# Understanding Jet Clustering: Why is only $p_{T}$ used?

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?

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.