Asymptotics of collision integral in infra-red and ultra-violet limits

Let us integrate Eq. (18) over $m_1$ and $m_2$.

$$\frac{\partial n_{\bm{p}}}{\partial t} = \frac{1}{k} \int \left(T^0_{1,2} - T^1_{2,0} - T^2_{0,1}\right) dk_1 dk_2 ,$$

$$T^0_{1,2} = k k_1 k_2 \vert V^{\bm{p}}_{\bm{p}_1\bm{p}_2}\vert^2 f^{\bm{p}}_{\bm{p}_1\bm{p}_2} / (\vert g^{0 \prime}_{1,2}\vert S^0_{1,2}) ,$$

$$g^{0 \prime}_{1,2}(k_1, k_2) = \left.\frac{d g^0_{1,2}(m_1)}{d m_1} \right\vert _{m_1 = m_1^{\ast}(k_1, k_2)},$$

$$g^0_{1,2}(m_1) = \frac{k}{m} - \frac{k_1}{\vert m_1\vert} - \frac{k_2}{\vert m - m_1\vert} \, ,$$ (9)

where $g^{0 \prime}_{1,2}$ appears owing to $\delta_{\omega_{\bm{p}} -\omega_{{\bm{p}_1}}-\omega_{{\bm{p}_2}}}$ and $m_1^{\ast}(k_1, k_2)$ is given by the resonant conditions (21)-(23).