Editing Invariant mass (section)

== As defined in particle physics ==
In [[particle physics]], the '''invariant mass''' {{math|''m''<sub>0</sub>}} is equal to the [[mass]] in the rest frame of the particle, and can be calculated by the particle's [[energy]]&nbsp;{{mvar|E}} and its [[momentum]]&nbsp;{{math|'''p'''}} as measured in ''any'' frame, by the [[energy–momentum relation]]:
<math display="block"> m_0^2 c^2 = \left( \frac{E}{c} \right) ^2 - \left\| \mathbf{p} \right\| ^2 </math>
or in [[natural units]] where {{math|1=''c'' = 1}},
<math display="block"> m_0^2 = E^2 - \left\| \mathbf{p} \right\| ^2 .</math>

This invariant mass is the same in all [[frame of reference|frames of reference]] (see also [[special relativity]]). This equation says that the invariant mass is the pseudo-Euclidean length of the [[four-vector]] {{math|(''E'', '''p''')}}, calculated using the [[pseudo-Euclidean space|relativistic version of the Pythagorean theorem]] which has a different sign for the space and time dimensions. This length is preserved under any Lorentz boost or rotation in four dimensions, just like the ordinary length of a vector is preserved under rotations. In quantum theory the invariant mass is a parameter in the relativistic [[Dirac equation]] for an elementary particle. The Dirac [[quantum operator]] corresponds to the particle four-momentum vector.

Since the invariant mass is determined from quantities which are conserved during a decay, the invariant mass calculated using the energy and momentum of the decay products of a single particle is equal to the mass of the particle that decayed.
The mass of a system of particles can be calculated from the general formula:
<math display="block"> \left( W c^2 \right) ^2 = \left( \sum E \right) ^2 - \left\| \sum \mathbf{p} c \right\| ^2 ,</math>
where
* <math>W</math> is the invariant mass of the system of particles, equal to the mass of the decay particle.
* <math display="inline">\sum E</math> is the sum of the energies of the particles
* <math display="inline">\sum \mathbf{p}</math> is the vector sum of the [[momentum]] of the particles (includes both magnitude and direction of the momenta)

The term invariant mass is also used in inelastic scattering experiments. Given an inelastic reaction with total incoming energy larger than the total detected energy (i.e. not all outgoing particles are detected in the experiment), the invariant mass (also known as the "missing mass") {{mvar|W}} of the reaction is defined as follows (in natural units):
<math display="block"> W^2 = \left( \sum E_\text{in} - \sum E_\text{out} \right) ^2 - \left\| \sum \mathbf{p}_\text{in} - \sum \mathbf{p}_\text{out} \right\| ^2 .</math>

If there is one dominant particle which was not detected during an experiment, a plot of the invariant mass will show a sharp peak at the mass of the missing particle.

In those cases when the momentum along one direction cannot be measured (i.e. in the case of a neutrino, whose presence is only inferred from the [[missing energy]]) the [[transverse mass]] is used.