# Relation between scaling dimension and critical exponents for harmonic peturbations in $O(N)$ Wilson-Fisher (WF) in an old paper

I am reading the paper "Harmonic perturbations of generalized Heisenberg spin systems" (D J Wallace and R K P Zia, 1975) - https://iopscience.iop.org/article/10.1088/0022-3719/8/6/014/meta . The authors consider an $$O(N)$$ Wilson-Fisher fixed point. They compute to order $$\epsilon^3$$ a critical exponent which determines whether an harmonic perturbation of the kind $$\phi_{i_1}\phi_{i_2}\ldots\phi_{i_k}N_{i_1\ldots i_k}$$ is relevant or irrelevant, where $$\phi_i$$ is the magnetization and $$N_{i_1\ldots i_n}$$ projects onto traceless symmetric tensors. I would like to relate the latter to the scaling dimension of the corresponding CFT operator (or, analogously, its RG dimension), but unfortunately I am unable to determine the precise relation from their explanation.

In detail, they claim that the free energy as a function of $$\phi_i$$ and and the inverse susceptibility $$r$$ takes the form: $$F(\phi,r)=\phi^2\Gamma^{(2)}(q=0,r)+ \phi^4\Gamma^{(4)}(0,r)+\ldots+ \phi_{i_1}\phi_{i_2}\ldots\phi_{i_k}\Gamma^{(k)}_{i_1\ldots i_k}(0,r)+\ldots.$$ Here $$\phi^2=\phi_i\phi_i$$. At the critical poitn $$r=\Gamma^{(2)}(0)\rightarrow 0$$ and $$\Gamma^{(k)}_{i_1\ldots i_k}(0,r)\propto r^{\alpha_k}N_{i_1\ldots i_k}.$$ Then they literally say "The relevance or otherwise of the perturbation is obtained by replacing $$\phi$$ by the equivalent power $$(2-\epsilon+\eta)/[2(2-\eta)]$$ of r according to scaling laws and writing" $$F(\phi,r)=\phi^2\Gamma^{(2)}(1+vN_{i_1\ldots i_k}\phi_{i_1}\phi_{i_2}\ldots\phi_{i_k}r^{\psi_k}\phi^{-k}+\ldots)$$ where $$v$$ is some coupling and $$\psi_k=\alpha_k-1+\frac{(k-2)(2-\epsilon+\eta)}{2(2-\eta)}.$$ The perturbation is relevant if $$\psi_k<0$$, irrelevant otherwise.

I would like to understand the relation between $$\alpha_k$$ or, equivalently, $$\psi_k$$ and $$\Delta_k$$, the scaling dimension of the field, related to the RG dimension (I believe) as $$y_k=d-\Delta_k$$. Can someone help me?