I am studying this book (Fundamentals of Physics of solids: Vol 3). On page 14 there is a calculation regarding change of fermi surfaces due to correlation of electrons. To illustrate that a discontinuity remains in the momentum distribution function they are calculating momentum distribution. The average number of electrons in correlated ground state $|\psi>$ with momentum $k$ and spin $\sigma$ is given as (equ. 28.2.1): $$<n_{k\sigma}>=<\psi|c_{k\sigma}^\dagger c_{k\sigma}|\psi>$$ where ground state $|\psi>$ can be written as (equ 28.2.2): $$|\psi>=|\psi_{FS}>+\frac{U}{V} \sum_{k_1,k_2,q}\frac{1}{\epsilon_{k_1}+\epsilon_{k_2}-\epsilon_{k_1+q}-\epsilon_{k_2-q}}c_{k_1+q\uparrow}c_{k_2-q\downarrow}c_{k_2\downarrow}c_{k_1\uparrow}|\psi_{FS}>$$ where $|\psi_{FS}>$ is ground state of uncorrelated system and other term is perturbation due to correlation. Correlation can be taken as two particles with wave vectors $k_1$ and $k_2$ are scattered into states with wave vectors $k_1+q $ and $k_2-q$. So far I am understanding what's happening.

After that in next paragraph they are discussing the case $|k|>k_F$. For non-vanishing perturbative momentum distribution $k$ should be either equal to (depending on the spin) $k_1+q$ or $k_2-q$. My first question is why? Going on, there is this sentence that is confusing me:

"The conditions that $k_1$ and $k_2$ are inside the Fermi sphere, while $k_1+k_2-k$ are outside, can be expressed by the fermi distribution function."

My second question is what is meaning of these conditions and why are we taking these conditions?
Then they calculated average number of electrons in state $|\psi>$ that is: $$<n_{k\sigma}>_{|k|>k_F}=\frac{U^2}{V^2} \sum_{k_1,k_2}\frac{[1-f_0(\epsilon_{k_1+k_2-k})]f_0(\epsilon_{k_1})f_0(\epsilon_{k_2})}{(\epsilon_{k_1}+\epsilon_{k_2}-\epsilon_{k}-\epsilon_{k_1+k_2-k})^2}$$ My third question is how they got this equation?

On the next page #15 they is a diagram (fig. 28.2) in which they are trying to give some explanation regarding allowed values of $k_2$ but my dumb brain is not able to understand it. NEED HELP.

I am posting pictures of relevant two pages here: enter image description here enter image description here


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