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Let's only consider iterative jet clustering algorithms. Famous ones are the $k_{T}$ ($p = 1$), anti-$k_{T}$ ($p = -1$) and Cambridge/Aachen ($p = 0$) jet reconstruction algorithms.

All these algorithms are based on the following quantity, which is a distance measure between the hadrons e. g.: $$d_{ij} \equiv \min\left\{ p^{2p}_{i, T}; p^{2p}_{j, T} \right\}\cdot \frac{\Delta^{2}_{ij}}{ R^{2} }.$$

Question: Why are only $p_{i, T}$ and $p_{j, T}$ considered, but not the three-momenta? After all, we don't know the initial $z$-component of the colliding partons, but we usually can measure the $z$-component of the hadrons in the ECAL/HCAL, cannot we?

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This is because pT is a better distinguisher, and the forward/backwards momentum isn't very useful information most of the time, especially in pp collisions. The initial state (existing before the collision took place) has some unknown and possibly large momentum along the beam line, but we are sure it has a very small pT. So pT tells us about the state (created DURING the collision) we are reconstructing, whereas the third component of momentum is difficult to disentangle from properties of the initial state.

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The experiment and observation of results happens in the laboratory frame. The Lorentz transformation to the center of mass does not change the $p_t$ measured in the lab. In accelerator experiments target at rest, and in cosmic rays, the $p_t$ distributions are a snapshot of the center of mass distributions, even though the interaction are with constituents . In colliders with the same momentum, the LHC for example, the data are in the center of mass of the two protons colliding, but the interaction happens with constituents which have momenta distributions, $p_t$ is again a snapshot of the center of mass of the individual collisions.

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