  The role of electron-phonon interactions in iron-based superconductor is
currently under debate with conflicting experimental reports on the isotope
effect. To address this important issue, we employ the renormalization-group
method to investigate the competition between electron-electron and
electron-phonon interactions in these materials. The renormalization-group
analysis shows that the ground state is a phonon-dressed unconventional
superconductor: the dominant electronic interactions account for pairing
mechanism while electron-phonon interactions are subdominant. Because of the
phonon dressing, the isotope effect of the critical temperature can be normal
or reversed, depending on whether the retarded intra- or inter-band
interactions are altered upon isotope substitutions. The connection between the
anomalous isotope effect and the unconventional pairing symmetry is discussed
at the end.
