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Massless particle

A massless particle, also known as a luxon, is an elementary particle in physics whose invariant mass (rest mass) is exactly zero, meaning it possesses no intrinsic mass when at rest and must always propagate at the speed of light in vacuum.^[1]^[2] This property arises from the principles of special relativity, where the energy-momentum relation E^2 = (pc)^2 + (m_0 c^2)^2 simplifies to E = pc for zero rest mass m_0, ensuring that such particles cannot be slowed down or accelerated without violating causality or becoming undetectable.^[2] In the Standard Model of particle physics, the confirmed massless particles are the photon, which mediates the electromagnetic force, and the gluon, which carries the strong nuclear force between quarks.^[1]^[3] These particles are gauge bosons, fundamental force carriers that do not interact with the Higgs field, thereby remaining massless unlike fermions such as quarks and leptons, which acquire mass through the Higgs mechanism.^[4] Gluons, however, are never observed in isolation due to color confinement in quantum chromodynamics, while photons are ubiquitous as quanta of light.^[1] Theoretically, the graviton is proposed as another massless spin-2 particle responsible for gravity in quantum gravity theories, though it remains undetected and unconfirmed.^[5] Massless particles exhibit unique quantum properties, such as helicity (a projection of spin along the direction of motion) that is fixed—right-handed or left-handed—due to their inability to change velocity direction without mass.^[6] Their relativistic invariance makes them crucial for understanding fundamental interactions, from electromagnetic radiation to the binding forces in atomic nuclei.^[7]

Theoretical Foundations

Definition and Invariant Mass

In special relativity, a massless particle is defined as one possessing zero invariant mass, denoted as m = 0, which serves as the rest mass in the particle's hypothetical rest frame.^[8] This contrasts with massive particles, for which the invariant mass remains a fixed, observer-independent scalar quantity that characterizes the particle's intrinsic inertia even when at rest.^[9] The invariant mass arises from the spacetime structure of special relativity, ensuring that it is Lorentz invariant under transformations between inertial frames.^[10] The concept of invariant mass is formalized using the four-momentum vector p^\mu = (E/c, \mathbf{p}), where E is the energy and \mathbf{p} is the three-momentum. In the mostly-minus metric signature (\eta_{\mu\nu} = \operatorname{diag}(+1, -1, -1, -1)), the invariant mass squared is given by the contraction

p^\mu p_\mu = \frac{E^2}{c^2} - \mathbf{p}^2 = m^2 c^2,

which holds for all observers. For a massless particle, m = 0 implies p^\mu p_\mu = 0, meaning the four-momentum is null (light-like) for real particles with positive energy. In natural units where c = 1 and \hbar = 1, this simplifies to p^2 = m^2 = 0.^[10] This null condition distinguishes massless particles from time-like (m > 0) or space-like trajectories in Minkowski spacetime. The foundational framework for massless particles emerged in Albert Einstein's 1905 paper on special relativity, which established the invariance of the speed of light c in vacuum and the relativity of simultaneity, laying the groundwork for treating light propagation without a rest frame.^[11] Early recognition of zero rest mass applied to electromagnetic waves, as their propagation at exactly c implied no inertial rest frame, consistent with the absence of invariant mass in the relativistic energy-momentum formalism.^[12] A key implication of zero invariant mass is that massless particles must travel at the speed of light c in vacuum in all inertial frames, precluding the existence of a rest frame where their three-velocity would be zero.^[9] Without a rest frame, concepts like proper time along the particle's worldline become degenerate, and their dynamics are governed solely by the light-like constraint E = |\mathbf{p}| c. This classification underpins the distinction between massive and massless particles in relativistic physics.^[8]

Energy-Momentum Relation

In relativistic mechanics, the total energy E of a particle is related to its three-momentum \vec{p} and rest mass m by the dispersion relation

E = \sqrt{(pc)^2 + (mc^2)^2},

where p = |\vec{p}| is the magnitude of the momentum and c is the speed of light in vacuum.^[13] This equation arises from the Lorentz invariance of the spacetime interval and the conservation of four-momentum in special relativity.^[13] For massless particles, where the invariant mass is zero, the rest mass term vanishes (m = 0), simplifying the relation to

E = pc.

