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Recently, Fukushima [Phys. Rev. B \textbf{78} 115105 (2008)] proposed asystematic derivation of the Gutzwiller approximation for the t-J model. In thepresent paper, using this approach we construct an effective single-particleHamiltonian, which leads to a renormalized mean-field theory (RMFT). We alsouse the method proposed by us recently and based on the maximum entropyprinciple (MaxEnt), which in turn, yields a consistent statistical descriptionof the problem. On the examples of non-magnetic superconducting d-waveresonating valence bond (dRVB) and normal staggered-flux (SF) solutions, wecompare two selections of the Gutzwiller renormalization schemes, i.e. the oneproposed by Fukushima with that used earlier by Sigrist et al. [Phys. Rev. B\textbf{49}, 12 058 (1994)]. We also confront the results coming from ourvariational solutions with the self-consistency conditions build in, with thoseof the non-variational approach based on the Bogoliubov-de Gennesself-consistent equations. Combination of the present variational approach withthe new renormalization scheme (taken from Fukushima's work) provides, for $t/J=3$, an upper critical hole concentration $x_{c} \approx 0.27$ for thedisappearance of the d-wave superconductivity. Also, the hole concentration $x\approx 0.125$ is obtained for the optimal doping. These results are in roughaccordance with experimental results for high-$T_c$ superconducting cuprates.

A statistically consistent variational approach to the renormalized  mean-field theory of the t-J model: critical hole concentrations for a paired  state