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Particle decay
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<h2 id="probability-of-survival-and-particle-lifetime">Probability of survival and particle lifetime</h2> <p>Particle decay is a <a href="/facts/Poisson_process/8uphjAUc">Poisson process</a>, and hence the probability that a particle survives for time t before decaying (the <a href="/facts/Survival_function/jkdCmDKq">survival function</a>) is given by an <a href="/facts/Exponential_distribution/yY3EdV0u">exponential distribution</a> whose time constant depends on the particle's velocity: </p><p> P ( t ) = exp ( − t γ τ ) {\displaystyle P(t)=\exp \left(-{\frac {t}{\gamma \tau }}\right)} </p> where τ {\displaystyle \tau } is the mean lifetime of the particle (when at rest), and γ = 1 1 − v 2 c 2 {\displaystyle \gamma ={\tfrac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}} is the <a href="/facts/Lorentz_factor/s5uyKNsA">Lorentz factor</a> of the particle. <h3>Table of some elementary and composite particle lifetimes</h3> <p>All data are from the <a href="/facts/Particle_Data_Group/e89W3u8f">Particle Data Group</a>. </p> <table><tbody><tr><th>Type</th><th>Name</th><th>Symbol</th><th><a href="/facts/Mass/HHdc8XmF">Mass</a> (<a href="/facts/MeV/YMT1fqX9">MeV</a>)</th><th>Mean lifetime</th></tr><tr><td rowspan="3">Lepton</td><td><a href="/facts/Electron/L0KBo5cW">Electron</a> / <a href="/facts/Positron/HgV4zBzA">Positron</a><a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></td><td> e − / e + {\displaystyle \mathrm {e} ^{-}\,/\,\mathrm {e} ^{+}} </td><td>0.511</td><td>>6.6×1028 years</td></tr><tr><td><a href="/facts/Muon/BP2fLVIR">Muon</a> / Antimuon</td><td> μ − / μ + {\displaystyle \mathrm {\mu } ^{-}\,/\,\mathrm {\mu } ^{+}} </td><td>105.7</td><td>2.2×10−6 seconds</td></tr><tr><td><a href="/facts/Tau_lepton/6DW8j6PC">Tau lepton</a> / Antitau</td><td> τ − / τ + {\displaystyle \mathrm {\tau } ^{-}\,/\,\mathrm {\tau } ^{+}} </td><td>1777</td><td>2.9×10−13 seconds</td></tr><tr><td rowspan="2">Meson</td><td>Neutral <a href="/facts/Pion/x5HamzG3">Pion</a></td><td> π 0 {\displaystyle \mathrm {\pi } ^{0}\,} </td><td>135</td><td>8.4×10−17 seconds</td></tr><tr><td>Charged <a href="/facts/Pion/x5HamzG3">Pion</a></td><td> π + / π − {\displaystyle \mathrm {\pi } ^{+}\,/\,\mathrm {\pi } ^{-}} </td><td>139.6</td><td>2.6×10−8 seconds</td></tr><tr><td rowspan="2">Baryon</td><td><a href="/facts/Proton/clCjepXP">Proton</a> / <a href="/facts/Antiproton/dLqJkhGQ">Antiproton</a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></td><td> p + / p − {\displaystyle \mathrm {p} ^{+}\,/\,\mathrm {p} ^{-}} </td><td>938.2</td><td>1.67×1034 years</td></tr><tr><td><a href="/facts/Neutron/oRCM1geb">Neutron</a> / <a href="/facts/Antineutron/NpWKxjhq">Antineutron</a></td><td> n / n ¯ {\displaystyle \mathrm {n} \,/\,\mathrm {\bar {n}} } </td><td>939.6</td><td>885.7 seconds</td></tr><tr><td rowspan="2">Boson</td><td><a href="/facts/W_and_Z_bosons/gAncY5XV">W boson</a></td><td> W + / W − {\displaystyle \mathrm {W} ^{+}\,/\,\mathrm {W} ^{-}} </td><td>80400</td><td>10−26 seconds</td></tr><tr><td><a href="/facts/W_and_Z_bosons/gAncY5XV">Z boson</a></td><td> Z 0 {\displaystyle \mathrm {Z} ^{0}\,} </td><td>91000</td><td>10−26 seconds</td></tr></tbody></table> <h2 id="decay-rate">Decay rate</h2> <p>This section uses <a href="/facts/Natural_units/nf8megYV">natural units</a>, where c = ℏ = 1. {\displaystyle c=\hbar =1.