Asymptotic freedom from the two loop term of the $\beta$-function in a cubic theory

Understand

We renormalize a six dimensional cubic theory to four loops in the MSbar scheme where the scalar is in a bi-adjoint representation.

• The underlying model was originally derived in a problem relating to gravity being a double copy of Yang-Mills theory.
• As a field theory in its own right we find that it has a curious property in that while unexpectedly there is no one loop contribution to the $\beta$-function the two loop coefficient is negative.
• It therefore represents an example where asymptotic freedom is determined by the two loop term of the $\beta$-function.