Say that you have a slow momentum \(P_{1}\) and a fast momentum \(P_{2}\) such that
$$ \Vert P_{1} \Vert^{2} = - m_{1}^{2}, \qquad \Vert P_{2} \Vert^{2} = + w_{2}^{2}. $$
That is,
$$ P_{1} = \begin{pmatrix} m_{1} \cosh{(\xi_{1})} & m_{1} \sinh{(\xi_{1})} \end{pmatrix}; $$
$$ P_{2} = \begin{pmatrix} w_{2} \sinh{(\rho_{2})} & w_{2} \cosh{(\rho_{2})} \end{pmatrix}. $$
You can introduce a 2-Mandelstam invariant:
$$ s_{12} = - \Vert P_{1} + P_{2} \Vert^{2}. $$
Now consider two null vectors \(N_{1}\) and \(N_{2}\):
$$ N_{1} = \begin{pmatrix} k_{1} & k_{1} \end{pmatrix}, \qquad N_{2} = \begin{pmatrix} -k_{2} & k_{2} \end{pmatrix}. $$
Note that
$$ N_{1} \cdot N_{2} = 2 k_{1} k_{2}. $$
This means that the 2-Sudakov invariant \(S_{12}\) is given by
$$ S_{12} = - \Vert N_{1} + N_{2} \Vert^{2} = - 2 (N_{1} \cdot N_{2}) = -4 k_{1} k_{2}. $$
The Sudakov two-body decomposition is
$$ P_{1} = N_{1} + \frac{m_{1}^{2}}{S_{12}} N_{2}, \qquad P_{2} = N_{2} - \frac{w_{2}^{2}}{S_{12}} N_{1}. $$
From here it follows that
$$ m_{1} \cosh{(\xi_{1})} = k_{1} + \frac{m_{1}^{2}}{4 k_{1}}, \qquad w_{2} \sinh{(\rho_{2})} = -k_{2} + \frac{w_{2}^{2}}{4 k_{2}}; $$
$$ m_{1} \sinh{(\xi_{1})} = k_{1} - \frac{m_{1}^{2}}{4 k_{1}}, \qquad w_{2} \cosh{(\rho_{2})} = k_{2} + \frac{w_{2}^{2}}{4 k_{2}}. $$
Then
$$ k_{1} = \frac{1}{2} m_{1} \exp{(\xi_{1})}, \qquad k_{2} = \frac{1}{2} w_{2} \exp{(-\rho_{2})}. $$
This means that
$$ N_{1} \cdot N_{2} = \frac{1}{2} m_{1} w_{2} \exp{(\xi_{1} - \rho_{2})} \quad \Longrightarrow \quad S_{12} = -m_{1} w_{2} \exp{(\xi_{1} - \rho_{2})}. $$
Define the Regge-Sudakov invariant \(R_{12}\) as
$$ R_{12} = - \frac{S_{12}}{m_{1} w_{2}} = \exp{(\xi_{1} - \rho_{2})}; $$
and the Regge-Mandelstam invariant \(r_{12}\) as
$$ r_{12} = \frac{P_{1} \cdot P_{2}}{m_{1} w_{2}} = \sinh{(\xi_{1} - \rho_{2})}. $$
There is a relationship between \(r_{12}\) and \(R_{12}\):
$$ 2 r_{12} = R_{12} - \frac{1}{R_{12}} \quad \Longrightarrow \quad R_{12} = r_{12} + \sqrt{r_{12}^{2} + 1}. $$
There is also a relationship between \(r_{12}\) and \(s_{12}\):
$$ r_{12} = \frac{s_{12} - m_{1}^{2} + w_{2}^{2}}{2 m_{1} w_{2}}. $$
Combining these you can find an expression for \(R_{12}\) in terms of \(s_{12}\) and the masses.
Since the exponential function is always positive, you have the condition
$$ R_{12} \geq 0. $$
However, the hyperbolic sine function can take any real value, so you have the condition
$$ -\infty < r_{12} < \infty. $$
