Pseudo-Nambu-Goldstone inflation with $Z_N$ symmetric waterfall fields

Abstract

We propose a hybrid inflation model where a pseudo-Nambu-Goldstone boson inflaton couples to $N$ waterfall scalar fields respecting a $Z_N$ symmetry. We identify the phases for the inflation and the consequent waterfall transition, concretely, in $Z_2$, $Z_3$ and $Z_4$ cases. From the Coleman-Weinberg potential for the inflaton, we show that the quadratically divergent corrections coming from the waterfall sector are cancelled due to the $Z_N$ symmetry, while the logarithmically divergent corrections are absent only for $N>2$, ensuring the radiative stability of the inflaton potential. We show the parameter space for a successful inflation with the loop-corrected inflaton potential in each model and compare the results between different discrete symmetries. We further analyze the vacuum structure of the models and the reheating process due to the $Z_N$-invariant Higgs-portal couplings for the waterfall fields. We find that the reheating temperature can be smaller than the mass of the waterfall field condensate such that the $Z_N$ symmetry is not restored after reheating and there is no domain wall problem in the models. We also comment on the possibility of multi-component dark matter from the $Z_N$ partners of the waterfall field condensate.