in a rather arbitrary manner (see, below Eq. (2.5). How- ever, the same constant enters the renormalized chemical potential µ $^{′}$, Eq. (2.14), and the renormalized thermody- namical potential Ω written below Eq. (2.14). Of course, the constant ¯ V should disappear from the final result for the thermodynamical potential, which can serve as a check of any computation. Actually, considering only perturbation theory in the interaction there is no neces- sity to introduce this constant. In such calculations the sign of ˜ V$_{r,r}$ ′ in Eq. (2.4) does not play an important role because one could decouple the interaction term with ˜ V integrating over purely imaginary HS. We have added ¯ V because we want to keep the HS fields real, keeping in mind a possibility of applying the method to numerical investigations. Using Eqs. (3.8,3.9) we easily integrate in Eq. (3.6) over the field φ$_{rσ}$ ( τ ) reducing the partition function Z to the form the functional integrals can be achieved introducing the superfields Ψ$_{r,r}$ ′ ( R ), R = { z, θ, θ $^{∗}$} , as follows where a$_{r,r}$ ′ ( z ) and f$_{r,r}$ ′ ( z ) are real fields of the coordi- nates r, r ′ and the variables z = ( τ, σ, u ). The hermi- tian conjugation means complex conjugation “ ∗ ” sup- plemented by the transposition, such that η + r,r ′ = η ∗ r, ′ $_{r}$. The variables θ , θ ∗ are artificially introduced Grassmann variables that help us to write the exponent in the com- pact form. As a result, we write the functional Z [ φ ], Eq. (2.31), in the form where S$_{0}$ [Ψ] is the bare part of the action The second term S ( uφ $^{)}$[Ψ] in the exponent in Eq. (3.6) is linear with respect to the HS field φ$_{rσ}$ ( τ ). Its explicit form reads with S $_{0}$[Ψ] given by Eq. (3.7) and The term S int [Ψ] in the action describes the interac- tion between the fields Ψ. The terms S 2 [Ψ], S $_{3}$[Ψ] and S $_{4}$[Ψ] contain quadratic, cubic and quartic in Ψ terms, respectively. They can be written in the form, It is interesting to remark here that the superfield Ψ ( R ) is anticommuting, which is a rather unusual feature of the field theory considered here. However, it is very important that this field describes bosons and not fermions. This follows from the periodic boundary con- dition Ψ ( τ ) = Ψ ( τ + β ). It is this condition that deter- mines unambiguously the statistics of the particles. The fact that Ψ ( R ) is anticommuting is only a formal prop- erty. The form of Eqs. (3.6-3.8) allows us to average im- mediately over the HS field φ$_{rσ}$ ( τ ). This integration is Gaussian and is specified by Eq. (2.18). The analytic supersymmetric field theory written here relies crucially on the