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 $\lambda$ . 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

\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})}}.

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.

References

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

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