The Z° Boson:
Invariant Mass

Particle physicists measure the masses of particles such as the W and Z bosons in order to investigate the predictions of the Standard Model and to make further predictions about new particles. To do this, they utilise a quantity called invariant mass.

The theory of special relativity, (developed by Albert Einstein) embodies several concepts which are very important to particle physicists. Amongst other things, the theory tells us that mass is a form of energy. You may also have heard that the mass of an object increases with its velocity. These effects are only significant at very high velocities (approaching the speed of light).

Frames of reference

The velocity of an object can only be measured relative to another object. No single point in the Universe is truly stationary. This means that when we measure the velocity (or mass!) of an object, we should say which frame of reference the measurement was taken in. To measure in an object's frame of reference means to measure velocities relative to that object, which is taken to be stationary. This frame of reference is called the object's rest frame.

The invariant mass

At the extreme velocities reached by particles in an accelerator, the effects of relativity become important and complicate measurements and comparisons of the masses and momenta of particles. Thankfully, a property of an object, or a system of objects, called the invariant mass is the same whatever frame of reference we measure it in.

The invariant mass is defined as the energy of an object in its rest frame (in appropriate units). This quantity is used when calculating and comparing masses and momenta in particle physics to remove the complications of relativity. It is particularly useful since:

The invariant mass of a particle's decay products equals the rest mass of the decaying particle.

For any system of particles:


equation for the invariant mass of a system

  • W is the invariant mass of the system
  • E is the sum of mass energies, E = .m.c, of particles in the system
  • p is the sum of linear momenta, p = .m.v, of particles in the system
  • = (1-(v/c))-
  • c is the speed of light
  • v is the magnitude of the velocity v


The in the expressions for mass energy and momentum are relativistic corrections and represent the effect of special relativity. Under normal, non-relativistic, conditions is very close to 1 and is usually omitted.

v must be measured in the same frame of reference for each particle.

The tasks on this site calculate the invariant mass for you.


Last modified Mon 21 January 2002 . View page history
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