SISCone: split-merge issues

# SISCone Documentation

### Version 2.0.6

## The split-merge variable issue:

SISCone is intended to be a cone algorithm that is as close as possible to the Tevatron run II cone algorithm, while also being infrared (IR) safe.

Though the main IR difficulties are related to the search for stable cones, and are solved by a seedless procedure, it turns out that the split-merge phase of the algorithm is also non-trivial. The Tevatron run II split-merge procedure requires that "protojets" be considered in a sequence determined by some ordering variable. A change in the sequence can leads to a change in the final jets. When trying to find the best suited variable for the ordering of protojets in the split--merge process of SISCone, we met a couple of physical issues. Their history goes as follows:

- the Et variable, suggested e.g. in the Tevatron Run II specification of the midpint algorithm, is not boost invariant.
- the pt variable, used e.g. in practical midpoint implementations (except for MCFM), suffers from problems with two back-to-back protojets. Due to momentum conservation, they can have the same pt. Their ordering is thus random and can be modified by the addition of a soft particle, leading to an IR unsafety problem.
- the mt variable does not directly suffer from that problem: two back-to-back jets only overlap if they cover a region in the eta-phi plane. In that case, their values of mt will differ and the ordering is safe. This choice (the default one for SISCone 1.1.0) suffers unfortunately from one remaining issue: though mt works for pure QCD, in the case of two back-to-back narrow-decay-width unstable particles (e.g. a pair of Higgs bosons), both pt and mt will be the same.
- As of SISCone 1.1.1, we use pttilde: the p-scheme pt (the sum of the moduli of transverse momenta of all particles in the protojet ) that interpolates between pt and mt according to the decay plane of the unstable massive particle. This will lift the degeneracy between decays of identical pairs of unstable particles, since the phase space for them to have identical decay planes vanishes.

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Contacts: Gregory Soyez Gavin Salam