Abstract
We renormalize various scalar field theories with a phi ^n self interaction such as n=5, 7 and 9 in their respective critical dimensions which are non-integer. The renormalization group functions for the O(N) symmetric extensions are also computed.
Highlights
Scalar quantum field theories have provided an excellent laboratory for many years to explore and test ideas in physical problems
A widely studied example is that of the Ising model which can be described by scalar φ4 theory
As has been noted for example in [15] if the critical dimension is close to an integer the use of d = Dn − 2 means that choosing a small value of in the -expansion of the first few terms of a critical exponent should yield accurate exponent estimates in that integer dimension
Summary
Scalar quantum field theories have provided an excellent laboratory for many years to explore and test ideas in physical problems. Given this resurgence of interest in scalar field theories with higher order potentials there is a clear need to complement the conformal field theory approach with explicit perturbative computations. As has been noted for example in [15] if the critical dimension is close to an integer the use of d = Dn − 2 means that choosing a small value of in the -expansion of the first few terms of a critical exponent should yield accurate exponent estimates in that integer dimension This is in contrast with the use of the -expansion in φ4 theory where d = 4 − 2 and d has to be 3 for Ising model predictions. Symmetry and compute the corresponding renormalization group functions This leads to an interesting prospect which may connect the O(N ) φ5 and O(N ) φ8 theories with potentially a generalization for higher order potentials.
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