content_image/411282.jpg content_image/411290.jpg content_image/411289.jpg content_image/411271.jpg content_image/411288.jpg content_image/411281.jpg
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]. 

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: 

<img src='content_image/411282.jpg'>

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 

<img src='content_image/411290.jpg'>

where the temperature-dependent floor functions s$_{B}$ and s$_{F}$ are given by 

<img src='content_image/411289.jpg'>

The differences in the flow equations for the ’superpo- tential’ describing the self-interaction of bosons and the 

## IV. FLOW EQUATIONS AT FINITE TEMPERATURE 

<img src='content_image/411271.jpg'>

’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 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 

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 

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 

<img src='content_image/411288.jpg'>

## A. Pressure 

<img src='content_image/411281.jpg'>

