Weak isospin & hypercharge

The two weak charges are now measured, not merely postulated. At LEP and the SLC, millions of $Z$ bosons produced on resonance measured the mixing angle to $\sin^2\theta_W^{\text{eff}}=0.23155(4)$ — a five-figure test of the very $T_3$/$Y_W$ couplings tabulated here, since the $Z$ couples to each fermion as $T_3-Q\sin^2\theta_W$. The hypercharge assignments are not decorative either: the vanishing of $\sum Q$ and $\sum Y_W$ over a generation is the anomaly-cancellation condition, so restrictive that it forces quarks (three colours, fractional charges) and leptons into matched families — the strongest hint that they are unified, as in the $SU(5)$ group where one multiplet fixes every hypercharge and explains $Q_p=-Q_e$ to one part in $10^{21}$. At the LHC the $W$-mass measurement (world average near $80.37\,$GeV, with the 2022 CDF value in unresolved tension) probes the electroweak fit at the part-per-thousand level, where any deviation would signal new weak-isospin or hypercharge content. Open questions stay sharp — why hypercharge is quantized at all, why three generations share these charges, and whether a right-handed neutrino (a pure singlet with $Y_W=0$) exists to give neutrinos mass.

Fix one generation and one chirality and the whole scheme is a ladder. The left-handed neutrino and electron are the two rungs of a single weak-isospin doublet, one unit of $T_3$ apart; a $W^\pm$ is the operator that climbs or descends that ladder, and because the rungs differ by one unit of electric charge, absorbing a $W$ necessarily changes the charge by one. The right-handed electron has no ladder to climb — it is a singlet, invisible to the $W$. The first figure draws that structure for the first generation and marks the exact quantum numbers on each rung.

The doublet fixes the weak isospin, but it does not fix the electric charge — the neutrino and electron sit at $T_3=\pm\tfrac12$ yet carry $Q=0$ and $Q=-1$, an asymmetry the SU(2) alone cannot produce. The missing ingredient is the second charge, weak hypercharge, added as an independent U(1)$_Y$ so that electric charge emerges as the combination $Q=T_3+\tfrac12 Y_W$. The second figure lays the entire first generation out in the plane of its two weak charges and shows that lines of constant electric charge are simply the diagonals of that plane — the visual content of the charge relation.

The starting point is the postulate of a local SU(2) symmetry acting on the left-handed fields. Demanding invariance under a position-dependent SU(2) rotation is what forces the gauge fields into existence, exactly as in QED and QCD. The transformation law of a left-handed doublet is

where: $\phi_L$ — a two-component left-handed weak-isospin doublet; $g_W$ — the SU(2) gauge coupling; $\boldsymbol\alpha(x)$ — three real gauge functions (one per generator); $\mathbf{T}=\tfrac12\boldsymbol\sigma$ — the three weak-isospin generators, half the Pauli matrices. Read it as: because there are three $2\times2$ generators, the field must be a two-component object and there must be three gauge parameters. The generators close into the SU(2) algebra, and the two off-diagonal combinations act as raising and lowering operators on the doublet. These are the algebraic seed of the physical charged bosons.

where: $\epsilon_{ijk}$ — the Levi-Civita symbol (the SU(2) structure constants); $T_\pm$ — ladder operators, with $T_+$ sending $e^-_L\!\to\nu_e$ and $T_-$ the reverse; $W^\pm_\mu$ — the physical charged gauge fields built from the real fields $W^{1,2}_\mu$. The $\mp$ sign convention fixes $W^+$ as the boson that raises electric charge. Weak hypercharge enters through a second, independent gauge field $B_\mu$ of a U(1)$_Y$. The full electroweak covariant derivative that replaces the ordinary $\partial_\mu$ carries both charges at once.

where: $\mathbf{W}_\mu=(W^1_\mu,W^2_\mu,W^3_\mu)$ — the SU(2) triplet of gauge fields; $B_\mu$ — the single U(1)$_Y$ gauge field; $g'$ — the hypercharge coupling; $Y_W$ — the weak hypercharge of whatever field $D_\mu$ acts on. For a right-handed singlet the $\mathbf{T}$ term vanishes and only the $B_\mu$ term survives — the singlet still feels hypercharge. The link between these abstract charges and the everyday electric charge is a single relation, structurally identical to Gell-Mann and Nishijima's formula for the strong isospin of hadrons. It is the definition of weak hypercharge.

