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Invariant mass

Invariant mass is a fundamental Lorentz-invariant quantity in special relativity and particle physics, representing the rest mass of a single particle or the effective mass of a multi-particle system, which remains unchanged regardless of the inertial reference frame.^[1] For a single particle, it is defined by the relation m^2 c^4 = E^2 - |\vec{p}|^2 c^2, where m is the invariant mass, E is the total energy, \vec{p} is the three-momentum vector, and c is the speed of light; in the particle's rest frame, where \vec{p} = 0, this simplifies to E = m c^2.^[2] In particle physics experiments, such as those at the Large Hadron Collider, invariant mass is calculated from the measured energies and momenta of decay products to reconstruct and identify unstable particles, like the Higgs boson or W boson, since it is conserved in decays and independent of boosts along the beam direction.^[3] For systems of particles, the invariant mass M of the total four-momentum P = \sum p_i satisfies M^2 c^4 = P^\mu P_\mu, allowing non-zero values even for combinations of massless particles, such as two photons colliding head-on, where M = 2E / c^2 and E is each photon's energy.^[4] This property underpins kinematic analyses, including Mandelstam variables for scattering processes and Dalitz plots for multi-body decays, enabling precise mass determinations and searches for new physics.^[1] Unlike the outdated concept of relativistic mass, which varies with velocity, invariant mass is the sole definition of mass in modern physics, emphasizing its role as an intrinsic property tied to the particle's four-momentum squared.^[1] Its frame-independence arises from the Minkowski spacetime metric, ensuring consistency across Lorentz transformations, and it plays a central role in verifying conservation laws and predicting thresholds for particle production in high-energy collisions.^[2]

Theoretical Foundations

Definition in Special Relativity

In special relativity, the invariant mass of a single particle or composite system is a Lorentz-invariant quantity that remains constant across all inertial reference frames, distinguishing it from the relativistic mass, which depends on the observer's frame and the particle's velocity.^[1] This invariance ensures that the intrinsic mass of the system, often referred to as the rest mass for a single particle at rest, provides a fundamental measure of its energy content independent of motion.^[5] The concept was introduced by Albert Einstein in his 1905 papers on special relativity and mass-energy equivalence, where he established the framework for understanding how mass relates to energy and momentum in a way that conserves these quantities across frames, laying the groundwork for the mass-energy equivalence principle.^[6] Einstein's work emphasized that while apparent mass effects vary with velocity, an underlying invariant property persists, crucial for the theory's consistency in describing physical laws.^[5] Central to this definition are prerequisite concepts from special relativity, including the four-momentum vector p^\mu = (E/c, \mathbf{p}), where E is the total energy, c is the speed of light, and \mathbf{p} is the three-momentum, combined within the Minkowski spacetime metric ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2.^[1] This metric, formalized by Hermann Minkowski in 1908, endows spacetime with a pseudo-Euclidean geometry where the invariant interval ds^2 is unchanged under Lorentz transformations, enabling the four-momentum's squared magnitude to yield a frame-independent scalar. For a single particle, the invariant mass m is derived from the four-momentum's norm via the relation

m = \sqrt{\frac{E^2}{c^4} - \frac{p^2}{c^2}},

where p = |\mathbf{p}| is the momentum magnitude; this formula follows directly from the invariance of p^\mu p_\mu = m^2 c^2 under the Minkowski metric.^[1] In the particle's rest frame, where \mathbf{p} = 0, E = m c^2, confirming the invariant mass as the rest mass.^[5] Units for invariant mass are typically expressed in kilograms (kg) in general contexts, aligning with classical mechanics, but in particle physics, it is conventional to use electronvolts per speed of light squared (eV/c²) to reflect energy scales directly.^[1]

Invariant Mass Formula

The invariant mass M of a system of particles is defined by the formula

M = \frac{1}{c^2} \sqrt{ \left( \sum_i E_i \right)^2 - c^2 \left| \sum_i \vec{p}_i \right|^2 },

