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We complement previous functional renormalization group (fRG) studies of thetwo-dimensional Hubbard model by mean-field calculations. The focus falls onVan Hove filling and the the hopping amplitude t'/t=0.341. The fRG data suggesta quantum critical point (QCP) in this region and in its vicinity a singularfermionic self-energy, Im $\Sigma(\omega)/\omega \sim |\omega|^{-\gamma}$ with$\gamma\approx 0.26$. Here we start a more detailed investigation of this QCPusing a bosonic formulation for the effective action, where the bosons coupleto the order parameter fields. To this end, we use the channel decomposition ofthe fermionic effective action developed in [Phys. Rev. B 79, 195125 (2009)],which allows to perform Hubbard-Stratonovich transformations for all relevantorder parameter fields at any given energy scale. We stop the flow at a scalewhere the correlations of the order parameter field are already pronounced, butthe flow is still regular, and derive the effective boson theory. It containsd-wave superconducting, magnetic, and density-density interactions. We analyzethe resulting phase diagram in the mean-field approximation. We show that thesingular fermionic self-energy suppresses gap formation both in thesuperconducting and magnetic channel already at the mean-field level, thusrounding a first-order transition (without self-energy) to a quantum phasetransition (with self-energy). We give a simple effective model that shows thegenerality of this effect. In the two-dimensional Hubbard model, the effectivedensity-density interaction is peaked at a nonzero frequency, so that solvingthe mean-field equations already involves a functional equation instead ofsimply a matrix equation (on a technical level, similar to incommensuratephases). Within a certain approximation, we show that such an interaction leadsto a short quasiparticle lifetime.

Low-Energy Effective Theory at a Quantum Critical Point of the Two-Dimensional Hubbard Model: Mean-Field Analysis