11. In the Cartesian coordinate system $x O y$, the coordinates of point $F$ are $(1,0)$, and points $A, B$ are on the parabola $y^{2}=4 x$, satisfying $\overrightarrow{O A} \cdot \overrightarrow{O B}=-4,|\overrightarrow{F A}|-|\overrightarrow{F B}|=4 \sqrt{3}$, find the value of $\overrightarrow{F A} \cdot \overrightarrow{F B}$.
Let $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$, according to the problem, we have $x_{1}=\frac{y_{1}^{2}}{4}, x_{2}=\frac{y_{2}^{2}}{4}$. Then $\overrightarrow{O A} \cdot \overrightarrow{O B}=x_{1} x_{2}+y_{1} y_{2}=\frac{y_{1}^{2} y_{2}^{2}}{16}+y_{1} y_{2}=-4$
$$
\begin{array}{l}
\Rightarrow y_{1}^{2} y_{2}^{2}+16 y_{1} y_{2}+64=0 \Rightarrow y_{1} y_{2}=-8 \Rightarrow x_{1} x_{2}=4 . \\
|\overrightarrow{F A}|-|\overrightarrow{F B}|=\left(x_{1}+1\right)-\left(x_{2}+1\right)=x_{1}-x_{2}=4 \sqrt{3} \\
\Rightarrow\left(x_{1}+x_{2}\right)^{2}-4 x_{1} x_{2}=48 \Rightarrow x_{1}+x_{2}=8 .
\end{array}
$$

Therefore, $\overrightarrow{F A} \cdot \overrightarrow{F B}=\left(x_{1}-1\right)\left(x_{2}-1\right)+y_{1} y_{2}$
$$
=x_{1} x_{2}+y_{1} y_{2}-\left(x_{1}+x_{2}\right)+1=-4-8+1=-11 \text {. }
$$