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Invariant mass
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<h2 id="sum-of-rest-masses">Sum of rest masses</h2> <p>The invariant mass of a system includes the mass of any kinetic energy of the system constituents that remains in the center of momentum frame, so the invariant mass of a system may be greater than sum of the invariant masses (rest masses) of its separate constituents. For example, rest mass and invariant mass are zero for individual photons even though they may add mass to the invariant mass of systems. For this reason, invariant mass is in general not an additive quantity (although there are a few rare situations where it may be, as is the case when massive particles in a system without potential or kinetic energy can be added to a total mass). </p><p>Consider the simple case of two-body system, where object A is moving towards another object B which is initially at rest (in any particular frame of reference). The magnitude of invariant mass of this two-body system (see definition below) is different from the sum of rest mass (i.e. their respective mass when stationary). Even if we consider the same system from center-of-momentum frame, where net momentum is zero, the magnitude of the system's invariant mass is not equal to the sum of the rest masses of the particles within it. </p><p>The kinetic energy of such particles and the potential energy of the force fields increase the total energy above the sum of the particle rest masses, and both terms contribute to the invariant mass of the system. The sum of the particle kinetic energies as calculated by an observer is smallest in the center of momentum frame (again, called the "rest frame" if the system is bound). </p><p>They will often also interact through one or more of the <a href="/facts/Fundamental_forces/WrFEJaFl">fundamental forces</a>, giving them a potential energy of interaction, possibly <a href="/facts/Negative_energy/2TgPAeto">negative</a>. </p> <h2 id="as-defined-in-particle-physics">As defined in particle physics</h2> <p>In <a href="/facts/Particle_physics/YgIuyj2t">particle physics</a>, the invariant mass <i>m</i>0 is equal to the <a href="/facts/Mass/HHdc8XmF">mass</a> in the rest frame of the particle, and can be calculated by the particle's <a href="/facts/Energy/wd8PrQM7">energy</a> E and its <a href="/facts/Momentum/TTkGv5eY">momentum</a> p as measured in <i>any</i> frame, by the <a href="/facts/Energy%E2%80%93momentum_relation/HPXngJQF">energy–momentum relation</a>: m 0 2 c 2 = ( E c ) 2 − ‖ p ‖ 2 {\displaystyle m_{0}^{2}c^{2}=\left({\frac {E}{c}}\right)^{2}-\left\|\mathbf {p} \right\|^{2}} or in <a href="/facts/Natural_units/nf8megYV">natural units</a> where <i>c</i> = 1, m 0 2 = E 2 − ‖ p ‖ 2 . {\displaystyle m_{0}^{2}=E^{2}-\left\|\mathbf {p} \right\|^{2}.} </p><p>This invariant mass is the same in all <a href="/facts/Frame_of_reference/15yYfB9M">frames of reference</a> (see also <a href="/facts/Special_relativity/vsFFc63E">special relativity</a>). This equation says that the invariant mass is the pseudo-Euclidean length of the <a href="/facts/Four-vector/89GpELMA">four-vector</a> (<i>E</i>, p), calculated using the <a href="/facts/Pseudo-Euclidean_space/UW6sbGeK">relativistic version of the Pythagorean theorem</a> which has a different sign for the space and time dimensions. This length is preserved under any Lorentz boost or rotation in four dimensions, just like the ordinary length of a vector is preserved under rotations. In quantum theory the invariant mass is a parameter in the relativistic <a href="/facts/Dirac_equation/DzFaXS4p">Dirac equation</a> for an elementary particle. The Dirac <a href="/facts/Quantum_operator/rCJw8UFz">quantum operator</a> corresponds to the particle four-momentum vector. </p><p>Since the invariant mass is determined from quantities which are conserved during a decay, the invariant mass calculated using the energy and momentum of the decay products of a single particle is equal to the mass of the particle that decayed. The mass of a system of particles can be calculated from the general formula: ( W c 2 ) 2 = ( ∑ E ) 2 − ‖ ∑ p c ‖ 2 , {\displaystyle \left(Wc^{2}\right)^{2}=\left(\sum E\right)^{2}-\left\|\sum \mathbf {p} c\right\|^{2},} where </p> <ul><li> W {\displaystyle W} is the invariant mass of the system of particles, equal to the mass of the decay particle.