infrared cut-off we get ł\_[h-1,i]{}=ł\_[0,i]{}+O(ł\_0\^2) Z\_h\^[()]{}=1+O(ł\_0\^2)\[x1\] and $\l_0=\l\hat v(0)+O(\l^2)$. This means that the effective couplings $\l_{h,i}$ converge to a line of fixed points (the beta function is asymptotically vanishing) and the critical exponent for the wave function renormalization is zero (contrary to what happens in the fermionic theory in which is positive).

The flow equation for the energy is given by =1+ał\_[1,h]{}+O(ł\_h\^2)=1+ał\_[2,h]{}+O(ł\_h\^2) with $a={1\over 2\pi}>0$ and by symmetry the contributions with different $\a$ do not mix, by the global phase symmetry valid at $E=0$. Therefore E\^[()]{}\_h= E 2\^[-\_h]{} with $\h_\a=a\hat v(0)\l+O(\l^2)$; this implies $2^{h^*_\a}=E^{1\over 1+\h_\a }$. For $h\ge h^*=\max (h^*_\phi, h^*_\psi)$ this makes clear why the second term in has the correct scaling dimension; indeed $
E^{(\a)}_h 2^{-h}$ can be bounded by $2^{(1+\h_\a)(h^*-h)}$ which is sufficient to make the dimension negative. For $h\le h^*$ the theory becomes purely fermionic or bosonic. Therefore = \_[h=h\^\*\_]{}\^(1+łF\_h) with $|F_h(\kk)|\le \l$ and $E^{(\psi)}_h=E$, $Z_h=1$ f or $h\ge 0$; we have used that the contributions from the scales $h \le h^*$ are summable. The density of states (with imaginary energy) is therefore bounded by \_[h=h\^\*\_]{}\^0 |E\^[()]{}\_h|+E\_[h=0]{}\^e\^[-2\^h]{} E\^[11+]{} where $C$ is a suitable constant and $\e$ is an extra ultraviolet cut-off ; that is the density 