Invariant mass

The center-of-mass energy or invariant mass \sqrt{s}(with the Mandelstam variable s) in particle physics is the total energy - i.e., the sum of the rest energies and the kinetic energies - of all particles involved with respect to their common center-of-mass coordinate system during a collision process. It is only a part of the total energy applied by the particle accelerator; the rest is contained in the co-motion of the center of gravity occurring in the laboratory system. Only the centre-of-mass energy is available to be converted into excitation energy or into the mass of new particles.

The coincidence of the two terms -energy and mass is due to the equivalence of mass and energy, since they differ only by a constant conversion factor .c^{2} This is often set to one in high energy physics and also in this article.

The special case of the invariant mass of a single particle is its physical mass itself.

Formula

When using natural units in particle physics, energy and mass have the same unit. The center-of-mass energy is then generally the square root of the total quadrature momentum:

\sqrt{s} = \sqrt{ \left( \sum_{i=1}^n{P_i} \right) ^2} ,

where by the square is meant the scalar product of the Minkowskimetric:

\sqrt{s} = \sqrt{ \left( \sum_{i=1}^n{P_{\mu, i}} \right) \cdot \left( \sum_{i=1}^n{P_i^\mu} \right) }.

Here is

  • nis the number of particles
  • P_{i}their quad pulses.

Properties

  • The center-of-mass energy is invariant under Lorentz transformations; hence the name invariant mass. This follows from the fact that the sum of four-vectors is a four-vector and the square of a four-vector is a Lorentz scalar, i.e. a scalar which remains invariant under Lorentz transformations. Correspondingly, the root of a Lorentz scalar is also a scalar.
  • The center-of-mass energy of all particles before a collision is equal to their center-of-mass energy after the collision (conservation quantity).

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