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First I'll try to give a summary of what I understand about the anti-##k_T## algorithm (as a continuation for a particle flow algorithm)...
The algorithm uses an iterative clustering with transverse momentum ##p_T## weighted distance parameter and is applying a selection on the "protojets". For particle flow jets every PF object is a protoject except for those that have been identified as electrons or muons. The anti-##k_T## algorithm takes the protojet with the highest energy and merges it with the next nearest one into a new protojet if the distance is within some boundary. Let's call that boundary set by us as ##L##. This is done until there are no more protojets left. Then the algorithm restarts with the highest of the remaining protojets as the seed.
The distance is defined as:
##d_{ij}= min(k_{Ti}^{2p}, k_{Tj}^{2p}) \frac{(\Delta R)^2}{r^2}##
Where ##r,p## are parameters ( for anti-##k_T## we set ##p=-1##), ##k_{Ti}## are the transverse momenta of the ##i##-th protojet, and ##\Delta R## are the protojets ##(\eta,\phi)## distance.
My Question:
Suppose I have the protojet with the highest energy ##k_{T1}=20GeV##.
And the nearest to it protojet has an energy ##k_{T2}=10GeV##.
In that case the momentum-weighed distance is:
##d_{12}= min (\frac{1}{400}, \frac{1}{100} ) \frac{(\Delta R)^2}{r^2} = \frac{(\Delta R)^2}{400 r^2}##
In order for this distance to be within the set boundary we have
##\frac{(\Delta R)^2}{(20r)^2}= d_{12} < L##
The problem:
I believe that ##\Delta R## should be set very small to keep this inequality. And the strange thing is that any protojet with energy smaller than 20GeV should satisfy this smallness for ##\Delta R##...I feel that this is unreasonable? For example if the 1 protojet has energy 20GeV and the 2nd nearest has 3GeV I don't find a reason they can't "exist" together in a larger cone to form the jet...In other words the energy of the 1st protojet is always going to rule out any other choice of "nearest" ones due to the min() function.
Where am I wrong?
The algorithm uses an iterative clustering with transverse momentum ##p_T## weighted distance parameter and is applying a selection on the "protojets". For particle flow jets every PF object is a protoject except for those that have been identified as electrons or muons. The anti-##k_T## algorithm takes the protojet with the highest energy and merges it with the next nearest one into a new protojet if the distance is within some boundary. Let's call that boundary set by us as ##L##. This is done until there are no more protojets left. Then the algorithm restarts with the highest of the remaining protojets as the seed.
The distance is defined as:
##d_{ij}= min(k_{Ti}^{2p}, k_{Tj}^{2p}) \frac{(\Delta R)^2}{r^2}##
Where ##r,p## are parameters ( for anti-##k_T## we set ##p=-1##), ##k_{Ti}## are the transverse momenta of the ##i##-th protojet, and ##\Delta R## are the protojets ##(\eta,\phi)## distance.
My Question:
Suppose I have the protojet with the highest energy ##k_{T1}=20GeV##.
And the nearest to it protojet has an energy ##k_{T2}=10GeV##.
In that case the momentum-weighed distance is:
##d_{12}= min (\frac{1}{400}, \frac{1}{100} ) \frac{(\Delta R)^2}{r^2} = \frac{(\Delta R)^2}{400 r^2}##
In order for this distance to be within the set boundary we have
##\frac{(\Delta R)^2}{(20r)^2}= d_{12} < L##
The problem:
I believe that ##\Delta R## should be set very small to keep this inequality. And the strange thing is that any protojet with energy smaller than 20GeV should satisfy this smallness for ##\Delta R##...I feel that this is unreasonable? For example if the 1 protojet has energy 20GeV and the 2nd nearest has 3GeV I don't find a reason they can't "exist" together in a larger cone to form the jet...In other words the energy of the 1st protojet is always going to rule out any other choice of "nearest" ones due to the min() function.
Where am I wrong?