Baryon number

In particle physics, the baryon number (denoted B) is an additive quantum number that labels particles based on their quark content, with quarks assigned B = +1/3 and antiquarks B = -1/3, while leptons and other non-quark particles have B = 0.^[1] This assignment ensures that baryons, composed of three quarks (such as the proton with quark content uud or the neutron with udd), have B = +1, antibaryons have B = -1, and mesons (quark-antiquark pairs) have B = 0.^[1] The concept of baryon number emerged in the quark model proposed independently by Murray Gell-Mann and George Zweig in 1964 to explain the observed patterns of hadron masses and quantum numbers.^[1] In the Standard Model of particle physics, baryon number is conserved at the classical level due to an accidental global U(1)_B symmetry in the Lagrangian, making it a key tool for classifying particles and predicting interaction outcomes.^[2] However, quantum anomalies associated with the electroweak interaction allow for processes that violate baryon number by multiples of three (\Delta B = 3n), though such effects are suppressed at low energies and not observed in experiments.^[2] Conservation of baryon number explains the stability of protons and neutrons against decay into non-baryonic particles and imposes strict selection rules in strong and electromagnetic interactions.^[2] Experimental searches for baryon number violation, such as proton decay (e.g., p \to e^+ \pi^0) or neutron-antineutron oscillations, have set stringent lower limits on lifetimes, with the proton lifetime exceeding $2.4 \times 10^{34} years at 90% confidence level from Super-Kamiokande data (as of 2020).^[2] Beyond the Standard Model, theories like Grand Unified Theories predict baryon number violation through unification of quarks and leptons, potentially linking it to the observed baryon asymmetry in the universe, where B - L (with lepton number L) remains conserved.^[2] These aspects underscore baryon number's role in probing fundamental symmetries and the matter-antimatter imbalance.^[2]

Fundamentals

Definition

The baryon number, denoted as B, is an additive quantum number in particle physics that is conserved in all known interactions within the Standard Model. It is assigned a value of +1 to baryons, -1 to antibaryons, and $0 to non-baryonic particles such as mesons and leptons.^[3] In the framework of the quark model, the baryon number is formally defined as
B = \frac{n_q - n_{\bar{q}}}{3},
where n_q is the number of quarks and n_{\bar{q}} is the number of antiquarks in the particle. Each quark carries a baryon number of +1/3, while each antiquark carries -1/3. For a composite particle composed of quarks and antiquarks, this is equivalently expressed as
B = \sum_i b_i,
where b_i = +1/3 for the i-th quark and b_i = -1/3 for the i-th antiquark.^[1] Unlike other quantum numbers such as electric charge, which couples to the electromagnetic field, or isospin, which approximates the symmetry between up and down quarks in strong interactions, the baryon number specifically quantifies the net quark content and serves as a key criterion for classifying hadrons into those with B = \pm 1 (baryons and antibaryons) and those with B = 0 (mesons).^[1] This conservation of B is observed in strong, weak, and electromagnetic interactions.^[3]

Historical Development

The recognition of baryon number conservation emerged in the pre-quark era during the 1930s, rooted in observations of nuclear reactions where the total number of nucleons (protons and neutrons) remained constant, even in processes like beta decay. This empirical pattern was formalized in 1938 by Ernst Stueckelberg, who proposed baryon number as a conserved quantum number to explain the stability of the proton, the lightest known baryon, preventing hypothetical decays such as p → e⁺ + γ that would otherwise be allowed by existing conservation laws. Stueckelberg assigned a baryon number of +1 to protons and neutrons (and later to other baryons) and -1 to their antiparticles, ensuring the total baryon number is invariant in interactions, a principle that extended the earlier nucleon conservation observed in nuclear physics.^[4] The concept evolved significantly in the 1950s with the discovery of strange particles in cosmic rays, such as the kaons and lambda baryon, which exhibited unexpectedly long lifetimes despite apparent strong interaction decays. To resolve this "strangeness problem," Abraham Pais introduced the quantum number of strangeness in 1952, conserved in strong interactions but violated in weak ones, a law independently formulated by T. Nakano and K. Nishijima in 1953 and refined by Murray Gell-Mann in 1956 through associated production mechanisms. This conservation law, integrated with isospin symmetry, highlighted patterns in particle multiplicities and masses, setting the stage for broader classifications.^[5] By the early 1960s, these developments culminated in the Eightfold Way scheme, proposed by Gell-Mann and Yuval Ne'eman in 1961–1962, which organized hadrons into SU(3) flavor symmetry multiplets—such as the baryon octet and decuplet—incorporating strangeness, baryon number, and charge to predict particle properties and relations like the Gell-Mann–Okubo mass formula. This classification resolved puzzles in hadron spectroscopy, including the existence of high-strangeness particles, and influenced the introduction of the quark model. In 1964, Gell-Mann and George Zweig independently proposed quarks as fundamental constituents with fractional baryon numbers of +1/3, explaining hadron multiplicities as combinations of three quarks for baryons (e.g., the Δ++ as three up quarks, fitting the decuplet's isospin 3/2 structure) and quark-antiquark pairs for mesons, while preserving overall integer baryon numbers. This framework, building directly on the Eightfold Way and strangeness conservation, provided a unified explanation for the observed particle spectrum and decay patterns.^[6]^[7] The quark model's assignment of fractional baryon numbers addressed longstanding issues, such as the symmetric quantum numbers of the Δ++ baryon, which challenged field theory antisymmetry requirements for identical fermions; although color charge was later invoked to fully resolve wavefunction symmetries, the 1964 proposal established baryon number as an additive quark property central to hadron structure. This historical progression from empirical nucleon conservation to the quark-based formalization underpins the modern understanding of baryon number in particle physics.^[8]

