We may alternatively consider a superpotential of the form [12]: ${\mathcal{W}}_{\varphi \overline{\varphi }}=\kappa \widehat{S}({\widehat{\varphi }}^n{\widehat{\overline{\varphi }}}^n/M^{\prime 2n-2}-M^2)$, with $n>1$, where $\varphi ,\overline{\varphi }$ can now transform nontrivially under a continuous symmetry group, e.g. under a global B-L symmetry. The choice $n=1$ would typically not allow positive square masses for both waterfall fields during inflation. The choice $n>1$ also allows “smooth” hybrid inflation driven by the $S$ field [13].

Henceforth we will consider one of the singlet sneutrinos as the inflaton for simplicity and drop family indices.

More precisely $S$ acquires a vev of order the soft masses after inflation and can be used to generate an effective $\mu$-term by adding a coupling $\widehat{S}{\widehat{H}}_{\mathrm{u}}{\widehat{H}}_{\mathrm{d}}$ to the superpotential, where ${\widehat{H}}_{\mathrm{u}}$ and ${\widehat{H}}_{\mathrm{d}}$ contain the Higgs doublets coupling to the up-type quarks and down-type quarks [16].

The dots in Eq. (1) include $Z_4$-violating higher-dimensional operators such as, e.g., $\widehat{S}{\widehat{\varphi }}^5/M^{\prime 3}$ (or even $\widehat{S}{\widehat{\varphi }}^5/m_{\mathrm{P}}^3$) which lift the degeneracy of the true vacuum and effectively blow away potential domain wall networks associated with $Z_4$-breaking after inflation.

Supergravity corrections to the inflaton mass can easily be of the order of the Hubble parameter, inducing $\eta =O(1)$. In our example, we are going to solve this “$\eta$-problem” by explicit tuning. An alternative route could be an effective no-scale supergravity theory leading to zero soft scalar masses at high energy (see e.g. [19]).

We have replaced the superfields in $\mathcal{W}$ and $\mathcal{K}$ by their bosonic components [3].

In general, after the second order phase transition ending inflation both involved fields, the waterfall field $\varphi$ and the sneutrino inflaton $\tilde{N}$, oscillate and can be relevant for leptogenesis and reheating. The sneutrino dominates leptogenesis and reheating if it decays later than $\varphi$. Leptogenesis and reheating via the decay of $\varphi$ have been discussed e.g. in [24].

For $T_{\mathrm{RH}}\approx {10}^6\mathrm{GeV}$, successful leptogenesis requires the upper bound on $\epsilon$ to be saturated. For quasidegenerate light neutrinos via an additional mass term from a triplet seesaw, $T_{\mathrm{RH}}\approx {10}^5\mathrm{GeV}$ is possible since the bound on the decay asymmetry is relaxed [28].