Four-point Kinematics: Momentum Basis

We have four external momentum vectors:\[k_{1} \qquad k_{2} \qquad k_{3} \qquad k_{4}\]In the $s$-channel we take $k_{1}$ and $k_{2}$ to be incoming. The external momenta are constrained by the conservation condition:\[k_{1} + k_{2} = k_{3} + k_{4}\]and the on-shell conditions:\[k_{1}^{2} = -m_{1}^{2} \qquad k_{2}^{2} = -m_{2}^{2} \qquad k_{3}^{2} = -m_{3}^{2} \qquad k_{4}^{2} = -m_{4}^{2}\]The Mandelstam invariants are defined as\[s \equiv - (k_{1} + k_{2})^{2} \qquad t \equiv - (k_{1} - k_{4})^{2} \qquad u \equiv - (k_{1} - k_{3})^{2}\]Note that these three invariants are not independent,\[s + t + u = m_{1}^{2} + m_{2}^{2} + m_{3}^{2} + m_{4}^{2}\]Using momentum conservation, we can also write\[s = -(k_{3} + k_{4})^{2} \qquad t = -(k_{3} - k_{2})^{2} \qquad u = -(k_{4} - k_{2})^{2}\]We now write all other inner products in terms of the massess and the three Mandelstam invariants:\[k_{1} \cdot k_{2} = \frac{1}{2} \left[ m_{1}^{2} + m_{2}^{2} - s \right] \qquad k_{1} \cdot k_{3} = \frac{1}{2} \left[ u - m_{1}^{2} - m_{3}^{2} \right] \qquad k_{1} \cdot k_{4} = \frac{1}{2} \left[ t - m_{1}^{2} - m_{4}^{2} \right]\]\[k_{2} \cdot k_{3} = \frac{1}{2} \left[ t - m_{2}^{2} - m_{3}^{2} \right] \qquad k_{2} \cdot k_{4} = \frac{1}{2} \left[ u - m_{2}^{2} - m_{4}^{2} \right]  \qquad k_{3} \cdot k_{4} = \frac{1}{2} \left[m_{3}^{2} + m_{4}^{2} - s \right]\]