This post is part of a series on few-body dynamics in the forward-JWKB approximation.
In previous posts I introduced the Scalars Toy Model, listed a set of one-loop contributions to an elastic four-point amplitude, and discussed kinematics for this elastic four-point amplitude. Here I would like to revisit the outgoing two-body Sudakov decomposition and adjust it to the elastic kinematics.
Recall that in the outgoing two-body Sudakov decomposition you write the outgoing momenta as
$$ p_{3} = n_{3} + \frac{m_{1}^{2}}{S_{34}} n_{4}, $$
$$ p_{4} = n_{4} + \frac{m_{2}^{2}}{S_{34}} n_{3}; $$
and the incoming momenta as
$$ p_{1} = P_{1} + \nu_{13} n_{3} + \nu_{14} n_{4}, $$
$$ p_{2} = P_{2} + \nu_{23} n_{3} + \nu_{24} n_{4}; $$
with
$$ P_{1} \cdot n_{3} = P_{1} \cdot n_{4} = 0, $$
$$ P_{2} \cdot n_{3} = P_{2} \cdot n_{4} = 0. $$
Due to the conservation constraint
$$ p_{1} + p_{2} = p_{3} + p_{4}, $$
you have
$$ P_{1} + P_{2} = 0, $$
$$ \nu_{13} + \nu_{23} = 1 + \frac{m_{2}^{2}}{S_{34}}, $$
$$ \nu_{14} + \nu_{24} = 1 + \frac{m_{1}^{2}}{S_{34}}. $$
Thus, it is sufficient to determine \(\left\Vert P_{1} \right\Vert^{2}\), \(\nu_{13}\), and \(\nu_{14}\). After accounting for the elasticity of the problem, you have
$$ \left\Vert P_{1} \right\Vert^{2} = \left( \nu_{13} - 1 \right) \left( \nu_{14} - \frac{m_{1}^{2}}{S_{34}} \right) S_{34} - t, $$
$$ \nu_{13} = \frac{S_{34} \left( m_{1}^{2} + m_{2}^{2} - u \right) - m_{2}^{2} \left( 2m_{1}^{2} - t \right)}{\left( S_{34} - m_{1} m_{2} \right) \left( S_{34} + m_{1} m_{2} \right)}, $$
$$ \nu_{14} = \frac{S_{34} \left( 2m_{1}^{2} - t \right) - m_{1}^{2} \left( m_{1}^{2} + m_{2}^{2} - u \right)}{\left( S_{34} - m_{1} m_{2} \right) \left( S_{34} + m_{1} m_{2} \right)}. $$
Using
$$ m_{1}^{2} + m_{2}^{2} - u = s + t - m_{1}^{2} - m_{2}^{2}, $$
and
$$ \left( S_{34} - m_{1} m_{2} \right) \left( S_{34} + m_{1} m_{2} \right) = S_{34} \sqrt{ \Lambda_{12} }, $$
you find
$$ \nu_{13} = \frac{S_{34} \left( s - m_{1}^{2} - m_{2}^{2} \right) - 2 m_{1}^{2} m_{2}^{2} + t \left( S_{34} + m_{2}^{2} \right) }{S_{34} \sqrt{\Lambda_{12}}}, $$
$$ \nu_{14} = \frac{m_{1}^{2}\left(2 S_{34} - s + m_{1}^{2} + m_{2}^{2} \right) - t \left( S_{34} + m_{1}^{2} \right) }{S_{34} \sqrt{\Lambda_{12}}}. $$
Then
$$ \nu_{13} - 1 = \frac{S_{34} \left( s - m_{1}^{2} - m_{2}^{2} - \sqrt{\Lambda_{12}} \right) - 2 m_{1}^{2} m_{2}^{2} + t \left( S_{34} + m_{2}^{2} \right) }{S_{34} \sqrt{\Lambda_{12}}}, $$