This holds exactly, as the energy-momentum four-vector satisfies the null condition.^[14] Step-by-step, the derivation begins with the general form, which combines the classical kinetic energy in the non-relativistic limit (E \approx mc^2 + \frac{p^2}{2m}) and extends to high speeds via the Lorentz factor \gamma = 1/\sqrt{1 - v^2/c^2}.^[13] As m \to 0, the rest energy mc^2 approaches zero, leaving only the relativistic kinetic contribution, which is linear in momentum: the particle's energy is entirely kinetic and proportional to p, with the constant of proportionality c.^[15] This linear dispersion distinguishes massless particles from massive ones, where E > pc even at high energies. The four-momentum of a massless particle is the contravariant vector p^\mu = (E/c, \vec{p}) in Minkowski space with metric signature (+,-,-,-).^[14] The invariant mass squared is given by the contraction p^\mu p_\mu = (E/c)^2 - p^2 = 0, confirming E = pc and implying that the particle's worldline is light-like (null geodesic in spacetime).^[14] Such particles trace paths at the invariant speed c, with their trajectories confined to the light cone. In particle detectors at scattering experiments, the relation E = pc is applied to reconstruct the four-momentum of candidate massless particles, such as photons, from calorimeter energy deposits and shower positions that provide directional information for \vec{p}.^[16] This assumption allows computation of invariant masses for decay products; for instance, pairs of reconstructed photons yielding near-zero invariant mass confirm their massless nature, distinguishing them from massive particle pairs (like \pi^0 mesons) where E > pc leads to non-zero reconstructed masses.^[16] Such analyses in collider events, like those at the LHC, enable precise kinematic fits and signal-background separation.^[16]

Physical Properties

Speed and Dispersion

In special relativity, massless particles are required to propagate at the speed of light c in vacuum, as their invariant mass is zero, precluding any rest frame and thus forbidding subluminal or superluminal velocities for real particles.^[17]^[2] This universal speed arises directly from the energy-momentum relation E = pc, where the absence of a mass term mandates v = c.^[18] The dispersion relation for massless particles is linear, given by \omega = c k, where \omega is the angular frequency and k is the wave number. This linearity implies that the phase velocity \omega / k = c equals the group velocity d\omega / dk = c, resulting in non-dispersive propagation where wave packets do not spread over distance. The invariance of c was indirectly supported by the Michelson-Morley experiment of 1887, which detected no variation in light speed due to Earth's motion through a presumed luminiferous aether, consistent with relativistic predictions.^[19] Subsequent direct measurements of photon speeds in vacuum have confirmed this invariance to high precision, aligning with the theoretically mandated value of c = 299{,}792{,}458 m/s.^[20] In material media, massless particles exhibit an effective phase velocity reduced by the refractive index n > 1, such that v = c / n, due to interactions with the medium; however, their intrinsic propagation speed in vacuum remains c, preserving masslessness.^[21]

Helicity and Polarization

In quantum field theory, helicity is defined as the projection of a particle's intrinsic spin angular momentum onto the direction of its linear momentum. For massless particles, which travel exclusively at the speed of light and possess no rest frame, helicity emerges as a Lorentz-invariant quantum number that characterizes their one-particle states. This invariance arises because boosts along the momentum direction leave the helicity unchanged, unlike for massive particles where such transformations can alter the spin projection.^[22] The classification of massless particle representations under the Poincaré group, as developed by Wigner, relies on the little group ISO(2), whose rotational subgroup generates the helicity operator. Massless particles of total spin s are thus described by irreducible representations with definite helicity eigenvalues h = \pm s, yielding two distinct helicity states for non-zero integer or half-integer spin. For instance, spin-1 bosons occupy the h = \pm 1 states, consistent with the spin-statistics theorem that dictates Bose-Einstein statistics for integer-spin particles. This restriction to extreme helicity values stems from the structure of the massless little group, where intermediate projections are not realized in finite-dimensional unitary representations.^[22]^[23] Due to the absence of a rest frame, massless particles cannot undergo spin flips that would change their helicity without acquiring mass, rendering helicity a fixed property intrinsic to each particle species. In free propagation, this conservation holds exactly, as the helicity operator commutes with the Hamiltonian for massless fields. Interactions may preserve or violate helicity depending on the theory's symmetries, but the fundamental states remain helicity eigenstates.^[24] For massless vector particles (spin-1), gauge invariance in quantum field theory mandates transverse polarization, where the polarization vectors \boldsymbol{\epsilon} are orthogonal to the four-momentum p^\mu. This orthogonality condition, \epsilon \cdot p = 0, eliminates the three longitudinal and timelike modes present in the massive case, leaving only the two transverse helicity states h = \pm 1. Longitudinal polarizations are forbidden for massless vectors, as they would violate unitarity or introduce ghost states with negative norm, incompatible with relativistic causality. This transverse structure ensures the consistency of massless gauge theories, such as those underlying the Standard Model.^[25]

