General quartic β-function at three loops

A bstract We determine the three-loop $$ \overline{\textrm{MS}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>MS</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> quartic β -function for the most general renormalisable four-dimensional theories. A general parametrisation of the β -function is compared to known β -functions for specific theories to fix all coefficients. Three-loop β -functions for the cubic coupling and scalar mass terms also follow from the result, which is made available in software packages.