Assignment to Particles

Baryons and Antibaryons

Baryons are composite particles consisting of three quarks (qqq) arranged in a color-singlet state, and they are assigned a baryon number of B = +1.^[1] The proton, with quark content uud, exemplifies this assignment, carrying B = +1.^[1] Similarly, the neutron (udd) and the hyperon \Lambda (uds) each have B = +1, reflecting their three-quark composition where each quark contributes B = +1/3.^[1] Antibaryons, the antiparticles of baryons, are composed of three antiquarks (\bar{q}\bar{q}\bar{q}) and carry a baryon number of B = -1.^[1] For instance, the antiproton (\bar{u}\bar{u}\bar{d}) has B = -1, as each antiquark contributes B = -1/3.^[1] This assignment ensures that baryon number is additive in particle-antiparticle pairs, maintaining overall conservation in interactions.^[1] Exotic baryons, such as pentaquarks with a quark content of four quarks and one antiquark (qqqq\bar{q}), also possess B = +1, arising from the net three quarks after accounting for the antiquark.^[9] The LHCb collaboration observed such hidden-charm pentaquark states in 2019 through the decay \Lambda_b^0 \to J/\psi \, [p](/page/P′′) \, K^-, including P_c(4312)^+, P_c(4440)^+, and P_c(4457)^+, with masses around 4312 MeV, 4440 MeV, and 4457 MeV, respectively.^[10] In 2023, LHCb also reported the observation of a neutral state P_c(4338)^0 in the same decay mode, with mass 4338 MeV.^[10] These pentaquarks consist of two up quarks, one down quark, one charm quark, and one anticharm quark, confirming their baryonic nature.

Quarks and Mesons

In the quark model, the fundamental constituents of hadrons, the six types of quarks—up (u), down (d), strange (s), charm (c), bottom (b), and top (t)—each possess a baryon number B = +\frac{1}{3}.^[1] Their corresponding antiquarks carry B = -\frac{1}{3}.^[1] These fractional assignments reflect the additive nature of baryon number under the strong interaction, ensuring that composite particles achieve integer values consistent with experimental observations. Mesons, which are bound states of a quark and an antiquark (q\bar{q}), thus have a total baryon number of B = 0, as the contributions from the quark and antiquark cancel.^[1] For instance, the charged pion \pi^+ consists of a u\bar{d} pair, yielding B = \frac{1}{3} - \frac{1}{3} = 0, which distinguishes mesons from baryonic matter and explains their role as mediators in strong decays rather than stable nucleons like the proton.^[1] The baryon number relates directly to the net quark content, defined as B = \frac{1}{3}(N_q - N_{\bar{q}}), where N_q is the number of quarks and N_{\bar{q}} is the number of antiquarks in the hadron.^[1] This formulation resolves the need for integer baryon numbers in observed hadrons while accommodating the fractional values of their constituents.

