I'm trying to calculate the Altarelli-Parisi splitting function in the collinear limit for a gluon splitting into a quark-antiquark pair, but I keep getting stuck. Let $p$ and $k$ be the momenta for the quark and the antiquark, respectively, and $t$ the momentum of the gluon, I have this Sudakov decomposition: $$\begin{align} p^\mu &= zt^\mu - \frac{k_\perp^2}{2z(p\cdot\eta)} + k_\perp \\ k^\mu &= (1-z)t^\mu - \frac{k_\perp^2}{2(1-z)(p\cdot\eta)} - k_\perp \end{align}$$ where $\eta = (1, 0, 0, -1),\ t=(t_0, 0, 0, t_0),\ k_\perp = (0, k_x, k_y, 0) $.
The amplitude squared for the splitting should be $$ |A|^2 = g_s^2\ T_R\ M_\mu M^*_\nu\ \frac{1}{(2p\cdot k)^2}4\left[p^\mu k^\nu + p^\nu k^\mu - g^{\mu\nu}(p\cdot k)\right] $$ where $g_s$ is the strong coupling constant, $T_R$ is the normalization of the generators of the $SU(3)$ algebra (normally, $T_R = \frac{1}{2}$) and $M^\mu$ is whatever is the process that has created the gluon before the splitting.
From here, I don't know what to do. I've noticed that I can collect the term $p\cdot k$ to get something that resembles a sum over polarizations of the gluon, but after that I don't know what to do. Also, I've tried substituting $p^\mu \approx zt^\mu$ and $k^\mu \approx (1-z)t^\mu$, because I'm working in the collinear limit ($k_\perp \rightarrow 0$), but again, I'm stuck after that. Any help?
Bonus question: how do I transform the phase space $$ \frac{d^3k}{(2\pi)^3 2k_0}\frac{d^3p}{(2\pi)^3 2p_0} $$ so that is uses the new variables $t,\ k_\perp,\ z$? I know how to do it with one particle, but with two I'm a bit confused.
