# Combinatorial background subtraction for e+e- signals

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It is common to use the formula 2*sqrt(N++ N--) to model the combinatorial background when studying e+e- signals, e.g. for J/psi and Upsilon analyses.  We can obtain this formula in the following way.

Assume we have an event in which there are Nsig particles that decay into e+e- pairs.  Since each decay generates one + and one - particle, the total number of unlike sign combinations we can make is N+- = Nsig2. To obtain the total number of pairs that are just random combinations, we subtract the number of pairs that came from a real decay.  So we have

N+-comb=Nsig2-Nsig=Nsig(Nsig-1)

For the number of like-sign combinations, for example for the ++ combinations, there will be a total of (Nsig-1) pairs that can be made by the first positron, then (Nsig-2) that can be made by the second positron, and so on.  So the total number of ++ combinations will be

N++ = (Nsig-1) + (Nsig - 2) + ... + (Nsig - (Nsig-1)) + (Nsig-Nsig)

Where there are Nsig terms. Factoring, we get:

N++ = Nsig2 - (1+2+...+Nsig) = Nsig2 - (Nsig(Nsig+1))/2 = (Nsig2 - Nsig)/2=Nsig(Nsig-1)/2

Similarly,

N-- = Nsig(Nsig-1)/2

If there are no acceptance effects, either the N++ or the N-- combinations can be used to model the combinatorial background by simply multiplying them by 2.  The geometric average also works:

2*sqrt(N++ N--) = 2*Nsig(Nsig-1)/sqrt(4) = Nsig(Nsig-1) = N+-comb.

The geometric average can also work for cases where there are acceptance differences, with the addition of a multiplicative correction factor R to take the relative acceptance of ++ and -- pairs into account. So the geometric average is for the case R=1 (similar acceptance for ++ and --).