  We study the influence of quantum fluctuations on the electron self energy in
the normal state of iron-pnictide superconductors using a five orbital tight
binding model with generalized Hubbard on-site interactions. Within a one-loop
treatment, we find that an overdamped collective mode develops at low frequency
in channels associated with quasi-1D $d_{xz}$ and $d_{yz}$ bands. When the
critical point for the $C_4$ symmetry broken phase (structural phase
transition) is approached, the overdamped collective modes soften, and acquire
increased spectral weight, resulting in non-Fermi liquid behavior at the Fermi
surface characterized by the frequency dependence of the imaginary part of
electron self energy of the form $\omega^\lambda$, $0<\lambda<1$. We argue that
this non-Fermi liquid behavior is responsible for the recently observed
zero-bias enhancement in the tunneling signal in quantum point contact
spectroscopy. A key experimental test of this proposal is the absence of the
non-Fermi liquid behaviour in the hole-doped materials. Our result suggests
that quantum criticality plays an important role in understanding the normal
state properties of iron-pnictide superconductors.