</li> <li> ∑ E {\textstyle \sum E} is the sum of the energies of the particles</li> <li> ∑ p {\textstyle \sum \mathbf {p} } is the vector sum of the <a href="/facts/Momentum/TTkGv5eY">momentum</a> of the particles (includes both magnitude and direction of the momenta)</li></ul> <p>The term invariant mass is also used in inelastic scattering experiments. Given an inelastic reaction with total incoming energy larger than the total detected energy (i.e. not all outgoing particles are detected in the experiment), the invariant mass (also known as the "missing mass") W of the reaction is defined as follows (in natural units): W 2 = ( ∑ E in − ∑ E out ) 2 − ‖ ∑ p in − ∑ p out ‖ 2 . {\displaystyle W^{2}=\left(\sum E_{\text{in}}-\sum E_{\text{out}}\right)^{2}-\left\|\sum \mathbf {p} _{\text{in}}-\sum \mathbf {p} _{\text{out}}\right\|^{2}.} </p><p>If there is one dominant particle which was not detected during an experiment, a plot of the invariant mass will show a sharp peak at the mass of the missing particle. </p><p>In those cases when the momentum along one direction cannot be measured (i.e. in the case of a neutrino, whose presence is only inferred from the <a href="/facts/Missing_energy/XnIXEv6F">missing energy</a>) the <a href="/facts/Transverse_mass/dk7VG4qJ">transverse mass</a> is used. </p> <h2 id="example-two-particle-collision">Example: two-particle collision</h2> <p>In a two-particle collision (or a two-particle decay) the square of the invariant mass (in <a href="/facts/Natural_units/nf8megYV">natural units</a>) is M 2 = ( E 1 + E 2 ) 2 − ‖ p 1 + p 2 ‖ 2 = m 1 2 + m 2 2 + 2 ( E 1 E 2 − p 1 ⋅ p 2 ) . {\displaystyle {\begin{aligned}M^{2}&=(E_{1}+E_{2})^{2}-\left\|\mathbf {p} _{1}+\mathbf {p} _{2}\right\|^{2}\\&=m_{1}^{2}+m_{2}^{2}+2\left(E_{1}E_{2}-\mathbf {p} _{1}\cdot \mathbf {p} _{2}\right).\end{aligned}}} </p> <h3>Massless particles</h3> <p>The invariant mass of a system made of two massless particles whose momenta form an angle θ {\displaystyle \theta } has a convenient expression: M 2 = ( E 1 + E 2 ) 2 − ‖ p 1 + p 2 ‖ 2 = [ ( p 1 , 0 , 0 , p 1 ) + ( p 2 , 0 , p 2 sin θ , p 2 cos θ ) ] 2 = ( p 1 + p 2 ) 2 − p 2 2 sin 2 θ − ( p 1 + p 2 cos θ ) 2 = 2 p 1 p 2 ( 1 − cos θ ) . {\displaystyle {\begin{aligned}M^{2}&=(E_{1}+E_{2})^{2}-\left\|{\textbf {p}}_{1}+{\textbf {p}}_{2}\right\|^{2}\\&=[(p_{1},0,0,p_{1})+(p_{2},0,p_{2}\sin \theta ,p_{2}\cos \theta )]^{2}\\&=(p_{1}+p_{2})^{2}-p_{2}^{2}\sin ^{2}\theta -(p_{1}+p_{2}\cos \theta )^{2}\\&=2p_{1}p_{2}(1-\cos \theta ).\end{aligned}}} </p> <h3>Collider experiments</h3> <p>In particle collider experiments, one often defines the angular position of a particle in terms of an azimuthal angle ϕ {\displaystyle \phi } and <a href="/facts/Pseudorapidity/adj99xAJ">pseudorapidity</a> η {\displaystyle \eta } . Additionally the transverse momentum, p T {\displaystyle p_{T}} , is usually measured. In this case if the particles are massless, or highly relativistic ( E ≫ m {\displaystyle E\gg m} ) then the invariant mass becomes: M 2 = 2 p T 1 p T 2 ( cosh ( η 1 − η 2 ) − cos ( ϕ 1 − ϕ 2 ) ) . {\displaystyle M^{2}=2p_{T1}p_{T2}(\cosh(\eta _{1}-\eta _{2})-\cos(\phi _{1}-\phi _{2})).} </p> <h2 id="rest-energy">Rest energy</h2> <p>Rest energy (also called rest mass energy) is the energy associated with a particle's invariant mass.