where: $Q$ — electric charge in units of $e$; $T_3$ — the third component of weak isospin ($\pm\tfrac12$ for doublet members, $0$ for singlets); $Y_W$ — weak hypercharge. Read it as: neither $T_3$ nor $Y_W$ is the electric charge alone — $Q$ is the specific diagonal combination that will remain unbroken after the symmetry breaks. The three SU(2) generators produce three weak currents; the ladder combinations are the charged currents and the diagonal one is a neutral current.

where: $j^\mu_k$ ($k=1,2,3$) — the three SU(2) currents; $j^\mu_\pm$ — the charged currents that couple to $W^\pm$ and carry the familiar V$-$A form; $j^\mu_3$ — the diagonal neutral current built from $\sigma_3$. The charged currents step between doublet members; the neutral $j^\mu_3$ does not change the particle's identity. Finally, the neutral $W^3_\mu$ and the hypercharge $B_\mu$ are not themselves the observed neutral bosons: they mix by an angle $\theta_W$ to give the massless photon and the massive $Z$. This is developed fully in the Weinberg-angle and electroweak-unification articles and recalled here by the coupling relation it enforces.

where: $e$ — the electromagnetic coupling ($\sqrt{4\pi\alpha}$); $\theta_W$ — the weak mixing (Weinberg) angle, $\sin^2\theta_W\approx0.231$; $g_W,g'$ — the SU(2) and U(1)$_Y$ couplings. Read it as: electric charge is a derived quantity, a projection of the two more fundamental gauge couplings onto the unbroken direction.

1 · Hypercharge of the whole first generation (fully worked)

Using only $Q=T_3+\tfrac12 Y_W$, assign the weak hypercharge to every fermion multiplet of the first generation, and verify the assignment reproduces the known electric charges.

Step 1. Lepton doublet $(\nu_e,e^-)_L$. The doublet carries a single hypercharge. Apply $Y_W=2(Q-T_3)$ to either member: using the electron, $Y_W=2(-1-(-\tfrac12))=2(-\tfrac12)=\mathbf{-1}$; using the neutrino, $Y_W=2(0-\tfrac12)=-1$. Both agree, confirming they belong in one doublet.

Step 2. Right-handed electron $e^-_R$ (singlet, $T_3=0$, $Q=-1$): $Y_W=2(-1-0)=\mathbf{-2}$. Note this differs from the left-handed electron's $-1$: the two chiralities carry different hypercharge, the signature that U(1)$_Y$ is not electromagnetism.

Step 3. Quark doublet $(u,d)_L$ ($Q=+\tfrac23,-\tfrac13$; $T_3=\pm\tfrac12$): $Y_W=2(\tfrac23-\tfrac12)=2(\tfrac16)=\mathbf{+\tfrac13}$, identical for both members. Right-handed quark singlets: $Y_W(u_R)=2(\tfrac23)=\mathbf{+\tfrac43}$ and $Y_W(d_R)=2(-\tfrac13)=\mathbf{-\tfrac23}$.

Step 4. Verify by running the relation forward. For $u_L$: $Q=T_3+\tfrac12 Y_W=+\tfrac12+\tfrac12(\tfrac13)=+\tfrac12+\tfrac16=+\tfrac23$ ✓. For $e_R$: $Q=0+\tfrac12(-2)=-1$ ✓. Every state checks. As a global test, the generation charge sum $\sum Q=0+(-1)+3(\tfrac23)+3(-\tfrac13)=\mathbf{0}$ — anomaly cancellation. The hypercharges are $-1,\,-2,\,+\tfrac13,\,+\tfrac43,\,-\tfrac23$; all reproduce $Q$ via $Q=T_3+\tfrac12 Y_W$, and both $\sum Q$ and $\sum Y_W$ vanish.

2 · Weak isospin bookkeeping in beta decay

At the quark level, neutron beta decay is $d\to u+e^-+\bar\nu_e$. Show that the $W$ emission steps consistently through weak isospin and charge.

Step 1. The quark transition $d\to u$ is a step up the doublet ladder: $T_3(d_L)=-\tfrac12\to T_3(u_L)=+\tfrac12$, i.e. $\Delta T_3=+1$. The charge changes by $\Delta Q=(+\tfrac23)-(-\tfrac13)=+1$, so $\Delta T_3=\Delta Q=+1$ — the quark emits a $W^-$ to shed the extra positive charge.

Step 2. The leptonic side absorbs that $W^-$ by creating $e^-\bar\nu_e$. In the current $\bar u_e\gamma^\mu\tfrac12(1-\gamma^5)v_{\bar\nu}$ the $T_-$ operator connects $\nu_L$ ($T_3=+\tfrac12$) down to $e_L$ ($T_3=-\tfrac12$), again $\Delta T_3=-1$ paired with $\Delta Q=-1$. Charge balances across the vertex.

