D = ∂ ∂ ¯ α + i ( γ $^{µ}$α ) ∂$_{µ}$, and ¯ D = − ∂ ∂α − i (¯ αγ $^{µ}$) ∂$_{µ}$. (4) The anticommutator of two supercharges yields { Q $_{α}$, ¯ Q β } = 2( γ $^{µ}$)$_{α}$$^{β}$∂$_{µ}$ . The supercovariant derivatives are 2 corrections to fermionic and bosonic masses, has been determined in the real-time formulation in [22]. The in- evitable breaking of supersymmetry at finite temperature has sometimes been called spontaneous collapse of super- symmetry [23]. Up to a sign they obey the same anticommutation rela- tion as the supercharges In Sect. IV we derive the RG flow equations at finite temperature. In addition to the momentum integrals we are confronted with sums over Matsubara frequencies. For the three-dimensional Wess-Zumino model and for a particular regulator the thermal sums can be calculated analytically. Related sums have been discussed in earlier works on finite-temperature renormalization group flow equations, for example in [24–27, 29–32]. We observe that the Wess-Zumino model in three dimensions at finite temperature in the GLYPH<CI> 2 symmetric phase behaves similarly to a gas of massless bosons. In particular we show in Sect. IV A that it obeys the Stefan-Boltzmann law in three dimensions. For high temperatures the fermions do not contribute to the flow equations since they do not have a thermal zero-mode. On the other hand we observe dimensional reduction in the bosonic part of the model due to the presence of a thermal zero-mode. We show in Sect. IV B how this is manifested in our RG framework. In a similar way dimensional reduction has been observed in O ( N )-models at finite temperature in [33, 34]. Finally we compute the phase diagram for the restoration of the global Z$_{2}$ symmetry at finite temperature in Sect. IV C. The complete off-shell Lagrange density L off = L $_{kin}$+ L int takes then the simple form The interaction term is the D term of 2 W (Φ) = ¯ αα L $_{int}$+ . . . and contains a Yukawa term, As kinetic term we choose the D term of ¯ D Φ D Φ = 2¯ αα L kin + . . . which reads {D $_{α}$, ¯ D $^{β}$} = − 2( γ )$_{α}$ $^{β}$∂$_{µ}$ . (5) L kin = 1 2 ∂$_{µ}$φ∂ $^{µ}$φ − i 2 ¯ ψ / ∂ψ + 1 2 F $^{2}$. (6) L int = F W $^{′}$( φ ) − 1 2 W $^{′′}$( φ ) ¯ ψψ. (7) L off = 1 2 ∂$_{µ}$φ∂ $^{µ}$φ − i 2 ¯ ψ / ∂ψ + 1 2 F $^{2}$+ F W $^{′}$( φ ) − 1 2 W $^{′′}$( φ ) ¯ ψψ. (8) L on = 1 2 ∂$_{µ}$φ∂ $^{µ}$φ − i 2 ¯ ψ / ∂ψ − 1 2 W ′ $^{2}$( φ ) − 1 2 W $^{′′}$( φ ) ¯ ψψ. (9) Eliminating the auxiliary field via its equation of motion F = − W $^{′}$( φ ), we end up with the on-shell density From this expression we read off that W ′ $^{2}$( φ ) acts as a self-interaction potential for the scalar fields. For a polynomial superpotential W ( φ ) in which the power of the leading term is even, W ( φ ) = cφ 2 n + O ( φ 2 $^{n}$), we do not observe supersymmetry breaking in our present non- perturbative renormalization group study $^{1}$. On the other hand spontaneous supersymmetry breaking is definitely possible for a superpotential in which the power of the leading term is odd. In the explicit calculations we shall use a Majorana representation for the γ -matrices, γ 0 = σ$_{2}$, γ 1 = iσ$_{3}$ and γ 2 = iσ$_{1}$. There are many works on the supersymmetric Wess- Zumino models in both four and two space-time di- mensions. Actually the two-dimensional model with N = 2 supersymmetries is just the toroidal compactifi- cation of the four-dimensional N = 1 model. The three- dimensional model with N = 1 supersymmetry, on the other hand, cannot be obtained by dimensional reduction of a local field theory in four dimensions. Thus it may be useful to recall the construction of the three-dimensional model starting from the real superfield II. THE N = 1 WESS-ZUMINO MODEL IN THREE DIMENSIONS AT T = 0 III. FLOW EQUATION AT ZERO TEMPERATURE with real (pseudo)scalar fields φ, F and Majorana spinor- field ψ . The supersymmetry variations are generated by the supercharge We use the metric tensor ( η$_{µν}$ ) = diag(1 , − 1 − 1) to lower Lorentz indices. With the aid of the symmetry relations for Majorana spinors ¯ ψχ = ¯ χψ, ¯ ψγ $^{µ}$χ = − ¯ χγ $^{µ}$ψ and the particular Fierz identity α ¯ = − ¯ αα GLYPH<BD> / 2 the transforma- tion laws for the component fields follow from Eq. (2): Φ( x, α ) = φ ( x ) + ¯ αψ ( x ) + 1 2 ¯ ααF ( x ) (1) δ$_{β}$ Φ = i ¯ β Q Φ , Q = − i ∂ ∂ ¯ α − ( γ $^{µ}$α ) ∂$_{µ}$ . (2) δφ = ¯ βψ, δψ = ( F + i / ∂φ ) β, δF = i ¯ β / ∂ψ . (3) 1 In a two-loop calculation a ground state with broken super- symmetry has been found in Ref. [36]. Since we neglect higher F -terms in our non-perturbative study it is not possible to check whether the findings of this perturbative analysis of the Wess- Zumino model hold when higher-order corrections are taken into account. To find a manifestly supersymmetric flow equation in the off-shell formulation we extend our earlier results on