g+\bar h,\bar A+ \bar a,\bar g,\bar A]= \breve{\Gamma}_k[\bar g, \bar A,\bar g,\bar A]+O(\bar h,\bar a)+\Gamma^{\textrm{quad}}_k[\bar h, \bar a;\bar g,\bar A]+{\cal O}(\{\bar h,\bar a\}^3)$$ Explicitly, $\Gamma^{\textrm{quad}}_k$ is the sum of the following terms which reflect the block structure of $\breve{\Gamma}^{(2)}_k$ in $(\bar h,\bar a)$-space: $$\begin{aligned}
\left(\Gamma^{\textrm{quad}}_k\right)_{\bar h \bar h}&=Z_N \kappa^2 \int {\textrm{d}^d x\:}\sqrt{\bar g}\,\bar h_{\chi\xi}\bigg(\left(U^{\chi\xi}_{\quad\eta\zeta}-\bar g^{\rho \sigma}\bar D_\rho \bar D_\sigma K^{\chi\xi}_{\quad\eta\zeta}\right)+\nonumber \\&\qquad + \left(1-\frac{1}{\alpha_{\textrm{D}}}\right)L^{\rho\sigma\chi\xi}_{\quad\quad\eta\zeta}\bar D_\rho\bar D_\sigma+ \frac{Z_F}{2 Z_N \kappa^2} N^{\mu\nu\rho\sigma\chi\xi}_{\qquad\quad\eta\zeta}\frac 1 4 \bar{F}^a_{\mu\nu}\bar{F}^a_{\rho\sigma}\bigg)\bar h^{\eta\zeta}\\
\left(\Gamma^{\textrm{quad}}_k\right)_{\bar a \bar a}&=\frac{Z_F}{2\,\hat{g}^2_{\textrm{YM}}}\int {\textrm{d}^d x\:}\sqrt{\bar g}\, \bar a^a_\xi\bigg(-\delta^{ab}\delta^{\xi}_{\eta}\bar g^{\rho\mu}\bar{\mathcal D}_\rho \bar{\mathcal D}_\mu +2 \bar g^{\xi\rho} f^{abc}\bar{F}^c_{\rho\eta}+\delta^{ab}\bar g^{\xi\rho}\bar R_{\rho\eta}+\nonumber \\ &\qquad +\left(1-\frac{1}{\alpha_{\textrm{YM}}}\right) \delta^{ab} \bar g^{\xi\rho}\bar{\mathcal D}_\rho \bar{\mathcal D}_\eta\bigg)\bar a^{b\eta}\\
\left(\Gamma^{\textrm{quad}}_k\right)_{\bar h \bar a}&=\frac{Z_F}{2\,\hat{g}^2_{\textrm{YM}}}\int {\textrm{d}^d x\:}\sqrt{\bar g}\,\bar h_{\eta\zeta}\left(\left(\frac 1 2 \delta_\xi^{\sigma}\bar g^{\eta\zeta}\bar g^{\mu\rho} +\delta_\xi^{\eta}\bar g^{\zeta\rho}\bar g^{\sigma\mu}+\delta_\xi^{\rho}\bar g^{\sigma\zeta}\bar g^{\mu\eta}\right)\bar F^a_{\rho\sigma}\bar{\mathcal D}_\mu\right)\bar a^{a\xi}\\
\left(\Gamma^{\textrm{quad}}_k\right)_{\bar a \bar h}&=\left(\Gamma^{\textrm{quad}}\right)_{\bar h \bar a}\end{aligned}$$ The above quadratic functionals contain the kernels $$\begin{aligned}
K^{\chi\xi}_{\quad\eta\zeta}&=\frac{1}{4}\left(\delta^\chi_\eta\delta^\xi_\zeta+\delta^\xi_\eta\delta^\chi_\zeta-\bar g^{\chi\xi}\bar g_{\eta\zeta}\right)\\
U^{\chi\xi}_{\quad\eta\zeta}&=\frac{1}{4}\left(\delta^\chi_\eta\delta^\xi_\zeta+\delta^\xi_\eta\delta^\chi_\zeta-\bar g^{\chi\xi}\bar g_{\eta\zeta}\right)\left(\bar R -2\bar\lambda\right)+\bar g^{\chi\xi}\bar R_{\eta\zeta}-\delta^\chi_\eta\bar R^\xi_{\;\zeta}-\bar R_{\zeta\ \eta}^{\ \chi\ \xi}\\
L^{\rho\sigma\chi\xi}_{\quad\quad\eta\zeta}&=\left(\frac 1 4 \bar g^{\chi\xi}\bar g^{\rho\sigma}\bar g_{\eta\zeta}-\frac 1 2 \delta^\rho_\eta\delta^\sigma_\zeta\bar g^{\chi\xi}-\frac 1 2 \bar g^{\chi\rho}\bar g^{\xi\sigma}\bar g_{\eta\zeta}+\delta^\chi_\eta\delta^\sigma_\zeta\bar g^{\xi\rho}\right)\\
N^{\mu\nu\rho\sigma\chi\xi}_{\qquad\quad\eta\zeta}&=\frac 1 2\left(\frac 1 2 \bar g^{\chi\xi}\bar g_{\eta\zeta}-\delta^\chi_\eta\delta^\xi_\zeta\right)\bar g^{\mu\rho}\bar g^{\nu\sigma}+\nonumber \\ &\qquad+2\left(\delta^\mu_\eta\delta^\rho_\zeta\bar g^{\nu\chi}\bar g^{\sigma\xi}-\delta^\mu_\eta\delta^\rho_\zeta\bar g^{\chi\xi}\bar g^{\nu\sigma}+2\delta^\xi_\eta\delta^\rho_\zeta\bar g^{\mu\chi}\bar g^{\sigma\nu}\right)\end{aligned}$$

Using these formulae it can be checked that the $\mathcal Z$-factors 