Vanishing and non-negativity of the first normal Hilbert coefficient

Let $(R,\mathfrak{m})$ be a Noetherian local ring such that $\widehat{R}$ is reduced. We prove that, when $\widehat{R}$ is $S_2$, if there exists a parameter ideal $Q\subseteq R$ such that ${\bar{e}}_1(Q)=0$, then $R$ is regular and $\nu (\mathfrak{m}\mathrm{/}Q)\le 1$. This leads to an affirmative answer to a problem raised by Goto-Hong-Mandal. We also give an alternative proof (in fact a strengthening) of their main result. In particular, we show that if $\widehat{R}$ is equidimensional, then ${\bar{e}}_1(Q)\ge 0$ for all parameter ideals $Q\subseteq R$, and in characteristic $p>0$, we actually have $e_1^{\ast }(Q)\ge 0$. Our proofs rely on the existence of big Cohen-Macaulay algebras.