I am currently working through some basic renormalisation group problems, and have come to one about derivative interactions. It has been a while since I have studied QFT formally so bear with me please. The question is as follows:

Integrating out high energy degrees of freedom can generate derivative interactions in the low energy theory. For example, the expansion of an exponential term of the form $$ e^{-\int d^d x \lambda \phi^2 \hat{\phi}^2}, $$ where $\phi$ is the low-energy degree of freedom and $\hat{\phi}$ is the high-energy degree of freedom. Expanding in $\lambda$ we find $$ e^{-\int d^d x \lambda \phi^2 \hat{\phi}^2}=1-\lambda \int d^d x \phi^2 \hat{\phi}^2+\frac{\lambda^2}{2 !} \int d^d x d^d y \phi^2(x) \hat{\phi}^2(x) \phi^2(y) \hat{\phi}^2(y)+\ldots $$ Show that integrating out $\hat{\phi}$ will generate a term of the form $\phi^2\left(\partial_\mu \phi\right)^2$. You will need to use $\langle\hat{\phi}(k) \hat{\phi}(p)\rangle \sim \frac{\delta^d(k+p)}{k^2}$.

So the only part of the exponential expansion which can contribute such a term is the quadratic. By using the inverse Fourier Transform to decompose the fields in position space to momentum space, we get an integral of the form, for the term $O\left(\lambda^2\right)$, as follows (according to the answer):

$$ I=\int d x d y d k_1 d k_2 d q_1 d q_2 e^{i\left(k_1+k_2+q_1+q_2\right) \cdot x} \\ \Phi\left(k_1\right) \Phi\left(k_2\right) \hat{\phi}\left(q_1\right) \phi\left(q_2\right) \\ e^{i\left(k_3+k_4+q_3+q_4\right) \cdot y} <br/> d k_3 d k_4 d q_3 d q_4 \\ \phi\left(k_3\right) \phi\left(k_4\right) \hat{\phi}\left(q_3\right) \hat{\phi}\left(q_4\right) $$ ignoring coefficients and using $d x$ instead of $d^d x$. $$ \begin{aligned} & \Rightarrow \\ & I=\int d k_1 d k_2 d k_3 d k_4 d q_1 d q_2 d q_3 d q_4 \\ & \delta \left(k_1+k_2+q_1+q_2\right) \delta \left(k_3+k_4+q_3+q_4\right) \\ & \phi\left(K_1\right) \phi\left(K_2\right) \phi\left(K_3\right) \phi\left(K_4\right) \\ & \hat{\phi}(q_1) \hat{\phi}(q_2) \hat{\phi}(q_3) \hat{\phi}(q_4) \\ & \end{aligned} $$

As is my understanding, the task is to integrate over the momentums of the high energy field. My confusion is that the next part of the answer seems to take the expectation value of I, which then seemingly is only applied to the high energy field as such:

$$ < \hat{\phi}(q_1) \hat{\phi}(q_2) \hat{\phi}(q_3) \hat{\phi}(q_4) > $$ and yields:

$$ \frac{\delta^d(q_1 + q_3)}{q_1^2} \\\ \frac{\delta^d(q_2 + q_4)}{q_2^2} $$

So I have 4 questions:

1. Why do we take the expectation value when integrating out?
2. Why is the expectation value of the entire integral only performed over the high-energy field?
3. Why does this only consider the combination sof 1,3 and 2,4 instead of say 1,4 etc?
4. Is there a particular reason why it would be $q_1$ instead of $q_3$ in the denominator or is this just an arbitrary choice?

Any help is appreciate as always.