Two-Body Decay Kinematics

Two-Body Decay Kinematics

When a particle $A$ of mass $M$ decays at rest into two daughters $B$ and $C$, conservation of 4-momentum uniquely fixes the kinematics. Working in natural units ($c = 1$, all quantities in GeV):

Momentum of the Daughters

Both daughters recoil back-to-back with equal and opposite momenta of magnitude $p^*$:

$p^* = \frac{\sqrt{\bigl[M^2 - (m_B + m_C)^2\bigr]\bigl[M^2 - (m_B - m_C)^2\bigr]}}{2M}$

This is real only when $M \geq m_B + m_C$ (energy conservation requires the parent to be heavier than the sum of daughters).

Energies of the Daughters

$E_B^* = \frac{M^2 + m_B^2 - m_C^2}{2M}, \qquad E_C^* = \frac{M^2 + m_C^2 - m_B^2}{2M}$

Note that $E_B^* + E_C^* = M$ (total energy equals parent rest mass) and $E_B^{*2} - p^{*2} = m_B^2$ as required.

Example: $\pi^0 \to \gamma\gamma$

With $M = 0.135$ GeV and $m_B = m_C = 0$ (massless photons):

$p^* = E_B^* = E_C^* = \frac{M}{2} = 0.0675 \text{ GeV}$

Your Task

Implement (all masses and energies in GeV, natural units):

• decay_momentum(M_GeV, mB_GeV, mC_GeV) — returns $p^*$
• decay_energy_B(M_GeV, mB_GeV, mC_GeV) — returns $E_B^*$
• decay_energy_C(M_GeV, mB_GeV, mC_GeV) — returns $E_C^*$