In a collision, how do I label the particles to calculate the Mandelstam variables $u$ and $t$?

Question

I want to calculate the Mandelstam variables of a 2 --> 2 collision.

The Mandelstam variables u and t are defined as: \begin{equation}t = (p_1 - p_3)^2 = (p_4 - p_2)^2\end{equation} \begin{equation}u = (p_1-p_4)^2 = (p_3-p_2)^2\end{equation}

Consider the collision $p + p \to p' + \Delta^+$

How do I decide how to label the four momentum of the proton and delta-baryon or more concrete, should I assign $p_3$ to the delta baryon and $p_4$ to $p'$ or $p_3$ to p' and $p_4$ to delta?

Answer 1

It can be either, provided you adopt a definition, state it and stick to it. In a typical experiment, in the lab frame there is a beam particle, a target particle, a scattered particle and a recoil particle: $t$ is taken from the difference between the beam particle and the scattered particle and it will be clear whether 3 or 4 is the 'scattered' particle either from the experimental apparatus (3 might be detected at small angles in a hodoscope some distance away, 4 might be a slow particle emerging from the target at large angles) or from the particles (for $\pi^- p \to \pi^+ \Delta^-$ the $\pi^+$ is clearly scattered and the $\Delta$ is the recoil). With your particles (and without further details of the detector) it's ambigious and not clear so you can make your own decision.