Center-of-mass energy

The center-of-mass energy of a relativistic system is the norm of the total four-momentum:

∣ptot∣=s=(∑iEi)2−(∑ipi)2\lvert p_{\text{tot}} \rvert=\sqrt{s} =\sqrt{\left(\sum\limits_{i}E_{i}\right)^{2}-\left(\sum\limits_{i}\mathbf{p}_{i}\right)^{2}}

It is a relativistic invariant. It is related to the invariant mass of the system.

The center-of-mass energy is most commonly used to determine the amount of energy that is available to a particle scattering or particle decay process, especially as a measure of the power of a collider.

Examples

> In the special case of a particle with much more [[kinetic energy]] than [[Relativistic energy|rest energy]], we can say $E_{1}\gg m_{1}$ and so > $$\sqrt{ s }\simeq \sqrt{2E_{1}m_{2}} > If the special case of particles with much more kinetic energy than rest energy, we have $E_{1}\simeq p_{1}$ and $E_{2}\simeq p_{2}$, therefore > $$\sqrt{s}\simeq\sqrt{2E_{1}E_{2}-2E_{1}E_{2}\cos\theta}=\sqrt{2E_{1}E_{2}(1-\cos\theta)}

If the collision is head-on (θ≃0\theta\simeq0), then we have s≃2E1E2\sqrt{s}\simeq\sqrt{2E_{1}E_{2}}.