In this post I will consider a problem with two independent momentum vectors, each of which describes a distinct massive body. That is, you do not have any conservation constraint relating these two massive momenta. But you still have two massive on-shell constraints:
$$ m_{i}^{2} = -\left\Vert p_{i} \right\Vert^{2}. $$
You can introduce a 2-Mandelstam invariant:
$$ s_{12} \equiv -\left\Vert p_{1} + p_{2} \right\Vert^{2}. $$
You can also introduce the corresponding Regge-Mandelstam invariant:
$$ r_{12} \equiv \frac{s_{12} - m_{1}^{2} - m_{2}^{2}}{2 m_{1} m_{2}}. $$
There is one 2-Gram invariant:
$$ G_{12} \equiv - \det{ \begin{pmatrix} \left\Vert p_{1} \right\Vert^{2} & p_{1} \cdot p_{2} \\ p_{2} \cdot p_{1} & \left\Vert p_{2} \right\Vert^{2} \end{pmatrix} } = \frac{1}{4} \left[ \left( s_{12} - m_{1}^{2} - m_{2}^{2} \right)^{2} - 4 m_{1}^{2} m_{2}^{2} \right] = m_{1}^{2} m_{2}^{2} \left( r_{12}^{2} - 1 \right). $$
It is useful to introduce a Källén function:
$$ \Lambda_{12} \equiv \left( s_{12} - m_{1}^{2} - m_{2}^{2} \right)^{2} - 4 m_{1}^{2} m_{2}^{2}. $$
All of these invariants are present in any two-body problem.
The two-body Sudakov decomposition is given by
$$ p_{1} = n_{1} + \frac{m_{1}^{2}}{S_{12}} n_{2}, $$
$$ p_{2} = n_{2} + \frac{m_{2}^{2}}{S_{12}} n_{1}. $$
Here \(n_{1}\) and \(n_{2}\) are (null) Sudakov vectors, and I have introduced a 2-Sudakov invariant:
$$ S_{12} \equiv -\left\Vert n_{1} + n_{2} \right\Vert^{2} = -2 \left( n_{1} \cdot n_{2} \right).$$
This decomposition can be thought of as a way to explicitly solve the on-shell constraints. The labeling of the Sudakov vectors is such that in a massless limit \(m_{i} \rightarrow 0\) the momentum \(p_{i}\) becomes the Sudakov vector \(n_{i}\). At first glance, this might lead you to believe that \(n_{1}\) is only associated with \(p_{1}\), but that is not the case. In matrix form, the above decomposition is
$$ \begin{pmatrix} p_{1} \\ p_{2} \end{pmatrix} = \mathcal{S}_{2} \cdot \begin{pmatrix} n_{1} \\ n_{2} \end{pmatrix}, \qquad \mathcal{S}_{2} \equiv \begin{pmatrix} 1 & \dfrac{m_{1}^{2}}{S_{12}} \\ \dfrac{m_{2}^{2}}{S_{12}} & 1 \end{pmatrix}. $$
The determinant of \(\mathcal{S}_{2}\) is
$$ \det{ \left( \mathcal{S}_{2} \right) } = 1 - \frac{m_{1}^{2} m_{2}^{2}}{S_{12}^{2}}. $$
The inverse of \(\mathcal{S}_{2}\) is
$$ \operatorname{inv}{\left( \mathcal{S}_{2} \right)} = \frac{1}{\det{ \left( \mathcal{S}_{2} \right) }} \begin{pmatrix} 1 & -\dfrac{m_{1}^{2}}{S_{12}} \\ -\dfrac{m_{2}^{2}}{S_{12}} & 1 \end{pmatrix}. $$
From here it is clear that both \(n_{1}\) and \(n_{2}\) can be written as linear combinations of \(p_{1}\) and \(p_{2}\). Thus, the "individual content" from each quantum is carried by both Sudakov vectors.
My goal is to write \(S_{12}\) in terms of \(s_{12}\) and the masses. It is helpful to introduce a Regge-Sudakov invariant:
$$ R_{12} \equiv \frac{S_{12}}{m_{1} m_{2}}. $$
Using the identity
$$ r_{12} = -\frac{p_{1} \cdot p_{2}}{m_{1} m_{2}}, $$
you find the relation
$$ 2r_{12} = R_{12} + \frac{1}{R_{12}}. $$
This is an univariate Laurent polynomial in \(R_{12}\). Solving this equation for \(R_{12}\) gives
$$ R_{12} = r_{12} + \sqrt{r_{12}^{2} - 1}. $$
Thus,
$$ S_{12} = \frac{1}{2} \left( s_{12} - m_{1}^{2} - m_{2}^{2} + \sqrt{ \Lambda_{12} } \right). $$
Note that in order for \(R_{12}\) to be real, you need
$$ r_{12} \leq -1 \text{ or } r_{12} \geq 1. $$
That is,
$$ s_{12} \leq \left( m_{1} - m_{2} \right)^{2} \text{ or } s_{12} \geq \left( m_{1} + m_{2} \right)^{2}. $$
This is equivalent to \(G_{12} \geq 0\).
