content_image/391746.jpg content_image/391745.jpg content_image/391743.jpg content_image/391742.jpg content_image/391740.jpg content_image/391747.jpg content_image/391750.jpg content_image/391748.jpg content_image/391749.jpg content_image/391741.jpg content_image/391751.jpg content_image/391744.jpg
Here, the momentum-dependent regulator function R$_{k}$ provides an IR cutoff for all modes and has to sat- isfy three conditions: (i) R$_{k}$ ( p ) | p $^{2}$/k$^{2}$→ 0 > 0 which im- plements the IR regularization, (ii) R$_{k}$ ( p ) | k $^{2}$/p$^{2}$→ 0 = 0 which ensures that the regulator vanishes for k → 0, (iii) R$_{k}$ ( p ) | k → Λ →∞ → ∞ which ensures that the path integral is dominated by the stationary point for k → Λ → ∞ . Different choices for R$_{k}$ define different RG trajecto- ries manifesting the RG scheme dependence, but the IR physics Γ$_{k}$$_{→}$$_{0}$ → Γ should remain invariant provided the truncation captures all relevant operators for the physi- cal observables under investigation. In turn, a variation of the regulator function may lead to more insight on the truncation dependence of our results. 

Next, we define the scale-dependent effective action: 

In order to properly formulate the flow equations in Minkowski-space we have to take k 2 ∼ p$_{µ}$p µ as flow parameter. Therefore we define ∂$_{t}$ = 2 k $^{2}$∂$_{k}$ 2 to be the derivative with respect to RG ’time’ t = ln k $^{2}$. Taking the derivative of Γ$_{k}$[ φ ] with respect to t yields 

<img src='content_image/391746.jpg'>

where we have used that the source is independent of k . For the derivative of W$_{k}$ we obtain 

<img src='content_image/391745.jpg'>

Using the definition of the Schwinger functional, Z$_{k}$ [ J ] = e − iW k , we find 

<img src='content_image/391743.jpg'>

Now we rewrite the integrand by making use of 

<img src='content_image/391742.jpg'>

then Eq. (A2) can be rewritten as follows: 

<img src='content_image/391740.jpg'>

<img src='content_image/391747.jpg'>

With this relation the variation of the effective action Eq. (A1) takes the form 

The second functional derivative of W with respect to the source J can be written in terms of the effective action: 

<img src='content_image/391750.jpg'>

Making use of 

<img src='content_image/391748.jpg'>

we obtain the Wetterich equation in Minkowski-Space: 

<img src='content_image/391749.jpg'>

## Appendix B: Derivation of the flow equations at finite temperature 

In order to preserve supersymmetry in the RG flow for vanishing temperature, we must choose a regulator function which regularizes the theory in the time-like di- rection and the space-like directions in the same way. In order to make apparent how soft SUSY-breaking due to finite temperature emerges, we use the same regulator for our finite-temperature and zero-temperature studies. It is given by: 

In the LPA, the finite-temperature flow equations can be obtained straightforwardly from the zero-temperature flow equations by replacing p$_{0}$ by the Matsubara modes ν$_{n}$ and ω$_{n}$ of fermion and boson fields, respectively, and replacing the integration over p$_{0}$ by a summation over the Matsubara modes. The contribution of the bosons to the RG flow of our Wess-Zumino model then reads: 

<img src='content_image/391741.jpg'>

<img src='content_image/391751.jpg'>

<img src='content_image/391744.jpg'>