where \sum_i E_i is the total energy of the system, \sum_i \vec{p}_i is the vector sum of the three-momenta, and c is the speed of light.^[7] This expression arises from the invariance of the four-momentum in special relativity. The four-momentum of a single particle is the four-vector p^\mu = (E/c, \vec{p}), and its Minkowski inner product with itself is p^\mu p_\mu = (E/c)^2 - |\vec{p}|^2 = (m c)^2, where m is the rest mass and the metric signature is (+, -, -, -).^[8] For a system of multiple particles, the total four-momentum is the sum P^\mu = \sum_i p_i^\mu, so the invariant is P^\mu P_\mu = (M c)^2, yielding the general formula upon expansion.^[8] For a single particle at rest, where \vec{p} = 0, the formula reduces to m = E / c^2, consistent with the rest energy E = m c^2.^[7] The invariance of M under Lorentz transformations follows from the property of the Minkowski metric: if P'^\mu = \Lambda^\mu{}_\nu P^\nu under a Lorentz boost \Lambda, then P'^\mu P'_\mu = P^\mu P_\mu because \Lambda^\mu{}_\rho \Lambda^\nu{}_\sigma \eta_{\mu\nu} = \eta_{\rho\sigma}, where \eta_{\mu\nu} is the metric tensor.^[8] For example, a boost along the x-direction transforms energy and momentum as E' = \gamma (E - v p_x c) and p_x' = \gamma (p_x - v E / c^2), with \gamma = 1 / \sqrt{1 - v^2/c^2}; substituting these into the invariant shows it remains M^2 c^4.^[7]^[8] As a numerical example, consider a hypothetical single particle at rest with total energy E = 100 GeV; then \vec{p} = 0, so m = 100 GeV/c^2.^[7]

Particle Systems

Sum of Rest Masses vs. Invariant Mass

In special relativity, the invariant mass M of a multi-particle system is generally not equal to the sum of the individual rest masses \sum m_i, as it incorporates contributions from the total energy and momentum of the system.^[9] This distinction arises because the invariant mass is defined via the system's four-momentum, reflecting relativistic effects that the simple sum of rest masses overlooks.^[10] For unbound systems, such as particles with relative motion, the invariant mass exceeds the sum of rest masses due to kinetic energy contributions. The total energy of the system is E_\text{total} = \sum \sqrt{m_i^2 c^4 + p_i^2 c^2}, which is strictly greater than \sum m_i c^2 unless all momenta p_i = 0, as each term \sqrt{m_i^2 c^4 + p_i^2 c^2} > m_i c^2 for nonzero momentum.^[9] In the center-of-momentum frame, where the total momentum vanishes, the invariant mass satisfies M c^2 = E_\text{total}, which thus surpasses the sum of rest energies owing to internal kinetic energies.^[10] For instance, two particles each of rest mass m_0 approaching with speed v yield a system invariant mass of $2 \gamma m_0, where \gamma = (1 - v^2/c^2)^{-1/2} > 1.^[9] In bound systems, the invariant mass can be smaller than the sum of rest masses because binding energy reduces the total energy available. The binding energy E_b accounts for the work needed to separate the components, leading to M = \sum m_i - E_b / c^2.^[11] A representative example is positronium, an electron-positron bound state analogous to hydrogen, where the invariant mass is slightly less than $2 m_e (with m_e the electron rest mass) due to the few eV binding energy.^[12] A common misconception is that the invariant mass always equals the sum of rest masses, but this holds only in the non-relativistic limit where particle velocities are much less than c and binding or kinetic energies are negligible compared to rest energies.^[10] In this approximation, relativistic effects vanish, and M \approx \sum m_i.^[9]

Two-Particle Collision Example

Consider a two-particle collision in special relativity, where two particles with rest masses m_1 and m_2 possess initial four-momenta p_1^\mu = (E_1/c, \vec{p_1}) and p_2^\mu = (E_2/c, \vec{p_2}), respectively, and collide to form a composite system or decay into products.^[13] The invariant mass M of the incoming system characterizes the total effective rest mass of the pair, independent of the reference frame, and is calculated from the total four-momentum p^\mu = p_1^\mu + p_2^\mu.^[14] The formula for the invariant mass is given by

M = \sqrt{ \frac{(E_1 + E_2)^2}{c^4} - \frac{|\vec{p_1} + \vec{p_2}|^2}{c^2} },

where E_1 and E_2 are the total energies, and c is the speed of light.^[13] This quantity M remains constant before and after the collision because the total four-momentum is conserved in both elastic and inelastic processes, ensuring the invariant mass of the system is preserved across interaction stages.^[14] In the center-of-momentum (CM) frame, where the total momentum \vec{p_1} + \vec{p_2} = 0, the calculation simplifies significantly, as the particles approach with equal and opposite momenta. For identical particles with equal energies E, the invariant mass satisfies M c^2 = 2E, highlighting how the total energy in this frame directly yields the effective rest energy of the system.^[14] This frame is particularly useful for analyzing collision dynamics, as it underscores the invariance under Lorentz transformations. A key application is determining the threshold energy required in the laboratory frame, where one particle (say, with mass m_2) is at rest, to produce a particle or resonance of rest mass M. The minimum total energy E_1 of the incoming particle (mass m_1) is

