Hypercharge
In particle physics, hypercharge (Y) is an additive quantum number that classifies hadrons and accounts for conservation laws in strong and electromagnetic interactions, particularly those involving particles with strangeness.[1] It is defined as the sum of the baryon number (B) and strangeness (S), such that Y = B + S, with extensions to include other flavor quantum numbers like charm and bottomness in the full quark model.[2][3] This quantum number remains conserved under strong and electromagnetic forces but is violated in weak interactions, explaining the relatively long lifetimes of strange particles like the lambda baryon (Λ⁰), which cannot decay via the dominant strong force due to hypercharge mismatch.[4][5] Introduced in the mid-1950s amid discoveries of strange particles in cosmic rays and accelerators, hypercharge provided a framework to organize hadrons into multiplets under the approximate SU(3) flavor symmetry, alongside isospin, as part of the "eightfold way" developed by Murray Gell-Mann and Yuval Ne'eman.[6] In the quark model proposed in 1964, hypercharge arises naturally from the properties of quarks: up and down quarks have Y = 1/3, while strange quarks have Y = -2/3, enabling the construction of baryons and mesons with integer or zero hypercharge values.[2] This classification predicted the existence of particles like the Ω⁻ baryon, later discovered experimentally, validating the model.[6] Distinct from the original flavor hypercharge, weak hypercharge (Y_W) emerged in the 1960s electroweak theory as a fundamental quantum number associated with the U(1)_Y gauge symmetry in the Standard Model's SU(3)_c × SU(2)_L × U(1)_Y structure.[7] It relates electric charge (Q), the third component of weak isospin (T_3), and itself via the Gell-Mann–Nishijima formula Q = T_3 + Y_W/2, ensuring consistency across left-handed fermion doublets and right-handed singlets.[8] For example, the left-handed electron-neutrino doublet has Y_W = -1, while right-handed electrons have Y_W = -2.[8] Weak hypercharge couples particles to the B boson, which mixes with the W^3 boson to form the photon and Z boson after electroweak symmetry breaking.[7] This unification, proposed by Sheldon Glashow, Abdus Salam, and Steven Weinberg, earned the 1979 Nobel Prize and underpins modern predictions of particle interactions.[9]Core Concepts
Definition
Hypercharge (Y) is a fundamental quantum number in particle physics that is conserved under strong interactions, originally introduced to account for patterns observed in the spectroscopy of hadrons containing strange quarks.[10] It serves as one of the key labels in the classification of particles within flavor symmetry groups, extending beyond the isospin quantum number to incorporate additional flavor degrees of freedom.[11] The term "hypercharge" was coined by Murray Gell-Mann in 1961 as part of his development of the eightfold way, a scheme based on SU(3) flavor symmetry that organizes baryons and mesons into multiplets.[12] In its original formulation, applicable to the three lightest quark flavors (up, down, and strange), hypercharge is defined mathematically as Y = B + S, where B is the baryon number (measuring the number of quarks minus antiquarks divided by three) and S is the strangeness quantum number (assigning -1 to each strange quark and +1 to each antistrange quark).[11] This definition ensures that Y takes integer or half-integer values consistent with the observed hadron multiplets and remains invariant under strong processes.[13] With the discovery of heavier quarks, the definition of hypercharge has been extended to encompass all six quark flavors while preserving its role as a conserved quantity under strong interactions. The generalized form is Y = B + S + C + B' + T, where C denotes charm (positive for charm quarks), B' denotes bottomness (negative for bottom quarks), and T denotes topness (positive for top quarks).[13] The specific signs in this expression follow conventions for the flavor quantum numbers that align with their definitions—such as the negative assignment for strangeness in strange quarks—to guarantee conservation in strong interactions, which do not alter quark flavors, thereby maintaining Y as an additive combination across generations.[14] Hypercharge is an additive quantum number for composite particles, meaning the total Y of a hadron is the sum of the Y values of its constituent quarks.[14] Gauge bosons mediating strong and electromagnetic interactions, such as gluons and photons, carry zero hypercharge, reflecting their lack of baryon number and flavor content.[13]Relation to Electric Charge and Isospin
