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Mandelstam variables
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<h2 id="feynman-diagrams">Feynman diagrams</h2> <p>The letters <i>s,t,u</i> are also used in the terms s-channel (timelike channel), t-channel, and u-channel (both spacelike channels). These channels represent different <a href="/facts/Feynman_diagram/BRFNvDdr">Feynman diagrams</a> or different possible scattering events where the interaction involves the exchange of an intermediate particle whose squared four-momentum equals <i>s,t,u</i>, respectively. </p> <table><tbody><tr><td></td><td></td><td></td></tr><tr><td>s-channel</td><td>t-channel</td><td>u-channel</td></tr></tbody></table> <p>For example, the s-channel corresponds to the particles 1,2 joining into an intermediate particle that eventually splits into 3,4: the s-channel is the only way that <a href="/facts/Resonance_(quantum_field_theory)/aBEXCbzT">resonances</a> and new <a href="/facts/Unstable_particle/aBEXCbzT">unstable particles</a> may be discovered provided their lifetimes are long enough that they are directly detectable. The t-channel represents the process in which the particle 1 emits the intermediate particle and becomes the final particle 3, while the particle 2 absorbs the intermediate particle and becomes 4. The u-channel is the t-channel with the role of the particles 3,4 interchanged. </p><p>When evaluating a Feynman amplitude one often finds scalar products of the external four momenta. One can use the Mandelstam variables to simplify these: </p><p> p 1 ⋅ p 2 = s / c 2 − m 1 2 − m 2 2 2 {\displaystyle p_{1}\cdot p_{2}={\frac {s/c^{2}-m_{1}^{2}-m_{2}^{2}}{2}}} </p><p> p 1 ⋅ p 3 = m 1 2 + m 3 2 − t / c 2 2 {\displaystyle p_{1}\cdot p_{3}={\frac {m_{1}^{2}+m_{3}^{2}-t/c^{2}}{2}}} </p><p> p 1 ⋅ p 4 = m 1 2 + m 4 2 − u / c 2 2 {\displaystyle p_{1}\cdot p_{4}={\frac {m_{1}^{2}+m_{4}^{2}-u/c^{2}}{2}}} </p><p>Where m i {\displaystyle m_{i}} is the mass of the particle with corresponding momentum p i {\displaystyle p_{i}} . </p> <h2 id="sum">Sum</h2> <p>Note that </p> ( s + t + u ) / c 4 = m 1 2 + m 2 2 + m 3 2 + m 4 2 {\displaystyle (s+t+u)/c^{4}=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}} <p>where <i>m</i><i>i</i> is the mass of particle <i>i</i>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> <table><tbody><tr><th>Proof</th></tr><tr><td><p>To prove this, we need to use two facts:</p><ul><li>The square of a particle's four momentum is the square of its mass,</li></ul> p i 2 = m i 2 ( 1 ) {\displaystyle p_{i}^{2}=m_{i}^{2}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (1)} <ul><li>And conservation of four-momentum,</li></ul> p 1 + p 2 = p 3 + p 4 {\displaystyle p_{1}+p_{2}=p_{3}+p_{4}} p 1 = − p 2 + p 3 + p 4 ( 2 ) {\displaystyle p_{1}=-p_{2}+p_{3}+p_{4}\quad \quad \quad \quad \quad \quad \,\,(2)} <p>So, to begin,</p> s / c 2 = ( p 1 + p 2 ) 2 = p 1 2 + p 2 2 + 2 p 1 ⋅ p 2 {\displaystyle s/c^{2}=(p_{1}+p_{2})^{2}=p_{1}^{2}+p_{2}^{2}+2p_{1}\cdot p_{2}} t / c 2 = ( p 1 − p 3 ) 2 = p 1 2 + p 3 2 − 2 p 1 ⋅ p 3 {\displaystyle t/c^{2}=(p_{1}-p_{3})^{2}=p_{1}^{2}+p_{3}^{2}-2p_{1}\cdot p_{3}} u / c 2 = ( p 1 − p 4 ) 2 = p 1 2 + p 4 2 − 2 p 1 ⋅ p 4 {\displaystyle u/c^{2}=(p_{1}-p_{4})^{2}=p_{1}^{2}+p_{4}^{2}-2p_{1}\cdot p_{4}} <p>Then adding the three while inserting squared masses leads to,</p> ( s + t + u ) / c 2 = m 1 2 + m 2 2 + m 3 2 + m 4 2 + 2 p 1 2 + 2 p 1 ⋅ p 2 − 2 p 1 ⋅ p 3 − 2 p 1 ⋅ p 4 {\displaystyle (s+t+u)/c^{2}=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}+2p_{1}^{2}+2p_{1}\cdot p_{2}-2p_{1}\cdot p_{3}-2p_{1}\cdot p_{4}} <p>Then note that the last four terms add up to zero using conservation of four-momentum,</p> 2 p 1 2 + 2 p 1 ⋅ p 2 − 2 p 1 ⋅ p 3 − 2 p 1 ⋅ p 4 = 2 p 1 ⋅ ( p 1 + p 2 − p 3 − p 4 ) = 0 {\displaystyle 2p_{1}^{2}+2p_{1}\cdot p_{2}-2p_{1}\cdot p_{3}-2p_{1}\cdot p_{4}=2p_{1}\cdot (p_{1}+p_{2}-p_{3}-p_{4})=0} <p>So finally,</p> ( s + t + u ) / c 2 = m 1 2 + m 2 2 + m 3 2 + m 4 2 {\displaystyle (s+t+u)/c^{2}=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}} .