Conformal primaries in momentum space

Question

Consider the Fourier transform of a conformal primary $O$ $$\tilde{O}(k) = \int d^dx e^{ik\cdot x} O(x)$$

Now consider the transformation of the momenta $k \to \lambda k$, so that the above reads $$\tilde{O}(\lambda k) = \int d^dx e^{i\lambda k\cdot x} O(x) = \lambda^{-d}\int d^dx'e^{ik\cdot x'} O(x'/\lambda)$$ where $x' = \lambda x$. Now by using the properties of the conformal primary $O(x/\lambda) = \lambda^{\Delta} O(x)$ we obtain, $$\tilde{O}(\lambda k) = \int d^dx e^{i\lambda k\cdot x} O(x) = \lambda^{\Delta-d}\int d^dx'e^{ik\cdot x'} O(x') = \lambda^{\Delta-d}\tilde{O}(k)$$ Hence, in the momentum space the conformal primary behaves as $$ \tilde{O}(\lambda k) = \lambda^{\Delta-d}\tilde{O}(k)$$, however, the two-point function in the momentum space goes like $$\langle\tilde{O}\tilde{O}\rangle \sim k^{2\Delta-d}$$ which is inconsistent. How does one resolve this enigma ?

Answer 1

Consider the two point function

$$\langle \tilde{\mathcal{O}}(-k) \tilde{\mathcal{O}}(k) \rangle = \int d^d x e^{i k \cdot (x_1-x_2) }\left\langle\mathcal{O}(x_1) \mathcal{O}(x_2)\right\rangle,$$

where the integral is over the differences $x=x_1-x_2$. Performing the rescaling $k\rightarrow \lambda k$ we have

$$\left\langle \tilde{\mathcal{O}}\left(-k \lambda \right) \tilde{\mathcal{O}}\left(\lambda k\right) \right\rangle = \int d^d x e^{i \lambda k \cdot (x_1-x_2) }\left\langle\mathcal{O}(x_1) \mathcal{O}(x_2)\right\rangle=\lambda^{-d} \int d^d x e^{i k \cdot (x_1-x_2) }\left\langle\mathcal{O}\left( \frac{ x_1}{\lambda}\right) \mathcal{O}\left(\frac{x_2}{\lambda}\right)\right\rangle\\ =\lambda^{2\Delta-d} \int d^d x e^{i k \cdot (x_1-x_2) }\left\langle\mathcal{O}(x_1) \mathcal{O}(x_2)\right\rangle=\lambda^{2\Delta-d} \left\langle \tilde{\mathcal{O}}\left(-k \right) \tilde{\mathcal{O}}\left(k\right) \right\rangle.$$

From here we can deduce that $$\left\langle \tilde{\mathcal{O}} \tilde{\mathcal{O}} \right\rangle \sim k^{2\Delta -d}.$$