Relativistic Collisions

Relativistic Collisions

In relativistic collisions, 4-momentum is conserved:

$p_1^\mu + p_2^\mu = p_3^\mu + p_4^\mu$

Center-of-Mass Energy

The Mandelstam variable $s$ gives the square of the center-of-mass energy:

$s = (E_1 + E_2)^2 - (\vec{p}_1 + \vec{p}_2)^2 c^2$

This is Lorentz-invariant — it takes the same value in every reference frame.

Threshold Energy

The minimum beam energy needed to create a new particle of mass $M$ when hitting a stationary target of mass $m$ (both initial particles also have mass $m$) is:

$E_{\text{threshold}} = \frac{(4mM + M^2)\,c^2}{2m}$

This comes from requiring enough center-of-mass energy to produce all final-state particles at rest: $\sqrt{s_{\min}} = (2m + M)c^2$.

Your Task

Implement:

• cm_energy_sq(E1, p1, E2, p2) — computes $s = (E_1+E_2)^2 - (p_1+p_2)^2 c^2$ (1D momenta)
• threshold_energy(m, M) — returns the threshold beam energy in J

Use $c = 299792458.0$ m/s defined inside each function.