D. F -and D -flat directions of MSSM Field configurations satisfying simultaneously: D a ≡ $_{X}$†$_{T}$ $^{a}$X = 0 , F$_{X}$$_{i}$ ≡ ∂W ∂X$_{i}$ = 0 . (120) for N chiral superfields X$_{i}$ , are called respectively D -flat and F -flat. D -flat directions are parameterized by gauge invariant monomials of the chiral superfields. A powerful tool for finding the flat directions has been developed in [324–326, 373, 405–407], for a review see [91, 92] where the correspondence between gauge invariance and flat directions has been employed. A single flat direction necessarily carries a global U (1) quantum number, which corre- sponds to an invariance of the effective Lagrangian for the order parameter φ under phase rotation φ → e $^{iθ}$φ . In the MSSM the global U (1) symmetry is B − L . For example, the LH$_{u}$ - direction (see below) has B − L = − 1. A flat direction can be represented by a composite gauge invariant operator, X$_{m}$ , formed from the product of k chiral superfields Φ$_{i}$ making up the flat direction: X$_{m}$ = Φ$_{1}$Φ$_{2}$ · · · Φ$_{m}$. The scalar component of the superfield X$_{m}$ is related to the order parameter φ through X$_{m}$ = cφ $^{m}$. An example of a D -and F -flat direction is provided by H$_{u}$ = 1 √$_{2}$   0 $_{φ}$  , L = 1 √$_{2}$   φ 0   , (121) where φ is a complex field parameterizing the flat direction, or the order parameter, or the AD field. All the other fields are set to zero. In terms of the composite gauge invariant operators, we would write X$_{m}$ = LH$_{u}$ ( m = 2). From Eq. 312 one clearly obtains F ∗ H$_{u}$ = λ$_{u}$Qu + µH$_{d}$ = F ∗ L = λ$_{d}$H$_{d}$e ≡ 0 for all φ . However there exists a non-zero F-component given by F ∗ H$_{d}$ = µH$_{u}$ . Since µ can not be much larger than the electroweak scale M$_{W}$ ∼ O (1) TeV, this contribution is of the same order as the soft SUSY breaking masses, which are going to lift the degeneracy. Therefore, following [325], one may nevertheless consider LH$_{u}$ to correspond to a F-flat direction. The relevant D -terms read D a SU (2) = $_{H}$† $_{u}$τ$_{3}$ H$_{u}$ + $_{L}$†$_{τ}$$_{3}$$_{L}$ = 1 2 | φ | 2 − 1 2 | φ | 2 ≡ 0 . (122) Therefore the LH$_{u}$ direction is also D -flat. 56