i k\_0 k(k\_0\^2+ k\^2)\^2]{}= 0. \[xxx\] A similar analysis can be repeated for the other terms to show that the scaling dimension is always negative. The conclusion is that the effective potential is uniformly bounded in $N$ and that there are [*no ultraviolet divergences*]{} even when the momentum cut-off is removed, that is for $N\to\io$.

Ward Identities and cancellation of the anomalies
-------------------------------------------------

A crucial role is played by [*Ward Identities*]{}, which can be obtained by performing in with $E=0$ the [*chiral*]{} local phase transformation $
\Psi^\pm_{\o,\a,\xx}\to e^{\pm i a_{\o,\a,\xx}} \Psi^\pm_{\o,\a,\xx}
$ and performing a derivative with respect to $a_{\o,\a,\xx}$ and the external fields; due to the presence of cut-offs the Jacobian is equal to $1$ but, with respect to the formal Ward Identities obtained neglecting cut-offs, one has an extra term; indeed it is found &&D\_ø()&lt;\_[,ø,]{}\^-\_[’,ø’]{}\^+\_[’,ø’,+]{}&gt;\_[T,0,N]{}= \_[,’]{}\_[ø,ø’]{}\[&lt;\^-\_[’,ø’,]{}\^+\_[’,ø’,]{}&gt;\_[T,0,N]{}\
&& -&lt;\^-\_[’,ø’,+]{}\^+\_[’,ø’,+]{}&gt;\_[T,0,N]{}\] +&lt;\_[,ø,]{}\^-\_[’,ø’,]{}\^+\_[’,ø’,+]{}&gt;\_[T,0,N]{}\[pp\] where $D_\o(\kk)=i k_1-\o k_2$, with $\o=\pm $, $\r_{\a,\o,\xx}=\Psi^+_{\a,\o,\xx}\Psi^-_{\a,\o,\xx}$, \_[,ø,]{}=dC\_[N]{}(,)\^+\_[,ø,]{}\^-\_[,ø,+]{} with C\_[N]{}(,)=\[(+ )\^[-1]{} - 1\] D\_[ø]{}(+) - \[ (()\^[-1]{} - 1\] D\_[ø]{}()\[fonddd1\]. The last term in is due the the presence of the momentum cut-off which breaks the [*local*]{} chiral invariance. Remarkably, such term it is not vanishing even removing the ultraviolet cut-off, but the following identity holds &&\_[T,0,N]{}=\
&& 