Mandelstam variable

The Mandelstam variables s , t and u (after Stanley Mandelstam , who introduced them in 1958) are abbreviations for terms that often appear in particle physics when calculating scattering processes with two incoming and two outgoing particles.

If the four impulses of the two incoming particles are denoted by and those of the outgoing particles are denoted by, the Mandelstam variables are given by: ${\displaystyle p_{1},p_{2}}$ ${\displaystyle p_{3},p_{4}}$

${\displaystyle s=(p_{1}+p_{2})^{2}=(p_{3}+p_{4})^{2}}$is equal to the square of the center of mass energy of the system (s-channel).

${\displaystyle t=(p_{1}-p_{3})^{2}=(p_{2}-p_{4})^{2}}$ is equal to the square of the four-pulse carry in a common scattering process such as electron-nucleon scattering (t-channel).

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

The square of quadruple impulses appearing in these definitions is - as is usual in relativistic physics - defined as (see quad vector ). The Mandelstam variables are thus Lorentz-variant scalars just like the scattering amplitude itself, which is supposed to be expressed by them in a relativistically invariant way. ${\displaystyle p^{2}:=p_{\mu }p^{\mu }}$

The three Mandelstam variables are not independent of each other: their sum is equal to the sum of the squares of the masses of the particles involved:

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

where, as usual in particle physics, the dimensionless value c = 1 is assumed for the speed of light ( natural units ).

In general, the scattering amplitude, since it is a relativistic invariant, should depend on the relativistic invariants (i = 1, 2, 3, 4) and the six possible independent (relativistic) scalar products - the Mandelstam variables s, t, u are also derived from these as Composite linear combination . They are not variables because of (the outer legs of the Feynman diagrams are on the shell ). Because of the conservation of the four-momentum (which results in four equations, since they each have four components), only two of the six scalar products are independent. So only two of the Mandelstam variables are independent, the third results from the above. Total. ${\displaystyle {p_{i}}^{2}}$ ${\displaystyle p_{1}\cdot p_{2},p_{1}\cdot p_{3},p_{1}\cdot p_{4},p_{2}\cdot p_{3},p_{2}\cdot p_{4},p_{3}\cdot p_{4}}$ ${\displaystyle {p_{i}}^{2}}$ ${\displaystyle {p_{i}}^{2}={m_{i}}^{2}}$ ${\displaystyle p_{1}+p_{2}=p_{3}+p_{4}}$ ${\displaystyle p_{i}}$ ${\displaystyle p_{i}\cdot p_{j}}$

s, t and u channels

The contributions to the scattering process in which the respective Mandelstam variables appear in their calculation are referred to as the s, t and u channels. The associated Feynman diagrams are shown in the following figure.

s channel

t channel

u channel

The representation follows the convention that the incoming particles are shown as lines coming from the left and the scatter products as lines coming out to the right. The dashed intermediate line represents a virtual particle ; the square of its quadruple momentum is equal to s , t or u according to the respective diagram .

For example, the map for the S channel is an electron - positron - annihilation on the left side with the formation of a virtual photon and electron-positron pair production on the right again. The formation of an unstable intermediate state ( resonance ) when two particles interact is also represented in this way. Ordinary electron-electron scattering is represented by the diagram of the t-channel (whereby the u-channel must also be taken into account, in which the outer legs 3, 4 of the diagram are reversed).

In the scattering amplitude, according to the rules of quantum mechanics , all possible processes with virtual particles of type s, t, u are added up, since only the initial states 1,2 and final states 3,4, but not the virtual particles of the intermediate states, are observed.