8 -22 -20 -18 -16 -14 -12 -10 -14 -13 -12 -11 -10 -9 -8 -7 -6 ln(( a 2 ) T=0 -( a 2 ) T ) ln(T) f(x)=1.49*x-0.68 results from the flow equation O ( N ) models, is that the unstable direction does not feed back into the fixed-point equation $^{8}$. Independent of the value of the coupling at the UV (ultraviolet) cutoff scale Λ, the second derivative of the superpotential flows always into its IR fixed point corresponding to a confor- mally invariant theory. In O ( N ) models, on the other hand, the unstable direction in the broken regime feeds back into the fixed-point equations for the couplings. Ap- proaching the IR fixed point of O ( N ) models requires therefore fine tuning of the UV parameters [16]. IV. FLOW EQUATIONS AT FINITE TEMPERATURE FIG. 5. Double logarithmic plot of the temperature depen- dent minimum of the superpotential versus the temperature. ’superpotential’ describing the Yukawa-type interaction between fermions and bosons originates in the supersym- metry breaking induced by the different thermal bound- ary conditions for the bosonic and fermionic fields. In this section we study finite temperature effects in the three-dimensional Wess-Zumino model. To this end we restrict ourselves to the LPA which we expect to provide already a quantitative insight into the finite- temperature phase structure as shown in Ref. [32]. The finite-temperature flow equation in LPA can be obtained straightforwardly from the zero-temperature equations (20) by replacing the momentum integration in time-like direction by a summation over Matsubara frequencies: with frequencies ω$_{n}$ = 2 πnT for bosonic fields and ν$_{n}$ = (2 n + 1) πT for fermionic fields. We refer to Appendix B for a detailed derivation of the finite-temperature flow equations. Here we simply note that we can perform the Matsubara sums explicitly for the regulator functions (18). The flow equations read In the previous section we have shown that the boson mass tends to zero for k → 0 in the phase with bro- ken supersymmetry (restored GLYPH<CI> 2 symmetry). Thus we expect that the thermodynamic properties of the three- dimensional model in the phase with restored GLYPH<CI> 2 symme- try are similar to that of a gas of massless bosons. The pressure of such a gas in 2 + 1 dimensions is given by A. Pressure Δ p = ζ (3) 2 π T $^{3}$. (46) p$_{0}$ −→ { ν$_{n}$ ω$_{n}$ } , ∫ dp$_{0}$ 2 π . . . −→ T ∞ ∑ n = −∞ . . . (42) The (normalized) pressure Δ p for a given temperature is determined by the negated difference of the interaction potential for a given temperature evaluated at the min- imum and its corresponding zero-temperature value. In our model this corresponds to the height of the barrier in the double-well potential ( W $^{′}$) 2 for a theory with broken Z$_{2}$ symmetry or the minimum of ( W $^{′}$) 2 with unbroken Z$_{2}$ symmetry. For small temperatures we therefore expect where the temperature-dependent floor functions s$_{B}$ and s$_{F}$ are given by The differences in the flow equations for the ’superpo- tential’ describing the self-interaction of bosons and the 8 Note that this is only true for finite N . In the large N limit, the running of the higher-order couplings is also independent of the running of the vacuum expectation value of the field [16] which corresponds to a 2 in our study of the N = 1 Wess-Zumino model. for the temperature dependence of the minimum of W $^{′}$( φ ). In Fig. 5 we show the double-logarithmic plot of the minimum of the superpotential versus temperature as obtained from a truncation of the potential with n = 4, i. e. an expansion of w ′ up to order ϕ 8 ∼ φ $^{8}$, see also Eq. (24). The linear fit to the double-logarithmic plot Δ( λa $^{2}$) ≡ ( λa $^{2}$)$_{T}$$_{=0}$ − ( λa $^{2}$)$_{T}$$_{̸}$$_{=0}$ = √ ζ (3) 2 π · T 3 2 ≃ exp( − 0 . 83) T 3 2 (47) s$_{B}$ = ⌊ k 2 πT ⌋ and s$_{F}$ = ⌊ k 2 πT + 1 2 ⌋ . (45) ∂$_{k}$W ′ k ferm = − k 2 8 π 2 ( k 2 − W ′′ 2 k ) W ′′′ k ( k 2 + W ′′ 2 k ) 2 ( 1 − ( 1 − 2 s$_{F}$πT k ) 2 ) , (44)