Dynamical Supersymmetry Breaking in Vector-like Gauge Theories

We provide vector-like gauge theories which break supersymmetry dynamically. ⋆ JSPS Research Fellow. There is a piece of folklore which holds that vector-like gauge theories cannot break supersymmetry dynamically. In this letter, we point out remarkable exceptions to this piece of folklore. Let us consider a supersymmetric SU(2) gauge theory with four doublet chiral superfields Qi. We also introduce six singlet chiral superfields Z ij = −Z. Here i and j denote the flavor indices (i, j = 1, · · · , 4). The tree-level superpotential of our model is given by † Wtree = λ kl ijZ QkQl, (1) where λ ij denote generic coupling constants with λ kl ij = −λ kl ji = −λ lk ij . The pecuriarity of this superpotential resides in that it raises all the D-flat directions in the doublets Qi, which is a necessary condition for supersymmetry to break down. [1] Of course, supersymmetry remains unbroken perturbatively in this model. The exact effective superpotential of the model, which takes into account the full nonperturbative effects, may be written in terms of gauge-invariant low-energy degrees of freedom [2] Vij = −Vji ∼ QiQj (2) as follows: Weff = X(PfVij − Λ ) + λ ijZ Vkl, (3) where X is an additional chiral superfield, PfVij denotes the Pfaffian of the antisymmetric matrix Vij, and Λ is a dynamical scale of the SU(2) gauge interaction. [2,3] † This tree-level superpotential is natural since it possesses two global symmetries. One is an axial U(1) symmetry associated with a Qi phase transformation and the other is an anomaly-free R symmetry. 2 This is none other than a superpotential of the O’Raifeartaigh type. [4] Namely, this effective superpotential yields conditions for supersymmetric vacua PfVij = Λ , λ ijVkl = 0, (4) which cannot be satisfied simultaneously as far as Λ 6= 0. Therefore we conclude that supersymmetry is dynamically broken in our model. We note that this conclusion is not in contradiction with the index argument. [5] The doublets Qi cannot be decoupled by means of mass terms m QiQj since the apparent masses may be absorbed in the shifts of the singlets Z . It is straightforward to generalize the above model to an Sp(N) gauge theory [6] with 2(N+1) chiral superfields in the 2N representation. Here we adopt a notation Sp(1) = SU(2). These vector-like models might serve as a supersymmetry-breaking mechanism in the hidden [7] or visible sector.