content_image/328765.jpg content_image/328766.jpg content_image/328767.jpg
For example consider the case where ϕ is a flat direction classified by the udd monomial, and Ψ = H$_{d}$ . In this case Σ$_{1}$$_{,}$$_{2}$ are Q superfields and λ$_{1}$$_{,}$$_{2}$ correspond to λ$_{u}$ and λ$_{d}$ respectively. Then with the help of X$_{1}$$_{,}$$_{2}$ = ( H$_{u}$ ± Ψ) $_{/}$√$_{2, part of MSSM superpotential can be written}$ as: W ⊃ λ 1 √$_{2}$ Xϕ Σ$_{1}$ + λ 2 √$_{2}$ Xϕ Σ$_{2}$. This results in [123]: 

<img src='content_image/328765.jpg'>

where we have again considered the real part of χ . Note that the first generation of (s)leptons and (s)quarks have a Yukawa coupling ∼ O (10 − 6 − 10 − $^{5}$), while the rest of the SM Yukawa couplings are: λ ≥ 3 × 10 − $^{4}$. 

The most important point is to note that 60 − 70 e-foldings of inflation is sufficient for the MSSM flat directions to take large VEVs during and after inflation by virtue of stochastic jumps during inflation, see the discussion in Sec. V E 1 and the Refs. [91, 92]. This however requires that the MSSM flat directions do not obtain positive Hubble induced corrections during inflation. 

## 1. kinematical blocking of preheating 

In order to understand the preheating dynamics it is important to take into account of χ coupling to the inflaton φ , as well as to the MSSM flat direction, ϕ , which is displaced away from its minimum (towards large VEVs) during inflation. The governing potential can be obtained from Eqs. (417,420): 

<img src='content_image/328766.jpg'>

As mentioned in the previous section, we generically have λ ≥ 3 × 10 − $^{4}$, and g can be as large as ∼ O (1). After mode decomposition of the field χ , the energy of the mode with momentum k , denoted by χ$_{k}$ , is given by: 

<img src='content_image/328767.jpg'>

Let us freeze the expansion of the universe first. Including the expansion will not change our conclusions anyway. Let us even consider the most opportunistic case for preheating with a large inflaton VEV, i.e. 〈 φ 〉 > M$_{P}$ . Therefore if g > 10 − $^{6}$, the inflaton induces a large mass g 〈 φ 〉 > H$_{inf}$ for χ during inflation. As a result, χ , quickly settles down to the minimum, i.e. 

