first term in the r.h.s. has negative dimension while regarding the other term one has simply to use that the bound for $\hat W^{(h)}_{4,0;a}$ as an extra ${E_h^{(\a)}\over 2^h}$.

We use now the following symmetries of the propagator at $E=0$ (g\^[(k)]{}\_ø)\^\*(k\_1,k\_2)=g\_[ø]{}\^[(h)]{}(-k\_1,k\_2)g\^[(h)]{}\_ø(k\_1,k\_2)=-iøg\^[(h)]{}\_[ø]{}(k\_2,-k\_1)\[bb1\] and that at $E=0$ there is global phase invariance $
\Psi^\pm_{\a,\o}\to e^{\pm i \a_{\a,\o}}\Psi^\pm_{\a,\o}
$. Therefore

1.  The local part of the terms with four fields with the same $\o$ is vanishing; indeed if $n$ is the order there are $n-2$ $(\o)$-propagators and $n$ $(-\o)$-propagators; then by $\hat W^{(h)}_{4,0;a}(\underline k_1,\underline k_2)=(i\o)^{-2} W^{(h)}_{4,0}(-\underline k_2,\underline k_1)$ so that $\hat W^{(h)}_{4,0;a}
    (\underline 0,\underline 0)=-\hat W^{(h)}_{4,0;a}(\underline 0,\underline 0)=0$; moreover by global phase invariance there is an even number of fields with the same $(\a,\o)$.

2.  The quartic terms are real. Indeed by $(\hat W^{(h)}_{4,0;a})^*(\underline k_1,\underline k_2)=
    \hat W^{(h)}_{4,0;a}(-\underline k_1,\underline k_2)$, so that the local part is real

3.  The local part of the terms with two external line and the same $\o$ is vanishing by the parity of the propagator.

4.  Finally $\partial_1 \hat W^{(h)}_{2,0;a}(0)=i\o \partial_2  \hat W^{(h)}_{2,0:a}(0)$

The only quadratic terms in $\LL\VV^{(h)}$ are the one multiplying $\partial W^{(h)}_{2,0;\e,\e;a}({\bf 0})$ and $ W^{(h)}_{2,0:\e,-\e}({\bf 0})$ producing respectively a renormalization of $Z^{(\a)}_h$ and $E^{(\a)}_h$. 