It is mentioned in first page of this paper by Seiberg and Komargodski that the Lagrangian in superspace of a $U(1)$ gauge SUSY theory with FI terms is not gauge invariant. However, the FI terms in superspace is $$ \xi \int d^4 \theta V $$ where $V$ is a vector superfield. Now if we do a gauge transformation on $V$, i.e., $$ V\longrightarrow V+ i (\Phi-\bar{\Phi}) $$ the FI term remains invariant since $$ \int d^4 \theta \Phi=\int d^4 \theta \bar{\Phi}=0. $$

So what is the source of the gauge non-invariance in superspace ?

It is mentioned in first page of this paper by Seiberg and Komargodski that the Lagrangian in superspace of a $U(1)$ gauge SUSY theory with FI terms is not gauge invariant. However, the FI terms in superspace is $$ \xi \int d^4 \theta V $$ where $V$ is a vector superfield. Now if we do a gauge transformation on $V$, i.e., $$ V\longrightarrow V+ i (\Phi-\bar{\Phi}) $$ the FI term remains invariant since $$ \int d^4 \theta \Phi=\int d^4 \theta \bar{\Phi}=0. $$

So what is the source of the gauge non-invariance in superspace ?