requires a U (1) symmetry. However, this mechanism does not work in the MSSM because some of the squarks and sleptons will get non-zero VEVs which may break color, electro- magnetism, and/or lepton number without breaking SUSY. Therefore the contribution from the Fayet-Iliopoulos (FI) term should be negligible at low scales. There are models of SUSY breaking by F -terms, known as O’Raifeartaigh models [371], where the idea is to pick a set of chiral supermultiplets Φ$_{i}$ ⊃ ( φ$_{i}$, ψ$_{i}$F$_{i}$ ) and a superpotential W in such a way that F$_{i}$ = − δW/δφ ∗ i = 0 have no simultaneous solution. The model requires a linear gauge singlet superfield in the superpotential. Such singlet chiral supermultiplet is not present in the MSSM. The scale of SUSY breaking has to be set by hand. The only mechanism of SUSY breaking where the breaking scale is not introduced either at the level of superpotential or in the gauge sector is through dynamical SUSY breaking [372–375], see also [376]. In these models a small SUSY breaking scale arises by dimensional transmutation. It is customary to treat the SUSY breaking sector as a hidden sector which has no direct couplings to the visible sector represented by the chiral supermultiplets of the MSSM. The only allowed interactions are those which mediate the SUSY breaking in the hidden sector to the visible sector. The main contenders are gravity mediated SUSY breaking, which is associated with new physics which includes gravity at the string scale or at the Planck scale [31, 377, 378], and gauge mediated SUSY breaking, which is transmitted to the visible sector by the ordinary electroweak and QCD gauge interactions [374, 379–381]. There are other variants of SUSY breaking based upon ideas on gravity and gauge mediation with some extensions, such as dynamical SUSY breaking (see [37], and references therein), and anomaly mediation (see [36, 382]). The simplest extension of the MSSM can be obtained by adding a new gauge-singlet chiral supermultiplet with even matter parity. The superpotential reads as [383–386], see also [387]: 3. Next to MSSM (NMSSM) where S is the new chiral supermultiplet. It is often called the next-to-minimal SUSY standard model (NMSSM). The NMSSM introduces extra coefficients, by choosing them W$_{NMSSM}$ = W$_{MSSM}$ + λSH$_{u}$H$_{d}$ + 1 3 κS 3 + 1 2 µ$_{S}$S $^{2}$, (108) 50