-- MichaelEdwardNelson - 2016-10-16

Combined Jet Mass

This TWiki summarises the combined jet mass definition, available for analysers in the ATLAS experiment. This new jet mass definition has been developed by the Jet Substructure and Jet-by-Jet tagging subgroup of the JetEtMiss CP group, and is available in the latest tag of JetCalibTools.

1. What's a Combined Jet Mass ?

In ATLAS, the jet mass is one of the most important substructure variables we have when we want to describe and study hadronic jets. The standard jet mass definition has, for some time, been the calorimeter jet mass, m_{calo}. This is the invariant mass of the sum of the four-momenta of the individual calorimeter topo-clusters which are associated to the jet via a jet reclustering algorithm (by default the Anti-k_{t} algorithm, in ATLAS). This calorimeter mass is calculated at both the EM and LC scales, and for different jet sizes (where the jet size is quantified in terms of a jet radius, R).

In order to take advantage of regions of phase space where the calorimeter resolution is sub-optimal, a second jet mass definition has recently been added to ATLAS derivations: the track-assisted jet mass, m_{TA}. In the track-assisted approach, tracks from the inner detector are first ghost-associated to jets in the calorimeter, and the sum of the masses of the individual tracks associated to a calorimeter jet yields the track mass, m_{track}, for that jet. The track-assisted mass is then calculated by multiplying the track mass with the "charged/neutral" fraction p_{T,calo}/p_{T,track}: m_{TA} = m_{track} \times p_{T,calo}/p_{T,track}.

The calorimeter and track-assisted jet mass definitions will give the smallest jet mass resolution at different values of mass and transverse momentum, and the exact behaviour of their resolutions will vary from jet topology to jet topology. Is it possible to find a way to optimise the jet mass, by taking a linear combination of the m_{calo} and m_{TA}, such that the final jet mass has a lower jet mass resolution than the individual calorimeter and track-assisted masses? Yes! Hence the combined jet mass.

The combined jet mass, m_{comb}, is the linear combination of m_{calo} and m_{TA} which minimises the jet mass resolution: m_{comb} = a \times m_{calo} + b \times m_{TA}, where the weights a and b are to be found.

2. Determining the a and b Weights

2.1 Neglible Response Correlations

Using the constraint a + b = 1, and minimising the combined mass resolution, the master equations for a and b follow immediately. The exact form of the master equations depends on the correlation, ρ, between the calorimeter and track-assisted jet mass response, m_{calo}/m_{truth} and m_{TA}/m_{truth} respectively. Assuming negatigible correlation, the master equations become:

a = σ_{calo}^{-2}/(σ_{calo}^{-2} + σ_{TA}^{-2})

b = σ_{TA}^{-2}/(σ_{calo}^{-2} + σ_{TA}^{-2})

Here the σ-values refer to the different jet mass resolutions. The mass resolution is defined to be 68 % confidence interval of the interquantile range of the jet mass response distrbution. Therefore, in order to determine the weights, one must first calculate the jet mass resolution. The jet mass resolutions for the calorimeter and the track-assisted masses are determined as a functon of the p_{T,calo}, and m_{reco}/p_{T,calo} (a single |η| bin is used). Two resolution maps are required: the calorimeter resolution map (m_{reco = calo}) and the track-assisted resolution map (m_{reco = TA}). The weights, binned in p_{T,calo}, m_{reco}/p_{T,calo}, then follow immediately from these maps.

2.2 Non-neglible Response Correlations

If the correlation is non-negligible (a working definition of non-negligible is a |ρ| > 0.3 between the calorimeter and track-assisted mass responses), then the a and b weights must be calculated using three maps: the two resolution maps, and a correlation map. The correlation map is binned in p_{T,calo},m_{TA}/p_{T,calo}. Since the correlations are a second-order effect, the map has coarser binning compared to the resolution maps. The final, correlated a and b are then given by:

a = (σ_{TA}^{2} - ρσ_{calo}σ_{TA})/(σ_{calo}^{2} + σ_{TA}^{2} -2ρσ_{calo}σ_{TA})

b = 1 - a

Adding in correlations can give rise to negative weights. The sum of the two weights (for a given p_{T},m/p_{T} bin) remains unity. It can be shown that for correlations ρ > σ_{TA}/σ_{calo}, a negative calorimeter weight is obtained.

3. Recommended Resolution and Correlation Maps

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Topic revision: r1 - 2016-10-16 - MichaelEdwardNelson
 
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