Particle decay is the spontaneous process of one elementary particle transforming into other elementary particles. During this process, an elementary particle becomes a different particle with less mass and an intermediate particle such as W boson in muon decay. The intermediate particle then transforms into other particles. If the particles created are not stable, the decay process can continue.
Note that this article uses natural units, where
Table of particle lifetimesEdit
All data are from the Particle Data Group.
Type Name Symbol Mass (MeV/c2) Mean lifetime Lepton Electron / Positron 0.511 Muon / Antimuon 105.6 Tau lepton / Antitau 1777 Meson Neutral Pion 135 Charged Pion 139.6 Baryon Proton / Antiproton 938.2 Neutron / Antineutron 939.6 Boson W boson 80,400 Z boson 91,000
Probability of survivalEdit
The mean lifetime of a particle is labeled , and thus the probability that a particle survives for a time greater than t before decaying is given by the relation
- is the Lorentz factor of the particle.
For a particle of a mass M, the decay rate is given by the general formula
- n is the number of particles created by the decay of the original,
- is the invariant matrix element that connects the initial state to the final state,
- is an element of the phase space, and
- is the four-momentum of particle i.
The phase space can be determined from
- is a four-dimensional Dirac delta function.
- Further information: Resonance#Resonances in quantum mechanics
The mass of an unstable particle is formally a complex number, with the real part being its mass in the usual sense, and the imaginary part being its decay rate in natural units. When the imaginary part is large compared to the real part, the particle is usually thought of as a resonance more than a particle. This is because in quantum field theory a particle of mass M (a real number) is often exchanged between two other particles when there is not enough energy to create it, if the time to travel between these other particles is short enough, of order 1/M, according to the uncertainty principle. For a particle of mass , the particle can travel for time 1/M, but decays after time of order of . If then the particle usually decays before it completes its travel.
As an example, the phase space element of one particle decaying into three is
- Main article: Four-momentum
The four-momentum of one particle is also known as its invariant mass.
The square of the four-momentum of one particle is defined as the difference between the square of its energy and the square of its three-momentum (note that all units from here on are chosen such at the speed of light is equal to 1):
The square of the four momentum of two particles is
Conservation of four-momentumEdit
Four-momentum must be conserved in all decays and all particle interactions, so
In two-body decaysEdit
If a parent particle of mass M decays into two particles (labeled 1 and 2), then the condition of four-momentum conservation becomes
Re-arrange this to
and then square both sides
Now use the very definition of the square of four-momentum, eq (1), to see
If we enter the rest frame of the parent particle, then
- , and
Plug these into eq (2):
Now we have arrived at the formula for the energy of particle 1 as seen in the rest frame of the parent particle,
Similarly, the energy of particle 2 as seen in the rest frame of the parent particle is
|File:2-body Particle Decay-CoM.svg||File:2-body Particle Decay-Lab.svg|
|In the Center of Momentum Frame the decay of a particle into two equal mass particles results in them being emitted with an angle of 180 degrees between them.||...while in the Lab Frame the parent particle is probably moving at a speed close to the speed of light so the two emitted particles would come out at angles different than that of in the center of momentum frame.|
From two different framesEdit
The angle of an emitted particle in the lab frame is related to the angle it's emitted in the center of momentum frame by the equation
Say a parent particle of mass M decays into two particles, labeled 1 and 2. In the rest frame of the parent particle,
Also, in spherical coordinates,
Use this with knowledge of the phase-space element for a two-body decay, to see that the decay rate in the frame of the parent particle is
- J.D. Jackson (2004). "Kinematics". Particle Data Group, http://pdg.lbl.gov/2005/reviews/kinemarpp.pdf. - See page 2.
- Particle Data Group.
- "The Particle Adventure" Particle Data Group, Lawrence Berkeley National Laboratory.ar:اضمحلال الجسيم