Mandelstam variables

In theoretical physics, the Mandelstam variables are numerical quantities that encode the energy, momentum, and angles of particles in a scattering process in a Lorentz-invariant fashion. They are used for scattering processes of two particles to two particles. The Mandelstam variables were first introduced by physicist Stanley Mandelstam in 1958.

If the Minkowski metric is chosen to be ${\displaystyle \mathrm {diag} (1,-1,-1,-1)}$, the Mandelstam variables ${\displaystyle s,t,u}$ are then defined by

• ${\displaystyle s=(p_{1}+p_{2})^{2}c^{2}=(p_{3}+p_{4})^{2}c^{2}}$
• ${\displaystyle t=(p_{1}-p_{3})^{2}c^{2}=(p_{4}-p_{2})^{2}c^{2}}$
• ${\displaystyle u=(p_{1}-p_{4})^{2}c^{2}=(p_{3}-p_{2})^{2}c^{2}}$,

where p₁ and p₂ are the four-momenta of the incoming particles and p₃ and p₄ are the four-momenta of the outgoing particles.

${\displaystyle s}$ is also known as the square of the center-of-mass energy (invariant mass) and ${\displaystyle t}$ as the square of the four-momentum transfer.