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 Anders Olof Gunnar Källén (13 February 1926 in Kristianstad, Sweden – 13 October 1968 in Hanover, West Germany in a plane accident) was a leading Swedish theoretical physicist and a professor at Lund University until his death at the age of 42.
...
, who introduced it as a short-hand in his textbook ''Elementary Particle Physics''.
[G. Källén, ''Elementary Particle Physics'', (Addison-Wesley, 1964)]
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
In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is,
:A = \sqrt.
It is named after first-centur ...
.
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
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 ...
.
[E. Byckling, K. Kajantie, ''Particle Kinematics'', (John Wiley & Sons Ltd, 1973)]
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
:
:
References
{{DEFAULTSORT:Kallen function
Kinematics (particle physics)
Mathematical concepts