The hypercharge Y provides a fundamental relation to the electric charge Q and the third component of isospin I_3 through the Gell-Mann–Nishijima formula: Q = I_3 + \frac{1}{2} Y This equation, proposed independently by Kazuhiko Nishijima (in collaboration with Tadao Nakano) and Murray Gell-Mann in 1953, expresses the electric charge as a linear combination of conserved quantum numbers under strong interactions.[15][16] Intuitively, the formula arises from the structure of isospin doublets, where particles differing only in charge form pairs with I = \frac{1}{2}. For the upper member of such a doublet (I_3 = +\frac{1}{2}), Q = \frac{1}{2} + \frac{1}{2} Y; for the lower (I_3 = -\frac{1}{2}), Q = -\frac{1}{2} + \frac{1}{2} Y. Solving for Y yields the same value for both, ensuring the doublet shares a common hypercharge while accommodating the charge difference of 1.[10] In isospin multiplets of arbitrary size, the average electric charge \bar{Q} across the members (where the average I_3 = 0) directly determines the hypercharge via Y = 2 \bar{Q}. This relation underscores hypercharge as the "center-of-charge" offset for the multiplet, independent of the specific isospin projection.[10] The formula predates the full development of SU(3) flavor symmetry, serving as a model-agnostic tool to classify particles and predict charge assignments based on observed isospin patterns.[16] The conservation properties of these quantum numbers have profound implications for interaction symmetries. Strong interactions preserve both I_3 (via approximate SU(2) isospin invariance) and Y (via absence of flavor-changing processes), thereby ensuring the conservation of electric charge Q. This additive conservation holds universally in strong processes, distinguishing them from weak interactions where I_3 and Y can change. At the quark level, the hypercharge of composite hadrons follows an additive rule based on quark content, without requiring detailed derivations from group theory. Specifically, Y = \frac{1}{3}(n_u + n_d) - \frac{2}{3}(n_s + n_b) + \frac{4}{3}(n_c + n_t), where n_i denotes the net number of quarks of flavor i (quarks minus antiquarks). This composite expression aligns with individual quark hypercharges—Y = \frac{1}{3} for u, d; Y = -\frac{2}{3} for s, b; Y = \frac{4}{3} for c, t—and reproduces the Gell-Mann–Nishijima relation when combined with isospin assignments.Hypercharge in Strong Interactions
Role in SU(3) Flavor Symmetry
The SU(3) flavor symmetry extends the SU(2) isospin symmetry to incorporate the strangeness quantum number, treating the up, down, and strange quarks as transforming under the fundamental representation 3 of the group, with the third component of isospin I_3 and hypercharge Y serving as key quantum numbers for particle classification.[17] This framework, proposed independently by Gell-Mann and Ne'eman in 1961, organizes hadrons into irreducible representations where hypercharge, defined as Y = B + S for light quarks (with B the baryon number and S the strangeness), combines baryon number conservation with strangeness to form a conserved quantity under strong interactions.[14] In weight diagrams, particles are plotted in the I_3-Y plane, with hypercharge along the vertical axis, revealing the structure of multiplets such as the octet (dimension 8) or decuplet (dimension 10), where non-strange states occupy Y=0 levels and strange states Y=-1.[14] These diagrams arise from the eigenvalues of the Cartan subalgebra generators of SU(3), specifically I_3 \propto \lambda_3/2 and Y \propto \lambda_8 / \sqrt{3}, where \lambda_3 and \lambda_8 are Gell-Mann matrices, allowing the identification of symmetry-related states within each representation.[17] Representations are labeled by Dynkin coefficients (\lambda_1, \lambda_2), such as (1,1) for the octet and (3,0) for the decuplet, providing a systematic way to enumerate possible hadron configurations.[14] Electromagnetic and weak interactions break the SU(3) flavor symmetry, while strong interactions approximately preserve it, conserving hypercharge as an additive quantum number.[14] Prior to the quark model, this symmetry proved invaluable in 1960s hadron spectroscopy for explaining mass splittings and decay patterns through first-order perturbations in the symmetry-breaking Hamiltonian.[17] Although quantum chromodynamics (QCD) later superseded SU(3) as the fundamental theory of strong interactions, hypercharge remains a useful approximate quantum number for phenomenological analyses of light hadron spectra.[14]Assignments and Examples
