Single Field Inflation in Supergravity with a $U(1)$ Gauge Symmetry

A single field inflation based on a supergravity model with a shift symmetry and $U(1)$ extension of the MSSM is analyzed. We show that one of the real components of the two $U(1)$ charged scalar fields plays the role of inflaton {with} an effective scalar potential similar to the "new chaotic inflation" scenario. Both non-anomalous and anomalous (with Fayet-Iliopoulos term) $U(1)$ are studied. We show that the non-anomalous $U(1)$ scenario is consistent with data of the cosmic microwave background and recent astrophysical measurements. A possible kinetic mixing between $U(1)$ {and} $U(1)_{B-L}$ is considered in order to allow for natural decay channels of the inflaton, leading to a reheating epoch. Upper limits on the reheating temperature thus turn out to favour an intermediate ($\sim {\cal O}(10^{13})$ GeV) scale $B-L$ symmetry breaking.