The following relations are useful:
$$ S_{12} + m_{1}^{2} = \frac{1}{2} \left( s_{12} + m_{1}^{2} - m_{2}^{2} + \sqrt{ \Lambda_{12} } \right), $$
$$ S_{12} + m_{2}^{2} = \frac{1}{2} \left( s_{12} - m_{1}^{2} + m_{2}^{2} + \sqrt{ \Lambda_{12} } \right);$$
$$ \frac{\left(S_{12} + m_{1}^{2} \right) \left(S_{12} + m_{2}^{2} \right)}{S_{12}} = s_{12} ;$$
$$ S_{12} = \frac{2 m_{1}^{2} m_{2}^{2}}{s_{12} - m_{1}^{2} - m_{2}^{2} - \sqrt{ \Lambda_{12} }};$$
$$ \frac{m_{1}^{2} m_{2}^{2}}{S_{12}} = \frac{1}{2} \left( s_{12} - m_{1}^{2} - m_{2}^{2} - \sqrt{ \Lambda_{12} } \right);$$
$$ \frac{m_{1}^{2} m_{2}^{2}}{S_{12}} + m_{1}^{2} = \frac{1}{2} \left( s_{12} + m_{1}^{2} - m_{2}^{2} - \sqrt{ \Lambda_{12} } \right), $$
$$ \frac{m_{1}^{2} m_{2}^{2}}{S_{12}} + m_{2}^{2} = \frac{1}{2} \left( s_{12} - m_{1}^{2} + m_{2}^{2} - \sqrt{ \Lambda_{12} } \right);$$
$$ S_{12} + \frac{m_{1}^{2} m_{2}^{2}}{S_{12}} = s_{12} - m_{1}^{2} - m_{2}^{2}, $$
$$ S_{12} - \frac{m_{1}^{2} m_{2}^{2}}{S_{12}} = \sqrt{ \Lambda_{12} }; $$
$$ S_{12} + 2 m_{1}^{2} + \frac{m_{1}^{2} m_{2}^{2}}{S_{12}} = s_{12} + m_{1}^{2} - m_{2}^{2}, $$
$$ S_{12} + 2 m_{2}^{2} + \frac{m_{1}^{2} m_{2}^{2}}{S_{12}} = s_{12} - m_{1}^{2} + m_{2}^{2}; $$
$$ \frac{S_{12} \left( S_{12} + m_{1}^{2} \right)}{m_{1}^{2} \left( S_{12} + m_{2}^{2} \right)} = \frac{s_{12} + m_{1}^{2} - m_{2}^{2} + \sqrt{ \Lambda_{12} }}{s_{12} + m_{1}^{2} - m_{2}^{2} - \sqrt{ \Lambda_{12} }}; $$
$$ \frac{S_{12} \left( S_{12} + m_{2}^{2} \right)}{m_{2}^{2} \left( S_{12} + m_{1}^{2} \right)} = \frac{s_{12} - m_{1}^{2} + m_{2}^{2} + \sqrt{ \Lambda_{12} }}{s_{12} - m_{1}^{2} + m_{2}^{2} - \sqrt{ \Lambda_{12} }}; $$
$$ S_{12}^{2} = m_{1}^{2} m_{2}^{2} \left( \frac{s_{12} - m_{1}^{2} - m_{2}^{2} + \sqrt{ \Lambda_{12} }}{s_{12} - m_{1}^{2} - m_{2}^{2} - \sqrt{ \Lambda_{12} }} \right); $$
$$ S_{12}^{2} + m_{1}^{2} m_{2}^{2} = 2m_{1}^{2} m_{2}^{2} \left( \frac{s_{12} - m_{1}^{2} - m_{2}^{2}}{s_{12} - m_{1}^{2} - m_{2}^{2} - \sqrt{ \Lambda_{12} }} \right); $$
$$ S_{12}^{2} - m_{1}^{2} m_{2}^{2} = 2m_{1}^{2} m_{2}^{2} \left( \frac{\sqrt{ \Lambda_{12} }}{s_{12} - m_{1}^{2} - m_{2}^{2} - \sqrt{ \Lambda_{12} }} \right); $$
$$ \det{\left( \mathcal{S}_{2} \right)} = \frac{2 \sqrt{ \Lambda_{12} } }{s_{12} - m_{1}^{2} - m_{2}^{2} + \sqrt{ \Lambda_{12} }}. $$
The logarithm of some of these expressions appear as rapidities in the center-of-momentum frame for either a 2-to-\(N\) or an \(N\)-to-2 scattering problem.
