Recently I learned a cool trick. It allows you to write a set of massive momenta in terms of null vectors. By massive momentum I mean an energy-momentum vector that describes a quantum that moves slower than light. For concreteness, consider a 2-to-2 scattering process:
$$ \Phi_{1}(p_{1}) + \Phi_{2}(p_{2}) \longrightarrow \Phi_{3}(p_{3}) + \Phi_{4}(p_{4}). $$
You have the conservation constraint,
$$p_{1} + p_{2} = p_{3} + p_{4}, $$
and the four massive on-shell constraints,
$$ m_{i}^{2} = -\left\Vert p_{i} \right\Vert^{2}. $$
Traditionally, you use the 2-Mandelstam invariants to describe the kinematics of this process:
$$ s = -\left\Vert p_{1} + p_{2} \right\Vert^{2} = -\left\Vert p_{3} + p_{4} \right\Vert^{2}, $$
$$ t = -\left\Vert p_{1} - p_{3} \right\Vert^{2} = -\left\Vert p_{4} - p_{2} \right\Vert^{2}, $$
$$ u = -\left\Vert p_{1} - p_{4} \right\Vert^{2} = -\left\Vert p_{3} - p_{2} \right\Vert^{2}. $$
Due to the conservation constraint, the three 2-Mandelstam invariants satisfy a linear constraint:
$$ s + t + u = m_{1}^{2} + m_{2}^{2} + m_{3}^{2} + m_{4}^{2}. $$
I will refer to the process of writing all physically-relevant quantities (e.g. energy, spatial momentum, speed, rapidity, scattering angles) in terms of the masses and the 2-Mandelstam invariants as the Mandelstam kinematic analysis.
What I call the Sudakov kinematic analysis consist of decomposing the incoming/outgoing massive momenta into incoming/outgoing null vectors. That is, let \(n_{1}\), \(n_{2}\), \(n_{3}\), and \(n_{4}\) be null vectors and consider the following Sudakov decomposition:
$$ p_{1} = n_{1} - \frac{m_{1}^{2}}{2 \left( n_{1} \cdot n_{2} \right)} n_{2}, $$
$$ p_{2} = n_{2} - \frac{m_{2}^{2}}{2 \left( n_{1} \cdot n_{2} \right)} n_{1}, $$
$$ p_{3} = n_{3} - \frac{m_{3}^{2}}{2 \left( n_{3} \cdot n_{4} \right)} n_{4}, $$
$$ p_{4} = n_{4} - \frac{m_{4}^{2}}{2 \left( n_{3} \cdot n_{4} \right)} n_{3}. $$
This is valid because, in general, a linear combination of two null vectors is not null. The null vectors \(n_{i}\) are the Sudakov vectors. Note that
$$ p_{1} + p_{2} = \left[ 1 - \frac{m_{2}^{2}}{2 \left( n_{1} \cdot n_{2} \right)} \right] n_{1} + \left[ 1 - \frac{m_{1}^{2}}{2 \left( n_{1} \cdot n_{2} \right)} \right] n_{2}, $$
$$ p_{3} + p_{4} = \left[ 1 - \frac{m_{4}^{2}}{2 \left( n_{3} \cdot n_{4} \right)} \right] n_{3} + \left[ 1 - \frac{m_{3}^{2}}{2 \left( n_{3} \cdot n_{4} \right)} \right] n_{4}. $$
You can introduce six 2-Sudakov invariants:
$$ S_{12} \equiv -\left\Vert n_{1} + n_{2} \right\Vert^{2} = -2 \left( n_{1} \cdot n_{2} \right), $$
$$ T_{13} \equiv -\left\Vert n_{1} - n_{3} \right\Vert^{2} = 2 \left( n_{1} \cdot n_{3} \right), $$
$$ U_{14} \equiv -\left\Vert n_{1} - n_{4} \right\Vert^{2} = 2 \left( n_{1} \cdot n_{4} \right), $$
$$ U_{23} \equiv -\left\Vert n_{3} - n_{2} \right\Vert^{2} = 2 \left( n_{2} \cdot n_{3} \right), $$
$$ T_{24} \equiv -\left\Vert n_{4} - n_{2} \right\Vert^{2} = 2 \left( n_{2} \cdot n_{4} \right), $$