Conservation

Theoretical Basis

In the Standard Model of particle physics, baryon number conservation arises as a consequence of a global U(1)_B symmetry in the Lagrangian, under which quarks transform with phase e^{i \alpha / 3} while leptons remain invariant.^[11] This symmetry commutes with the strong, weak, and electromagnetic interactions, ensuring that baryon number is preserved in all perturbative processes described by the theory.^[12] The U(1)_B transformation acts separately on left- and right-handed quark fields, reflecting the vector-like nature of the symmetry in quantum chromodynamics (QCD) and the electroweak sector.^[13] Unlike the electromagnetic charge, which corresponds to a gauged U(1)_{EM} symmetry, U(1)_B remains a global symmetry and is not promoted to a local gauge interaction in the Standard Model.^[11] This global status renders it an accidental symmetry, emerging from the specific separation of quarks (which carry baryon number) from leptons (which do not) in the fermion content of the theory, without being fundamentally imposed by the gauge structure.^[12] In the absence of higher-dimensional operators or new physics, this accidental nature protects baryon number at the renormalizable level of the Lagrangian.^[14] The conserved quantity associated with U(1)_B is encoded in the baryon number current, given by

J^\mu_B = \frac{1}{3} \sum_f \bar{q}_f \gamma^\mu q_f,

summed over quark flavors. In perturbation theory, the divergence of this current vanishes, \partial_\mu J^\mu_B = 0, confirming exact conservation at tree level and in loop diagrams without non-perturbative effects.^[11] However, non-perturbative effects introduce violations of baryon number through anomalies in the electroweak sector. Specifically, sphaleron processes—topological transitions in the SU(2)_L gauge fields—mediate baryon number violation while preserving B - L, as first detailed by Kuzmin, Rubakov, and Shaposhnikov.^[15] These processes are highly suppressed at low energies, far below the electroweak scale, due to the exponential Boltzmann factor e^{-E_{\rm sph}/T}, where E_{\rm sph} \approx 9-12 TeV is the sphaleron energy.^[12] Thus, baryon number remains effectively conserved in observable phenomena within the Standard Model at accessible energies.^[11]

Experimental Verification

The conservation of baryon number was empirically established in early nuclear and particle physics experiments, where no violations were observed in processes such as beta decay, in which a neutron transforms into a proton, electron, and antineutrino while maintaining total baryon number B = 1.^[11] Similarly, interactions involving pions, such as the reaction \pi^- + p \to n + \pi^0 or pion absorption in nuclei, consistently preserved baryon number in cosmic ray observations and early accelerator experiments at facilities like Berkeley's Bevatron. Searches for baryon number violation through proton decay, a \Delta B = 1 process predicted in some grand unified theories, have yielded stringent lower limits on the proton lifetime. The Super-Kamiokande experiment, using a 0.417 megaton-year exposure of water Cherenkov detection, has set a lower limit of \tau / B(p \to e^+ \pi^0) > 2.4 \times 10^{34} years at 90% confidence level, with no candidate events observed. As reported in 2020, this remains one of the most sensitive probes for \Delta B = 1 violations, far exceeding the age of the universe and constraining models beyond the Standard Model.^[16] Experiments probing \Delta B = 2 processes, such as neutron-antineutron oscillations, have also confirmed the robustness of baryon number conservation. The Institut Laue-Langevin (ILL) experiment in the early 1990s, using a high-flux cold neutron beam of intensity $10^{11} n/s and a magnetically shielded storage volume, established a lower limit on the oscillation time of \tau_{n\bar{n}} > 8.6 \times 10^7 seconds at 90% confidence level, corresponding to an oscillation probability below $10^{-7}.^[17] More recently, Super-Kamiokande has set a limit \tau_{n\bar{n}} > 4.7 \times 10^8 seconds (90% CL) inferred from bound neutron searches.^[18] No oscillations were detected, providing bounds on physics at energy scales around 100-500 TeV depending on the underlying model. At high-energy colliders, proton-proton collisions at the Large Hadron Collider (LHC) have shown no evidence of \Delta B \neq 0 processes, upholding conservation up to TeV scales. A 2024 CMS search using 138 fb^{-1} of data at \sqrt{s} = 13 TeV analyzed top quark production and decay for baryon number-violating signatures, such as same-sign dileptons with multiple jets, setting limits on effective operators with coefficients below $10^{-9} GeV^{-2} and excluding certain models up to mediator masses of several TeV.^[19] These null results from ATLAS and CMS collaborations reinforce baryon number as an approximate symmetry in high-energy environments.

