Invariant mass
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In particle physics, the mathematical combination of a particle's energy and its momentum to give a value for the mass of the particle at rest. The invariant mass is the same for all frames of reference (see Special Relativity).
The invariant mass of a system of decay particles is related to the rest mass of the original particle by the following equation:
- <math>\mbox{W}^2\mbox{c}^4=(\Sigma \mbox{E})^2-(\Sigma \mbox{pc})^2</math>
Where:
- <math>W</math> is the invariant mass of the system of particles
- <math>\Sigma E</math> is the sum of the energies of the particles
- <math>\Sigma pc</math> is the vector sum of the momenta of the particles (includes both magnitude and direction of the momenta) times the speed of light, <math>c</math>
A simple way of deriving this relation is by using the momentum four-vector (in natural units):
- <math>p_i^\mu=\left(E_i,\mathbf{p}_i\right)</math>
- <math>P^\mu=\left(\Sigma E_i,\Sigma \mathbf{p}_i\right)</math>
- <math>P^\mu P_\mu=\eta_{\mu\nu}P^\mu P^\nu=(\Sigma E_i)^2-(\Sigma \mathbf{p}_i)^2=W^2</math>, since the norm of any four-vector is invariant.
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