Question

Define Mandelstam variables

Answer 1

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.

If the Minkowski metric is chosen to be

$${\mathrm {diag}}(1,-1,-1,-1)$$

, the Mandelstam variables s,t,u are then defined by

$$s=(p_1+p_2)^2=(p_3+p_4)^2\\t=(p_1-p_3)^2=(p_4-p_2)^2\\u=(p_1-p_4)^2=(p_3-p_2)^2$$

Where $p_1\ and\ p_2$ are the four-momenta of the incoming particles and $p_3\ and\ p_4$ are the four-momenta of the outgoing particles, and we are using relativistic units (c=1).

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