Källén function

From Wikipedia
https://en.wikipedia.org/wiki/Källén_function

The Källén function, also known as triangle function, is a polynomial function in three variables, which appears in geometry and particle physics. In the latter field it is usually denoted by the symbol . It is named after the theoretical physicist Gunnar Källén, who introduced it as a short-hand in his textbook Elementary Particle Physics. [1]

Definition

The function is given by a quadratic polynomial in three variables

Applications

In geometry the function describes the area of a triangle with side lengths :

See also Heron's formula.

The function appears naturally in the Kinematics of relativistic particles, e.g. when expressing the energy and momentum components in the center of mass frame by Mandelstam variables. [2]

Properties

The function is (obviously) symmetric in permutations of its arguments, as well as independent of a common sign flip of its arguments:

If the polynomial factorizes into two factors

If the polynomial factorizes into four factors

Its most condensed form is

Interesting special cases are [2]:eqns. (II.6.8–9)

References

  1. ^ G. Källén, Elementary Particle Physics, (Addison-Wesley, 1964)
  2. ^ a b E. Byckling, K. Kajantie, Particle Kinematics, (John Wiley & Sons Ltd, 1973)