Confirmed Examples

Photon

The photon is the quantum of the electromagnetic field, representing the fundamental excitation of the quantized electromagnetic field in quantum electrodynamics (QED). As a massless particle, it travels at the speed of light in vacuum and obeys the energy-momentum relation E = pc, where p is momentum and c is the speed of light. Experimental constraints confirm its rest mass is effectively zero, with upper limits set below $10^{-18} eV/c^2 from high-precision tests in particle accelerators and other facilities, where any nonzero mass would alter photon propagation and dispersion relations in electromagnetic interactions.^[26] The concept of the photon emerged from early 20th-century developments in quantum theory. In 1900, Max Planck introduced the idea of energy quantization to resolve the blackbody radiation problem, proposing that electromagnetic radiation is emitted and absorbed in discrete units proportional to frequency, though he initially viewed this as a mathematical artifice rather than physical reality. Albert Einstein extended this in 1905 by applying the quantum hypothesis to light itself, postulating discrete light particles—later termed photons—to explain the photoelectric effect, where light ejects electrons from metals only above a frequency threshold, with electron kinetic energy linearly dependent on photon frequency. For this explanation, Einstein received the 1921 Nobel Prize in Physics. In QED, the photon serves as the gauge boson mediating the electromagnetic force between charged particles via virtual photon exchange, enabling both attractive and repulsive interactions over infinite distances due to its masslessness. Unlike charged particles, the photon carries no electric charge, allowing it to interact solely with charged matter without self-interaction. This masslessness ensures the force's infinite range, as a nonzero photon mass would introduce a Yukawa-like exponential decay, shortening the interaction distance. Direct evidence for the photon's masslessness comes from precision tests of electrostatics and astrophysical phenomena. The inverse-square dependence in Coulomb's law, verified to high accuracy in laboratory experiments (e.g., torsion balance measurements showing deviations below $10^{-16} relative to $1/r^2), implies an infinite-range force consistent only with a massless mediator. Astrophysical observations further tighten bounds; for instance, analyses of galactic magnetic fields yield upper limits on the photon mass below $10^{-18} eV/c^2, while pulsar timing dispersion provides complementary bounds around $10^{-10} eV/c^2 or tighter from other tests, as any mass would cause frequency-dependent delays in photon arrival times from distant sources.^[26]^[27]

Gluons

Gluons are the massless gauge bosons that mediate the strong nuclear force within quantum chromodynamics (QCD), the theory describing the interactions of quarks and gluons. There are eight gluons, forming an octet that corresponds to the generators of the SU(3)_c color symmetry group, and unlike photons, they carry color charge, allowing them to interact with quarks and each other. This color charge renders gluons colored particles, with each gluon possessing a specific combination of color and anticolor quantum numbers, such as red-antiblue or green-antigreen. As spin-1 bosons, gluons are massless in the QCD Lagrangian, a property that ensures the strong force can exhibit long-range perturbative behavior at high energies and short distances, where the theory becomes tractable. The masslessness of gluons is fundamental to asymptotic freedom, a key feature of QCD where the effective strength of the strong coupling constant diminishes as the energy scale increases, enabling quarks and gluons to behave as nearly free particles at very short distances. This phenomenon arises because the non-Abelian structure of QCD leads to gluon self-interactions that cause an antiscreening effect, reducing the coupling at high momenta. Asymptotic freedom was theoretically established in 1973 through independent calculations showing that the beta function for the strong coupling is negative, confirming the weakening of interactions at short distances. Experimental confirmation came in the 1970s from deep inelastic scattering experiments at SLAC and CERN, which observed approximate scaling in structure functions—indicative of point-like partons—and subsequent logarithmic deviations in moments of these functions that matched QCD predictions for asymptotic freedom. Despite their masslessness, gluons are confined and do not propagate as free particles in nature, a consequence of QCD's non-Abelian gauge symmetry that permits gluon self-interactions and generates a linearly rising potential between color charges at large distances. These self-interactions, absent in the Abelian quantum electrodynamics, lead to the formation of flux tubes between quarks, ultimately binding them into color-neutral hadrons such as protons and mesons. Confinement ensures that only color-singlet states are observed at low energies, with gluons contributing to the internal structure of hadrons rather than existing in isolation. Indirect experimental evidence for gluons and their massless propagation supports QCD's predictions, particularly through the observation of jet events in high-energy particle collisions. The discovery of three-jet events in electron-positron annihilation at the PETRA collider in 1979 provided the first clear signature of gluon emission from quarks, with the angular distribution and energy sharing consistent with massless spin-1 bosons. At the Large Hadron Collider (LHC), extensive measurements of jet production rates, shapes, and substructure in proton-proton collisions align precisely with perturbative QCD calculations assuming massless gluons, including the ratio of quark to gluon jet cross-sections and their momentum distributions up to TeV scales. These observations validate the massless gluon propagator in Feynman diagrams for hard scattering processes.