In the SU(3) flavor symmetry of strong interactions, hypercharge Y is assigned to quarks based on their baryon number B and strangeness S, with Y = B + S. The up quark u and down quark d each have B = 1/3 and S = 0, yielding Y = +1/3 for both. The strange quark s has B = 1/3 and S = -1, resulting in Y = -2/3.[18][19] These quark hypercharges combine additively to determine the values for hadrons. For instance, the proton (uud) has B = 1 and S = 0, so Y = 1; similarly, the neutron (udd) has Y = 1. The lambda baryon \Lambda (uds) has B = 1 and S = -1, giving Y = 0. Among mesons, the neutral pion \pi^0 (a combination including u\bar{u}) has B = 0 and S = 0, so Y = 0; the positively charged kaon K^+ (u\bar{s}) has B = 0 and S = +1, yielding Y = 1.[18][19] In the baryon octet, the nucleons (proton and neutron) have Y = 1; the sigma baryons \Sigma (e.g., uus) and lambda \Lambda (uds) have Y = 0; and the xi baryons \Xi (e.g., uss) have Y = -1. For the baryon decuplet, the delta resonances \Delta have Y = 1; the sigma-star \Sigma^* have Y = 0; the xi-star \Xi^* have Y = -1; and the omega-minus \Omega^- (sss) has B = 1 and S = -3, so Y = -2. The meson nonet includes pions and rho mesons with Y = 0; kaons K with Y = +1; and anti-kaons \bar{K} with Y = -1.[18][19] Hypercharge conservation in strong processes is illustrated by decays such as \Delta \to N \pi, where the initial delta has Y = 1, and the final nucleon N and pion \pi each contribute Y = 1 and Y = 0, respectively, preserving the total Y = 1.[18] Extensions to heavy quarks modify the hypercharge formula to Y = B + S - C/3, where C is the charm quantum number, to accommodate SU(4) flavor considerations while maintaining approximate symmetry patterns. The charm quark c is assigned C = +1, leading to charmed hadrons like the lambda-charmed baryon \Lambda_c^+ (udc) with B = 1, S = 0, and C = 1, yielding Y = 2/3. Similarly, up-type quarks c and top t are treated analogously to u with effective Y = +1/3 in light-flavor limits, while the down quark d also has Y = +1/3, and bottom-type b aligns with s at Y = -2/3.[18]Weak Hypercharge in Electroweak Theory
Definition and Particle Assignments
In the electroweak sector of the Standard Model, weak hypercharge Y_W serves as the quantum number associated with the abelian U(1)_Y gauge symmetry, forming part of the full gauge group SU(2)_L \times U(1)_Y. This local gauge symmetry is mediated by the neutral gauge boson B_\mu, which, following electroweak symmetry breaking, mixes with the third component of the SU(2)_L gauge boson triplet W^3_\mu to produce the massless photon A_\mu and the massive Z boson.[20] Unlike the flavor hypercharge employed in the global SU(3) symmetry of strong interactions—which classifies quarks and hadrons based on baryon number and strangeness—weak hypercharge is a gauged quantity specific to electroweak processes, with its normalization chosen such that the U(1)_Y coupling constant g' enters interactions proportional to Y_W / 2. The electric charge Q of particles relates to the third component of weak isospin I_3 and weak hypercharge via the electroweak analogue of the Gell-Mann–Nishijima formula: Q = I_3 + Y_W / 2.[20] Specific assignments of Y_W to the fermionic and scalar fields ensure consistency with observed charges and enable the unification of weak and electromagnetic interactions. These values are identical across the three generations of fermions.| Field | SU(2)_L Representation | Y_W |
|---|---|---|
| Left-handed quark doublet Q_L = (u_L, d_L) (and analogs for c, s; t, b) | Doublet | $1/3 |
| Right-handed up-type quark u_R (and c_R, t_R) | Singlet | $4/3 |
| Right-handed down-type quark d_R (and s_R, b_R) | Singlet | -2/3 |
| Left-handed lepton doublet L_L = (\nu_e_L, e_L) (and analogs for \mu, \tau) | Doublet | -1 |
| Right-handed charged lepton e_R (and \mu_R, \tau_R) | Singlet | -2 |
| Higgs doublet H = (H^+, H^0) | Doublet | $1 |