Center of Mass Energy Calculator

Center-of-mass energy, written √s, is the total energy available in a particle collision as measured in the reference frame where the total momentum is zero (the center-of-mass frame). It represents the maximum energy that can go into creating new particles, since none of it is 'wasted' on bulk motion of the collision products.

√s = 2E, where E is the energy of each beam, valid exactly whenever two particles of equal mass and equal energy collide head-on with opposite momenta, regardless of how relativistic they are. This is why collider design energies are simply doubled from the per-beam energy.

s = m₁² + m₂² + 2E₁m₂ (in natural units where c=1), where E₁ is the total energy of the incoming beam particle, m₁ is its rest mass, and m₂ is the rest mass of the stationary target particle. Taking the square root gives √s.

In a fixed-target setup, momentum conservation forces the collision products to move forward in a fast-moving 'jet,' carrying away much of the beam's energy as bulk motion rather than making it available for new particle creation. A symmetric collider has zero net momentum, so essentially all of both beams' energy goes into √s, dramatically increasing reach for discovering high-mass particles at the same beam energy.

The LHC collides two proton beams head-on, each carrying 6.5 TeV of energy. Since this is a symmetric collider configuration, √s = 2 × 6.5 TeV = 13 TeV exactly, matching the LHC's well-known design center-of-mass energy.

By conservation of energy (via E=mc²), the total rest mass energy of everything produced in a collision cannot exceed the available center-of-mass energy √s. This is why discovering heavier particles, like the Higgs boson or hypothetical new physics, requires progressively higher √s from more powerful colliders.

Particle physicists conventionally use electronvolt-based units: GeV (giga-electronvolts) for lower-energy experiments and TeV (tera-electronvolts, 1000 GeV) for the highest-energy colliders like the LHC. This calculator uses GeV as its base unit.

Yes significantly, a heavier target particle absorbs more of the beam's momentum, making more energy available in the center-of-mass frame. This is why fixed-target experiments sometimes use heavy nuclei as targets to boost the effective √s reachable at a given beam energy.

No, √s is always at least as large as m₁+m₂ (with equality only in the limit of particles at rest relative to each other), since the center-of-mass energy must be enough to account for both particles' intrinsic rest mass energy plus any kinetic energy contribution.

Yes, when physicists report a collider running 'at √s = 13 TeV' or similar, they mean exactly this quantity, the total energy available in the collision's center-of-mass frame, the standard way collision energy is quoted across all of high-energy physics.