In chapter 6 of the book about condensed matter physics written by Gerald Mahan, the self-energy of the conduction electron is calculated to the third order in $J$ (the Kondo coupling) to show that the imaginary part of the self-energy diverges logarithmically as the energy scale (or temperature) decreases.
What I am wondering is whether this 'the third order expansion in $J$' can be derived from the Anderson's impurity model.
Kondo problem, modelled by a Hamiltonian $$ H_{K}= \sum_k \varepsilon_k {c^\dagger _k} {c_k}+J\boldsymbol{S}\cdot\boldsymbol{s}(0), $$ where $\boldsymbol{s}(0)$ is the spin-1/2 operator of the conduction electron at the origin and $\boldsymbol{S}$ is the spin of the localized electron $f$, is derivable from the Andersons's impurity model: $$ H_{A}=\sum_k \varepsilon_k {c^\dagger _k} {c_k}+ V\sum_k ({f^\dagger} c_k+{c^\dagger_k} f)+Un_{f,\uparrow} n_{f,\downarrow} $$ through Schrieffer-Wolff transformation. In this derivation, $J$ in $H_K$ is related to $V$ and $U$ in $H_A$ by $J \propto V^2/U$.
However, in the transformation, higher-order corrections proportional to, for example, $V^{6}/U^3$ are neglected and only the first order correction (proportional to $J \propto V^2/U$) from the high energy excitaion is kept. Therefore, it seems to me that if $V^{6}/U^3$ correction in Schrieffer-Wolff transformation is kept, the result of the third order expansion (derived in the book by Mahan) will look different.
If this is the case, why are such higher order corrections usually neglected in Schrieffer-Wolff transformation to derive Kondo problem even if we need to calculate the perturbation series beyond the first order in $J$ to solve Kondo problem?