\,} </p><p>The lifetime of a particle is given by the inverse of its decay rate, Γ, the probability per unit time that the particle will decay. For a particle of a mass M and <a href="/facts/Four-momentum/t1F5Lmbr">four-momentum</a> P decaying into particles with momenta pi, the differential decay rate is given by the general formula (expressing <a href="/facts/Fermi%2527s_golden_rule/knMrlpV9">Fermi's golden rule</a>) d Γ n = S | M | 2 2 M d Φ n ( P ; p 1 , p 2 , … , p n ) {\displaystyle d\Gamma _{n}={\frac {S\left|{\mathcal {M}}\right|^{2}}{2M}}d\Phi _{n}(P;p_{1},p_{2},\dots ,p_{n})\,} </p> 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), M {\displaystyle {\mathcal {M}}\,} is the <i>invariant matrix element</i> or <a href="/facts/Probability_amplitude/ryPgJvs8">amplitude</a> connecting the initial state to the final state (usually calculated using <a href="/facts/Feynman_diagrams/BRFNvDdr">Feynman diagrams</a>), d Φ n {\displaystyle d\Phi _{n}\,} is an element of the <a href="/facts/Phase_space/qPe4Fq57">phase space</a>, and pi is the <a href="/facts/Four-momentum/t1F5Lmbr">four-momentum</a> of particle i. <p>The factor S is given by S = ∏ j = 1 m 1 k j ! {\displaystyle S=\prod _{j=1}^{m}{\frac {1}{k_{j}!}}\,} </p> where m is the number of sets of indistinguishable particles in the final state, and kj is the number of particles of type j, so that ∑ j = 1 m k j = n . {\displaystyle \sum _{j=1}^{m}k_{j}=n\,.} <p>The phase space can be determined from d Φ n ( P ; p 1 , p 2 , … , p n ) = ( 2 π ) 4 δ 4 ( P − ∑ i = 1 n p i ) ∏ i = 1 n d 3 p → i 2 ( 2 π ) 3 E i {\displaystyle d\Phi _{n}(P;p_{1},p_{2},\dots ,p_{n})=(2\pi )^{4}\delta ^{4}\left(P-\sum _{i=1}^{n}p_{i}\right)\prod _{i=1}^{n}{\frac {d^{3}{\vec {p}}_{i}}{2(2\pi )^{3}E_{i}}}} </p> where δ 4 {\displaystyle \delta ^{4}\,} is a four-dimensional <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a>, p → i {\displaystyle {\vec {p}}_{i}\,} is the (three-)momentum of particle i, and E i {\displaystyle E_{i}\,} is the energy of particle i. <p>One may integrate over the phase space to obtain the total decay rate for the specified final state. </p><p>If a particle has multiple decay branches or <i>modes</i> with different final states, its full decay rate is obtained by summing the decay rates for all branches. The <a href="/facts/Branching_ratio/QrbbS9PI">branching ratio</a> for each mode is given by its decay rate divided by the full decay rate. </p> <h2 id="two-body-decay">Two-body decay</h2> <p>This section uses <a href="/facts/Natural_units/nf8megYV">natural units</a>, where c = ℏ = 1. {\displaystyle c=\hbar =1.\,} </p> <h3>Decay rate</h3> <p>Say a parent particle of mass M decays into two particles, labeled 1 and 2. In the rest frame of the parent particle, | p → 1 | = | p → 2 | = [ M 2 − ( m 1 + m 2 ) 2 ] [ M 2 − ( m 1 − m 2 ) 2 ] 2 M , {\displaystyle |{\vec {p}}_{1}|=|{\vec {p}}_{2}|={\frac {\sqrt {[M^{2}-(m_{1}+m_{2})^{2}][M^{2}-(m_{1}-m_{2})^{2}]}}{2M}},\,} which is obtained by requiring that <a href="/facts/Four-momentum/t1F5Lmbr">four-momentum</a> be conserved in the decay, i.e. ( M , 0 → ) = ( E 1 , p → 1 ) + ( E 2 , p → 2 ) . {\displaystyle (M,{\vec {0}})=(E_{1},{\vec {p}}_{1})+(E_{2},{\vec {p}}_{2}).\,} </p><p>Also, in spherical coordinates, d 3 p → = | p → | 2 d | p → | d ϕ d ( cos θ ) . {\displaystyle d^{3}{\vec {p}}=|{\vec {p}}\,|^{2}\,d|{\vec {p}}\,|\,d\phi \,d\left(\cos \theta \right).\,} </p><p>Using the delta function to perform the d 3 p → 2 {\displaystyle d^{3}{\vec {p}}_{2}} and d | p → 1 | {\displaystyle d|{\vec {p}}_{1}|\,} 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 </p><p> d Γ = | M | 2 32 π 2 | p → 1 | M 2 d ϕ 1 d ( cos θ 1 ) . {\displaystyle d\Gamma ={\frac {\left|{\mathcal {M}}\right|^{2}}{32\pi ^{2}}}{\frac {|{\vec {p}}_{1}|}{M^{2}}}\,d\phi _{1}\,d\left(\cos \theta _{1}\right).\,} </p> <h3>From two different frames</h3> <p>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 θ ′ = sin θ γ ( β / β ′ + cos θ ) {\displaystyle \tan {\theta '}={\frac {\sin {\theta }}{\gamma \left(\beta /\beta '+\cos {\theta }\right)}}} </p> <h2 id="complex-mass-and-decay-rate">Complex mass and decay rate</h2> <p class="note">Further information: <a href="/facts/Resonance/q7VLALUz">Resonance § Atomic, particle, and molecular resonance</a>; and <a href="/facts/Resonance_(particle_physics)/aBEXCbzT">Resonance (particle physics)</a></p> <p>This section uses <a href="/facts/Natural_units/nf8megYV">natural units</a>, where c = ℏ = 1. {\displaystyle c=\hbar =1.