content_image/334288.jpg content_image/334289.jpg content_image/334290.jpg
where the dimensionality of the effective scalar operator d = mk , and h is a coupling constant which could be complex with | h | ∼ O (1). Here M is some large mass, typically of the order of the Planck mass or the string scale (in the heterotic case M ∼ M$_{GUT}$ ). The lowest value of k is 1 or 2, depending on whether the flat direction is even or odd under R -parity. 

A second type of term lifting the flat direction would be of the form [325, 326] 

<img src='content_image/334288.jpg'>

where ψ is not contained in X$_{m}$ . The superpotential term Eq. (126) spoils F-flatness through F$_{ψ}$ ̸ = 0. An example is provided by the direction u$_{1}$u$_{2}$u$_{3}$e$_{1}$e$_{2}$ , which is lifted by the non- renormalizable term W = ( h ′ /M ) u$_{1}$u$_{2}$d$_{2}$e$_{1}$ . This superpotential term gives a non-zero con- tribution F ∗ d 2 = ( h ′ /M ) u$_{1}$u$_{2}$e$_{1}$ ∼ ( h ′ /M ) φ 3 along the flat direction. 

Assuming minimal kinetic terms, both types discussed above in Eqs. (125,126) yield a generic non-renormalizable potential contribution that can be written as 

<img src='content_image/334289.jpg'>

where we have defined the coupling | λ | 2 ≡ | h | $^{2}$+ | h $^{′}$| $^{2}$. By virtue of an accidental R -symmetry under which φ has a charge R = 2 /d , the potential Eq. (127) conserves the U (1) symmetry carried by the flat direction, in spite of the fact that at the superpotential level it is violated, see Eqs. (125,126). 

All the non-renormalizable operators can be generated from SM gauge monomials with R - parity constraint which allows only even number of odd matter parity fields ( Q, L, u, d, e ) to be present in each superpotential term. At each dimension d , the various F = 0 constraints are separately imposed in order to construct the basis for monomials. 

As an example, consider flat directions involving the Higgs fields such as H$_{u}$H$_{d}$ and LH$_{u}$ directions. Even though they are already lifted by the µ term, since µ is of the order of SUSY breaking scale, for cosmological purposes they can be considered flat. At the d = 4 level the superpotential reads [325, 326, 407] 

<img src='content_image/334290.jpg'>

Let us assume λ, λ$_{ij}$ ̸ = 0. Note that F$_{H}$$_{d}$ = 0 constraint implies λH α $_{u}$( H$_{u}$H$_{d}$ ) = 0, which acts as a basis for the monomials. An additional constraint can be obtained by contracting F$_{H}$$_{d}$ = 0 by ϵ$_{αβ}$H β $_{d}$, which forms the polynomial H$_{u}$H$_{d}$ = 0 in the same monomial basis. 