presence of a bath, whose function is to break the translational symmetry of the system. In order to take the bath into account, we model the interaction in the same way as in Eq. (2.37). The pair correlation of the fields with the distribution W can be written as where [ , ] stands for the commutator. Integration over R in Eq. (3.12a-3.12c) implies summation over σ and inte- gration over u, τ, θ, θ $^{∗}$. The bare action S 0 and the inter- action term S $_{4}$, being invariant under the transformation of the fields Ψ ( κ and κ ∗ being anticommuting variables) are fully su- persymmetric in the sense of Ref. 31, whereas the terms S 2 and S 3 break this invariance. The invariance under the transformation (3.13) is stronger than the standard BRST symmetry for stochastic field equations [invariance under the transformation Ψ ( θ $^{∗}$) → Ψ ( θ ∗ + κ $^{∗}$)] $^{31}$, and reflects additional symmetries of Eqs. (3.7,3.12c). where V total 0 and V (1) ,total r,r ′ are defined from V$_{0}$ and V (1) according to Eqn. (2.37). The coupling constants V$_{0}$ and V (1) contain the constant ¯ V that has been introduced 〈 σσ $^{′}$φ$_{rσ}$ ( τ ) φ ′ r ′ σ ′ ( τ $^{′}$) 〉 W = U$_{r,r}$ ′ ( R, R $^{′}$) (3.9) U$_{r,r}$ ′ ( R, R $^{′}$) = δ ( τ − τ $^{′}$) ( σσ $^{′}$V total 0 δ$_{r,r}$ ′ + V (1) ,total r,r ′ ) , Ψ$_{r,r}$ ′ ( θ, θ $^{∗}$) → Ψ$_{r,r}$ ′ ( θ + κ, θ ∗ + κ $^{∗}$) (3.13) S 4 = 1 2 ∫ ∑ r,r ′ ,r$_{1}$,r$_{1}$ ′ Ψ$_{r}$ ′ $_{,r}$( R )Ψ$_{r,r}$ ′ ( R ) U$_{r,r}$$_{1}$ ( R, R$_{1}$ ) × Ψ$_{r}$ ′ $_{$_{1}$,r$_{1}$}$( R$_{1}$ )Ψ$_{r}$$_{1}$$_{,r}$ ′ $_{1}$( R$_{1}$ ) u$_{1}$udRdR$_{1}$ (3.12c) S 3 = ∫ ∑ r,r ′ ,r$_{1}$ Ψ$_{r}$ ′ $_{,r}$( R )Ψ$_{r,r}$ ′ ( R ) U$_{r,r}$$_{1}$ ( R, R$_{1}$ ) × { i Ψ$_{r}$$_{1}$$_{,r}$$_{1}$( R$_{1}$ ) θ ∗ 1 − [Ψ ( R$_{1}$ ) , ˆ n ]$_{r}$$_{1}$$_{,r}$$_{1}$ u$_{1}$θ$_{1}$ } udRdR$_{1}$, (3.12b) S 2 = 1 2 ∫ ∑ r,r$_{1}$ { i Ψ$_{r,r}$( R ) θ ∗ − [Ψ ( R ) , ˆ n ]$_{rr}$ uθ } U$_{r,r}$$_{1}$ ( R, R$_{1}$ ) × { i Ψ$_{r}$$_{1}$$_{,r}$$_{1}$( R$_{1}$ ) θ ∗ 1 − [Ψ ( R$_{1}$ ) , ˆ n ]$_{r}$$_{1}$$_{,r}$$_{1}$ u$_{1}$θ$_{1}$ } dRdR$_{1}$, (3.12a) S ( uφ $^{)}$[Ψ] = − i ∑ r,r ′ ∫ φ$_{rσ}$ ( τ ) [ u (Ψ$_{r}$ ′ ,r ( R ) − n$_{r}$ ′ $_{,r}$θ ) × (Ψ$_{r,r}$ ′ ( R ) − n$_{r,r}$ ′ θ ) − iσδ$_{r,r}$ ′ Ψ$_{r,r}$ ( R ) θ ∗ ] dR (3.8) S 0 = i 2 ∑ r,r ′ ∫ [ Ψ$_{r}$ ′ ,r ( R ) ( ∂ ∂τ + (ˆ ε$_{r}$ − ˆ ε$_{r}$ ′ ) Ψ$_{r,r}$ ′ ( R ) ] dR (3.7) Z [ φ ] = Z$_{0}$ ∫ exp ( −S $_{0}$[Ψ] − S ( uφ ) [Ψ] ) D Ψ (3.6) Z = Z$_{0}$ ∫ exp ( −S $_{0}$[Ψ] − S int [Ψ]) D Ψ (3.10) S int [Ψ] = S $_{2}$[Ψ] + S $_{3}$[Ψ] + S $_{4}$[Ψ] . (3.11) Ψ$_{r,r}$ ′ ( R ) = a$_{r,r}$ ′ ( z ) θ + f T r,r ′ ( z ) θ $^{∗}$+ η$_{r,r}$ ′ ( z )+ η + r,r ′ ( z ) θ $^{∗}$θ (3.5) 10