Weak hypercharge
In particle physics, weak hypercharge is a conserved quantum number associated with the abelian U(1)Y gauge symmetry in the electroweak sector of the Standard Model. It characterizes how fermions and bosons transform under this symmetry and is related to other fundamental quantities by the Gell-Mann–Nishijima formula adapted for electroweak interactions: the electric charge Q of a particle equals the third component of its weak isospin T^3 plus half its weak hypercharge Y/2, or Q = T^3 + Y/2.[1] Weak hypercharge assignments differ for left-handed and right-handed chiral fermion fields, underscoring the parity-violating nature of the weak interaction; for instance, the left-handed lepton doublet (νeL, eL) carries Y = -1, the left-handed quark doublet (uL, dL) has Y = 1/3, the right-handed electron has Y = -2, and the right-handed up quark has Y = 4/3.[2] The Higgs boson doublet, responsible for electroweak symmetry breaking, is assigned Y = 1.[1] These values ensure anomaly cancellation in the theory and determine the coupling strengths to the neutral gauge boson B_\mu via the term g' (Y/2) B_\mu in the covariant derivative, where g' is the U(1)Y coupling constant.[3] In the broader electroweak theory, weak hypercharge unifies with the non-abelian SU(2)L weak isospin symmetry to form the SU(2)L × U(1)Y gauge group, which is spontaneously broken by the Higgs vacuum expectation value to the U(1)EM electromagnetic symmetry.[4] This breaking generates masses for the W± and Z bosons (approximately 80 GeV and 91 GeV, respectively) while leaving the photon massless, with the Z boson coupling involving a combination of weak isospin and hypercharge currents.[3] Weak hypercharge thus underpins neutral weak currents, observed in processes like neutrino scattering, and is essential for the renormalizability and consistency of the Standard Model.[1]Definition and Formulation
Core Definition
In the Standard Model of particle physics, weak hypercharge, denoted Y_W, serves as an additive quantum number assigned to elementary particles, quantifying their interaction strength with the U(1)_Y gauge field and serving as the quantum number for the abelian U(1)_Y symmetry that complements the SU(2)_L weak isospin.[1] This quantum number is essential for classifying particles within the electroweak sector, where it complements weak isospin to determine overall charge properties.[1] The mathematical definition of weak hypercharge is given by the relation Y_W = 2(Q - T_3), where Q is the electric charge of the particle and T_3 is the third component of its weak isospin.[1] This formula arises from the structure of the electroweak gauge theory and ensures consistency with observed charge assignments.[1] Weak hypercharge originated in the Glashow-Weinberg-Salam model developed in the 1960s, which posits that the weak and electromagnetic interactions are unified under the non-Abelian gauge group SU(2)_L × U(1)_Y, with the U(1)_Y factor directly associated with weak hypercharge.[5][6] In this framework, Glashow introduced the SU(2) × U(1) structure in 1961 to extend symmetries of weak interactions, while Weinberg and Salam independently developed the full unification in 1967–1968, incorporating spontaneous symmetry breaking.[5][6] Through the Higgs mechanism, weak hypercharge facilitates the unification of electromagnetic and weak forces by enabling electroweak symmetry breaking, where the Higgs field—a complex SU(2)_L doublet with Y_W = 1—acquires a vacuum expectation value, generating masses for the W and Z bosons while leaving the photon massless.[1] This process preserves the conservation of weak hypercharge in interactions mediated by the unbroken U(1)_\text{em} subgroup.[1]Normalization Conventions
In the electroweak theory, the standard normalization of weak hypercharge Y_W is defined through the relation Q = T^3 + \frac{Y_W}{2}, where Q is the electric charge and T^3 is the third component of the weak isospin. This convention assigns Y_W = 1 to the Higgs doublet, ensuring that the upper component has Q = 1 and the lower component has Q = 0. The U(1)_Y gauge coupling g' enters the theory such that the hypercharge contribution to the fermion kinetic term in the Lagrangian is \overline{\psi} \gamma^\mu \left( \frac{g'}{2} Y_W \right) \psi B_\mu, where B_\mu is the hypercharge gauge field. An alternative half-scale convention defines Y' = \frac{Y_W}{2}, so that Q = T^3 + Y', with the Higgs doublet assigned Y' = \frac{1}{2}. In this framework, the covariant derivative includes the term -i g' Y' B_\mu, simplifying the notation for particle assignments: for instance, the left-handed quark doublet has Y' = \frac{1}{6} and the left-handed lepton doublet has Y' = -\frac{1}{2}. This choice is common in pedagogical treatments and aligns the hypercharge values more closely with the weak isospin components, facilitating the Higgs vacuum expectation value expression as \langle \phi^0 \rangle = \frac{v}{\sqrt{2}}, where v \approx 246 GeV is the electroweak scale.