Implications and Extensions

Role in Particle Physics

In particle physics, the baryon number plays a central role in classifying hadrons within the framework of the quark model and the Standard Model. The Particle Data Group (PDG) utilizes baryon number to categorize baryons, assigning B = 1 to all members of the baryon octet (such as the proton, neutron, \Lambda, \Sigma, and \Xi) and the baryon decuplet (including the \Delta, \Sigma^*, \Xi^*, and \Omega). These groupings arise from the SU(3) flavor symmetry of light quarks (u, d, s), where the octet consists of spin-1/2 states and the decuplet of spin-3/2 states, ensuring that baryon number distinguishes these composite particles from mesons, which have B = 0.^[1] Baryon number also imposes strict selection rules on particle decays and interactions, prohibiting processes that would violate its conservation. For instance, the decay n \to e^- + \pi^+ is forbidden because it changes the baryon number by \Delta B = 1 (from B = 1 for the neutron to B = 0 for the electron and pion) while also violating lepton number conservation by \Delta L = 1. In contrast, weak decays of baryons, such as the neutron beta decay n \to p + e^- + \bar{\nu}_e and semileptonic decays of heavier baryons, are allowed as they preserve baryon number (\Delta B = 0). These selection rules are empirically verified and form a cornerstone of hadron spectroscopy.^[2] The baryon number is integrated with other quantum numbers in the Gell-Mann–Nishijima formula, which relates electric charge Q, isospin third component I_3, and hypercharge Y: Q = I_3 + \frac{Y}{2}, where Y = B + S + C + B' + T (with S, C, B', and T denoting strangeness, charm, bottomness, and topness, respectively). This relation, originally formulated for strange particles and extended to include baryon number, enables the systematic assignment of charges to hadrons and unifies the classification of baryons and mesons within flavor SU(3) multiplets. While baryon number is conserved in all Standard Model interactions, extensions beyond it, such as grand unified theories (GUTs), suggest scenarios where B may not be strictly conserved, but the combination B - L (baryon minus lepton number) often remains invariant. In models like SO(10) GUTs, B - L emerges as a gauged symmetry, protecting against rapid proton decay while allowing processes like neutrino Majorana mass generation that violate L but preserve B - L. This feature hints at subtle non-conservation of B in high-energy regimes, motivating searches for rare processes.^[12]

Cosmological Applications

In cosmology, the conservation of baryon number plays a pivotal role in explaining the observed matter-antimatter asymmetry in the universe, quantified by the baryon-to-photon ratio \eta \approx 6.1 \times 10^{-10}, which indicates that for every billion photons, there is roughly one baryon.^[20] This asymmetry, essential for the existence of ordinary matter, requires mechanisms beyond the standard model to generate a net baryon number during the early universe, as the initial conditions of the Big Bang would otherwise produce equal amounts of matter and antimatter that annihilate.^[21] Baryogenesis refers to these processes, which must satisfy Sakharov's three conditions: baryon number violation, charge conjugation (C) and charge-parity (CP) violation, and departure from thermal equilibrium, allowing irreversible interactions to preferentially produce baryons over antibaryons.^[22] Big Bang Nucleosynthesis (BBN), occurring minutes after the Big Bang, provides a key probe of baryon number conservation by linking the primordial baryon density to the abundances of light elements like deuterium, helium-4, and lithium-7. The baryon density parameter \Omega_b h^2, derived from BBN, is constrained to approximately 0.0224, reflecting the freeze-out of weak interactions that set the neutron-to-proton ratio and subsequent nuclear reactions.^[20] This value aligns closely with independent measurements from the cosmic microwave background (CMB), confirming that baryons constitute about 4-5% of the universe's total energy density, with the remainder dominated by dark matter and dark energy.^[23] Violations of baryon number conservation at BBN scales are tightly bounded, as they would alter predicted element abundances inconsistent with observations.^[24] Baryon acoustic oscillations (BAO) represent another cosmological application, arising from pressure waves in the primordial baryon-photon plasma that imprinted a characteristic scale of about 150 Mpc on the distribution of galaxies and large-scale structure.^[25] These oscillations, frozen after recombination, serve as a "standard ruler" for measuring the universe's expansion history and the equation of state of dark energy, with baryon number conservation ensuring the stability of this fossilized signal across cosmic time.^[26] Surveys like the Sloan Digital Sky Survey have detected BAO peaks with high precision, yielding distance measurements accurate to 1-2% and reinforcing the \LambdaCDM model's predictions for baryon contributions to structure formation.^[25] Overall, these applications underscore baryon number's role in unifying early-universe physics with late-time observations, constraining extensions to the standard model that might relax its conservation.^[21]