4. The line $l$ intersects the curve $C: x^{2}-y^{2}=1(x>0)$ at points $A$ and $B$. Then the minimum value of $f=\overrightarrow{O A} \cdot \overrightarrow{O B}$ is $\qquad$
4. 1 .

Let $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$. Then
$$
\begin{aligned}
f & =x_{1} x_{2}+y_{1} y_{2} \\
& =\frac{1}{2}\left[\left(x_{1}+y_{1}\right)\left(x_{2}+y_{2}\right)+\left(x_{1}-y_{1}\right)\left(x_{2}-y_{2}\right)\right] \\
& \geqslant 1 .
\end{aligned}
$$

When points $A$ and $B$ are symmetric with respect to the $x$-axis, the equality holds. Therefore, $f_{\min }=1$.