Is this correlation zero?

Question

Let $\alpha(\vec{x})$ be a random function and $\vec{\beta}(\vec{x})$ be a continuous random vector such that $\alpha(\vec{x})\in (0, \infty)$, and mean of $\vec{\beta}$ is 0, i.e. $\langle\vec{\beta}\rangle=0$. Both $\alpha$ and $\vec{\beta}$ have Gaussian probability distribution, centered at 0. I am interested in the following correlation $$C(\vec{x}_1, \vec{x}_2)=\langle \vec{\nabla}_{\vec{x}_1}\alpha(\vec{x}_1)\cdot\vec{\beta}(\vec{x}_2)\rangle~,$$ where $\nabla$ os the gradient and $\vec{x}_1$, and $\vec{x}_2$ are two dimensional vector positions. My question is: can I write $$C(\vec{x}_1, \vec{x}_2)=\vec{\nabla}_{\vec{x}_1}\cdot\langle\alpha(\vec{x}_1)\vec{\beta}(\vec{x}_2)\rangle~?$$ If yes, does this mean that $C(\vec{x}_1, \vec{x}_2)=0$, as $\langle\vec{\beta}\rangle=0$, while $\alpha$ is purely positive?