Harmonic superspace

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.

Particle decay is also used to refer to the decay of hadrons. However, the term is not typically used to describe radioactive decay, in which an unstable atomic nucleus is transformed into a lighter nucleus accompanied by the emission of particles or radiation, although the two are conceptually similar.

Note that this article uses natural units, where $c=\hslash =1.$

Probability of survival and particle lifetime

Particle decay is a Poisson process, and hence the probability that a particle survives for time t before decaying is given by an exponential distribution whose time constant depends on the particle's velocity:

$P(t)=e^{-t/(\gamma \tau )}$

where

$\tau$ is the mean lifetime of the particle (when at rest), and

$\gamma =\frac{1}{\sqrt{1-v^2/c^2}}$ is the Lorentz factor of the particle.

Table of some elementary and composite particle lifetimes

All data is from the Particle Data Group.

Type	Name	Symbol	Energy (MeV)	Mean lifetime
Lepton	Electron / Positron	$e^{-}/e^{+}$	0.511	$>4.6\times 10^{26}\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{s}$
	Muon / Antimuon	${\mu }^{-}/{\mu }^{+}$	105.7	$2.2\times 10^{-6}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{s}$
	Tau lepton / Antitau	${\tau }^{-}/{\tau }^{+}$	1777	$2.9\times 10^{-13}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{s}$
Meson	Neutral Pion	${\pi }^0$	135	$8.4\times 10^{-17}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{s}$
	Charged Pion	${\pi }^{+}/{\pi }^{-}$	139.6	$2.6\times 10^{-8}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{s}$
Baryon	Proton / Antiproton	$p^{+}/p^{-}$	938.2	$>10^{29}\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{s}$
	Neutron / Antineutron	$n/\overline{n}$	939.6	$885.7\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{s}$
Boson	W boson	$W^{+}/W^{-}$	80,400	$10^{-25}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{s}$
	Z boson	$Z^0$	91,000	$10^{-25}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{s}$

Decay rate

The lifetime of a particle is given by the inverse of its decay rate, $\mathrm{\Gamma }$, the probability per unit time that the particle will decay. For a particle of a mass M and four-momentum P decaying into particles with momenta $p_i$, the differential decay rate is given by the general formula (see also Fermi's golden rule)

$d{\mathrm{\Gamma }}_n=\frac{S{\left|\mathcal{ℳ}\right|}^2}{2M}d{\mathrm{\Phi }}_n(P;p_1,p_2,\dots ,p_n)$

where

n is the number of particles created by the decay of the original,

S is a combinatorial factor to account for indistinguishable final states (see below),

$\mathcal{ℳ}$ is the invariant matrix element or amplitude connecting the initial state to the final state (usually calculated using Feynman diagrams),

$d{\mathrm{\Phi }}_n$ is an element of the phase space, and

$p_i$ is the four-momentum of particle i.

The factor S is given by

$S=\prod _{j=1}^m\frac{1}{k_j!}$

where

m is the number of sets of indistinguishable particles in the final state, and

$k_j$ is the number of particles of type j, so that $\sum _{j=1}^m{k}_j=n$.

The phase space can be determined from

$d{\mathrm{\Phi }}_n(P;p_1,p_2,\dots ,p_n)=(2\pi {)}^4{\delta }^4(P-\sum _{i=1}^n{p}_i)\prod _{i=1}^n\frac{d^3{\overrightarrow{p}}_i}{2(2\pi {)}^3{E}_i}$

where

${\delta }^4$ is a four-dimensional Dirac delta function,

${\overrightarrow{p}}_i$ is the (three-)momentum of particle i, and

$E_i$ is the energy of particle i.

One may integrate over the phase space to obtain the total decay rate for the specified final state.

If a particle has multiple decay branches or modes with different final states, its full decay rate is obtained by summing the decay rates for all branches. The branching ratio for each mode is given by its decay rate divided by the full decay rate.

Two-body decay

Decay rate

Say a parent particle of mass M decays into two particles, labeled 1 and 2. In the rest frame of the parent particle,

$\left|{\overrightarrow{p}}_1\right|=\left|\overrightarrow{p_2}\right|=\frac{[(M^2-(m_1+m_2{)}^2)(M^2-(m_1-m_2{)}^2){]}^{1/2}}{2M},$

which is obtained by requiring that four-momentum be conserved in the decay, i.e.

$(M,\overrightarrow{0})=(E_1,{\overrightarrow{p}}_1)+(E_2,{\overrightarrow{p}}_2).$

Also, in spherical coordinates,

$d^3\overrightarrow{p}=\left|\overrightarrow{p}{|}^{2}d\right|\overrightarrow{p}|d\varphi d(\cos \theta ).$

Using the delta function to perform the $d^3{\overrightarrow{p}}_2$ and $d\left|{\overrightarrow{p}}_1\right|$ integrals in the phase-space for a two-body final state, one finds that the decay rate in the rest frame of the parent particle is

$d\mathrm{\Gamma }=\frac{{\left|\mathcal{ℳ}\right|}^2}{32{\pi }^2}\frac{\left|{\overrightarrow{p}}_1\right|}{M^2}d{\varphi }_{1}d(\cos {\theta }_1).$

From two different frames

The angle of an emitted particle in the lab frame is related to the angle it has emitted in the center of momentum frame by the equation

$\tan {\theta }^{\prime }=\frac{\sin \theta }{\gamma (\beta /{\beta }^{\prime }+\cos \theta )}$

3-body decay

The phase space element of one particle decaying into three is

$d{\mathrm{\Phi }}_3=\frac{1}{(2\pi {)}^5}{\delta }^4(P-p_1-p_2-p_3)\frac{d^3{\overrightarrow{p}}_1}{2E_1}\frac{d^3{\overrightarrow{p}}_2}{2E_2}\frac{d^3{\overrightarrow{p}}_3}{2E_3}$

Complex mass and decay rate

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 $M+i\mathrm{\Gamma }$, the particle can travel for time 1/M, but decays after time of order of $1/\mathrm{\Gamma }$. If $\mathrm{\Gamma }>M$ then the particle usually decays before it completes its travel.

See also

• Relativistic Breit-Wigner distribution
• Particle physics
• List of particles
• Weak interaction

References

• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - See page 2.
• Particle Data Group.
• "The Particle Adventure" Particle Data Group, Lawrence Berkeley National Laboratory.