Einstein-Hermitian connection on twisted Higgs bundles

Let $X$ be a smooth projective variety over $\mathbb{C}$. We prove that a twisted Higgs vector bundle $(\text{calE},\theta )$ on $X$ admits an Einstein–Hermitian connection if and only if $(\text{calE},\theta )$ is polystable. A similar result for twisted vector bundles (no Higgs fields) was proved by S. Wang in \cite{Wa}. Our approach is simpler.