$$ \nu_{14} - \frac{m_{1}^{2}}{S_{34}} = \frac{m_{1}^{2}\left(2 S_{34} - s + m_{1}^{2} + m_{2}^{2} - \sqrt{\Lambda_{12}} \right) - t \left( S_{34} + m_{1}^{2} \right) }{S_{34} \sqrt{\Lambda_{12}}}. $$
Using
$$ \frac{2 m_{1}^{2} m_{2}^{2}}{S_{34}} = s - m_{1}^{2} - m_{2}^{2} - \sqrt{\Lambda_{12}}, $$
you get
$$ \nu_{13} - 1 = \frac{t \left( S_{34} + m_{2}^{2} \right)}{S_{34} \sqrt{\Lambda_{12}}}, $$
$$ \nu_{14} - \frac{m_{1}^{2}}{S_{34}} = - \frac{t \left( S_{34} + m_{1}^{2} \right)}{S_{34} \sqrt{\Lambda_{12}}}. $$
Thus
$$ \left\Vert P_{1} \right\Vert^{2} = -t \left(1 + \frac{s t}{\Lambda_{12}} \right). $$
In terms of the cosine of the scattering angle in the center-of-momentum frame, you have
$$ \left\Vert P_{1} \right\Vert^{2} = -t \left( \frac{z_{13} + 1}{2} \right). $$
If \(z_{13} = \cos{(\theta_{13})}\), then
$$\frac{z_{13} + 1}{2} = \cos^{2}{\left( \frac{\theta_{13}}{2} \right)}. $$
For future reference, here are the four Sudakov moduli:
$$ \nu_{13} = 1 + \frac{t \left( S_{34} + m_{2}^{2} \right)}{S_{34} \sqrt{\Lambda_{12}}}, $$
$$ \nu_{14} = \frac{m_{1}^{2}}{S_{34}} - \frac{t \left( S_{34} + m_{1}^{2} \right)}{S_{34} \sqrt{\Lambda_{12}}}, $$
$$ \nu_{23} = \frac{m_{2}^{2}}{S_{34}} - \frac{t \left( S_{34} + m_{2}^{2} \right)}{S_{34} \sqrt{\Lambda_{12}}}, $$
$$ \nu_{24} = 1 + \frac{t \left( S_{34} + m_{1}^{2} \right)}{S_{34} \sqrt{\Lambda_{12}}}. $$
Forward-JWKB Scattering
The Regge-Sudakov invariant \(R_{34}\) is a function of the Regge-Mandelstam invariant:
$$ R_{34} \equiv \frac{S_{34}}{m_{1} m_{2}} = r_{12} + \sqrt{r_{12}^{2} - 1}. $$
The Regge-Mandelstam invariant \(r_{12}\) is a function of \(\xi_{1}\) and \(\xi_{2}\). Thus, in the forward-JWKB regime \(R_{34}\) is kept fixed. The Sudakov moduli \(\nu_{13}\) and \(\nu_{14}\) are functions of \(\xi_{1}\), \(\xi_{2}\), and \(\xi_{3}\). In the forward-JWKB approximation you have
$$ \nu_{13} \rightarrow 1, $$
$$ \nu_{14} \rightarrow \frac{m_{1}^{2}}{S_{34}}; $$
and thus
$$ \left\Vert P_{1} \right\Vert^{2} \rightarrow -t. $$
That is, the vector \(p_{1} - p_{3}\) becomes orthogonal to the Sudakov subspace:
$$ p_{1} - p_{3} \rightarrow P_{1}. $$
Due to the conservation constraint, you also have
$$ \nu_{23} \rightarrow \frac{m_{2}^{2}}{S_{34}} , $$
$$ \nu_{24} \rightarrow 1. $$
The vector \(p_{1} - p_{4}\) maintains a nontrivial component along the Sudakov subspace:
$$ p_{1} - p_{4} \rightarrow P_{1} + \left( 1 - \frac{m_{2}^{2}}{S_{34}} \right) n_{3} - \left( 1 - \frac{m_{1}^{2}}{S_{34}} \right) n_{4}. $$
These results will be important when evaluating one-loop contributions.
