content_image/411399.jpg content_image/411420.jpg content_image/411412.jpg content_image/411418.jpg content_image/411419.jpg content_image/411415.jpg content_image/411414.jpg content_image/411413.jpg
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## C. Zero-temperature phase diagram and scaling of the mass 

The breaking of supersymmetry is driven by the un- stable direction a $^{2}$, similarly as in two dimensions [13]. In the plane spanned by the values of the dimensionless couplings λ and λa 2 given at the cutoff-scale Λ we find a transition line for supersymmetry breaking, see Fig. 3 (right panel). 

From the effective potential V = lim$_{k}$$_{→}$$_{0}$ W ′ 2 k / 2, we read off the mass of the boson, 

<img src='content_image/411412.jpg'>

where the field φ$_{0}$ minimizes V . In the case of broken supersymmetry (and unbroken GLYPH<CI> $_{2}$) we have φ$_{0}$ = 0 and 

the mass is given by 

<img src='content_image/411418.jpg'>

whereas for unbroken supersymmetry it is given by 

<img src='content_image/411419.jpg'>

In the broken phase a polynomial expansion of the su- perpotential is justified 7 and we find 

From the scaling of the couplings for k → 0, 3 

we read off that the mass scales as 

This scaling behavior is demonstrated in Fig. 4 for a trun- cation with n = 4, i. e. an expansion of w ′ up to order ϕ 8 ∼ φ $^{8}$, see also Eq. (24). Due to our analysis of the convergence of the fixed-point values of the couplings λ and b$_{2}$$_{i}$ as well as the anomalous dimension η$_{∗}$ , see Tab- ular I, II and III, we expect that the truncation order n = 4 is already sufficient to capture qualitatively and quantitatively most of the features of the zero- and finite- temperature phase diagram, see also Sect. IV B. From a linear fit to the double-logarithmic plot of m ( k ) we find m ( k ) ∼ k 0 . 23 which is indeed reasonably close to the expected scaling given in Eq. (41). The reason for this behavior, which is very different from the one found in 

7 Note that for unbroken supersymmetry a polynomial expansion is bound to fail and one needs to solve the partial differential equation in order to determine the boson mass. 

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