Exclusive Jet Study With Kt, AntiKt and Cambridge Aachen Jet Finders

Legend Info:

  • Signal - Purple
  • NP0 - Black
  • NP1 - Red
  • NP2 - Green (Also Sum of the Backgrounds)
  • NP3 - Blue

Kt Jet Finder

The Kt jet finding algorithm can be broken down into three steps.

1) For each particle i, j in the event the Kt distance is calculated dij = min(P^2ti,P^2tj) ΔR ^2ij / R

Where ΔR ij= (Δetai - Δetaj)^2 + (Δphii - Δphij)^2

The Kt distance from particle i to the beam is also calculated

dij = P^2iB

2) The next step is to find the dmin value for all dij and diB. (Note there are differences between the Inclusive and Exclusive Kt Jet algorithms)

  • INCLUSIVE Kt Jet Algorithm:

If dij is the dmin value then the i, j particles are combined ( the particles four momenta are summed together).

if diB is the dmin value then the particle i is a final jet and the particle i is removed from the list.

  • Exclusive Kt Jet Algorithm:

If dij is the dmin value and is less than dcut then the i, j particles are combined ( the particles four momenta are summed together).

if diB is the dmin value and is less than dcut then the particle i combined with the beam jet.

3)

  • INCLUSIVE Kt Jet Algorithm:

This process is repeated until no more particles are left.

  • Exclusive Kt Jet Algorithm:

This process ends when all dij and diB are above the dcut value.

The dcut variable again varies when considering an inclusive or exclusive event.

When looking at an exclusive n-jet event.

dcut = Ptmin^(2)

This maintains the jet algorithm ability to factorize the initial-state of the collinear singularities, which allows you to obtain a finite cross-section.

When looking at an inclusive jet event.

dcut = Et^(2)

  • Recombination Scheme used:

The recombination scheme is the method the user chooses to combine the particles to one and other. There are a number of schemes to choose from, I have chosen the E-Scheme.

E-scheme:

The E-scheme combines the four momenta of the particles vectorially. Therefore the true rapidity is used instead of the pseudorapidity (since the particles have mass not equaling zero) . This procedure must be used to maintain the longitudinal boost invariance of the recombination procedure.

We need the recombination procedure to be longitudinal boost invariant so that corrections to hadronization won't cause strong effects to the recombination procedure (large jump in jet multiplicity).

Details of R values used in Kt Algorithm (This will be changed so we can have a direct comparison with the anti-kt algorithm)

dmin_01 -> R = 0.1, dmin_12 -> R = 0.2, dmin_23 -> R = 0.3, dmin_34 -> R = 0.4, dmin_45 -> R = 0.5, dmin_56 -> R = 0.6, dmin_67 -> R = 0.7, dmin_78 -> R = 0.8 and dmin_89 -> R = 0.9

These chosen R values will need to be changed to more suitable values, especially 01 to 23, as the R values are far too small for the corresponding jet multiplicity. Also, I will need to change these to R = 1 to compare with the anti-kt algorithm.

It will also be an interesting study to see the effects on the chosen value of R (how it effects the dmin value, does it increase the discriminant against the background dmins???).

Note:

Most study's look at jets with R = 0.4 and 0.6.

Signal and Background Comparisons (Bin Width 100 GeV^2)

Signal and Background Comparisons Not Normalized
Fig1.1 AllDminSigBckKtUnNorm.png
Signal and Background Comparisons Normalized (1/sigma dsigma/dDmin [GeV-2] )
Fig2.1 AllDminSigBckKtNorm.png
Signal and Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] )
Fig3.1 AllDminSigBckKtNormLuminosityBinWidth.png
Signal and Sum of Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] )
Fig4.1 AllDminSigBckKtNormLuminosityBinWidth.png

Signal and Background Comparisons (Binned in Log(Dmin))

Signal and Background Comparisons Not Normalized
Fig5.1 AllDminSigBckktUnNormLog10.png

