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 ${\displaystyle \lambda }$. 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

${\displaystyle \lambda (x,y,z)\equiv x^{2}+y^{2}+z^{2}-2xy-2yz-2zx.}$

## Applications

In geometry the function describes the area ${\displaystyle A}$ of a triangle with side lengths ${\displaystyle a,b,c}$:

${\displaystyle A={\frac {1}{4}}{\sqrt {-\lambda (a^{2},b^{2},c^{2})}}.}$

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:

${\displaystyle \lambda (-x,-y,-z)=\lambda (x,y,z).}$

If ${\displaystyle y,z>0}$ the polynomial factorizes into two factors

${\displaystyle \lambda (x,y,z)=(x-({\sqrt {y}}+{\sqrt {z}})^{2})(x-({\sqrt {y}}-{\sqrt {z}})^{2}).}$

If ${\displaystyle x,y,z>0}$ the polynomial factorizes into four factors

${\displaystyle \lambda (x,y,z)=-({\sqrt {x}}+{\sqrt {y}}+{\sqrt {z}})(-{\sqrt {x}}+{\sqrt {y}}+{\sqrt {z}})({\sqrt {x}}-{\sqrt {y}}+{\sqrt {z}})({\sqrt {x}}+{\sqrt {y}}-{\sqrt {z}}).}$

Its most condensed form is

${\displaystyle \lambda (x,y,z)=(x-y-z)^{2}-4yz.}$

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

${\displaystyle \lambda (x,y,y)=x(x-4y)\,,}$
${\displaystyle \lambda (x,y,0)=(x-y)^{2}\,.}$

## References

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