Gradient terms in quantum-critical theories of itinerant fermions

We investigate the origin and renormalization of the gradient ($Q^2$) term in the propagator of soft bosonic fluctuations in theories of itinerant fermions near a quantum critical point (QCP) with $Q =0$. A common belief is that (i) the $Q^2$ term comes from fermions with high energies (roughly of order of the bandwidth) and, as such, should be included into the bare bosonic propagator of the effective low-energy model, and (ii) fluctuations within the low-energy model generate Landau damping of soft bosons, but affect the $Q^2$ term only weakly. We argue that the situation is in fact more complex. First, we found that the high- and low-energy contributions to the $Q^2$ term are of the same order. Second, we computed the high-energy contributions to the $Q^2$ term in two microscopic models (a Fermi gas with Coulomb interaction and the Hubbard model) and found that in all cases these contributions are numerically much smaller than the low-energy ones, blue especially in 2D. This last result is relevant for the behavior of observables at low energies, because the low-energy part of the $Q^2$ term is expected to flow when the effective mass diverges near QCP. If this term is the dominant one, its flow has to be computed self-consistently, which gives rise to a novel quantum-critical behavior. Following up on these results, we discuss two possible ways of formulating the theory of a QCP with $Q=0$.