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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:
- <math>p_1+p_2 = -p_3-p_4<math>
The outgoing four-momenta <math>p_3,p_4<math> are taken to have a negative time-like component. The Mandelstam variables <math>s,t,u<math> are then defined by
- <math>s=(p_1+p_2)^2=(p_3+p_4)^2,\qquad
t=(p_1+p_3)^2=(p_2+p_4)^2,\qquad
u=(p_1+p_4)^2=(p_2+p_3)^2<math>
Note that
- <math>s+t+u = \sum_{i=1}^4 m_i^2<math>
The letters <math>s,t,u<math> 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 <math>s,t,u<math>, respectively.
Stu1.png
Image:stu1.png
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.
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