The main purpose of a Sudakov decomposition is to identify particular null vectors (the Sudakov vectors) and decompose any other given vectors into components along the Sudakov subspace and components orthogonal to the Sudakov subspace. An important example of this is a two-body decomposition in a four-point process of the form
$$ \Phi_{1}(p_{1}) + \Phi_{2}(p_{2}) \longrightarrow \Phi_{3}(p_{3}) + \Phi_{4}(p_{4}). $$
Each massive momentum vector is subject to an on-shell constraint:
$$ m_{i}^{2} = -\left\Vert p_{i} \right\Vert^{2}; $$
and the four momenta satisfy a conservation constraint:
$$ p_{1} + p_{2} = p_{3} + p_{4}. $$
You have three 2-Mandelstam invariants:
$$ s \equiv -\left\Vert p_{1} + p_{2} \right\Vert^{2} = -\left\Vert p_{3} + p_{4} \right\Vert^{2}, $$
$$ t \equiv -\left\Vert p_{1} - p_{3} \right\Vert^{2} = -\left\Vert p_{4} - p_{2} \right\Vert^{2}, $$
$$ u \equiv -\left\Vert p_{1} - p_{4} \right\Vert^{2} = -\left\Vert p_{3} - p_{2} \right\Vert^{2}. $$
Due to the conservation constraint, these invariants satisfy the relation
$$ s + t + u = m_{1}^{2} + m_{2}^{2} + m_{3}^{2} + m_{4}^{2}. $$
The outgoing two-body Sudakov decomposition is
$$ p_{3} = n_{3} + \frac{m_{3}^{2}}{S_{34}} n_{4}, $$
$$ p_{4} = n_{4} + \frac{m_{4}^{2}}{S_{34}} n_{3}. $$
Here \(n_{3}\) and \(n_{4}\) are (null) Sudakov vectors, and \(S_{34}\) is a 2-Sudakov invariant:
$$ S_{34} \equiv -\left\Vert n_{3} + n_{4} \right\Vert^{2} = -2 \left( n_{3} \cdot n_{4}\right). $$
As usual, from the identity
$$ -\frac{p_{3} \cdot p_{4}}{m_{3} m_{4}} = \frac{s - m_{3}^{2} - m_{4}^{2}}{2 m_{3} m_{4}}, $$
it follows that
$$ S_{34} = \frac{1}{2} \left( s - m_{3}^{2} - m_{4}^{2} + \sqrt{ \Lambda_{34}} \right), $$
where \(\Lambda_{34}\) is a Källén function:
$$ \Lambda_{34} \equiv \left( s - m_{3}^{2} - m_{4}^{2} \right)^{2} - 4 m_{3}^{2} m_{4}^{2}. $$
Thus, from the point of view of orthogonal decompositions, \(p_{3}\) and \(p_{4}\) live entirely in the Sudakov subspace spanned by \(n_{3}\) and \(n_{4}\). However, the same cannot be said about the incoming momenta. In general, you have
$$ p_{1} = P_{1} + \nu_{13} n_{3} + \nu_{14} n_{4}, $$
$$ p_{2} = P_{2} + \nu_{23} n_{3} + \nu_{24} n_{4}; $$
with
$$ P_{1} \cdot n_{3} = P_{1} \cdot n_{4} = 0, $$
$$ P_{2} \cdot n_{3} = P_{2} \cdot n_{4} = 0. $$
That is, \(P_{1}\) and \(P_{2}\) are orthogonal to the Sudakov subspace. My goal is to find the values of \(\left\Vert P_{1} \right\Vert^{2}\), \(\left\Vert P_{2} \right\Vert^{2}\), and the four \(\nu_{ij}\) coefficients (which I will call Sudakov moduli) in terms of the masses and the 2-Mandelstam invariants.
