S=1/2 to account for identical particles and then multiply by 2, again to account for identical particles, and the two factors would cancel out in this particular case. Is that correct?