Invariant Mass

Invariant Mass

One of the most powerful concepts in special relativity is the invariant mass (or rest mass energy) of a particle or system. Unlike energy and momentum individually, the combination

$M^2 = E^2 - |\mathbf{p}|^2 c^2$

is the same in every inertial frame. In natural units ($c = 1$) this becomes simply:

$M^2 = E^2 - |\mathbf{p}|^2$

Energy–Momentum Relation

For a single particle of rest mass $m$ with 3-momentum $p$:

$E = \sqrt{p^2 + m^2}$

This replaces the non-relativistic $E = p^2 / 2m$.

Lorentz Factor from Energy

The Lorentz factor can be recovered from a particle's total energy and rest mass:

$\gamma = \frac{E}{mc^2} = \frac{E}{m} \quad (\text{natural units})$

Reconstructing Resonances

Detectors measure the 4-momenta of decay products. The invariant mass of the system reveals the parent particle's rest mass — this is how the Higgs boson was discovered in 2012.

Your Task

Implement the following functions (all quantities in GeV, natural units, $c = 1$):

• invariant_mass(E_GeV, px_GeV, py_GeV, pz_GeV) — returns $M = \sqrt{\max(0,\, E^2 - p_x^2 - p_y^2 - p_z^2)}$
• particle_energy(m_GeV, p_GeV) — returns $E = \sqrt{m^2 + p^2}$
• lorentz_gamma_from_E(E_GeV, m_GeV) — returns $\gamma = E/m$