Hypothetical Examples

Graviton

The graviton is a hypothetical elementary particle proposed as the quantum mediator of the gravitational force, described as a massless spin-2 tensor boson.^[28] Unlike vector bosons such as photons, which have two polarization states, the graviton possesses only two helicity states, +2 and -2, due to its massless nature and the constraints of special relativity on higher-spin particles.^[29] This structure ensures compatibility with general relativity's description of gravity as an infinite-range interaction, where the force law falls off as 1/r², requiring a massless mediator to avoid exponential damping.^[30] The theoretical foundation for the graviton arises from the quantization of linearized general relativity, where the metric perturbation is treated as a spin-2 field propagating on a flat background.^[31] In this framework, the graviton propagator in momentum space scales as 1/q², directly implying a zero mass for consistency with the long-range nature of gravity.^[30] The concept was first introduced in the 1930s by Léon Rosenfeld, who applied perturbative quantization techniques to the gravitational field, laying the groundwork for viewing gravity as exchanged via massless quanta.^[32] A major challenge in graviton theory stems from the non-renormalizability of general relativity at the quantum level, where perturbative expansions lead to infinities that cannot be absorbed by finite counterterms beyond one loop.^[33] Despite this, approaches like string theory resolve these issues by embedding the graviton as a massless closed-string excitation in a higher-dimensional, UV-complete framework.^[34] Similarly, loop quantum gravity, a non-perturbative approach, recovers general relativity at low energies but does not rely on perturbative graviton excitations, instead quantizing spacetime geometry directly.^[35] Indirect evidence for the graviton comes from the observation of gravitational waves by the LIGO collaboration in 2015, which detected signals from binary black hole mergers propagating at the speed of light without observable dispersion.^[36] This massless, non-dispersive propagation over billions of light-years aligns with predictions for spin-2 gravitons in quantum gravity, ruling out massive alternatives that would introduce frequency-dependent delays.^[37]

Axion-like Particles

Axion-like particles (ALPs) are hypothetical pseudoscalar bosons that generalize the axion, a particle originally proposed to resolve the strong CP problem in quantum chromodynamics (QCD), where the theory predicts a small but non-zero CP-violating parameter θ that is experimentally unobserved to high precision. In the Peccei-Quinn (PQ) theory introduced in 1977, a global U(1) symmetry dynamically adjusts θ to zero, with the axion emerging as the pseudo-Nambu-Goldstone boson associated with this spontaneously broken symmetry; in the original formulation, the axion was massless, but subsequent models incorporate a small explicit breaking via QCD effects, yielding an effective mass on the order of 10^{-5} eV or less. Ultra-light axions and ALPs, with masses approaching the massless limit (m_a ≲ 10^{-22} eV), maintain this pseudoscalar nature while exhibiting weak couplings to Standard Model fields, particularly photons via the term g_{aγγ} a F μν\tilde{F}^{μν}, where a is the axion field.^[38] This photon coupling enables axion-photon oscillations in the presence of magnetic fields, a process predicted by the PQ mechanism and central to detection strategies, where axions convert into detectable X-ray photons in strong laboratory or astrophysical magnetic fields. Beyond solving the strong CP problem—arising from the non-observation of CP violation in the neutral pion decay despite allowance in the QCD θ-term—axions and ALPs serve as motivated dark matter candidates, as their production via the misalignment mechanism in the early universe populates the cosmic axion density, with the massless limit ensuring long-term cosmological stability without decay or annihilation.^[39] In this context, their interactions with gluons, inherited from the QCD anomaly breaking the PQ symmetry, tie them to the strong sector while keeping couplings feeble enough to evade early universe dilution.^[38] Experimental searches for axions and ALPs have yielded no detections to date, imposing stringent constraints on their parameter space. The CERN Axion Solar Telescope (CAST) experiment, utilizing a 9-Tesla LHC dipole magnet to search for solar axions via photon conversion, has set limits on the axion-photon coupling g_{aγγ} < 5.8 × 10^{-11} GeV^{-1} (95% CL, as of 2024) for masses below 0.02 eV, excluding much of the original PQ axion model.^[40] Astrophysical observations further bound ultra-light axions, with supernova cooling arguments excluding ALPs in the approximate mass range 10^{-12} eV to 10^{-9} eV as dark matter candidates due to excess energy loss, while globular cluster dynamics and cosmic microwave background data impose additional constraints on couplings and masses.^[39] These bounds highlight the effectively massless regime where ALPs could still constitute all or part of dark matter, motivating ongoing haloscope and helioscope efforts. As of 2025, no axions or ALPs have been detected, with experiments like CAST and ADMX continuing to probe the parameter space.^[39]

