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Invariant mass

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The invariant mass or intrinsic mass or proper mass or rest mass or just mass is a measurement or calculation of the mass of an object that is the same for all frames of reference. For any frame of reference, the invariant mass may be determined from a calculation involving an object's total energy and momentum.

The definition of the invariant mass of a system of particles shows that it is equal to total system energy (divided by c) when the system is viewed from an inertial reference frame which minimizes the total system energy. This reference frame is that in which the velocity of the system's center of mass is zero, and the system's total momentum is zero. This frame is also known as the "center of mass" or "center of momentum" frame.

Particle physics

In particle physics, the mass is often calculated as a mathematical combination of a particle's energy and its momentum to give a value for the mass of the particle that is the same for all observers. This invariant mass is the same for all frames of reference (see Special Relativity).

The mass for a particle is "m" in the equation

m 2 c 4 = ( E ) 2 − ( pc ) 2 {\displaystyle {\mbox{m}}^{2}{\mbox{c}}^{4}=({\mbox{E}})^{2}-({\mbox{pc}})^{2}\,}

The invariant mass of a system of decay particles which originate from a single originating particle, is related to the mass of the original particle by a similar equation:

W 2 c 4 = ( Σ E ) 2 − ( Σ pc ) 2 {\displaystyle {\mbox{W}}^{2}{\mbox{c}}^{4}=(\Sigma {\mbox{E}})^{2}-(\Sigma {\mbox{pc}})^{2}\,}

Where:

W {\displaystyle W} is the invariant mass of the system of particles, equal to the mass of the decay particle.
Σ E {\displaystyle \Sigma E} is the sum of the energies of the particles
Σ p c {\displaystyle \Sigma pc} is the vector sum of the momenta of the particles (includes both magnitude and direction of the momenta) times the speed of light, c {\displaystyle c}

A simple way of deriving this relation is by using the momentum four-vector (in natural units):

p i μ = ( E i , p i ) {\displaystyle p_{i}^{\mu }=\left(E_{i},\mathbf {p} _{i}\right)}
p μ = ( Σ E i , Σ p i ) {\displaystyle p^{\mu }=\left(\Sigma E_{i},\Sigma \mathbf {p} _{i}\right)}
p μ p μ = η μ ν p μ p ν = ( Σ E i ) 2 − ( Σ p i ) 2 = W 2 {\displaystyle p^{\mu }p_{\mu }=\eta _{\mu \nu }p^{\mu }p^{\nu }=(\Sigma E_{i})^{2}-(\Sigma \mathbf {p} _{i})^{2}=W^{2}} , since the norm of any four-vector is invariant.


Example two particle collision

In a two particle collision (or a two particle decay) the square of the mass (in natural units) is

M 2 {\displaystyle M^{2}\,} = ( p 1 + p 2 ) 2 {\displaystyle =(p_{1}+p_{2})^{2}\,}
= p 1 2 + p 2 2 + 2 p 1 p 2 {\displaystyle =p_{1}^{2}+p_{2}^{2}+2p_{1}p_{2}\,}
= m 1 2 + m 2 2 + 2 ( E 1 E 2 − 2 p → 1 ⋅ p → 2 ) {\displaystyle =m_{1}^{2}+m_{2}^{2}+2\left(E_{1}E_{2}-2{\vec {p}}_{1}\cdot {\vec {p}}_{2}\right)\,}

See also

References

  • Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0471887412.{{cite book}}: CS1 maint: multiple names: authors list (link)
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