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

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

The Mandelstam variables u and t are defined as: $$t = (p_1 - p_3)^2 = (p_4 - p_2)^2$$ $$u = (p_1-p_4)^2 = (p_3-p_2)^2$$

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?

• $p_1$ and $p_2$ are incoming momenta. $p_3$ and $p_4$ are outgoing momenta. It does not matter which momentum you assign to which particle. Choose your convention (make sure to make it clear which you are using) and do your calculation. – Prahar Jun 4 '18 at 12:44
• But depending on the assignment, wouldn't this give me a different t and u? – Eren Jun 4 '18 at 12:47
• Yes, and that's OK. As I said, choose what you want, as long as you make it clear at the time of choice. – Prahar Jun 4 '18 at 12:48

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.