Available energy (particle collision)

In particle physics, the available energy, often denoted as √s or the center-of-mass energy, represents the total energy available to create new particles in a collision between two particles. It's a Lorentz-invariant quantity, meaning its value remains the same regardless of the observer's frame of reference.

The available energy is not simply the sum of the kinetic energies of the two colliding particles in the laboratory frame. A significant portion of the initial kinetic energy can be "locked up" in the form of kinetic energy of the overall system, which must be conserved. Only the energy remaining in the center-of-mass frame is available to be converted into the mass-energy of newly created particles according to E=mc².

More specifically, in a two-particle collision, the available energy squared (s) is defined as the square of the four-momentum sum of the two incoming particles:

s = (p₁ + p₂)² = p₁² + p₂² + 2(p₁ ⋅ p₂)

where p₁ and p₂ are the four-momenta of the two particles, and the dot product is a relativistic dot product. Expanding further, we get:

s = m₁²c⁴ + m₂²c⁴ + 2(E₁E₂/c² - p₁ ⋅ p₂)

where m₁ and m₂ are the rest masses of the particles, E₁ and E₂ are their energies, and p₁ and p₂ are their three-momenta.

For the specific case of a fixed-target experiment, where one particle (e.g., a proton) is accelerated and collides with a stationary target particle (e.g., another proton at rest), the equation simplifies. If particle 2 is at rest in the lab frame, then E₂ = m₂c² and p₂ = 0. The available energy squared then becomes:

s = m₁²c⁴ + m₂²c⁴ + 2E₁m₂c²

In this fixed-target scenario, as the energy E₁ of the accelerated particle increases, the available energy √s increases much slower than E₁ because √s is proportional to the square root of E₁.

In contrast, for a collider experiment, where two beams of particles collide head-on, the situation is much more efficient. If the particles have equal mass (m₁ = m₂ = m) and equal and opposite momenta (so p₁ = -p₂, and E₁ = E₂ = E), then:

s = 4E²

Thus, the available energy √s is directly proportional to the energy E of each beam. This is why collider experiments are favored for exploring the highest energy scales in particle physics. The Large Hadron Collider (LHC), for example, is a collider, allowing for a much higher available energy than would be possible with a fixed-target experiment of similar beam energy.

The available energy is a crucial parameter in particle physics experiments, as it determines the mass of the heaviest particles that can be created in the collision. Experiments are designed to achieve the necessary available energy to probe specific physics phenomena.