\,} </p><p>The mass of an unstable particle is formally a <a href="/facts/Complex_number/2w7mqI7e">complex number</a>, with the real part being its mass in the usual sense, and the imaginary part being its decay rate in <a href="/facts/Natural_units/nf8megYV">natural units</a>. When the imaginary part is large compared to the real part, the particle is usually thought of as a <a href="/facts/Resonance_(particle_physics)/aBEXCbzT">resonance</a> more than a particle. This is because in <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a> a particle of mass M (a <a href="/facts/Real_number/R02gw5Pb">real number</a>) 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 , {\displaystyle {\tfrac {1}{M}},} according to the <a href="/facts/Uncertainty_principle/UUSoSp4T">uncertainty principle</a>. For a particle of mass M + i Γ {\displaystyle M+i\Gamma } , the particle can travel for time 1 M , {\displaystyle {\tfrac {1}{M}},} but decays after time of order of 1 Γ . {\displaystyle {\tfrac {1}{\Gamma }}.} If Γ > M {\displaystyle \Gamma >M} then the particle usually decays before it completes its travel.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Relativistic_Breit-Wigner_distribution/wxcUXbjv">Relativistic Breit-Wigner distribution</a></li> <li><a href="/facts/Particle_physics/YgIuyj2t">Particle physics</a></li> <li><a href="/facts/Particle_radiation/62YazDtS">Particle radiation</a></li> <li><a href="/facts/List_of_particles/FwR0Ab6u">List of particles</a></li> <li><a href="/facts/Weak_interaction/f6upmKMs">Weak interaction</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/John_David_Jackson_(physicist)/C6F5I1kU">J. D. Jackson</a> (2004). <a href="https://web.archive.org/web/20141121115205/http://pdg.lbl.gov/2005/reviews/kinemarpp.pdf">"Kinematics"</a> (PDF). <i><a href="/facts/Particle_Data_Group/e89W3u8f">Particle Data Group</a></i>. Archived from <a href="http://pdg.lbl.gov/2005/reviews/kinemarpp.pdf">the original</a> (PDF) on 2014-11-21. Retrieved 2006-11-26. (See page 2).</li> <li><a href="http://pdg.lbl.gov/">Particle Data Group</a>.</li> <li>"<a href="https://web.archive.org/web/20190719141632/http://particleadventure.org/">The Particle Adventure</a>" <a href="/facts/Particle_Data_Group/e89W3u8f">Particle Data Group</a>, Lawrence Berkeley National Laboratory.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Electron lifetime is at least 66,000 yottayears – Physics World". 9 December 2015. <a href="https://physicsworld.com/a/electron-lifetime-is-at-least-66000-yottayears/" target="_blank">https://physicsworld.com/a/electron-lifetime-is-at-least-66000-yottayears/</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Bajc, Borut; Hisano, Junji; Kuwahara, Takumi; Omura, Yuji (2016). "Threshold corrections to dimension-six proton decay operators in non-minimal SUSY SU (5) GUTs". Nuclear Physics B. 910: 1–22. arXiv:1603.03568. Bibcode:2016NuPhB.910....1B. doi:10.1016/j.nuclphysb.2016.06.017. S2CID 119212168. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>"How Certain Are We That Protons Don't Decay?". Forbes. <a href="https://www.forbes.com/sites/startswithabang/2020/01/03/how-certain-are-we-that-protons-dont-decay/" target="_blank">https://www.forbes.com/sites/startswithabang/2020/01/03/how-certain-are-we-that-protons-dont-decay/</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>"The Particle Adventures" <a href="http://particleadventure.org/mediator.html" target="_blank">http://particleadventure.org/mediator.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>