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 , the Mandelstam variables are then defined by
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 rest mass can be neglected, so for example,
because and It is reminded that by relativistic limit one means that the momentum (speed) is so large that in the relativistic energy-momentum equation the energy becomes essentially the momentum norm (e.g. becomes ).
where is the mass of particle .
To prove this, we need to use two facts:
- The square of a particle's four momentum is the square of its mass,
- And conservation of four-momentum,
So, to begin,
First, use (1) to re-write these,
Then add them
Then use eq (2) to simplify further,
- Mandelstam, S. (1958). "Determination of the Pion-Nucleon Scattering Amplitude from Dispersion Relations and Unitarity". Physical Review 112 (4): 1344. Bibcode:1958PhRv..112.1344M. doi:10.1103/PhysRev.112.1344.
- Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2.
- Perkins, Donald H. (2000). Introduction to High Energy Physics (4th ed.). Cambridge University Press. ISBN 0-521-62196-8.