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.

For the usual case of two particles going to two particles, the momentum conservation implies:

$p_{1}+p_{2}=p_{3}+p_{4}$

The outgoing four-momenta $p_{3}$ and $p_{4}$ are taken to have a negative time-like component. The Mandelstam variables $s,t,u$ are then defined by

$s=(p_{1}+p_{2})^{2}=(p_{3}+p_{4})^{2},$

$t=(p_{1}-p_{3})^{2}=(p_{2}-p_{4})^{2},$

$u=(p_{1}-p_{4})^{2}=(p_{2}-p_{3})^{2}$

Note that

$s+t+u=\sum _{i=1}^{4}p_{i}^{2}=\sum _{i=1}^{4}m_{i}^{2}$

where $m_{i}$ is the mass of particle number $i$. s is the square of the center-of-mass energy and t is the square of the momentum transfer.

The letters $s,t,u$ 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 $s,t,u$, 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 how resonances and new unstable particles may be discovered unless their lifetime is 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.

The Mandelstam variables were first introduced by physicist Stanley Mandelstam in 1958.

References

Mandelstam, S. (1958). "Determination of the Pion-Nucleon Scattering Amplitude from Dispersion Relations and Unitarity". Phys. Rev. 112: 1344.