Källén function explained

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

λ(x,y,z)\equivx²+y²+z²-2xy-2yz-2zx.

Applications

In geometry the function describes the area

of a triangle with side lengths

a,b,c

A=	1
	4

\sqrt{-λ(a^2,b^2,c^2)}.

Properties

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

λ(-x,-y,-z)=λ(x,y,z).

y,z>0

the polynomial factorizes into two factors

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

x,y,z>0

the polynomial factorizes into four factors

λ(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

λ(x,y,z)=(x-y-z)^2-4yz.

Interesting special cases are^[2]

λ(x,y,y)=x(x-4y),

λ(x,y,0)=(x-y)^2.

Notes and References

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