This post is part of a series on few-body dynamics in the forward-JWKB approximation.
In this post I will discuss the kinematics of the four-point elastic process
$$ \Phi_{1}(p_{1}) + \Phi_{2}(p_{2}) \longrightarrow \Phi_{1}(p_{3}) + \Phi_{2}(p_{4}). $$
There are four on-shell constraints:
$$ m_{1}^{2} = -\left\Vert p_{1} \right\Vert^{2} = -\left\Vert p_{3} \right\Vert^{2}, $$
$$ m_{2}^{2} = -\left\Vert p_{2} \right\Vert^{2} = -\left\Vert p_{4} \right\Vert^{2}; $$
and a conservation constraint:
$$ p_{1} + p_{2} = p_{3} + p_{4}. $$
Mandelstam Invariants
There are 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
$$ s + t + u = 2 m_{1}^{2} + 2 m_{2}^{2}. $$
Regge-Mandelstam Invariants
There are four Regge-Mandelstam invariants:
$$ r_{12} = r_{34} = \frac{s - m_{1}^{2} - m_{2}^{2}}{2 m_{1} m_{2}}, $$
$$ r_{13} = \frac{2m_{1}^{2} - t}{2 m_{1}^{2}}, $$
$$ r_{24} = \frac{2m_{2}^{2} - t}{2 m_{2}^{2}}, $$
$$ r_{14} = r_{23} = \frac{m_{1}^{2} + m_{2}^{2} - u}{2 m_{1} m_{2}}. $$
Gram Invariants
The 2-Gram invariants are
$$ G_{12} = G_{34} = \frac{1}{4} \left[ \left(s - m_{1}^{2} - m_{2}^{2} \right)^{2} - 4 m_{1}^{2} m_{2}^{2} \right], $$
$$ G_{13} = \frac{1}{4} \left[ \left(2m_{1}^{2} - t \right)^{2} - 4 m_{1}^{4} \right], $$
$$ G_{24} = \frac{1}{4} \left[ \left(2m_{2}^{2} - t \right)^{2} - 4 m_{2}^{4} \right], $$
$$ G_{14} = G_{23} = \frac{1}{4} \left[ \left(m_{1}^{2} + m_{2}^{2} - u \right)^{2} - 4 m_{1}^{2} m_{2}^{2} \right]. $$
It will be useful to introduce four Källén functions:
$$ \Lambda_{12}(s) \equiv \left(s - m_{1}^{2} - m_{2}^{2} \right)^{2} - 4 m_{1}^{2} m_{2}^{2}, $$
$$ \Lambda_{13}(t) \equiv \left(2m_{1}^{2} - t \right)^{2} - 4 m_{1}^{4}, $$
$$ \Lambda_{24}(t) \equiv \left(2m_{2}^{2} - t \right)^{2} - 4 m_{2}^{4}, $$
$$ \Lambda_{14}(u) \equiv \left(m_{1}^{2} + m_{2}^{2} - u \right)^{2} - 4 m_{1}^{2} m_{2}^{2}. $$
The 3-Gram invariants are
$$ G_{123} = G_{124} = G_{134} = G_{234} = \frac{1}{4} t \left[ s u - \left( m_{1}^{2} - m_{2}^{2} \right)^{2} \right]. $$
Center-of-Momentum Frame
In the center-of-momentum frame you have
$$ p_{1} = \begin{pmatrix} E_{1} & \mathbf{p}_{1} \end{pmatrix}, $$
$$ p_{2} = \begin{pmatrix} E_{2} & -\mathbf{p}_{1} \end{pmatrix}, $$
$$ p_{3} = \begin{pmatrix} E_{3} & \mathbf{p}_{3} \end{pmatrix}, $$
$$ p_{4} = \begin{pmatrix} E_{4} & -\mathbf{p}_{3} \end{pmatrix}. $$
That is, the momentum is trivially conserved, but the four energies satisfy a conservation constraint:
$$ E_{1} + E_{2} = E_{3} + E_{4}. $$
All energies are positive in this reference frame.
