Mandelstam variables

In the Mandelstam variables s, t and u is shorthand for terms that appear frequently in the calculation of scattering processes with two incoming and two outgoing particles in particle physics. They are named after Stanley Mandelstam, who introduced it in 1958.

If the four-momenta of the two incoming particles with, and that of the outgoing particles denoted by, the Mandelstam variables are given by

The emerging in these definitions square of four pulses is as usual in relativistic physics as defined (see article four-momentum ). The Mandelstam variables are thus lorentzinvariante scalars as well as the scattering amplitude itself, to be expressed by them in relativistically invariant way. The term s is equal to the square of the center of mass energy of the system. The term t is equal to the square of the four- pulse - carry in an ordinary diffusion process as the electron - scattering nucleon ( T channel). The three Mandelstamvariablen are not independent: the sum of the three Mandelstam variables is equal to the sum of the squares of the mass of the particles involved, ie

Wherein as in the conventional particle is the dimensionless value of c = 1 is assumed for the speed of light.

General should the scattering amplitude, as it is a relativistic invariant, of the relativistic invariants (i = 1, 2, 3, 4) and the six possible independent ( relativistic ) scalar products depend - the Mandelstamvariablen s, t, u are from these composed as a linear combination. The are no variables because of ( the outer legs of the Feynman diagrams are on shell) and because of the four-momentum conservation (which results in four equations, because the four components have ) are only two of the six products independently. So only two of the Mandelstamvariablen are independent.

S-, t- and u - channel

The contributions to the scattering process, in which appear the corresponding Mandelstam variables in its calculation are referred to as s-, t-, and u - channel. The corresponding Feynman diagrams are shown in the following figure.

The presentation follows the convention that the incoming particles are shown from left coming lines and the scattered products as to the right outgoing lines. The dotted line represents a virtual intermediate particles; the square of its four-momentum is the relevant diagram corresponding to the same s, t or u

For example, the diagram for the s- channel one electron-positron pair annihilation on the left to form a virtual photon and an electron-positron pair production on the right side. Also the formation of an unstable intermediate state (resonance) in the interaction of two particles is reproduced. An ordinary electron-electron scattering is given by the graph of the T- channel ( in which the U-channel has to be considered with in which the outer legs 3,4 of the diagram are reversed). In the scattering amplitude to the rules of quantum mechanics are in accordance with all sorts of processes with virtual particles of type s, t, u summed, because only the initial states and final states are 1,2 3,4 observed that virtual particles of the intermediate states are not observed.

• Particle Physics