# Class of scalar field actions invariant under conformal transformations

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)?

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For a theory to be truly conformally invariant, the beta function $\beta=0$, and demonstrating that is somewhat laborious as it requires renormalization. –  JamalS May 25 at 18:07

## 1 Answer

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).

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Thanks for the answer. Is there a theorem that scale invariance implies invariance under SCTs, and vice versa? –  ramanujan_dirac May 26 at 19:22
Scale invariance does not always imply invariance under SCTs (that is scale does not imply conformal invariance). In 2-dimensions its been proven for unitary Lorentz invariant theories that scale->conformal and its possible it holds in higher dimensions under similar assumptions. There are examples of non-unitary scale invariant theories that are not conformally invariant. –  David Meltzer May 27 at 0:40
Invariance under SCTs always implies scale invariance. This can be seen by looking at the conformal algebra: en.wikipedia.org/wiki/Conformal_symmetry The commutator of K, the generator of SCTs, with P gives D, the dilation operator, so invariance under SCTs and translation must imply scale invariance. Also a special conformal transformation looks locally like a dilation, its just the scaling factor depends on position (so its a position dependent rescaling), so demanding invariance under an SCT is stronger then demanding invariance under a scale transformation. –  David Meltzer May 27 at 0:45