Step 3. Check hypercharge is untouched by the $W$. The $W$ carries $T_3=\pm1$ but $Y_W=0$, so weak hypercharge is conserved separately at the vertex; the quark doublet keeps $Y_W=+\tfrac13$ throughout. Only $T_3$ is stepped by the charged current ($\Delta T_3=\pm1$, always matched by $\Delta Q=\pm1$); hypercharge is a spectator, conserved at every charged-current vertex.

3 · From weak isospin to the electric charge: the Weinberg angle (fully worked)

Given the SU(2) coupling $g_W=0.653$ and $\sin^2\theta_W=0.231$, extract the hypercharge coupling $g'$ and the electromagnetic coupling $e$, and check the resulting fine-structure constant.

Step 1. From (6), $e=g_W\sin\theta_W$. With $\sin\theta_W=\sqrt{0.231}=0.4806$: $e=0.653\times0.4806=\mathbf{0.314}$.

Step 2. The hypercharge coupling follows from $e=g'\cos\theta_W$ with $\cos\theta_W=\sqrt{1-0.231}=0.877$: $g'=e/\cos\theta_W=0.314/0.877=\mathbf{0.358}$. Check the mixing directly: $\tan\theta_W=g'/g_W=0.358/0.653=0.548$, and $\sin^2\theta_W=\tan^2/(1+\tan^2)=0.300/1.300=0.231$ ✓.

Step 3. Form the fine-structure constant $\alpha=e^2/4\pi=(0.314)^2/12.566=0.0985/12.566=7.84\times10^{-3}=\mathbf{1/127.6}$. This is the running value near $q^2=M_Z^2$, not the low-energy $1/137$ — precisely the value electroweak fits use at the $Z$ pole. e = 0.314, g′ = 0.358, and α(M_Z) = 1/128, all following from the two weak couplings via the mixing angle — electric charge is a derived quantity.

Weak isospin and hypercharge are the two coordinates of one man's decisive idea: that the weak and electromagnetic forces are two faces of a single SU(2)$\times$U(1) gauge symmetry, with hypercharge the glue that makes electric charge come out right.

Texts. Thomson, Modern Particle Physics, §15.2 (the weak interaction gauge group) and §15.3 (electroweak unification), Eqs. (15.10)–(15.13) (the source this article grounds) · Halzen & Martin, Quarks and Leptons, §13.3–§13.6 (weak isospin, hypercharge, and the Weinberg–Salam model) · Griffiths, Introduction to Elementary Particles, 2nd ed., §9.6 · Peskin & Schroeder, An Introduction to QFT, §20.1–20.2 · Schwartz, QFT and the Standard Model, §29.1–29.2.

Provenance. The local-gauge machinery is recalled from aq-pp-local-gauge-principle; the chiral projections $\tfrac12(1\mp\gamma^5)$ from aq-pp-weyl-chirality and aq-pp-chiral-weak-structure; the charged-current V$-$A vertex from aq-pp-weak-charged-current; and the strong-isospin analogue of the charge formula from aq-pp-isospin. The $W^3$–$B$ mixing, the Weinberg angle, and the emergence of the photon and $Z$ are owned by aq-pp-electroweak-gauge-group, aq-pp-neutral-current, aq-pp-weinberg-angle, and aq-pp-electroweak-unification (forward references, recalled by name). A verifier should confirm: (1) the hypercharges $-1,-2,+\tfrac13,+\tfrac43,-\tfrac23$ reproduce every first-generation charge via $Q=T_3+\tfrac12 Y_W$; (2) $\sum Q=\sum Y_W=0$ per generation; (3) $e=g_W\sin\theta_W=0.314$, $g'=0.358$, $\alpha(M_Z)=1/128$ from $g_W=0.653$, $\sin^2\theta_W=0.231$; (4) $\Delta T_3=\Delta Q=\pm1$ at every charged-current vertex.

Prerequisites. The Local Gauge Principle · Weyl Spinors and Chirality · The Weak Charged Current · Isospin. Leads to. The Electroweak Gauge Group SU(2)×U(1) · The Weak Neutral Current · The Weinberg Angle · Electroweak Unification.

Ask the tutor: Ask me to assign weak isospin and hypercharge to any fermion and verify its charge from $Q=T_3+\tfrac12 Y_W$, or to walk the $W^3$–$B$ mixing that turns two weak charges into the photon and the $Z$.