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-invariantfashion. They are used for scattering processes of two particles to two particles. The Mandelstam variables were first introduced by physicistStanley Mandelstam in 1958. If the Minkowski metric is chosen to be, the Mandelstam variables are then defined by Where p1 and p2 are the four-momenta of the incoming particles and p3 and p4 are the four-momenta of the outgoing particles, and we are using relativistic units. s is also known as the square of the center-of-mass energy and t is also known as the square of the four-momentum transfer.
The letters are also used in the terms s-channel, t-channel, u-channel. These channels represent different Feynman diagrams or different possible scattering events where the interaction involves the exchange of an intermediate particle whose squared four-momentum equals, respectively. For example, the s-channel corresponds to the particles 1,2 joining into an intermediate particle that eventually splits into 3,4: the s-channel is the only way that resonances and new unstable particles may be discovered provided their lifetimes are long enough that they are directly detectable. The t-channel represents the process in which the particle 1 emits the intermediate particle and becomes the final particle 3, while the particle 2 absorbs the intermediate particle and becomes 4. The u-channel is the t-channel with the role of the particles 3,4 interchanged.
In the relativistic limit, the momentum is large, so using the relativistic energy-momentum equation, the energy becomes essentially the momentum norm. The rest mass can also be neglected. So for example, because and Thus,
To prove this, we need to use two facts: So, to begin, Then adding the three while inserting squared masses leads to, Then note that the last four terms add up to zero using conservation of four-momentum, So finally,