Energy and Momentum
The energies in the center-of-momentum frame are
$$ E_{1} = E_{3} = \frac{s + m_{1}^{2} - m_{2}^{2}}{2 \sqrt{s}}, $$
$$ E_{2} = E_{4} = \frac{s - m_{1}^{2} + m_{2}^{2}}{2 \sqrt{s}}. $$
The magnitude of the momenta in the center-of-momentum frame are
$$ \left\lvert \mathbf{p}_{1} \right\rvert = \left\lvert \mathbf{p}_{3} \right\rvert = \frac{\sqrt{ \Lambda_{12}(s) }}{2 \sqrt{s}}. $$
Scattering Angle
The scattering angle in the center-of-momentum frame is the angle between \(\mathbf{p}_{1}\) and \(\mathbf{p}_{3}\). The cosine of the scattering angle is
$$ z_{13} \equiv \frac{ \mathbf{p}_{1} \cdot \mathbf{p}_{3} }{\left\lvert \mathbf{p}_{1} \right\rvert \left\lvert \mathbf{p}_{3} \right\rvert} = \frac{ \left( m_{1}^{2} - m_{2}^{2} \right)^{2} - s \left(u - t\right) }{\left( m_{1}^{2} - m_{2}^{2} \right)^{2} - s \left(u + t\right)}. $$
In terms of \(s\), \(t\), and the masses, this can also be written as
$$ z_{13} = 1 + \frac{2 s t}{\Lambda_{12}(s)}. $$
Speed and Rapidity
The speed of each external quantum in the center-of-momentum frame is
$$ \left\lvert \mathbf{v}_{1} \right\rvert = \left\lvert \mathbf{v}_{3} \right\rvert = \frac{\sqrt{ \Lambda_{12}(s) }}{s + m_{1}^{2} - m_{2}^{2}}, $$
$$ \left\lvert \mathbf{v}_{2} \right\rvert = \left\lvert \mathbf{v}_{4} \right\rvert = \frac{\sqrt{ \Lambda_{12}(s) }}{s - m_{1}^{2} + m_{2}^{2}}. $$
The rapidity of each external quantum in the center-of-momentum frame is
$$ \rho_{1} = \rho_{3} = \frac{1}{2} \log{ \left[ \frac{ s + m_{1}^{2} - m_{2}^{2} + \sqrt{\Lambda_{12}(s)} }{ s + m_{1}^{2} - m_{2}^{2} - \sqrt{\Lambda_{12}(s)}} \right] }, $$
$$ \rho_{2} = \rho_{4} = \frac{1}{2} \log{ \left[ \frac{ s - m_{1}^{2} + m_{2}^{2} + \sqrt{\Lambda_{12}(s)} }{ s - m_{1}^{2} + m_{2}^{2} - \sqrt{\Lambda_{12}(s)}} \right] }. $$
Note that
$$ \rho_{1} + \rho_{2} = \rho_{3} + \rho_{4} = \frac{1}{2} \log{\left[ \frac{s - m_{1}^{2} - m_{2}^{2} + \sqrt{\Lambda_{12}(s)}}{s - m_{1}^{2} - m_{2}^{2} - \sqrt{\Lambda_{12}(s)}} \right]}. $$
This expression is similar to the logarithm of the two-body Regge-Sudakov invariant.
Physical Scattering Region
The physical scattering region in the center-of-momentum frame is
$$ s > \left( m_{1} + m_{2} \right)^{2}, $$
$$ t < 0, $$
$$ u < \left( m_{1} - m_{2} \right)^{2}. $$
In terms of the Regge-Mandelstam invariants, this region is
$$ r_{12} > 1, $$
$$ r_{13} > 1, $$
$$ r_{24} > 1, $$
$$ r_{14} > 1. $$
In this region all Gram invariants are positive.
Scattering Regimes
With two independent 2-Mandelstam invariants and two masses you can construct three independent dimensionless ratios:
$$ \xi_{1} \equiv \frac{s}{m_{1} m_{2}}, $$
$$ \xi_{2} \equiv \frac{m_{1}}{m_{2}}, $$
$$ \xi_{3} \equiv \frac{t}{m_{1} m_{2}}. $$
The value of these ratios determines the scattering regime used in the problem.
Rapidity Regimes
The rapidity (and also the speed) of each external quantum in the center-of-momentum frame is a function of \(\xi_{1}\) and \(\xi_{2}\). There are three rapidity regimes. Small-rapidity involves the limit
$$ s \rightarrow \left( m_{1} + m_{2} \right)^{2}. $$
This is the smallest physical value that \(s\) can take. In terms of dimensionless ratios, this means
$$ \xi_{1} \rightarrow \xi_{2} + 2 + \frac{1}{\xi_{2}}, $$
$$ \xi_{2} \text{ fixed}. $$
Small-rapidity is the same as small-speed (i.e. speeds approaching zero).
Fixed-rapidity is the regime where \(s\), \(m_{1}\), \(m_{2}\) take asymptotically large (or small) values, but the ratios are kept fixed:
$$ \xi_{1} \text{ fixed}, $$
$$ \xi_{2} \text{ fixed}. $$
Fixed-rapidity is the same as fixed-speed. Since \(\xi_{1}\) is fixed, you may have large \(s\) and large masses, or small \(s\) and small masses.
Large-rapidity involves the limit
$$ s \rightarrow \infty. $$
In terms of dimensionless ratios, this means
$$ \xi_{1} \rightarrow \infty, $$
$$ \xi_{2} \text{ fixed}. $$
That is, in this regime \(s\) is much larger than the masses. This is a high-energy regime. Indeed, in this regime the massive external quanta behaves as null.
Forward-JWKB Scattering
Forward-JWKB scattering is the regime of fixed-rapidity and small-cosine. Thus,
$$ \xi_{1} \text{ fixed}, $$
$$ \xi_{2} \text{ fixed}, $$
$$ \xi_{3} \rightarrow 0. $$
That is, \(t\) is much smaller than \(s\) or the masses. This is a high-energy regime, but also a heavy-matter (i.e. macroscopic) regime.
Regge Scattering
Regge scattering is the regime of fixed-rapidity and unphysical large-cosine. Thus,
$$ \xi_{1} \text{ fixed}, $$
$$ \xi_{2} \text{ fixed}, $$
$$ \xi_{3} \rightarrow \infty. $$
That is, \(t\) is much larger than \(s\) or the masses. In contrast with forward-JWKB scattering, Regge scattering is a low-energy regime, but also a light-matter (i.e. microscopic) regime.
Fixed-Angle Scattering
Fixed-angle scattering is the regime of large-rapidity and fixed-cosine. Thus,
$$ \xi_{1} \rightarrow \infty, $$
$$ \xi_{2} \text{ fixed}, $$
$$ \xi_{3} \rightarrow -\infty. $$
Indeed, in this regime you have
$$ \frac{\xi_{3}}{\xi_{1}} = \frac{t}{s} \text{ fixed}. $$
In this regime the masses are much smaller than both \(s\) and \(t\), but \(\xi_{2}\) is finite.
