- Thu 26 November 2015
- Physics
- #kinematics, #three-body
In this post I will consider the kinematics of problems with three independent momentum vectors, each of which describe a distinct massive body. That is, you do not have any conservation constraint relating these three momenta. But you still have three massive on-shell constraints:
$$ m_{i}^{2} = -\left\Vert p_{i} \right\Vert^{2}. $$
You can introduce three 2-Mandelstam invariants:
$$ s_{ij} \equiv -\left\Vert p_{i} + p_{j} \right\Vert^{2}; $$
and one 3-Mandelstam invariant:
$$ s_{123} \equiv -\left\Vert p_{1} + p_{2} + p_{3} \right\Vert^{2} = s_{12} + s_{23} + s_{31} - m_{1}^{2} - m_{2}^{2} - m_{3}^{2}. $$
You can also introduce three Regge-Mandelstam invariants:
$$ r_{ij} \equiv \frac{s_{ij} - m_{i}^{2} - m_{j}^{2}}{2 m_{i} m_{j}}. $$
There are three 2-Gram invariants:
$$ G_{ij} \equiv - \det{ \begin{pmatrix} \left\Vert p_{i} \right\Vert^{2} & p_{i} \cdot p_{j} \\ p_{j} \cdot p_{i} & \left\Vert p_{j} \right\Vert^{2} \end{pmatrix} } = \frac{1}{4} \left[ \left( s_{ij} - m_{i}^{2} - m_{j}^{2} \right)^{2} - 4 m_{i}^{2} m_{j}^{2} \right] = m_{i}^{2} m_{j}^{2} \left( r_{ij}^{2} - 1 \right); $$
and one 3-Gram invariant:
$$G_{123} = -\det{ \begin{pmatrix} \left\Vert p_{1} \right\Vert^{2} & p_{1} \cdot p_{2} & p_{1} \cdot p_{3} \\ p_{2} \cdot p_{1} & \left\Vert p_{2} \right\Vert^{2} & p_{2} \cdot p_{3} \\ p_{3} \cdot p_{1} & p_{3} \cdot p_{2} & \left\Vert p_{3} \right\Vert^{2} \end{pmatrix} } = m_{1}^{2} m_{2}^{2} m_{3}^{2} \left( 2 r_{12} r_{23} r_{31} - r_{12}^{2} - r_{23}^{2} - r_{31}^{2} + 1 \right).$$
More explicitly,
$$ G_{123} = \frac{1}{4} \left[ \left( s_{12} - m_{1}^{2} - m_{2}^{2} \right) \left( s_{23} - m_{2}^{2} - m_{3}^{2} \right) \left( s_{31} - m_{3}^{2} - m_{1}^{2} \right) - m_{1}^{2} \left( s_{23} - m_{2}^{2} - m_{3}^{2} \right)^{2} - m_{2}^{2} \left( s_{31} - m_{3}^{2} - m_{1}^{2} \right)^{2} - m_{3}^{2} \left( s_{12} - m_{1}^{2} - m_{2}^{2} \right)^{2} + 4 m_{1}^{2} m_{2}^{2} m_{3}^{2} \right]. $$
Note the massless limits of the Gram invariants:
$$ \lim_{m_{1} \rightarrow 0} G_{12} = \frac{1}{4} \left( s_{12} - m_{2}^{2} \right)^{2}, $$
$$ \lim_{m_{2} \rightarrow 0} G_{12} = \frac{1}{4} \left( s_{12} - m_{1}^{2} \right)^{2}; $$
$$ \lim_{m_{2} \rightarrow 0} G_{23} = \frac{1}{4} \left( s_{23} - m_{3}^{2} \right)^{2}, $$
$$ \lim_{m_{3} \rightarrow 0} G_{23} = \frac{1}{4} \left( s_{23} - m_{2}^{2} \right)^{2}; $$
$$ \lim_{m_{3} \rightarrow 0} G_{31} = \frac{1}{4} \left( s_{31} - m_{1}^{2} \right)^{2}, $$
$$ \lim_{m_{1} \rightarrow 0} G_{31} = \frac{1}{4} \left( s_{31} - m_{3}^{2} \right)^{2}; $$
$$ \lim_{m_{1} \rightarrow 0} G_{123} = \frac{1}{4} \left[ \left( s_{12} - m_{2}^{2} \right) \left( s_{23} - m_{2}^{2} - m_{3}^{2} \right) \left( s_{31} - m_{3}^{2} \right) - m_{2}^{2} \left( s_{31} - m_{3}^{2} \right)^{2} - m_{3}^{2} \left( s_{12} - m_{2}^{2} \right)^{2} \right], $$
$$ \lim_{m_{2} \rightarrow 0} G_{123} = \frac{1}{4} \left[ \left( s_{12} - m_{1}^{2} \right) \left( s_{23} - m_{3}^{2} \right) \left( s_{31} - m_{3}^{2} - m_{1}^{2} \right) - m_{1}^{2} \left( s_{23} - m_{3}^{2} \right)^{2} - m_{3}^{2} \left( s_{12} - m_{1}^{2} \right)^{2} \right], $$
$$ \lim_{m_{3} \rightarrow 0} G_{123} = \frac{1}{4} \left[ \left( s_{12} - m_{1}^{2} - m_{2}^{2} \right) \left( s_{23} - m_{2}^{2} \right) \left( s_{31} - m_{1}^{2} \right) - m_{1}^{2} \left( s_{23} - m_{2}^{2} \right)^{2} - m_{2}^{2} \left( s_{31} - m_{1}^{2} \right)^{2} \right]; $$
$$ \lim_{m_{1} \rightarrow 0} \lim_{m_{2} \rightarrow 0} G_{123} = \frac{1}{4} \left[ s_{12} \left( s_{23} - m_{3}^{2} \right) \left( s_{31} - m_{3}^{2} \right) - m_{3}^{2} s_{12}^{2} \right], $$
$$ \lim_{m_{2} \rightarrow 0} \lim_{m_{3} \rightarrow 0} G_{123} = \frac{1}{4} \left[ \left( s_{12} - m_{1}^{2} \right) s_{23} \left( s_{31} - m_{1}^{2} \right) - m_{1}^{2} s_{23}^{2} \right], $$
$$ \lim_{m_{3} \rightarrow 0} \lim_{m_{1} \rightarrow 0} G_{123} = \frac{1}{4} \left[ \left( s_{12} - m_{2}^{2} \right) \left( s_{23} - m_{2}^{2} \right) s_{31} - m_{2}^{2} s_{31}^{2} \right]. $$
For convenience, I will also introduce Regge-Gram invariants:
$$ g_{12} \equiv \frac{G_{12}}{m_{1}^{2} m_{2}^{2}}, $$
$$ g_{23} \equiv \frac{G_{23}}{m_{2}^{2} m_{3}^{2}}, $$
$$ g_{31} \equiv \frac{G_{31}}{m_{3}^{2} m_{1}^{2}}, $$
$$ g_{123} \equiv \frac{G_{123}}{m_{1}^{2} m_{2}^{2} m_{3}^{2}}. $$
Note that
$$ g_{123} + g_{12} = 2 r_{12} r_{23} r_{31} - r_{23}^{2} - r_{31}^{2}, $$
$$ g_{123} + g_{23} = 2 r_{12} r_{23} r_{31} - r_{12}^{2} - r_{31}^{2}, $$
$$ g_{123} + g_{31} = 2 r_{12} r_{23} r_{31} - r_{12}^{2} - r_{23}^{2}. $$
All of these invariants are present in any three-body problem.