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We propose a systematic approach to the systems of correlated electrons, theso-called $\mathbf{k}$-DE-GWF method, based on reciprocal-space($\mathbf{k}$-resolved) diagrammatic expansion of the variationalGutzwiller-type wave function for parametrized models of correlated fermions.The present approach, in contrast to either variational Monte-Carlo (VMC), orthe recently developed real-space diagrammatic expansion of the Gutzwiller-typewave function (direct-space DE-GWF technique), is applicable directly in thethermodynamic limit and thus is suitable for describing selected singularfeatures of the wave-vector-dependent quantities. We employ the$\mathbf{k}$-DE-GWF method to extract the non-analytic part of the two leadingmoments of the fermion spectral-density function across the (two-dimensional)Brillouin zone for the Hubbard model and away from the half-filling. Thosemoments are used to evaluate the nodal quasiparticle velocities and theirspectral weights in the correlated superconducting state. The two velocitiesdetermined in that manner exhibit scaling with the electron concentrationqualitatively different from that obtained earlier for the excited states ofthe high-$T_c$ cuprates within the projected quasi-particle ansatz, and theresults are in a very good quantitative agreement with experimental data ifinterpreted as those characterizing the spectrum below and above the observedkink. We provide a detailed discussion of the two gaps and two excitationbranches (two velocities) appearing naturally within our DE-GWF approach. Thetwo separate sets of characteristics distinguish the renormalized quasiparticlestates very close to the Fermi surface from the deeper correlated-stateproperties. Also, an enhancement of the $\mathbf{k}$-dependent magneticsusceptibility is shown to contain a spin-fluctuation contribution within ourlanguage.

Realistic estimates of superconducting properties for the cuprates: reciprocal-space diagrammatic expansion combined with variational approach