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

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 $β$-function the two loop coefficient is negative. It therefore represents an example where asymptotic freedom is determined by the two loop term of the $β$-function. We also examine a multi-adjoint cubic theory in order to see whether this is a more universal property of these models.

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