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?


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