$$ S_{34} \equiv -\left\Vert n_{3} + n_{4} \right\Vert^{2} = -2 \left( n_{3} \cdot n_{4} \right). $$
You can write the 2-Sudakov invariants in terms of the masses and the 2-Mandelstam invariants. Introduce six Regge-Mandelstam invariants
$$ r_{12} \equiv \frac{s - m_{1}^{2} - m_{2}^{2}}{2 m_{1} m_{2}}, $$
$$ r_{13} \equiv \frac{m_{1}^{2} + m_{3}^{2} - t}{2 m_{1} m_{3}}, $$
$$ r_{14} \equiv \frac{m_{1}^{2} + m_{4}^{2} - u}{2 m_{1} m_{4}}, $$
$$ r_{23} \equiv \frac{m_{2}^{2} + m_{3}^{2} - u}{2 m_{2} m_{3}}, $$
$$ r_{24} \equiv \frac{m_{2}^{2} + m_{4}^{2} - t}{2 m_{2} m_{4}}, $$
$$ r_{34} \equiv \frac{s - m_{3}^{2} - m_{4}^{2}}{2 m_{3} m_{4}}; $$
and six Regge-Sudakov invariants
$$ R_{12} \equiv \frac{S_{12}}{m_{1} m_{2}}, $$
$$ R_{13} \equiv -\frac{T_{13}}{m_{1} m_{3}}, $$
$$ R_{14} \equiv -\frac{U_{14}}{m_{1} m_{4}}, $$
$$ R_{23} \equiv -\frac{U_{23}}{m_{2} m_{3}}, $$
$$ R_{24} \equiv -\frac{T_{24}}{m_{2} m_{4}}, $$
$$ R_{34} \equiv \frac{S_{34}}{m_{3} m_{4}}. $$
Using the six relations
$$ r_{ij} = - \frac{p_{i} \cdot p_{j}}{m_{i} m_{j}}, $$
along with the Sudakov decomposition, you find the system of six equations to solve:
$$ 2 r_{12} = R_{12} + \frac{1}{R_{12}}, $$
$$ 2 r_{13} = R_{13} + \frac{R_{14}}{R_{34}} + \frac{R_{23}}{R_{12}} + \frac{R_{24}}{R_{12} R_{34}}, $$
$$ 2 r_{14} = R_{14} + \frac{R_{13}}{R_{34}} + \frac{R_{24}}{R_{12}} + \frac{R_{23}}{R_{12} R_{34}}, $$
$$ 2 r_{23} = R_{23} + \frac{R_{24}}{R_{34}} + \frac{R_{13}}{R_{12}} + \frac{R_{14}}{R_{12} R_{34}}, $$
$$ 2 r_{24} = R_{24} + \frac{R_{23}}{R_{34}} + \frac{R_{14}}{R_{12}} + \frac{R_{13}}{R_{12} R_{34}}, $$
$$ 2 r_{34} = R_{34} + \frac{1}{R_{34}}. $$
The solution is
$$ R_{12} = r_{12} + \sqrt{r_{12}^{2} - 1}, $$
$$ R_{13} = \frac{2 R_{12}^{2} R_{34}^{2}}{\left( R_{12}^{2} - 1 \right) \left( R_{34}^{2} - 1 \right)} \left[ r_{13} - \frac{r_{14}}{R_{34}} - \frac{r_{23}}{R_{12}} + \frac{r_{24}}{R_{12} R_{34}} \right], $$
$$ R_{14} = \frac{2 R_{12}^{2} R_{34}^{2}}{\left( R_{12}^{2} - 1 \right) \left( R_{34}^{2} - 1 \right)} \left[ r_{14} - \frac{r_{13}}{R_{34}} - \frac{r_{24}}{R_{12}} + \frac{r_{23}}{R_{12} R_{34}} \right], $$
$$ R_{23} = \frac{2 R_{12}^{2} R_{34}^{2}}{\left( R_{12}^{2} - 1 \right) \left( R_{34}^{2} - 1 \right)} \left[ r_{23} - \frac{r_{24}}{R_{34}} - \frac{r_{13}}{R_{12}} + \frac{r_{14}}{R_{12} R_{34}} \right], $$
$$ R_{24} = \frac{2 R_{12}^{2} R_{34}^{2}}{\left( R_{12}^{2} - 1 \right) \left( R_{34}^{2} - 1 \right)} \left[ r_{24} - \frac{r_{23}}{R_{34}} - \frac{r_{14}}{R_{12}} + \frac{r_{13}}{R_{12} R_{34}} \right], $$
$$ R_{34} = r_{34} + \sqrt{r_{34}^{2} - 1}. $$
In terms of the masses and the 2-Mandelstam invariants, you have
$$ S_{12} = \frac{1}{2} \left( s - m_{1}^{2} - m_{2}^{2} + \sqrt{ \left( s - m_{1}^{2} - m_{2}^{2} \right)^{2} - 4 m_{1}^{2} m_{2}^{2} } \right), $$
$$ S_{34} = \frac{1}{2} \left( s - m_{3}^{2} - m_{4}^{2} + \sqrt{ \left( s - m_{3}^{2} - m_{4}^{2} \right)^{2} - 4 m_{3}^{2} m_{4}^{2} } \right). $$
Note that in four spacetime dimensions all Sudakov invariants can be written in terms of twistor products.
