Mandelstam variables (s, t, u)

When two particles scatter — bounce off each other — you would like to summarize the encounter with a few numbers that everyone, regardless of their own motion, will agree on. The Mandelstam variables, named s, t, and u, are exactly that summary for a two-into-two collision. They are three invariant quantities built from the energies and momenta of the incoming and outgoing particles, and because they are invariant, they carry the physics of the scattering without depending on whose lab the experiment sits in.

Each one has an intuitive meaning. The variable s is the square of the total energy in the center-of-mass frame, so its square root is the total energy available — the bigger s is, the more violent and energetic the collision. The variable t measures how much momentum is transferred between the particles, which is closely tied to the scattering angle: a small t means a glancing, small-angle deflection, while a large t means a hard, large-angle bounce. The variable u is a similar momentum-transfer measure for the crossed pairing of particles. The three are not independent; they sum to the total of the squared masses involved.

Mandelstam variables are the natural language for describing and computing scattering. Theorists write the probability of a process as a function of s and t, and the same underlying formula can describe three related processes depending on which variable plays the role of energy — the s-, t-, and u-channels. Experimenters bin their data in t to study how the cross-section falls off with angle, which reveals the size and structure of the colliding objects. They turn a messy three-dimensional collision into two or three honest, frame-independent knobs.