  We study bond percolations on hierarchical scale-free networks with the open
bond probability of the shortcuts $\tilde{p}$ and that of the ordinary bonds
$p$. The system has a critical phase in which the percolating probability $P$
takes an intermediate value $0<P<1$. Using generating function approach, we
calculate the fractal exponent $\psi$ of the root clusters to show that $\psi$
varies continuously with $\tilde{p}$ in the critical phase. We confirm
numerically that the distribution $n_s$ of cluster size $s$ in the critical
phase obeys a power law $n_s \propto s^{-\tau}$, where $\tau$ satisfies the
scaling relation $\tau=1+\psi^{-1}$. In addition the critical exponent
$\beta(\tilde{p})$ of the order parameter varies as $\tilde{p}$, from
$\beta\simeq 0.164694$ at $\tilde{p}=0$ to infinity at
$\tilde{p}=\tilde{p}_c=5/32$.
