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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]. 

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. 

## II. THE N = 1 WESS-ZUMINO MODEL IN THREE DIMENSIONS AT T = 0 

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 

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): 

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The anticommutator of two supercharges yields { Q $_{α}$, ¯ Q β } = 2( γ $^{µ}$)$_{α}$$^{β}$∂$_{µ}$ . The supercovariant derivatives are 

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Up to a sign they obey the same anticommutation rela- tion as the supercharges 

As kinetic term we choose the D term of ¯ D Φ D Φ = 2¯ αα L kin + . . . which reads 

The interaction term is the D term of 2 W (Φ) = ¯ αα L $_{int}$+ . . . and contains a Yukawa term, 

The complete off-shell Lagrange density L off = L $_{kin}$+ L int takes then the simple form 

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Eliminating the auxiliary field via its equation of motion F = − W $^{′}$( φ ), we end up with the on-shell density 

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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$_{1}$. 

To find a manifestly supersymmetric flow equation in the off-shell formulation we extend our earlier results on 

## III. FLOW EQUATION AT ZERO TEMPERATURE 

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