[7][8] The two conventions differ only in the rescaling of the hypercharge quantum number, with a corresponding adjustment in the definition of g' to preserve physical predictions. In Lagrangian terms, the standard convention yields the neutral current interaction involving \frac{g'}{2} Y_W \overline{f} \gamma^\mu f B_\mu, while the half-scale version uses g' Y' \overline{f} \gamma^\mu f B_\mu; both lead to identical electroweak phenomenology after gauge boson mixing. These normalizations are selected to ensure consistency with the definition of the Weinberg angle \theta_W, where \sin^2 \theta_W = \frac{g'^2}{g^2 + g'^2} \approx 0.231, matching precision electroweak measurements from Z-pole observables.[9][8]Quantum Number Relations
Link to Electric Charge and Isospin
In the electroweak theory, the weak hypercharge Y_W is intrinsically linked to the electric charge Q and the third component of weak isospin T_3 through the fundamental relation Q = T_3 + \frac{Y_W}{2}.[10] This equation, analogous to the Gell-Mann–Nishijima formula in quantum chromodynamics, ensures the consistent assignment of electric charges to particles within the SU(2)_L × U(1)_Y gauge structure. It was first proposed by Sheldon Glashow in his 1961 model of partial symmetries for weak interactions, where Y_W was introduced as an additional quantum number to unify weak and electromagnetic processes while accommodating both left- and right-handed currents.[10] In this framework, the U(1)_Y gauge group associated with Y_W mixes with the SU(2)_L neutral component to form the photon and Z boson fields after electroweak symmetry breaking. For left-handed fermion doublets under SU(2)_L, the relation quantizes charges by assigning T_3 = +\frac{1}{2} to the upper component (up-type) and T_3 = -\frac{1}{2} to the lower component (down-type), with a common Y_W for the doublet. This yields fractional charges that match observed values, such as Q = +\frac{2}{3} for up-type and Q = -\frac{1}{3} for down-type quarks when Y_W = \frac{1}{3}, or Q = 0 for neutrinos and Q = -1 for charged leptons when Y_W = -1. Right-handed fermions, transforming as SU(2)_L singlets with T_3 = 0, have Y_W = 2Q, directly tying their hypercharge to electric charge—for instance, Y_W = \frac{4}{3} for right-handed up-type quarks and Y_W = -2 for right-handed charged leptons. These assignments preserve charge conservation across chiral sectors, distinguishing the theory from purely left-handed models. The relation also governs neutral current interactions mediated by the Z boson, whose coupling to fermions is proportional to g_V = T_3 - 2 Q \sin^2 \theta_W for the vector part and g_A = T_3 for the axial-vector part, where \theta_W is the weak mixing angle. Here, Y_W enters indirectly through the definition of \sin^2 \theta_W = \frac{g'^2}{g^2 + g'^2}, with g' the U(1)_Y coupling constant, ensuring the Z boson is neutral under electromagnetism (Q = 0). This structure predicted neutral currents before their experimental discovery in 1973, validating the role of Y_W in suppressing right-handed contributions to certain processes while allowing parity violation.Connection to Baryon and Lepton Numbers
In the Standard Model, the weak hypercharge Y_W for fermions is determined by the relation Y_W = 2(Q - T_3), where Q is the electric charge and T_3 is the third component of weak isospin; this assignment ensures consistent charge quantization across left- and right-handed fields.[11] The specific values of Y_W for chiral multiplets—such as Y_W = 1/3 for left-handed quark doublets Q_L, Y_W = -1 for left-handed lepton doublets L_L, Y_W = 4/3 for right-handed up-type quarks u_R, Y_W = -2/3 for right-handed down-type quarks d_R, and Y_W = -2 for right-handed electrons e_R—are crucial for canceling all gauge anomalies in the electroweak sector, including the [\mathrm{SU}(2)_L]^2 \mathrm{U}(1)_Y, \mathrm{U}(1)_Y^3, and mixed \mathrm{SU}(3)_c^2 \mathrm{U}(1)_Y triangle diagrams across three generations.