Signal and Background Comparisons Normalized (1/sigma dsigma/dDmin [GeV-2] )
Fig6.1 AllDminSigBckktNormLog10.png

Signal and Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] )
Fig7.1 AllDminSigBckktNormLuminosityBinWidthLog10.png
Signal and Sum of Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] )
Fig8.1 AllDminSigBckKtSumNormLuminosityBinWidthLog10.png

Signal and Sum of Background Comparisons Normalized (1/sigma dsigma/dDmin [GeV-2] )
Fig9.1 AllDminSigBckKtSumNormUnityLog10.png

AntiKt Jet Finder

The Anti-Kt Algorithm follows the same procedure as the Kt algorithm explained above. The differences between these two algorithms are the definitions for dij and diB.

dij = min(1/P^2ti,1/P^2tj) ΔR ^2ij / R

and

dij = 1/P^2iB

More detailed points on the behavior of the anti-kt algorithm:

If we have a hard particle and a soft particle we find the dij value is dominated by the hard particle. When considering the soft particles we find the dij value to be much larger. Therefore the harder particles will cluster with the softer particles long before they cluster with each other.

  • R value dependence on jet shape:

If a hard particle has no other hard particles within a distance of 2R then the hard particle will merge with all the surrounded soft particles producing a conical jet shape.

If we have 2 hard particles within R < Δ12 < 2R then we will have two hard jets. These will produce conical and partially conical jets (due to the overlapping getting taken away). The shapes of these two jets are determined by the jets Pt^2.

  • Pt1^2 >> Pt2^2 Then jet 1 will be conical and jet 2 will be partially conical (since it will miss the overlapping with jets)
  • Pt1^2 = Pt2^2 Neither jet is conical and the overlapping part of the jet will be split by a straight line equally between the two.

If we have 2 hard particles with Δ12 < R then there will be 1 jet produced. The conical jet will be centered around the jet with the highest Pt^2. However if the two gets have similar Pt^2 then the jet shape will be more complex. The shape will be a union of cones with a radius < R around each hard particle plus a cone centered around the final jet with a radius < R.

Getting the dmin value for n+1 jets into n jets from the anti-kt jet algorithm:

With the anti-kt algorithm the exclusive jet algorithm does not make physical sense. Since the exclusive jets undoes the last n steps of the clustering and returns whatever objects were left in the cluster sequence after undoing those steps. With the anti-kt algorithm those are often soft objects (low pt, hence large diB or dij, hence clustered late) and bear no relation to the hard structure of the event.

To over come this, I used the constituents that were clustered with the anti-kt algorithm and used them to be clustered with the kt-algorithm allowing me to access the dmin variables.

*************** These plots are for R=1 *******************************************

Signal and Background Comparisons (Bin Width 100 GeV^2)

Signal and Background Comparisons Not Normalized
Fig10.1 AllDminSigBckAntktUnNorm.png
Signal and Background Comparisons Normalized (1/sigma dsigma/dDmin [GeV-2] )
Fig11.1 AllDminSigBckAntktNorm.png
Signal and Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] )
Fig12.1 AllDminSigBckAntktNormLuminosityBinWidth.png
Signal and Sum of Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] )
Fig13.1 AllDminSigBckAntktSumNormLuminosityBinWidth.png

Signal and Background Comparisons (Binned in Log(Dmin))

Signal and Background Comparisons Not Normalized
Fig14.1 AllDminSigBckKtAntUnNormLog10.png
Signal and Background Comparisons Normalized (1/sigma dsigma/dDmin [GeV-2] )
Fig15.1 AllDminSigBckAntKtNormLog10.png
Signal and Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] )
Fig16.1 AllDminSigBckAntktNormLuminosityBinWidthLog10.png
Signal and Sum of Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] )
Fig17.1 AllDminSigBckAntktSumNormLuminosityBinWidthLog10.png

Signal and Sum of Background Comparisons Normalized (1/sigma dsigma/dDmin [GeV-2] )
Fig18.1 AllDminSigBckAntktSumNormUnityLog10.png

-- MichaelWright - 2010-10-06

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