done as in the previous section (the fact that $E\not=0$ plays no role in the ultraviolet regime). We consider now the integration of the negative infrared scales, in the $L\to\io$ limit. In this case there is no improvement with respect to the scaling dimension, and one has to define a [*renormalized*]{} multiscale integration in the following way. Assume that we have integrated the fields $\Psi^{(N)},..,\Psi^{(h)}$, $h\le 0$ obtaining e\^[(J,)]{} =P\_[Z\_h,E\_h]{}(d\^[(h)]{})e\^[V\^[(h)]{}(,,J)]{}where $P_{Z_h,E_h}(d\Psi^{(\le h)})$ is the gaussian integration with propagator, $\a=\psi,\phi$ g\_(,) =[1L\^2]{} \_\_h() e\^[i(-)]{} [1Z\^[()]{}\_h]{}

D\_+() & E\^[()]{}\_h\
E\^[()]{}\_h & D\_-()

\^[-1]{} and $\hat\chi_h(\kk)=\sum_{j=-\io}^h f_j(\kk)$ and again $V^{(h)}$ being a sum of integral of monomials with $n\ge 0$ $\Psi,\h$ and $m\ge 0$ $J$ fields multiplied by kernels $\hat W_{n,m}^{(h)}$. We decompose the kernels as W\_[n,m]{}\^[(h)]{}()=W\_[n,m;a]{}\^[(h)]{}( )+W\_[n,m;b]{}\^[(h)]{}( )+W\_[n,m;c]{}\^[(h)]{}( ) where $\hat W_{n,m;a}^{(h)}$ and $\hat W_{n,m;b}^{(h)}$ are respectively the zero-th and first order contribution in $E$ to $\hat W_{n,m}^{(h)}$ and $\hat W_{n,m;c}^{(h)}$ is the rest. We define a [*localization operator*]{} on the terms with positive scaling dimension $D=2-n/2-m$ in the following way &&W\^[(h)]{}\_[4,0]{}(\_1,\_2,\_3,\_4)=W\^[(h)]{}\_[4,0;a]{}([**0**]{}, [**0**]{},[**0**]{},[**0**]{})\
&&W\^[(h)]{}\_[2,1]{}(\_1,\_2,\_3)=W\^[(h)]{}\_[2,1;a]{}([**0**]{},[**0**]{},[**0**]{})\
&&W\^[(h)]{}\_[2,0;ø,ø]{}()= W\^[(h)]{}\_[2,0:ø,ø;a]{}([**0**]{})+ W\^[(h)]{}\_[2,0;ø,ø;a]{}([**0**]{})\
&&W\^[(h)]{}\_[2,0;ø,-ø]{}()=W\^[(h)]{}\_[2,0:ø,-ø;a]{}([**0**]{})+ W\^[(h)]{}\_[2,0:ø,-ø;b]{}([**0**]{}) and we write e\^[(J,)]{} =P\_[Z\_h,E\_h]{}(d\^[(h)]{})e\^[V\^[(h)]{}(,,J)+(1-)V\^[(h)]{}(,,J) ]{}\[fg\]The action of $(1-\LL)$ on the kernels improve their scaling dimension. For instance (1-)W\^[(h)]{}\_[4,0]{}= \[W\^[(h)]{}\_[4,0;a]{}()- W\^[(h)]{}\_[4,0;a]{}()\]+W\^[(h)]{}\_[4,0;b]{}()\[41\] and the 