content_image/359495.jpg content_image/359497.jpg content_image/359504.jpg content_image/359505.jpg content_image/359507.jpg
SUSY breaking and non-renormalizable superpotential terms of the type [91, 92] 

are included. Here Φ is a gauge invariant superfield which contains the flat direction. Within MSSM all the flat directions are lifted by non-renormalizable operators with 4 ≤ n ≤ 9 [407], where n depends on the flat direction. Let us focus on the lowest order superpotential term in Eq. (310) which lifts the flat direction. Softly broken SUSY induces a mass term for φ and an A -term so that the scalar potential along the flat direction reads [87, 88] 

<img src='content_image/359495.jpg'>

<img src='content_image/359497.jpg'>

Here φ and θ denote respectively the radial and the angular coordinates of the complex scalar field Φ = φ exp[ iθ ], while θ$_{A}$ is the phase of the A -term (thus A is a positive quantity with dimension of mass). Note that the first and third terms in Eq. (311) are positive definite, while the A -term leads to a negative contribution along the directions whenever cos( nθ + θ$_{A}$ ) < 0. 

## 1. Inflaton candidates 

As discussed in [87–90], among nearly 300 flat directions [407], there are two that can lead to a successful inflation along the lines discussed above. 

One is udd , up to an overall phase factor, which is parameterized by: 

<img src='content_image/359504.jpg'>

Here 1 ≤ α, β, γ ≤ 3 are color indices, and 1 ≤ i, j, k ≤ 3 denote the quark families. The flatness constraints require that α ̸ = β ̸ = γ and j ̸ = k . 

The other direction is LLe , parameterized by (again up to an overall phase factor) 

where 1 ≤ a, b ≤ 2 are the weak isospin indices and 1 ≤ i, j, k ≤ 3 denote the lepton families. The flatness constraints require that a ̸ = b and i ̸ = j ̸ = k . Both these flat directions are lifted by n = 6 non-renormalizable operators [407], 

<img src='content_image/359505.jpg'>

<img src='content_image/359507.jpg'>

