Center-of-Mass Energy

Center-of-Mass Energy

The center-of-mass energy $\sqrt{s}$ is the total available energy in the CM frame — the maximum energy that can go into creating new particles. The Mandelstam variable $s$ is:

$s = (p_1 + p_2)^2 = (E_1 + E_2)^2 - |\mathbf{p}_1 + \mathbf{p}_2|^2$

Collider Mode (head-on, equal beams)

When two equal-energy beams collide head-on, the momenta cancel:

$\sqrt{s} = 2 E_{\text{beam}}$

The LHC at 6.5 TeV per beam delivers $\sqrt{s} = 13$ TeV. This is why colliders are so powerful compared to fixed-target experiments.

Fixed-Target Mode

With a beam hitting a stationary target (natural units, $c = 1$):

$s = m_{\text{beam}}^2 + m_{\text{target}}^2 + 2 E_{\text{beam}}\, m_{\text{target}}$

The $\sqrt{s}$ only grows as $\sqrt{E_{\text{beam}}}$, far slower than the linear growth of the collider case.

Threshold Energy

The minimum beam kinetic energy $T_{\text{thresh}}$ to produce a set of final-state particles with total mass $M_{\text{prod}}$ is found by setting $s = M_{\text{prod}}^2$ at threshold:

$T_{\text{thresh}} = \frac{M_{\text{prod}}^2 - m_{\text{beam}}^2 - m_{\text{target}}^2}{2\, m_{\text{target}}} - m_{\text{beam}}$

Your Task

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

• collider_cm_energy(E_beam_GeV) — returns $\sqrt{s} = 2 E_{\text{beam}}$
• fixed_target_cm_energy(E_beam_GeV, m_beam_GeV, m_target_GeV) — returns $\sqrt{s}$
• threshold_kinetic_energy(m_beam_GeV, m_target_GeV, m_products_total_GeV) — returns minimum kinetic energy $T_{\text{thresh}}$