<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> </p><p>The rest energy E 0 {\displaystyle E_{0}} of a <a href="/facts/Elementary_particle/CTFDM97U">particle</a> is defined as: E 0 = m 0 c 2 , {\displaystyle E_{0}=m_{0}c^{2},} where c {\displaystyle c} is the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light in vacuum</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> In general, only differences in <a href="/facts/Energy/wd8PrQM7">energy</a> have physical significance.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p><p>The concept of rest energy follows from the <a href="/facts/Special_theory_of_relativity/vsFFc63E">special theory of relativity</a> that leads to Einstein's famous conclusion about equivalence of energy and mass. See <i><a href="/facts/Special_relativity/vsFFc63E">Special relativity § Relativistic dynamics and invariance</a></i>. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Mass_in_special_relativity/aRloR0nW">Mass in special relativity</a></li> <li><a href="/facts/Invariant_(physics)/UfHcehYs">Invariant (physics)</a></li> <li><a href="/facts/Transverse_mass/dk7VG4qJ">Transverse mass</a></li></ul> <ul><li>Landau, L.D.; Lifshitz, E.M. (1975). <i>The Classical Theory of Fields: 4-th revised English Edition: Course of Theoretical Physics Vol. 2</i>. Butterworth Heinemann. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7506-2768-9.</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></ul> <h2 id="citations">Citations</h2> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Lawrence S. Lerner. Physics for Scientists and Engineers, Volume 2, page 1073. 1997. <a href="https://books.google.com/books?id=Nv5GAyAdijoC&pg=PA1073" target="_blank">https://books.google.com/books?id=Nv5GAyAdijoC&pg=PA1073</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Nave, C.R. "Relativistic Energy". HyperPhysics. Georgia State University. Retrieved 28 August 2023. <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/releng.html#c2" target="_blank">http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/releng.html#c2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>"13.6 Relativistic Energy or E = m c^2". <a href="https://pressbooks.bccampus.ca/introductorygeneralphysics2phys1207opticsfirst/chapter/28-6-relativistic-energy/access-date=2023-09-07" target="_blank">https://pressbooks.bccampus.ca/introductorygeneralphysics2phys1207opticsfirst/chapter/28-6-relativistic-energy/access-date=2023-09-07</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Nave, C.R. "Relativistic Energy". HyperPhysics. Georgia State University. Retrieved 28 August 2023. <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/releng.html#c2" target="_blank">http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/releng.html#c2</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>"13.6 Relativistic Energy or E = m c^2". <a href="https://pressbooks.bccampus.ca/introductorygeneralphysics2phys1207opticsfirst/chapter/28-6-relativistic-energy/access-date=2023-09-07" target="_blank">https://pressbooks.bccampus.ca/introductorygeneralphysics2phys1207opticsfirst/chapter/28-6-relativistic-energy/access-date=2023-09-07</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Phillip L. Reu (March 2007). Development of the Doppler Electron Velocimeter—Theory (PDF) (Report). Sandia National Laboratories. SAND2006-6063. Archived from the original (PDF) on 2015-06-23. <a href="https://wayback.archive-it.org/all/20150623204045/http://www.prod.sandia.gov/cgi-bin/techlib/access-control.pl/2006/066063.pdf" target="_blank">https://wayback.archive-it.org/all/20150623204045/http://www.prod.sandia.gov/cgi-bin/techlib/access-control.pl/2006/066063.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Modell, Michael; Reid, Robert C. (1974). Thermodynamics and Its Applications. Englewood Cliffs, NJ: Prentice-Hall. ISBN 0-13-914861-2. <a href="0-13-914861-2" target="_blank">0-13-914861-2</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> </ol>