Unit vectors

$\overrightarrow{a},\ \overrightarrow{b},\ \overrightarrow{c},\ \overrightarrow{d}$ are unit vectors such that $\overrightarrow{a} \perp \overrightarrow{b}$ , $|\overrightarrow{c}-\overrightarrow{d}|=\sqrt{3}$ . Vector $\overrightarrow{p}=2\sqrt{2}(\cos^2 \theta \cdot \overrightarrow{a}+\sin^2 \theta \cdot \overrightarrow{b})$ where $\theta \in \mathbb R$ .

What is the minimum value of $(\overrightarrow{c}-\overrightarrow{p})\cdot(\overrightarrow{d}-\overrightarrow{p})$ ?

Note: $|\overrightarrow{n}|$ notes the length of the vector on the Euclidean plane. i.e. The Euclidean norm.