Abstract
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. We also examine a multi-adjoint cubic theory in order to see whether this is a more universal property of these models.
Highlights
Scalar φ3 theory in six dimensions has proved to be a useful laboratory or tool to explore major ideas in quantum field theory
After the discovery of asymptotic freedom in quantum chromodynamics (QCD) [1,2], it was used as a testing ground to study the implications of this property
It has proved useful as a toy model of Regge theory but in six dimensions where ladder diagrams were analyzed in order to gain insight into the Regge slope
Summary
Scalar φ3 theory in six dimensions has proved to be a useful laboratory or tool to explore major ideas in quantum field theory. For instance various decorations of the scalar field with different symmetries allowed the critical exponents that relate to percolation and the Lee-Yang edge singularity problems [9,10,11] to be determined very accurately in the ε expansion in integer dimensions below six More recently another perhaps surprising example of the connection a cubic scalar theory has with physics has emerged. Another direction that was followed in [18,19] was to study classical solutions of a linearized version of Yang-Mills theory and their relation to double copies of scalar fields in the bi-adjoint cubic theory These ideas were explored further in [20,21] where new. An appendix records full details of the renormalization group functions of the quartic adjoint model
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