Using the identities
$$ -\frac{p_{1} \cdot p_{3}}{m_{1} m_{3}} = \frac{m_{1}^{2} + m_{3}^{2} - t}{2 m_{1} m_{3}}, $$
$$ -\frac{p_{1} \cdot p_{4}}{m_{1} m_{4}} = \frac{m_{1}^{2} + m_{4}^{2} - u}{2 m_{1} m_{4}}, $$
$$ -\frac{p_{2} \cdot p_{3}}{m_{2} m_{3}} = \frac{m_{2}^{2} + m_{3}^{2} - u}{2 m_{2} m_{3}}, $$
$$ -\frac{p_{2} \cdot p_{4}}{m_{2} m_{4}} = \frac{m_{2}^{2} + m_{4}^{2} - t}{2 m_{2} m_{4}}; $$
you find the relations
$$ m_{1}^{2} + m_{3}^{2} - t = \nu_{13} m_{3}^{2} + \nu_{14} S_{34}, $$
$$ m_{1}^{2} + m_{4}^{2} - u = \nu_{13} S_{34} + \nu_{14} m_{4}^{2}, $$
$$ m_{2}^{2} + m_{3}^{2} - u = \nu_{23} m_{3}^{2} + \nu_{24} S_{34}, $$
$$ m_{2}^{2} + m_{4}^{2} - t = \nu_{23} S_{34} + \nu_{24} m_{4}^{2}. $$
Solving these leads to
$$ \nu_{13} = \frac{S_{34} \left( m_{1}^{2} + m_{4}^{2} - u \right) - m_{4}^{2} \left( m_{1}^{2} + m_{3}^{2} - t \right)}{\left( S_{34} - m_{3} m_{4} \right) \left( S_{34} + m_{3} m_{4} \right)}, $$
$$ \nu_{14} = \frac{S_{34} \left( m_{1}^{2} + m_{3}^{2} - t \right) - m_{3}^{2} \left( m_{1}^{2} + m_{4}^{2} - u \right)}{\left( S_{34} - m_{3} m_{4} \right) \left( S_{34} + m_{3} m_{4} \right)}, $$
$$ \nu_{23} = \frac{S_{34} \left( m_{2}^{2} + m_{4}^{2} - t \right) - m_{4}^{2} \left( m_{2}^{2} + m_{3}^{2} - u \right)}{\left( S_{34} - m_{3} m_{4} \right) \left( S_{34} + m_{3} m_{4} \right)}, $$
$$ \nu_{24} = \frac{S_{34} \left( m_{2}^{2} + m_{3}^{2} - u \right) - m_{3}^{2} \left( m_{2}^{2} + m_{4}^{2} - t \right)}{\left( S_{34} - m_{3} m_{4} \right) \left( S_{34} + m_{3} m_{4} \right)}. $$
Note that
$$ \nu_{13} + \nu_{23} = \frac{S_{34} \left(s - m_{3}^{2} + m_{4}^{2} \right) - m_{4}^{2} \left( s + m_{3}^{2} - m_{4}^{2} \right) }{\left( S_{34} - m_{3} m_{4} \right) \left( S_{34} + m_{3} m_{4} \right)}, $$
$$ \nu_{14} + \nu_{24} = \frac{S_{34} \left(s + m_{3}^{2} - m_{4}^{2} \right) - m_{3}^{2} \left( s - m_{3}^{2} + m_{4}^{2} \right) }{\left( S_{34} - m_{3} m_{4} \right) \left( S_{34} + m_{3} m_{4} \right)}. $$
That is, these linear combinations of Sudakov moduli only depend on \(s\) and the masses. Recall that
$$ s - m_{3}^{2} + m_{4}^{2} = S_{34} + 2m_{4}^{2} + \frac{m_{3}^{2} m_{4}^{2}}{S_{34}}, $$
$$ s + m_{3}^{2} - m_{4}^{2} = S_{34} + 2m_{3}^{2} + \frac{m_{3}^{2} m_{4}^{2}}{S_{34}}. $$
Thus, you find that
$$ \nu_{13} + \nu_{23} = 1 + \frac{m_{4}^{2}}{S_{34}}, $$
$$ \nu_{14} + \nu_{24} = 1 + \frac{m_{3}^{2}}{S_{34}}. $$
These relations follow directly from the conservation constraint, which also yields the relation \(P_{1} + P_{2} = 0\).
With the incoming momenta, you have
$$ m_{1}^{2} = -\left\Vert p_{1} \right\Vert^{2} \quad \Longrightarrow \quad \left\Vert P_{1} \right\Vert^{2} = \nu_{13} \nu_{14} S_{34} - m_{1}^{2}, $$
$$ m_{2}^{2} = -\left\Vert p_{2} \right\Vert^{2} \quad \Longrightarrow \quad \left\Vert P_{1} \right\Vert^{2} = \nu_{23} \nu_{24} S_{34} - m_{2}^{2}, $$
$$ -2 \left( p_{1} \cdot p_{2} \right) = s - m_{1}^{2} - m_{2}^{2} \quad \Longrightarrow \quad -2 \left\Vert P_{1} \right\Vert^{2} = \left( \nu_{13} \nu_{24} + \nu_{14} \nu_{23} \right) S_{34} - s + m_{1}^{2} + m_{2}^{2}. $$
Combining these, you find
$$ s = \left( \nu_{13} + \nu_{23} \right) \left( \nu_{14} + \nu_{24} \right) S_{34}. $$
You also have the vectors
$$ p_{1} - p_{3} = P_{1} + \left( \nu_{13} - 1 \right) n_{3} + \left( \nu_{14} - \frac{m_{3}^{2}}{S_{34}} \right) n_{4}, $$
$$ p_{1} - p_{4} = P_{1} + \left( \nu_{13} - \frac{m_{4}^{2}}{S_{34}} \right) n_{3} + \left( \nu_{14} - 1 \right) n_{4}, $$
$$ p_{3} - p_{2} = P_{1} + \left( 1 - \nu_{23} \right) n_{3} + \left( \frac{m_{3}^{2}}{S_{34}} - \nu_{24} \right) n_{4}, $$
$$ p_{4} - p_{2} = P_{1} + \left( \frac{m_{4}^{2}}{S_{34}} - \nu_{23} \right) n_{3} + \left( 1 - \nu_{24} \right) n_{4}; $$
which lead to more relations:
$$ \left\Vert P_{1} \right\Vert^{2} = \left( \nu_{13} - 1 \right) \left( \nu_{14} - \frac{m_{3}^{2}}{S_{34}} \right) S_{34} - t, $$
$$ \left\Vert P_{1} \right\Vert^{2} = \left( \nu_{13} - \frac{m_{4}^{2}}{S_{34}} \right) \left( \nu_{14} - 1 \right) S_{34} - u. $$