</td></tr></tbody></table> <h2 id="relativistic-limit">Relativistic limit</h2> <p>In the relativistic limit, the momentum (speed) is large, so using the <a href="/facts/Relativistic_energy-momentum_equation/HPXngJQF">relativistic energy-momentum equation</a>, the energy becomes essentially the momentum norm (e.g. E 2 = p ⋅ p + m 0 2 {\displaystyle E^{2}=\mathbf {p} \cdot \mathbf {p} +{m_{0}}^{2}} becomes E 2 ≈ p ⋅ p {\displaystyle E^{2}\approx \mathbf {p} \cdot \mathbf {p} } ). The rest mass can also be neglected. </p><p>So for example, </p> s / c 2 = ( p 1 + p 2 ) 2 = p 1 2 + p 2 2 + 2 p 1 ⋅ p 2 ≈ 2 p 1 ⋅ p 2 {\displaystyle s/c^{2}=(p_{1}+p_{2})^{2}=p_{1}^{2}+p_{2}^{2}+2p_{1}\cdot p_{2}\approx 2p_{1}\cdot p_{2}} <p>because p 1 2 = m 1 2 {\displaystyle p_{1}^{2}=m_{1}^{2}} and p 2 2 = m 2 2 {\displaystyle p_{2}^{2}=m_{2}^{2}} . </p><p>Thus, </p> <table><tbody><tr><td> s / c 2 ≈ {\displaystyle s/c^{2}\approx } </td><td> 2 p 1 ⋅ p 2 ≈ {\displaystyle 2p_{1}\cdot p_{2}\approx } </td><td> 2 p 3 ⋅ p 4 {\displaystyle 2p_{3}\cdot p_{4}} </td></tr><tr><td> t / c 2 ≈ {\displaystyle t/c^{2}\approx } </td><td> − 2 p 1 ⋅ p 3 ≈ {\displaystyle -2p_{1}\cdot p_{3}\approx } </td><td> − 2 p 2 ⋅ p 4 {\displaystyle -2p_{2}\cdot p_{4}} </td></tr><tr><td> u / c 2 ≈ {\displaystyle u/c^{2}\approx } </td><td> − 2 p 1 ⋅ p 4 ≈ {\displaystyle -2p_{1}\cdot p_{4}\approx } </td><td> − 2 p 3 ⋅ p 2 {\displaystyle -2p_{3}\cdot p_{2}} </td></tr></tbody></table> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Feynman_diagrams/BRFNvDdr">Feynman diagrams</a></li> <li><a href="/facts/Bhabha_scattering/hdTD5ESz">Bhabha scattering</a></li> <li><a href="/facts/M%C3%B8ller_scattering/EbVUCEBm">Møller scattering</a></li> <li><a href="/facts/Compton_scattering/Q98Ty8rz">Compton scattering</a></li></ul> <ul><li><a href="/facts/Stanley_Mandelstam/MtfqcXPS">Mandelstam, S.</a> (1958). <a href="https://web.archive.org/web/20000528212400/http://dbserv.ihep.su/hist/owa/hw.part2?s_c=MANDELSTAM+1958">"Determination of the Pion-Nucleon Scattering Amplitude from Dispersion Relations and Unitarity"</a>. <i><a href="/facts/Physical_Review/QxekSF11">Physical Review</a></i>. 112 (4): 1344. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1958PhRv..112.1344M">1958PhRv..112.1344M</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2FPhysRev.112.1344">10.1103/PhysRev.112.1344</a>. Archived from <a href="http://dbserv.ihep.su/hist/owa/hw.part2?s_c=MANDELSTAM+1958">the original</a> on 2000-05-28.</li> <li><a href="/facts/Francis_Halzen/s1pX4HUg">Halzen, Francis</a>; <a href="/facts/Alan_Martin_(physicist)/aJa3odKd">Martin, Alan</a> (1984). <a href="https://archive.org/details/quarksleptonsint0000halz"><i>Quarks & Leptons: An Introductory Course in Modern Particle Physics</i></a>. <a href="/facts/John_Wiley_%26_Sons/53g3LqJc">John Wiley & Sons</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-88741-2.</li> <li>Perkins, Donald H. (2000). <i>Introduction to High Energy Physics</i> (4th ed.). <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-62196-8.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Griffiths, David (2008). Introduction to Elementary Particles (2nd ed.). Wiley-VCH. p. 113. ISBN 978-3-527-40601-2. <a href="978-3-527-40601-2" target="_blank">978-3-527-40601-2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>