= Solution
Take $p_z>0$ and $E\gg m$. Then $\sqrt{E+p_z}\simeq\sqrt{2E}$ while $\sqrt{E-p_z}=m/\sqrt{E+p_z}\simeq m/\sqrt{2E}$. Hence
$$
\boxed{u_+(p)\longrightarrow\sqrt{2E}\begin{pmatrix}1\\0\\0\\0\end{pmatrix},\qquad
u_-(p)\longrightarrow\sqrt{2E}\begin{pmatrix}0\\0\\0\\1\end{pmatrix}.}
$$
The <helicity> operator is $\widehat h=\tfrac12\operatorname{diag}(\sigma_3,\sigma_3)$ for <momentum> along $+z$. It gives
$$
\boxed{h[u_+]=+\tfrac12,\qquad h[u_-]=-\tfrac12.}
$$
The positive-helicity state becomes entirely right chiral, while the negative-helicity state becomes entirely left chiral. The suppressed opposite-chirality amplitude is of relative order $m/(2E)$; at finite <mass> <chirality> is not the same as <helicity>. If $p_z<0$, <helicity> is measured along the reversed <momentum> and the labels of the two spin-axis states interchange.
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