E_{1,\text{threshold}} = \frac{(M^2 - m_1^2 - m_2^2) c^4}{2 m_2 c^2},

derived by setting the invariant mass of the incoming pair equal to M at the point where the produced system is at rest in the CM frame.^[15] Below this energy, the collision cannot generate the required invariant mass, preventing production. The invariance can be visualized through the vector addition of momenta: the individual three-momenta \vec{p_1} and \vec{p_2} combine to form the total \vec{p_{\text{tot}}} = \vec{p_1} + \vec{p_2}, whose magnitude relative to the total energy E_1 + E_2 determines M via the formula above; transforming to another frame alters both energy and momentum consistently, leaving M unchanged.^[13]

Massless Particles

In special relativity, a single massless particle, such as a photon, has zero invariant mass, as its energy E and momentum magnitude p satisfy the relation E = pc, where c is the speed of light, leading to the invariant mass m = 0 from the formula m c^2 = \sqrt{E^2 - (pc)^2}.^[16] This relation arises because massless particles travel at the speed of light, with their energy entirely kinetic and no rest mass contribution.^[17] For a system of multiple massless particles, the total invariant mass can be non-zero if their momenta are not perfectly collinear, as the vector sum of their four-momenta yields an effective rest mass for the composite system. In the case of two photons with energies E_1 and E_2 propagating at an angle \theta relative to each other, the invariant mass M of the pair is given by

M = \frac{\sqrt{2 E_1 E_2 (1 - \cos \theta)}}{c^2},

which demonstrates how relative motion between the particles generates an effective mass.^[18] This formula derives from the Minkowski inner product of the photons' four-momenta, where the non-zero angle prevents complete cancellation of the momentum components.^[19] A key example occurs in the decay of the neutral pion (\pi^0) into two photons, where the invariant mass of the photon pair precisely equals the rest mass of the pion, approximately 135 MeV/c^2, conserving the four-momentum in the process \pi^0 \to \gamma \gamma.^[19] In the pion's rest frame, the photons are emitted back-to-back with equal energies, but in the lab frame, their measured energies and opening angle allow reconstruction of this invariant mass, confirming the decay kinematics.^[20] Physically, massless particles contribute to the invariant mass of a system through their collective energy and the misalignment of their momenta, effectively binding the system as if it had a rest mass, even though individual components do not.^[4] This interpretation highlights how relative motion among massless constituents can produce observable effects akin to massive particles in composite systems. In high-energy physics, many particles behave approximately as massless in the ultra-relativistic limit, where their speeds approach c and the energy-momentum relation simplifies to E \approx pc, allowing treatments similar to truly massless particles like photons or gluons for calculating invariant masses in collisions or decays.^[20] This approximation is particularly useful for hadrons or leptons at energies much greater than their rest masses, simplifying kinematic analyses in particle detectors.^[14]

Experimental and Applied Contexts

Collider Experiments

In high-energy particle collider experiments, the invariant mass of a decaying particle is reconstructed by combining measurements from detector subsystems to determine the total four-momentum of its visible decay products. Charged particle trajectories are tracked in magnetic fields to measure their momenta, while electromagnetic and hadronic calorimeters record energy deposits from both charged and neutral particles. The summed total energy E and three-momentum \vec{p} of the system then yield the invariant mass via M = \sqrt{E^2 - |\vec{p}|^2 c^2}/c^2, enabling the identification of resonances as peaks in invariant mass distributions.^[21] At the Large Hadron Collider (LHC), the ATLAS and CMS collaborations employed this technique to confirm the 2012 discovery of the Higgs boson through invariant mass peaks in key decay channels. In the diphoton (H \to \gamma\gamma) mode, photons are reconstructed from calorimeter clusters, producing a narrow peak at approximately 125 GeV with a full width at half maximum (FWHM) resolution of about 3.9 GeV in ATLAS and approximately 2.6 GeV in CMS, standing out against smooth backgrounds from Drell-Yan and QCD processes. The four-lepton (H \to ZZ^* \to 4\ell) channel, involving electron and muon tracks, similarly revealed a resonance at 125 GeV with resolutions of 1.7–2.3 GeV in ATLAS and 1–2 GeV in CMS, where backgrounds like ZZ^* continuum and Z+jets were subtracted using Monte Carlo simulations and data-driven methods; post-2012 Run 2 data further refined these measurements, determining the Higgs mass to be 125.1 ± 0.1 GeV (as of 2024) with combined precision approaching 0.1%. As of 2025, Run 3 data collection continues, enabling further improvements in Higgs analyses. Invariant mass distributions in collider data are broadened by detector resolution effects, such as momentum smearing from multiple scattering and energy resolution limits in calorimeters, which convolute the true resonance shape with a Gaussian-like response. For the Z boson, reconstructed in dilepton (\ell^+\ell^-) events at the LHC, this results in a prominent peak at 91 GeV with typical resolutions of 1.5 GeV for muons and 2.5 GeV for electrons, allowing efficient separation from falling Drell-Yan backgrounds modeled via exponential or power-law fits.^[22] A pivotal historical application occurred at the e⁺e⁻ Large Electron-Positron (LEP) collider, operational from 1989 to 2000, where invariant mass techniques enabled precise Z and W boson mass determinations. For the Z, energy scans near the 91 GeV pole analyzed over 17 million decays to extract the mass from cross-section lineshapes, achieving 2.1 MeV precision via beam energy calibration to 2 MeV accuracy. W mass measurements from 1996–2000 runs at 161–209 GeV, using fully leptonic and semileptonic decays, reconstructed invariant masses of decay products to reach 33 MeV accuracy, informing electroweak radiative corrections.^[23] In events with incomplete reconstruction due to undetected particles like neutrinos, causing missing transverse energy, the transverse mass m_T—computed from visible transverse momenta and missing energy—is employed as a proxy. Fitting techniques, such as template matching or binned likelihoods, exploit the m_T distribution's endpoint (bounded by the parent mass) or shape to extract resonance parameters, with variants like m_{T2} handling symmetric decay topologies in supersymmetry searches.^[24]

