arXiv:1001.2399v2 [hep-th] 21 May 2010 N = 1 Wess Zumino Model in d = 3 at zero and finite temperature Supersymmetric renormalization group (RG) flow equations for the effective superpotential of the three-dimensional Wess-Zumino model are derived at zero and non-zero temperature. This model with fermions and bosons interacting via a Yukawa term possesses a supersymmetric ana- logue of the Wilson-Fisher fixed-point. At zero temperature we determine the phase-transition line in coupling-constant space separating the supersymmetric from the nonsupersymmetric phase. At finite temperature we encounter dimensional reduction from 3 to 2 dimensions in the infrared regime. We determine the finite-temperature phase diagram for the restoration of the global GLYPH<CI> 2 - symmetry and show that for temperatures above the GLYPH<CI> 2 phase-transition the pressure obeys the Stefan-Boltzmann law of a gas of massless bosons in 2 + 1 dimensions. PACS numbers: 05.10.Cc,12.60.Jv,11.30.Qc,11.10.Wx Franziska Synatschke, Jens Braun and Andreas Wipf Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit¨ at Jena, Max-Wien-Platz 1, D-07743 Jena, Germany I. INTRODUCTION In this paper we investigate the three-dimensional N = 1 Wess-Zumino model with general superpotential and explore the model beyond the realm of perturba- tive expansions. This quantum field theory describes Majorana fermions and uncharged bosons in interaction, with their spatial motion restricted to a two-dimensional layer. The selfcoupling of the bosons and the Yukawa coupling between fermions and bosons are such that the theory possesses one supersymmetry. There exists a class of superpotentials for which the three-dimensional mod- els are perturbatively renormalizable, in contrast to the four-dimensional models. For superpotentials of the form W ( φ ) ∼ λφ 2 n +1 there exists both a supersymmetric and a nonsupersymmetric phase. In this paper we shall cal- culate the phase-transition curve separating the super- symmetric from the nonsupersymmetric phase. Besides supersymmetry the action is invariant under φ → − φ , and at zero temperature the breaking of this global Z$_{2}$ symmetry is intimately linked to the breaking of super- symmetry. We shall see that there exists a finite phase- transition temperature at which Z$_{2}$ symmetry is restored, independent of our choice for the couplings at the cutoff- scale. Similarly as for other two-dimensional systems, e.g. surface science, heterostructures or electron gases, the physics in two space-dimensions is rather different from that in three space-dimensions. We employ the functional renormalization group (RG) to calculate the phase structure at zero and finite tem- perature, the scaling behavior of the mass with the RG- scale, the wave-function renormalization, critical expo- nents, the effective potential and the temperature depen- dence of the pressure. The method has previously been applied to a wide range of nonperturbative problems such as critical phenomena, fermionic systems, gauge theories and quantum gravity, see [1–8] for reviews. A number of conceptual studies of supersymmetric theories has al- ready been performed with the functional RG. The del- icate point here is, of course, the construction and use of a manifestly supersymmetry-preserving regulator. For The second part of this paper is devoted to the be- havior of the model at finite temperature. The fate of supersymmetry at finite temperature has been discussed extensively in the literature. For example, in previous works the KMS condition has been implemented directly in thermal superspace [19]. In [20, 21] the supersymme- try breaking has been studied on the level of thermal Green functions. The breaking of supersymmetry by fi- nite temperature corrections, for example the one-loop In the present work we first sketch the derivation of the manifestly supersymmetric flow equations in Sect. III. Since there exist no Majorana fermions in three-dimensional Euclidean spacetime we derive the flow equations in Minkowski spacetime and continue the result to imaginary time. We investigate the flow of the super- potential in Sect. III A, study the fixed-point structure in detail, and identify the supersymmetric analogue of the Wilson-Fisher fixed point of three-dimensional bosonic O ( N ) models with one unstable direction. Taking into account a nonzero anomalous dimension in Sect. III B yields a scaling relation between the critical exponent of the unstable direction and the anomalous dimension. In addition we determine the zero-temperature phase dia- gram for spontaneous breaking of supersymmetry. the four-dimensional Wess-Zumino model such a regu- lator has been presented in [9, 10]. Recently, general theories of a scalar superfield including the Wess-Zumino model have been investigated with a Polchinski-type RG equation in [11], yielding a new approach to supersym- metric nonrenormalization theorems. A Wilsonian effec- tive action for the Wess-Zumino model by perturbatively iterating the functional RG has been constructed in [12]. The present study builds on our earlier results on two- dimensional supersymmetric field theories at zero tem- perature [13, 14] as well as on supersymmetric quantum mechanics, where we have constructed a manifestly su- persymmetric functional RG flow, see [15]. The two- dimensional models possess an infinite series of fixed points described by two-dimensional super-conformal theories. On the contrary, supersymmetric models in three dimensions possess just one fixed point, similarly as three-dimensional O ( N ) models, see e. g. [2, 16–18].