- Wed 02 December 2015
- Physics
- #kinematics, #three-body
In a previous post I discussed the kinematics of three-body problems. In that post I only discussed invariants that only involve the momenta of individual quanta. That situation can also be called simple kinematics. You can also introduce 2-composite momenta:
$$ p_{12} \equiv p_{1} + p_{2}, \qquad p_{23} \equiv p_{2} + p_{3}, \qquad p_{31} \equiv p_{3} + p_{1}. $$
These vectors involve the momenta of two quanta. Note that
$$ s_{123} \equiv \left\Vert p_{1} + p_{2} + p_{3} \right\Vert^{2} = \left\Vert p_{12} + p_{3} \right\Vert^{2} = \left\Vert p_{23} + p_{1} \right\Vert^{2} = \left\Vert p_{31} + p_{2} \right\Vert^{2}. $$
For Regge-Mandelstam and 2-Gram invariants, you need two momentum vectors. Taking one vector to be 2-composite and the other to be simple but complementary, you have three composite Regge-Mandelstam invariants:
$$ r_{(12)3} \equiv \frac{s_{123} - s_{12} - m_{3}^{2}}{2 \sqrt{s_{12}} m_{3}} = \frac{s_{23} + s_{31} - m_{1}^{2} - m_{2}^{2} - 2 m_{3}^{2}}{2 \sqrt{s_{12}} m_{3}} = \frac{m_{2} r_{23} + m_{1} r_{31}}{\sqrt{s_{12}}}; $$
$$ r_{(23)1} \equiv \frac{s_{123} - s_{23} - m_{1}^{2}}{2 \sqrt{s_{23}} m_{1}} = \frac{s_{31} + s_{12} - m_{2}^{2} - m_{3}^{2} - 2 m_{1}^{2}}{2 \sqrt{s_{23}} m_{1}} = \frac{m_{3} r_{31} + m_{2} r_{12}}{\sqrt{s_{23}}}; $$
$$ r_{(31)2} \equiv \frac{s_{123} - s_{31} - m_{2}^{2}}{2 \sqrt{s_{31}} m_{2}} = \frac{s_{12} + s_{23} - m_{3}^{2} - m_{1}^{2} - 2 m_{2}^{2}}{2 \sqrt{s_{31}} m_{2}} = \frac{m_{1} r_{12} + m_{3} r_{23}}{\sqrt{s_{31}}}. $$
and three composite 2-Gram invariants:
$$ G_{(12)3} \equiv -\det{ \begin{pmatrix} \left\Vert p_{12} \right\Vert^{2} & p_{12} \cdot p_{3} \\ p_{3} \cdot p_{12} & \left\Vert p_{3} \right\Vert^{2} \end{pmatrix} } = \frac{1}{4} \left[ \left( s_{123} - s_{12} - m_{3}^{2} \right)^{2} - 4 s_{12} m_{3}^{2} \right] = \frac{1}{4} \left[ \left( s_{23} + s_{31} - m_{1}^{2} - m_{2}^{2} - 2 m_{3}^{2} \right)^{2} - 4 s_{12} m_{3}^{2} \right], $$
$$ G_{(23)1} \equiv -\det{ \begin{pmatrix} \left\Vert p_{23} \right\Vert^{2} & p_{23} \cdot p_{1} \\ p_{1} \cdot p_{23} & \left\Vert p_{1} \right\Vert^{2} \end{pmatrix} } = \frac{1}{4} \left[ \left( s_{123} - s_{23} - m_{1}^{2} \right)^{2} - 4 s_{23} m_{1}^{2} \right] = \frac{1}{4} \left[ \left( s_{31} + s_{12} - 2m_{1}^{2} - m_{2}^{2} - m_{3}^{2} \right)^{2} - 4 s_{23} m_{1}^{2} \right], $$
$$ G_{(31)2} \equiv -\det{ \begin{pmatrix} \left\Vert p_{31} \right\Vert^{2} & p_{31} \cdot p_{2} \\ p_{2} \cdot p_{31} & \left\Vert p_{2} \right\Vert^{2} \end{pmatrix} } = \frac{1}{4} \left[ \left( s_{123} - s_{31} - m_{2}^{2} \right)^{2} - 4 s_{31} m_{2}^{2} \right] = \frac{1}{4} \left[ \left( s_{12} + s_{23} - m_{1}^{2} - 2m_{2}^{2} - m_{3}^{2} \right)^{2} - 4 s_{31} m_{2}^{2} \right]. $$
In terms of simple Regge-Mandelstam and 2-Gram invariants you have
$$ G_{(12)3} = m_{3}^{2} \left[ \left( m_{2} r_{23} + m_{1} r_{31} \right)^{2} - \left(m_{1}^{2} + m_{2}^{2} + 2 m_{1} m_{2} r_{12} \right) \right] = G_{23} + G_{31} + 2 m_{1} m_{2} m_{3}^{2} \left( r_{23} r_{31} - r_{12} \right); $$
$$ G_{(23)1} = m_{1}^{2} \left[ \left( m_{3} r_{31} + m_{2} r_{12} \right)^{2} - \left(m_{2}^{2} + m_{3}^{2} + 2 m_{2} m_{3} r_{23} \right) \right] = G_{31} + G_{12} + 2 m_{2} m_{3} m_{1}^{2} \left( r_{31} r_{12} - r_{23} \right); $$
$$ G_{(31)2} = m_{2}^{2} \left[ \left( m_{1} r_{12} + m_{3} r_{23} \right)^{2} - \left(m_{3}^{2} + m_{1}^{2} + 2 m_{3} m_{1} r_{31} \right) \right] = G_{12} + G_{23} + 2 m_{3} m_{1} m_{2}^{2} \left( r_{12} r_{23} - r_{31} \right). $$
Other invariants involving 2-composite momenta are not complementary (i.e. they repeat bodies).