Quasiparticles and Effective Descriptions

Phonons

Phonons are quasiparticle excitations that represent the quantized collective vibrations of atoms in a crystal lattice, emerging from the harmonic approximation of lattice dynamics in solids. These vibrations can be decomposed into normal modes, each characterized by a wavevector \mathbf{k} and frequency \omega(\mathbf{k}), with phonons serving as the quanta of these modes. In the long-wavelength limit, acoustic phonons exhibit a linear dispersion relation \omega = v |\mathbf{k}|, where v is the speed of sound in the material, which closely mimics the relativistic energy-momentum relation E = pc for massless particles with p = \hbar k and c analogous to v. This behavior arises because the effective mass of phonons vanishes in this regime, allowing them to propagate without a rest energy gap.^[41] In crystals with multiple atoms per unit cell, the phonon spectrum includes both acoustic and optical branches. Acoustic phonons, corresponding to in-phase motions of atoms within the unit cell, are inherently massless and act as Goldstone modes resulting from the spontaneous breaking of continuous translation symmetry when the lattice forms from a uniform medium. Optical phonons, involving out-of-phase motions, typically acquire a frequency gap at \mathbf{[k](/page/K)} = 0 due to interatomic forces, rendering them massive quasiparticles. The Goldstone nature ensures that acoustic phonons remain gapless, preserving the linear dispersion essential for their massless character.^[42] As bosonic quasiparticles with integer spin (effectively spin-0 or spin-1 polarizations depending on the mode), phonons obey Bose-Einstein statistics, permitting arbitrary occupation numbers in each mode without Pauli exclusion. This statistical behavior underpins their role in thermal properties, as modeled by the Debye theory, which approximates the phonon density of states with a linear dispersion up to a cutoff frequency and computes the specific heat contribution from phonon excitations. Debye's 1912 model predicts a low-temperature specific heat C_V \propto T^3 in three dimensions, arising from the integration over the Bose-Einstein distribution of phonon modes, which has been foundational for understanding lattice thermal capacity in insulators.^[43] Experimentally, the linear dispersion of acoustic phonons has been confirmed through inelastic neutron scattering techniques, which probe the dynamic structure factor to map \omega(\mathbf{k}) across the Brillouin zone, revealing sound-like propagation speeds matching macroscopic measurements. In electrically insulating materials, where electronic contributions are negligible, phonon heat transport dominates thermal conductivity, with mean free paths determined by scattering from defects, boundaries, and anharmonic interactions, enabling applications in thermoelectric devices and thermal management.^[44]^[45]

Other Excitations

In addition to phonons, other collective excitations in condensed matter systems can manifest massless characteristics through linear dispersion relations, where the energy-momentum relation follows E = v k (with v as the velocity and k the wavevector), analogous to relativistic massless particles.^[5] These excitations arise from cooperative behaviors of many particles, such as spins or electrons, and are described as quasiparticles in effective theories. Magnons, the quanta of spin waves in magnetically ordered materials, often exhibit massless modes in certain systems. In two-dimensional van der Waals honeycomb ferromagnets like CrCl₃, inelastic neutron scattering reveals gapless Dirac magnons at the Brillouin zone center, characterized by linear dispersion cones that emerge from the lattice symmetry and spin interactions, without requiring topological protection.^[46] These bosonic excitations propagate at speeds on the order of kilometers per second and contribute to low-energy spin dynamics, enabling applications in spintronics.^[47] In bulk CrCl₃, the magnon spectrum remains gapless, contrasting with gapped modes in related materials like CrI₃, due to weaker spin-orbit coupling.^[47] Plasmons, collective oscillations of electron density in metals or semiconductors, can also appear massless under specific conditions. A notable example is Pines' demon, a neutral acoustic plasmon predicted in 1956 and observed in 2023 in the layered metal Sr₂RuO₄ using momentum-resolved electron energy-loss spectroscopy (EELS).^[48] This three-dimensional excitation involves out-of-phase oscillations between multiple electron bands, resulting in zero net charge and a linear dispersion with velocities matching Fermi surface tangencies, confirming its massless nature.^[48] Such modes may influence electron pairing in unconventional superconductors like Sr₂RuO₄, where they provide a neutral channel for momentum transfer.^[48] These excitations highlight how effective massless descriptions extend beyond phonons to diverse many-body systems, unifying phenomena across magnetism and electron correlations while respecting the underlying lattice and interaction scales.

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