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We consider the problem of fermions interacting with gapless long-wavelengthcollective bosonic modes. The theory describes, among other cases, aferromagnetic quantum-critical point (QCP) and a QCP towards nematic ordering.We construct a controllable expansion at the QCP in two steps: we first createa new, non Fermi-liquid ``zero-order'' Eliashberg-type theory, and thendemonstrate that the residual interaction effects are small. We prove that thisapproach is justified under two conditions: the interaction should be smallerthan the fermionic bandwidth, and either the band mass $m_B$ should be muchsmaller than $m = p_F/v_F$, or the number of fermionic flavors $N$ should belarge. For an SU(2) symmetric ferromagnetic QCP, we find that the Eliashbergtheory itself includes a set of singular renormalizations which can beunderstood as a consequence of an effective long-range dynamic interactionbetween quasi-particles, generated by the Landau damping term. These singularrenormalizations give rise to a negative non-analytic $q^{3/2}$ correction tothe static spin susceptibility, and destroy a ferromagnetic QCP. We demonstratethat this effect can be understood in the framework of the $\phi^4$ theory ofquantum criticality. We also show that the non-analytic $q^{3/2}$ correction tothe bosonic propagator is specific to the SU(2) symmetric case. For systemswith a scalar order parameter, the $q^{3/2}$ contributions from individualdiagrams cancel out in the full expression of the susceptibility, and the QCPremains stable.Comment: 37 pages, 10 fig

Quantum critical behavior in itinerant electron systems: Eliashberg theory and instability of a ferromagnetic quantum critical point