${\cal L}_{mix}$ breaks the SU(2)$_L$ gauge symmetry of the SM.

As previously mentioned, at perturbative level in vev insertions, there are also fermionic corrections to $Zb\bar b$ arising from the vev mixings between the fermions, Eq. (\[deltagf\]). The analysis is similar for both embeddings in Eqs. (\[bembedding1\]) and (\[bembedding2\]), $b_R$ mixes at leading order in vev insertions with the state $|{\cal P}_{b_L}q^{2cp}\rangle\equiv|D_2^{cp}\rangle$, and it also mixes with $|D_2^{cp'}\rangle$, that is the associated down custodian. Since these states have opposite SU(2)$_L$ charges, their contribution to $\delta g^{b_R}$ is partially cancelled, as can be checked using Eq. (\[deltagf\]) (they do not cancel exactly because their Yukawa couplings can be different, depending on the corresponding Clebsch-Gordan coefficients, and because the custodians can be lighter than the other resonances, due to the SU(2)$_R$ breaking by ${\cal L}_{mix}$, as discussed in section \[effmodel\]). At leading order in vev insertions $b_L$ mixes with the state $|{\cal P}_{b_R}b^{cp}\rangle\equiv|D_b^{cp}\rangle$, inducing a positive $\delta g^{b_L}$.

As a summary, we present in table \[tdeltagsym\] the perturbative contributions to $\delta g^{b_L}$ and $\delta g^{b_R}$, at leading order in vev insertions, for the different embeddings $[T1],\ [T2],\ [B1]$ and $[B2]$. We have included also the fermionic contribution to $\delta g^{b_L}$ arising from 