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

• Exclusive Jet Talk (Rough Idea): ExclusiveJetAnalysis_UK_Higgs_Meeting.ppt

• Reference used to calculate bin width with respect to Jet Energy Resoultion (Calculated using dijet sample): Jet_Energy_Resolution.eps (Jet_Energy_Resolution.eps was taken from the Jet energy Resolution Group URL https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-2010-054/ )

Legend Info:

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

Kt Jet Finder

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

1) For each particle i, j in the event the K_t distance is calculated d_ij = min(P^2_ti,P^2_tj) ΔR ^2_ij / R

Where ΔR _ij^2= (Δeta_i - Δeta_j)^2 + (Δphi_i - Δphi_j)^2

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

d_ij = P^2_iB

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

• INCLUSIVE K_t Jet Algorithm:

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

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

• Exclusive K_t Jet Algorithm:

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

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

3)

• INCLUSIVE K_t Jet Algorithm:

This process is repeated until no more particles are left.

• Exclusive K_t Jet Algorithm:

This process ends when all d_ij and d_iB 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 = P_tmin^(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 = E_t^(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).

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

Signal and Background Comparisons Not Normalized R=1

Fig1.1

• AllDminSigBckKtUnNorm.eps:

Signal and Background Comparisons Normalized (1/sigma dsigma/dDmin [GeV-2] ) R=1

Fig2.1

• AllDminSigBckKtNorm.eps:

Signal and Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] ) R=1

Fig3.1

• AllDminSigBckKtNormLuminosityBinWidth.eps:

Signal and Sum of Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] ) R=1

Fig4.1

• AllDminSigBckKtSumNormLuminosityBinWidth.eps:

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

Signal and Background Comparisons Not Normalized R=1

Fig5.1

• AllDminSigBckktUnNormLog10.eps

Signal and Background Comparisons Normalized (1/sigma dsigma/dDmin [GeV-2] ) R=1

Fig6.1

• AllDminSigBckktNormLog10.eps

Signal and Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] ) R=1

Fig7.1

• AllDminSigBckktNormLuminosityBinWidthLog10.eps

Signal and Sum of Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] ) R=1

Fig8.1

• AllDminSigBckKtSumNormLuminosityBinWidthLog10.eps

Signal and Sum of Background Comparisons Normalized (1/sigma dsigma/dDmin [GeV-2] ) R=1

Fig9.1

• AllDminSigBckKtSumNormUnityLog10.eps

Signal and Background Comparisons Not Normalized R=0.4

Fig10.1

• AllDminSigBckktUnNormLog10_R04.eps

Signal and Background Comparisons Normalized (1/sigma dsigma/dDmin [GeV-2] ) R=0.4

Fig11.1

• AllDminSigBckktNormLog10_R04.eps

Signal and Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] ) R=0.4

Fig12.1

• AllDminSigBckktNormLuminosityBinWidthLog10_R04.eps

Signal and Sum of Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] ) R=0.4

Fig13.1

• AllDminSigBckKtSumNormLuminosityBinWidthLog10_R04.eps

Signal and Sum of Background Comparisons Normalized (1/sigma dsigma/dDmin [GeV-2] ) R=0.4

Fig14.1

• AllDminSigBckKtSumNormUnityLog10_R04.eps

AntiKt Jet Finder

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

d_ij = min(1/P^2_ti,1/P^2_tj) ΔR ^2_ij / R

and

d_ij = 1/P^2_iB

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

If we have a hard particle and a soft particle we find the d_ij value is dominated by the hard particle. When considering the soft particles we find the d_ij 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 P_t^2.

• P_t1^2 >> P_t2^2 Then jet 1 will be conical and jet 2 will be partially conical (since it will miss the overlapping with jets)
• P_t1^2 = P_t2^2 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 P_t^2. However if the two gets have similar P_t^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.

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

Signal and Background Comparisons Not Normalized R=1

Fig15.1

• AllDminSigBckAntktUnNorm.eps:

Signal and Background Comparisons Normalized (1/sigma dsigma/dDmin [GeV-2] ) R=1

Fig16.1

• AllDminSigBckAntktNorm.eps:

Signal and Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] ) R=1

Fig17.1

• AllDminSigBckAntktNormLuminosityBinWidth.eps:

Signal and Sum of Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] ) R=1

Fig18.1

• AllDminSigBckAntktSumNormLuminosityBinWidth.eps:

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

Signal and Background Comparisons Not Normalized R=1

Fig19.1

• AllDminSigBckKtAntUnNormLog10.eps

Signal and Background Comparisons Normalized (1/sigma dsigma/dDmin [GeV-2] ) R=1

Fig20.1

• AllDminSigBckAntKtNormLog10.eps

Signal and Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] ) R=1

Fig21.1

• AllDminSigBckAntktNormLuminosityBinWidthLog10.eps

Signal and Sum of Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] ) R=1

Fig22.1

• AllDminSigBckAntktSumNormLuminosityBinWidthLog10.eps

Signal and Sum of Background Comparisons Normalized (1/sigma dsigma/dDmin [GeV-2] ) R=1

Fig23.1

• AllDminSigBckAntktSumNormUnityLog10.eps

Signal and Background Comparisons Not Normalized R=0.4

Fig24.1

• AllDminSigBckKtAntUnNormLog10_R04.eps

Signal and Background Comparisons Normalized (1/sigma dsigma/dDmin [GeV-2] ) R=0.4

Fig25.1

• AllDminSigBckAntKtNormLog10_R04.eps

Signal and Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] ) R=0.4

Fig26.1

• AllDminSigBckAntktNormLuminosityBinWidthLog10_R04.eps

Signal and Sum of Background Comparisons Normalized (dsigma/dDmin [pb/GeV^2] ) R=0.4

Fig27.1

• AllDminSigBckAntktSumNormLuminosityBinWidthLog10_R04.eps

Signal and Sum of Background Comparisons Normalized (1/sigma dsigma/dDmin [GeV-2] ) R=0.4

Fig28.1

• AllDminSigBckAntktSumNormUnityLog10_R04.eps

-- MichaelWright - 2010-10-06

• ExclusiveJetAnalysis_UK_Higgs_Meeting.pdf: ExclusiveJetAnalysis_UK_Higgs_Meeting.pdf

• AllDminSigBckAntKtNormLog10_R04_ttbb.eps: AllDminSigBckAntKtNormLog10_R04_ttbb.eps

• AllDminSigBckAntktSumNormUnityLog10_R04_ttbb.eps: AllDminSigBckAntktSumNormUnityLog10_R04_ttbb.eps