[11] This anomaly-free structure arises precisely from the interplay of these Y_W values with the representation content under \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y \times \mathrm{SU}(3)_c, rendering the theory quantum consistent without additional fields.[11] The connection to baryon number B and lepton number L emerges through the global \mathrm{U}(1)_B and \mathrm{U}(1)_L symmetries, which are vector-like classically but chiral at the quantum level due to the left-handed nature of electroweak interactions. Assigning B = 1/3 to quarks and L = 1 to leptons (with zeros otherwise), the mixed anomalies between these global currents and the electroweak gauge fields—particularly the Adler-Bell-Jackiw (ABJ) anomalies involving \mathrm{SU}(2)_L^2 \mathrm{U}(1)_B, \mathrm{SU}(2)_L^2 \mathrm{U}(1)_L, and \mathrm{U}(1)_Y^2 \mathrm{U}(1)_{B,L}—are nonzero and identical for B and L.[11] As a result, individual B and L are violated by instanton processes, with the anomaly equation taking the form \partial_\mu J^\mu_B = \frac{g_2^2}{32\pi^2} n_g \operatorname{Tr}(W_{\mu\nu} \tilde{W}^{\mu\nu}), where g_2 is the \mathrm{SU}(2)_L coupling, n_g = 3 counts generations, and W_{\mu\nu} is the weak field strength; an analogous equation holds for L.[11] However, the combination B - L experiences no such anomaly because the contributions cancel, making \mathrm{U}(1)_{B-L} an exact perturbative symmetry; the Y_W assignments ensure this cancellation by balancing the chiral contributions from quarks (with B = 1/3) and leptons (with L = 1) in the mixed gravitational-\mathrm{U}(1)_Y and other diagrams.[11] In contrast to the strong hypercharge Y in quantum chromodynamics, which incorporates flavor dependencies like Y = B + S + \frac{C + B' + T}{3} for classifying hadrons and accounting for strangeness S, the weak hypercharge Y_W is flavor-universal and ignores hyperflavor quantum numbers, applying the same Y_W values across all three generations without reference to strangeness or other internal symmetries. Non-perturbative effects further tie Y_W to B and L via sphaleron processes, which are static saddle-point solutions in the \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y gauge fields that mediate the transition between topologically distinct vacua. These processes violate B + L by \Delta (B + L) = 2 n_g = 6 units while conserving B - L, with the rate suppressed by a factor \exp(-8\pi^2 / g_2^2) at zero temperature but unsuppressed at electroweak-scale temperatures relevant to early-universe dynamics. In electroweak baryogenesis, such sphalerons equilibrate B + L unless perturbed by the phase transition, linking the hypercharge gauge dynamics directly to the observed baryon asymmetry through B - L preservation.Particle Assignments
Fermions
In the Standard Model, the weak hypercharge Y_W is assigned to fermions according to their chiral representations under the electroweak gauge group SU(2)_L \times U(1)_Y. Left-handed fermions transform as doublets under SU(2)_L, while right-handed fermions are singlets. These assignments ensure consistency with the electric charge formula Q = T_3 + Y_W / 2, where T_3 is the third component of weak isospin, and apply identically across all three generations of fermions.[12][2] For quarks, the left-handed fields form doublets (u_L, d_L) (and analogously for charm-strange and top-bottom pairs) with Y_W = +1/3. The right-handed up-type quarks u_R (and similarly c_R, t_R) have Y_W = +4/3, while right-handed down-type quarks d_R (and s_R, b_R) have Y_W = -2/3. Quarks also carry color charge under SU(3)_C, but Y_W is the same for each of the three color components and independent of color.[12][13] For leptons, the left-handed fields form doublets (\nu_L, e_L) (and similarly for muon and tau pairs) with Y_W = -1. The right-handed charged leptons e_R (and \mu_R, \tau_R) have Y_W = -2. The minimal Standard Model does not include right-handed neutrinos, but in extensions accommodating neutrino masses, sterile right-handed neutrinos \nu_R are singlets with Y_W = 0.[12][14] The following table summarizes the assignments for one generation of fermions (replicated for the other generations), verifying the charges via Q = T_3 + Y_W / 2. For quarks, the values apply per color triplet.| Fermion Field | SU(2)_L Rep. | T_3 | Y_W | Q |
|---|---|---|---|---|
| (u_L, d_L) doublet | 2 | +1/2 (u_L) -1/2 (d_L) | +1/3 | +2/3 (u_L) -1/3 (d_L) |
| u_R | 1 | 0 | +4/3 | +2/3 |
| d_R | 1 | 0 | -2/3 | -1/3 |
| (\nu_L, e_L) doublet | 2 | +1/2 (\nu_L) -1/2 (e_L) | -1 | 0 (\nu_L) -1 (e_L) |
| e_R | 1 | 0 | -2 | -1 |
| \nu_R (if present) | 1 | 0 | 0 | 0 |