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 $\lambda$ . It is named after the theoretical physicist Gunnar Källén, who introduced it as a short-hand in his textbook Elementary Particle Physics. 

## Definition

The function is given by a quadratic polynomial in three variables

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

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

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

## Properties

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

$\lambda (-x,-y,-z)=\lambda (x,y,z).$ If $y,z>0$ the polynomial factorizes into two factors

$\lambda (x,y,z)=(x-({\sqrt {y}}+{\sqrt {z}})^{2})(x-({\sqrt {y}}-{\sqrt {z}})^{2}).$ If $x,y,z>0$ the polynomial factorizes into four factors

$\lambda (x,y,z)=-({\sqrt {x}}+{\sqrt {y}}+{\sqrt {z}})(-{\sqrt {x}}+{\sqrt {y}}+{\sqrt {z}})({\sqrt {x}}-{\sqrt {y}}+{\sqrt {z}})({\sqrt {x}}+{\sqrt {y}}-{\sqrt {z}}).$ Its most condensed form is

$\lambda (x,y,z)=(x-y-z)^{2}-4yz.$ Interesting special cases are : eqns. (II.6.8–9)

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