What is Combinatorial Analysis

Combinatorial analysis is a common pattern where individual final state particles reconstructed from the event are combined in each possible way, and the resulting composite object information is stored along with the daughter particles in one tuple per combination, rather than one tuple per event record.

In FSA, this is accomplished by requesting the "final state" for which you wish to create ntuples. As an example, one might be interested in a 4 lepton state for a ZZ -> 4l (2l+2l') analysis. In this case your "final states" are "eeee", "eemm", "mmmm." FSA creates a separate TTree in your ntuples for each final state, and creates multiple "rows" in the TTree for an event with more than one combinatorially distinct final states. Consider the "eemm" state. For each event in the EDM file you are Ntuplizing, FSA will look for at least 2 electrons and 2 muons in the final state passing minimal selection. An event with less than 2 electrons or 2 muons will not be included. As an example, consider an event with 3 electron candidates and 3 muon candidates, e1, e2, e3, m1, m2, m3. FSA will create TTree rows for:

• e1+e2+m1+m2
• e1+e3+m1+m2
• e2+e3+m1+m2
• e1+e2+m1+m3
• e1+e3+m1+m3
• e2+e3+m1+m3
• e1+e2+m2+m3
• e1+e3+m2+m3
• e2+e3+m2+m3

Thus this single event appears 9 times in your final Ntuple. The advantage of this is that the computationally heavy combinatorics of the analysis have already been carried out, giving you access to composite candidates like Z = 2l, and h = ZZ = 4l. You must, however, include in your analysis code a technique to remove multiple entries for a single event by defining a "best" combinatorial state. For example, you might always choose the combination with m_l+l- closest to the Z mass.