Class of scalar field actions invariant under conformal transformations

Question

From the actions in $d$ dimensions given by

$$S = \int d^dx \,\, \partial_{\mu}\phi \partial^{\mu} \phi + g \phi^k$$.

What is the condition that needs to be $k$ so that the theory is invariant under conformal transformations?

Initially, I have been trying to tackle the special case of pure scale transformations $x^{\prime}= \lambda x$. After putting in the transformed measure and fields as $d^d x^{\prime} = \lambda ^d d^dx$ and $\phi^{\prime}(\lambda x)= \lambda ^{-\Delta}\phi(x)$, I got the following equation

$$\lambda^{-2-2\Delta} + \lambda ^{-k \Delta}=\lambda^{-d}$$

Can I solve this in general for k, in terms of $\Delta$ and $d$. How do I find the scaling dimension of a theory, or is it a parameter?

And how do I solve the general case of any conformal transformation (including SCTs)?

Answer 1

1) You are correct in how you transform the fields, but the condition you derived for scale invariance is incorrect. Each piece of the action must be invariant under scale transformations in order that the whole action is scale invariant. You should get $\lambda^{-2-2\Delta}=\lambda^{-d}$ and $\lambda^{-k\Delta}=\lambda^{-d}$. You can check you get the expected mass dimensions for a free scalar field in d-dimensions.

2) To test invariance under conformal transformations you either have to calculate how your fields transform under special conformal transformations or under inversion. Invariance under inversion will imply conformal invariance since K=I*P*I where K is the generator of SCTs, I is the inversion operator, and P is the translation operator. Inversion is a conformal transformation not smoothly connected to the identity and not all conformally invariant theories are inversion invariant (but these are theories that include spinors which you do not have to worry about here).