We investigate the origin and renormalization of the gradient (Q2) term in the propagator of soft bosonic fluctuations in theories of itinerant fermions near a quantum critical point (QCP) with ordering wavevector Q0=0. A common belief is that (i) the Q2 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 Q2 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 Q2 term are of the same order. Second, we computed the high-energy contributions to the Q2 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, especially in 2D. This last result is relevant for the behavior of observables at low energies, because the low-energy part of the Q2 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 Q0=0.
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We thank N. Prokofiev for his help with the numerical calculation, and E. Abrahams, C. Batista, S. Maiti, P. Wolfle, and V. A. Zyuzin for stimulating discussions. P.S. acknowledges support from the Institute for Fundamental Theory, University of Florida. The work of A.V.C. was supported by the NSF via Grant No. DMR-1523036. D.L.M. and A.V.C. acknowledge the hospitality of the Kavli Institute for Theoretical Physics, which is supported by the NSF via Grant No. NSF PHY11-25915.