What it does
At speeds near light, energy and momentum mix together as a four-momentum (E, px, py, pz). This page uses natural units, where c = 1, so energies, momenta and masses all share one unit (MeV or GeV). The one number every observer agrees on is the invariant mass, M² = E² − |p|². Everything here follows from that.
- Invariant mass: add up the four-momenta of a set of particles (for example two muons from a detector) and get the mass of whatever made them, plus the system’s total energy, momentum, transverse momentum, velocity, γ and rapidity.
- Two-body decay: a particle of mass M at rest splits into two. Their momentum is fixed: p* = √λ(M², m₁², m₂²) / 2M, where λ is the Källén function. If the parent is moving, the page gives each daughter’s lab energy range and the largest angle it can make with the parent’s direction, with a chart of lab angle against rest-frame angle.
- Lorentz boost: a four-momentum seen from a frame moving with any velocity β.
- Threshold: the least beam energy that can make a given set of products on a target at rest, and the energy per beam if two equal beams collide head-on instead.
How to use it
- Choose MeV or GeV. Wherever a mass is asked for, type a number or a particle name: e, mu, tau, pi+, pi0, K+, K0, eta, p, n, d, alpha, Lambda, J/psi, W, Z, H, top, and gamma or nu for massless particles.
- In the invariant-mass box, write one particle per line: mass (or name), px, py, pz, separated by spaces, commas or tabs.
Worked example
The page opens in GeV. Two made-up muons with momenta (30, 10, 20) and (−25, −5, −15) GeV have an invariant mass of 66.435 GeV. In the decay K⁺ → π⁺ π⁰, each pion has 0.20514 GeV of momentum in the kaon’s rest frame; for a kaon of 1 GeV momentum, the π⁺ has between 0.14496 and 0.97603 GeV in the lab and is never more than 46.52° from the kaon’s direction. A proton with 1 GeV of momentum along z, seen from a frame moving at β = 0.5 along z, has E = 1.006 GeV and p = (0, 0, 0.363) GeV. Making an antiproton by p + p → p + p + p + p̄ needs a beam kinetic energy of 5.6296 GeV on a fixed target (six proton masses), but only 0.93827 GeV per beam in a collider.
Tips
- Rapidity adds simply under boosts along z, which is why colliders plot it; pseudorapidity η is its massless version, computed from the angle alone.
- In a two-body decay the daughters’ momenta are always back to back and equal in the rest frame; only the boost spreads them out in the lab.
FAQ
Why c = 1? It saves writing c and c² everywhere. A mass of 938 MeV means 938 MeV/c², and a momentum of 1 GeV means 1 GeV/c.
Does the threshold check conservation laws? No, only energy and momentum. Charge, baryon number and other rules are up to you.
How accurate are the built-in masses? They are rounded general values, good to about five figures. Type your own numbers for precision work.
Where the numbers come from
- Special-relativistic kinematics (four-vectors, invariant mass, the Källén function for two-body decays, general Lorentz boosts, threshold energies): standard results, for example the Particle Data Group's Review of Particle Physics, Kinematics section, and Hagedorn, R. (1963), Relativistic Kinematics.
- Particle masses are rounded general values (Particle Data Group style), listed in the page's code.
- Checked against a NumPy Lorentz-matrix implementation written for the tests and against textbook thresholds (antiproton production, 5.63 GeV kinetic; π⁰ photoproduction, 144.7 MeV).
- The example particles and momenta are made up.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.