content_image/374237.jpg content_image/374238.jpg content_image/374239.jpg
where Ψ = H$_{u}$ if R$_{Φ}$ = +1 and Ψ = L if R$_{Φ}$ = − 1. Taking into account of the inflaton superpotential mass term: ( m$_{φ}$/ 2) ΦΦ, and defining X$_{1}$$_{,}$$_{2}$ = ( H$_{u}$ ± Ψ) $_{/}$√$_{2, the}$ renormalizable part of the potential , which is relevant for the inflaton decay into MSSM scalars is given by: 

<img src='content_image/374237.jpg'>

where χ denotes the scalar component of X$_{1}$$_{,}$$_{2}$ superfields, and we have only considered the real parts of the inflaton, φ , and χ field. Further note that the cubic interaction term appears with different signs for χ$_{1}$ and χ$_{2}$ , but this is irrelevant during inflaton oscillations. 

In addition to the terms in Eq. (417), there are also the self- and-cross-couplings, 

arising from the superpotential and D -terms respectively ( α is a gauge fine structure con- stant). Therefore even in the simplest SUSY set up the scalar potential is more involved than the non-SUSY case given in Eq. (378), which can alter the picture of preheating presented in the literature, see for the detailed discussion in Refs. [122, 123] $^{93}$. 

## D. MSSM flat directions, reheating and thermalization 

<img src='content_image/374238.jpg'>

The MSSM flat directions have important role to play in SUSY reheating and thermal- ization [122, 123]. Consider a MSSM flat direction, ϕ , with the corresponding superfield denoted by ϕ (only for flat directions we are denoting the superfield and the field with the same notation in this section). Note that since ϕ and X superfields are linear combinations of the MSSM superfields (defined in the earlier subsection after Eq. (416)), and hence are coupled through the MSSM superpotential in Eq. (103). 

<img src='content_image/374239.jpg'>

where Σ$_{1}$$_{,}$$_{2}$ are some MSSM superfields such that Σ$_{1}$ ̸ = Ψ and Σ$_{2}$ ̸ = H$_{u}$ , since ϕ is a non- gauge-singlet. 

