upper critical dimension in field theory

Question

Is there field theory which describe a second order phase transition without upper critical dimension ? Mermin-Wagner says something about lower critical dimension but nothing about upper dimension.

Answer 1

The upper critical dimension is the dimension where the statistical field theory is well described by a mean field theory. It is also the dimension where the fluctuation theory turns into a free field theory. You can avoid having an upper critical dimension by tuning the kinetic terms properly:

Consider the Euclidean action:

$$S= \int |q|^{2n} |\phi|^2 + \lambda \phi^4 d^n x$$

This field theory never has an upper critical dimension. But this is because the dimensional extrapolation is wrong. For any fixed power of q, there is an upper critical dimension.