With this relation the variation of the effective action Eq. (A1) takes the form 12 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. The second functional derivative of W with respect to the source J can be written in terms of the effective action: Next, we define the scale-dependent effective action: Making use of we obtain the Wetterich equation in Minkowski-Space: 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 where we have used that the source is independent of k . For the derivative of W$_{k}$ we obtain 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: Using the definition of the Schwinger functional, Z$_{k}$ [ J ] = e − iW k , we find Now we rewrite the integrand by making use of then Eq. (A2) can be rewritten as follows: 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: ∂$_{t}$W$_{k}$ [ J ] = 1 2 ∫ d $^{d}$q (2 π ) $_{d}$∂$_{t}$ R$_{k}$ ( − δW$_{k}$ δJ ︸ ︷︷ ︸ − φ δW$_{k}$ δJ ︸ ︷︷ ︸ − φ − i δ $^{2}$W$_{k}$ δJδJ ) = − Δ S$_{k}$ − i 2 ∫ d $^{d}$q (2 π ) $_{d}$∂$_{t}$ R$_{k}$ δ $^{2}$W$_{k}$ δJδJ . ∞ ∑ n = −∞ f ( n ) = ∞ ∑ ℓ = −∞ ∫ ∞ −∞ dqf ( q ) exp( − 2 πiℓq ) , δ $^{2}$e − iW k δJδJ = δ δJ e − iW k ( − i ) δW$_{k}$ δJ = e − iW k ( − i ) δW$_{k}$ δJ ( − i ) δW$_{k}$ δJ + e − iW k ( − i ) δ $^{2}$W$_{k}$ δJδJ , ∂$_{t}$W$_{k}$ [ J ] = e iW k 1 2 ∫ d $^{d}$q (2 π ) $_{d}$( ∂$_{t}$R$_{k}$ ) δ $^{2}$e − iW k δJδJ . (A2) r$_{2}$ = ( k | p | − 1 ) θ ( p 2 k 2 − 1 ) , r$_{1}$ = 0 . ∂$_{t}$W$_{k}$ [ J ] = i∂$_{t}$ ln Z [ J ] = i Z [ J ] ∫ D ϕ i 2 ∫ d $^{d}$q (2 π ) $_{d}$ϕ∂$_{t}$ R$_{k}$ϕe i ( S [ ϕ ]+ ∫$_{x}$ ϕJ +Δ S k [ ϕ ]) = − 1 2 Z [ J ] ∫ d $^{d}$q (2 π ) $_{d}$∂$_{t}$ R$_{k}$ ∫ D ϕ ϕ ϕ e i ( S [ ϕ ]+ ∫$_{x}$ ϕJ +Δ S k [ ϕ ]) = 1 2 Z [ J ] ∫ d $^{d}$q (2 π ) $_{d}$( ∂$_{t}$R$_{k}$ ) δ $^{2}$Z [ J ] δJδJ . ∂$_{t}$ Γ$_{k}$[ φ ] = − ∂$_{t}$W$_{k}$ [ J ] − ∂$_{t}$ ∫ d $^{4}$xJφ − ∂$_{t}$ Δ S$_{k}$ [ φ ] , (A1) Γ$_{k}$[ φ ] = − W$_{k}$ [ J ] − ∫ d $^{4}$x Jφ − Δ S$_{k}$ [ φ ] . δ ( q − q $^{′}$) = δφ ( q ) δφ ( q $^{′}$) = − δ δφ δW$_{k}$ δJ = − ∫ d $^{d}$q (2 π ) d δ $^{2}$W$_{k}$ δJδJ δJ δφ , ∂$_{t}$ Γ$_{k}$[ φ ] = i 2 Tr [ ( ∂$_{t}$R$_{k}$ ) ( δ $^{2}$Γ$_{k}$ δφδφ + R$_{k}$ ) − 1 ] . (A4) δ $^{2}$Γ$_{k}$ δφδφ = − δJ δφ − R$_{k}$ ⇒ δJ δφ = − ( δ $^{2}$W$_{k}$ δJδJ ) − 1 = − ( δ $^{2}$Γ$_{k}$ δφδφ + R$_{k}$ ) . (A3) ∂$_{t}$ Γ$_{k}$[ φ ] = i 2 ∫ d $^{d}$q (2 π ) $_{d}$( ∂$_{t}$R$_{k}$ ) δ $^{2}$W$_{k}$ δJδJ . Appendix B: Derivation of the flow equations at finite temperature