Rest Energy Equivalence

The invariant mass M of a system in special relativity is fundamentally linked to its rest energy through the relation E_{\text{rest}} = M c^2, where E_{\text{rest}} is the total energy in the system's center-of-momentum frame and c is the speed of light. This equation generalizes Einstein's mass-energy equivalence E = m c^2 from single particles to composite systems, where M represents the effective mass derived from the four-momentum invariant.^[25] For multi-particle systems, the invariant mass incorporates not only the rest masses of individual constituents but also their kinetic energies, potential energies, and interaction energies in the rest frame, making M a measure of the system's total internal energy content divided by c^2. This extension highlights that binding processes can alter the invariant mass: negative binding energy reduces M below the sum of individual rest masses, while positive contributions from excitations increase it.^[26] A classic example is the deuteron, a bound state of a proton and a neutron, whose invariant mass corresponds to a rest energy about 2.2 MeV less than the sum of the separate proton and neutron rest energies due to the nuclear binding energy.^[27] Specifically, the proton rest energy is 938.272 MeV, the neutron rest energy is 939.565 MeV, and the deuteron binding energy is 2.2245 MeV, yielding an invariant mass defect of \Delta m = -2.2245 \, \text{MeV}/c^2.^[28] This mass reduction underscores how interaction energies manifest as changes in the effective mass of the system.^[29] In cosmology and astrophysics, the invariant mass of large-scale systems like galaxy clusters is inferred from their total rest energy, encompassing gravitational binding and kinetic contributions from member galaxies, which provides constraints on dark matter and cosmic expansion. Similarly, for black hole binaries, the system's invariant mass determines the rest energy available for gravitational wave emission during inspiral and merger, influencing observable waveforms and energy release.^[30] In particle physics, natural units simplify calculations by setting c = 1 and \hbar = 1, rendering mass, energy, and inverse length interchangeable, with 1 GeV equivalent to approximately $1.78 \times 10^{-27} kg via m = E / c^2.^[31] This convention facilitates expressing particle masses in energy units (e.g., the proton mass as 0.938 GeV), emphasizing the equivalence without explicit factors of c.^[32]

References

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This gives us the 4-momentum $$p^\mu = (E,\vec p)\nonumber$$ and the invariant equation $$E^2 = p^2 + m^2\nonumber$$ Easy! The way we keep track of things is ...Space-Time And Invariants · Lorentz Contraction · Lorentz Transformations
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→p=m→v=m0→v√1−v2/c2E=mc2=√m20c4+c2→p2. ... E2−c2→p2=m20c4. That is to say, E2−c2→p2 depends only on the rest mass of the particle and the speed of light. It does ...Missing: four- | Show results with:four-
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The invariant mass of the two photons is related by, m. 2. = (q1 + q2). 2. = q. 2 ... α/2. 1 γ. 2 sin. 2 α/2 − 1 . (21). This distribution is peaked at αmin ...Missing: formula | Show results with:formula
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It is the length. (invariant mass) of the total energy-momentum 4-vector. The technique has been used to identify, for example, the ρ, ω, φ and J/ψ resonances.