Eqs. (3.10-3.12c) completely determine the new bosonic superfield theory. This model can be studied using standard methods of field theory. One can, e.g., expand in the interaction U using the Wick theorem with simple contraction rules following from the form of the bare action S $_{0}$. In the next subsection, we will demon- strate how such calculations can be carried out explicitly but now let us understand what one obtains neglecting the cubic and quartic in Ψ terms S 3 and S 4 in the action. In this approximation, one has a purely quadratic action and, making the Fourier transform with respect to time and space, we obtain easily for the partition function Z an RPA-like expression, and similarly one can obtain the Fourier transform of the interaction, Eqs. (3.12a-3.12c). According to Eq. (3.15), the bare propagator of the bosonized theory is given by wherein, g ( ω ; p , p $^{′}$) is the Green function in the momen- tum representation Using Wick’s theorem and the pairing rule, Eq. (3.16), one can calculate averages of Ψ-products arising in per- turbation theory expansions analytically in a standard manner. For a convenient diagrammatical representation, the propagator can be depicted in the form of an oriented double-line as shown in Fig. 2 (a), the first momentum p being carried by the right-facing upper line, the second momentum p ′ by the left-facing lower line. In the same spirit, the interaction vertices, Eqs. (3.12a- 3.12c), are diagrammatically represented as shown in Fig. 2 (b). Depending on whether a leg is depicted as “out-going” [right-facing] or “incoming” [left-facing], the momenta and frequencies are either reversed [momenta p and p ′ are interchanged and frequency ω changes the sign] or not. The interaction lines carrying the momen- tum q correspond to the propagator and are attached according to the analytical structure of Eqs. (3.12a-3.12c). In this way, the single lines within the double-line structure obey evident momentum con- servation rules, whereas the bosonic frequency of each double-line is conserved in the usual way and, therefore, not labelled in the diagram. With the rules formulated, we are now ready to discuss the perturbation theory for the thermodynamic poten- tial Ω. The same result can be obtained using Eqs. (2.18,2.31, 2.32) and neglecting the field φ$_{rσ}$ ( τ ) in the L.H.S. of Eq. (2.32). In Eq. (3.14), ( K − 1) is the contribution of non- interacting bosonic excitations. Considering their inter- action, one can fully describe the initial fermionic system. We note that the boson model, as formulated in this paper, does not appear convenient for a standard analyt- ical approach such as renormalization group procedures, where the low lying modes should have been singled out in the very beginning. However, the superfield model Eq. (3.11) and a proper low-energy formulation are for- mally quite similar. Therefore, this Section does not only serve as check of the superfield model but also as a guide for future analytical studies of low energy effects. In the next subsections, we demonstrate how the con- tributions of the conventional perturbation theory 1 in the original fermion language are reproduced in the boson superfield representation. We perform this check calcu- lating the perturbation series for the thermodynamic po- tential Ω up to the second order in the interaction. In order to develop the perturbation theory in the in- teraction, we Fourier transform the superfields As S 4 does not break the symmetry specified by Eq. (3.13), the first order contribution to Ω in the su- perfield theory comes only from S $_{2}$. However, T 〈S 2 〉 is not the only contribution to the first order correc- tion ΔΩ (1) in the interaction potential. In addition, one should renormalize the chemical potential thus us- ing µ $^{′}$, Eq. (2.14), and shift the thermodynamic po- tential Ω → Ω + (2 n ) 2 V $^{(1)}$N$_{d}$ / 2 in accordance with Eq. (2.13). This leads to a trivial first order contribution to the thermodynamic potential Ω that can be written where ρ = ( σ, u, θ, θ $^{∗}$) and ω stands for the bosonic Mat- subara frequencies. The momentum integration ( d p ) = d p / (2 π ) d extends over the first Brillouin zone. In the Fourier representation, the bare action reads S 0 = 1 2 i T ∑ ω ∫ Ψ$_{p}$ ′ p , − $_{ω}$( ρ ) { iω − [ ε$_{p}$ − ε$_{p}$ ′ ] } Ψ$_{pp}$ ′ $_{,ω}$( ρ ) × ( d p d p $^{′}$) dρ (3.15) Ψ$_{rr}$ ′ ( R ) = T ∑ ω ∫ Ψ$_{pp}$ ′ $_{,ω}$( ρ ) e i ( p r − p $^{′}$r $^{′}$) − iωτ ( d p d p $^{′}$) Z ≃ Z$_{0}$ exp [ − T 2 ∑ ω ∫ d $^{d}$k (2 π ) d ln K ] , (3.14) K = 1 + V$_{0}$ ∫ n$_{p}$$_{−}$$_{k}$$_{/}$$_{2}$ − n$_{p}$$_{+}$$_{k}$$_{/}$$_{2}$ iω + ε$_{p}$$_{−}$$_{k}$$_{/}$$_{2}$ − ε$_{p}$$_{+}$$_{k}$$_{/}$$_{2}$ d $^{d}$p (2 π ) $_{d}$. g ( ω ; p , p $^{′}$) = i [ iω − ( ε$_{p}$ − ε$_{p}$ ′ )] − 1 . 〈 Ψ$_{p}$$_{,}$$_{p}$ ′ ; $_{ω}$( ρ )Ψ$_{p}$$_{1}$$_{,}$$_{p}$ ′ $_{1}$; $_{ω$_{1}$}$( ρ$_{1}$ ) 〉 = ( θ − θ$_{1}$ )( θ ∗ − θ ∗ $_{1}$) δ ( u − u$_{1}$ ) × δ$_{σσ}$$_{1}$ (2 π ) 2 $^{d}$δ ( p − p ′ $_{1}$) δ ( p ′ − p$_{1}$ ) δ$_{ω,}$$_{−}$$_{ω}$$_{1}$ T g ( ω ; p , p $^{′}$) (3.16) U$_{q}$$_{,ω}$ ( σ, σ$_{1}$ ) = σσ$_{1}$V$_{0}$ + V (1) q (3.17) C. First order of the perturbation theory B. Wick theorem and diagrams 11