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Note that thermal effects of the first type require that fields which have Yukawa couplings to the flat direction are in full equilibrium, while those of the second type require the gauge fields (and gauginos) be in full equilibrium. There are two main possibilities: 

• u$_{1}$d$_{2}$d$_{3}$ : SU (2)$_{W}$ remains unbroken in this case. This implies that the correspon- ding gauge fields and gauginos, H$_{u}$ and H$_{d}$ (plus the Higgsinos) and the LH (s)leptons reach full thermal equilibrium. The back-reaction of H$_{u}$ results in a thermal correction λ 2 $_{1}$T $^{2}$| ϕ | $^{2}$, see Eq. (447), with λ$_{1}$ ∼ 10 − $^{5}$. The free energy of the SU (2)$_{W}$ gauge fields result in a thermal correction ∼ + α$_{W}$T $^{4}$ln ( | ϕ | 2 ) . Note that the sign is positive since the flat direction induces a mass which is > T (through the d ) for the LH (s)quarks but not the SU (2)$_{W}$ gauge fields and gauginos. 

• Q$_{1}$Ld : An SU (2) subgroup of SU (3)$_{C}$ is unbroken in this case. Therefore only the corresponding gauge fields and gauginos plus some of the (s)quark fields reach full ther- mal equilibrium. Then the back-reaction of u$_{1}$ and d$_{1}$ results in a thermal correction ( λ 2 1 + λ 2 $_{2}$) T $^{2}$| ϕ | 2 according to Eq.(447), where λ$_{1}$ ∼ λ$_{2}$ ∼ 10 − $^{5}$. Note that decoupling of a number of gluons (and gluinos) from the running of strong gauge coupling results in a negative contribution to the free energy of the unbroken part of SU (3)$_{C}$. 

Therefore, within MSSM and from the point of view of thermal effects, the u$_{1}$d$_{2}$d$_{3}$ flat direction is the most suitable curvaton candidate. The L$_{2}$L$_{3}$e$_{1}$ flat direction can obtain a large VEV . The SU (3)$_{C}$ part of the SM gauge symmetry remains unbroken for this flat direction, and hence gluons, gluinos and (s)quarks will reach full equilibrium. We note that neither of L and e are coupled to the color degrees of freedom. This implies that there will be no T $^{2}$ϕ2 or logarithmic correction to the flat direction potential in this case. This excludes the L$_{2}$L$_{3}$e$_{1}$ flat direction from being a successful curvaton candidate. 

## 4. u$_{1}$d$_{2}$d$_{3}$ as the MSSM curvaton 

From the discussion in the previous section, it follows that thermal effects will lead to a following potential: 

where λ$_{1}$ ∼ 10 − 5 and α$_{W}$ ∼ 10 − $^{2}$. Typically at initial (larger) temperatures the first term dominates and at later (lower) temperatures the second term dominates. Note that the 

<img src='content_image/361975.jpg'>

