  We study the wetting transition and the directed polymer delocalization
transition on diamond hierarchical lattices.These two phase transitions with
frozen disorder correspond to the critical points of quadratic renormalizations
of the partition function.(These exact renormalizations on diamond lattices can
also be considered as approximate Migdal-Kadanoff renormalizations for
hypercubic lattices). In terms of the rescaled partition function
$z=Z/Z_{typ}$,we find that the critical point corresponds to a fixed point
distribution with a power-law tail $P_c(z) \sim \Phi(\ln z)/z^{1+\mu}$ as $z
\to +\infty$ (up to some sub-leading logarithmic correction $\Phi(\ln z)$), so
that all moments $z^{n}$ with $n>\mu$ diverge. For the wetting transition, the
first moment diverges $\bar{z}=+\infty$ (case $0<\mu<1$), and the critical
temperature is strictly below the annealed temperature $T_c<T_{ann}$. For the
directed polymer case, the second moment diverges $\bar{z^2}=+\infty$ (case
$1<\mu<2$), and the critical temperature is strictly below the exactly known
transition temperature $T_2$ of the second moment.We then consider the
correlation length exponent $\nu$ :the linearized renormalization around the
fixed point distribution coincides with the transfer matrix describing a
directed polymer on the Cayley tree, but the random weights determined by the
fixed point distribution $P_c(z)$ are broadly distributed. This induces some
changes in the travelling wave solutions with respect to the usual case of more
narrow distributions.
