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 $s,t,u$ are then defined by

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

Where p₁ and p₂ are the four-momentum of the incoming particles and p₃ and p₄ are the four-momentum of the outgoing particles.

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 momentum transfer.

Details

Note that

$s+t+u=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}\,$

where $m_{i}$ is the mass of particle $i$.

Proof

We use the fact that the square of a particle's four momentum is the square of it's mass

$p_{i}^{2}=m_{i}^{2}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (1)\,$

So, to begin,

$s=(p_{1}+p_{2})^{2}=p_{1}^{2}+p_{2}^{2}+2p_{1}\cdot p_{2}\,$

$t=(p_{1}-p_{3})^{2}=p_{1}^{2}+p_{3}^{2}-2p_{1}\cdot p_{3}\,$

$u=(p_{1}-p_{4})^{2}=p_{1}^{2}+p_{4}^{2}-2p_{1}\cdot p_{4}\,$

First, use (1) to re-write these,

$s=m_{1}^{2}+m_{2}^{2}+2p_{1}\cdot p_{2}\,$

$t=m_{1}^{2}+m_{3}^{2}-2p_{1}\cdot p_{3}\,$

$u=m_{1}^{2}+m_{4}^{2}-2p_{1}\cdot p_{4}\,$

Then add them

$s+t+u\,$	$=3m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}+2p_{1}\cdot p_{2}-2p_{1}\cdot p_{3}-2p_{1}\cdot p_{4}\,$
	$=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}+2\left(m_{1}^{2}+p_{1}\cdot p_{2}-p_{1}\cdot p_{3}-p_{1}\cdot p_{4}\right)\,$

Feynman diagrams

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.