Electroweak Gauge and Matter Structure

The electroweak sector is a chiral $SU(2)_L\times U(1)_Y$ gauge theory: left-handed quarks and leptons occupy weak doublets, their right-handed partners are weak singlets, and one scalar doublet permits gauge-invariant Yukawa interactions. With the convention $Q=T_3+Y$, the hypercharges below reproduce the observed charge pattern and pass the one-generation anomaly checks.

Required background. The Yang–Mills action and self-interaction supplies the non-Abelian covariant derivative and field strength. Compact Lie groups, roots, and weights supplies weights and representation labels.

Helpful background. Perturbative chiral gauge anomalies owns the quantum consistency derivation previewed here.

The structural chain is summarized below. Representations and the Higgs orbit determine the unbroken charge, mass eigenstates, currents, Yukawa rotations, low-energy matching, and finally a renormalized observable; every arrow has an algebraic check.

The electroweak construction closes when the vacuum is electrically neutral, the photon remains massless, Yukawa terms reproduce the charges, rotations preserve kinetic terms, and observables are independent of input coordinates. The diagram is schematic.

One-generation field content

Use Hermitian $SU(2)$ generators $T^a=\tau^a/2$, where $\tau^a$ are Pauli matrices, and

$$Q=T_3+Y, \qquad D_\mu=\partial_\mu-i g_sG_\mu^AT_c^A-i gW_\mu^aT^a-i g'YB_\mu.$$

Terms for groups under which a field is a singlet are omitted. The minimal one-generation assignments are

Field	Chirality	$(SU(3)_c,SU(2)_L)_Y$	Component charges from $T_3+Y$
$Q_L=(u_L,d_L)^{\mathsf T}$	left	$(\mathbf3,\mathbf2)_{1/6}$	$(2/3,-1/3)$
$u_R$	right	$(\mathbf3,\mathbf1)_{2/3}$	$2/3$
$d_R$	right	$(\mathbf3,\mathbf1)_{-1/3}$	$-1/3$
$L_L=(\nu_L,e_L)^{\mathsf T}$	left	$(\mathbf1,\mathbf2)_{-1/2}$	$(0,-1)$
$e_R$	right	$(\mathbf1,\mathbf1)_{-1}$	$-1$
$H=(H^+,H^0)^{\mathsf T}$	scalar	$(\mathbf1,\mathbf2)_{1/2}$	$(1,0)$

This chiral gauge-and-scalar organization is the renormalizable lepton model introduced in Weinberg 1967, pp. 1264–1266, extended here by the quark and color representations.

The minimal renormalizable model has no right-handed neutrino. Adding one with $Y=0$ is a consistent extension of the local gauge representation but changes the neutrino-mass sector and is treated separately.

For example,

$$D_\mu Q_L= \left(\partial_\mu-i g_sG_\mu^AT_c^A -i gW_\mu^a\frac{\tau^a}{2} -i g'\frac16B_\mu\right)Q_L,$$

whereas

$$D_\mu e_R=(\partial_\mu+i g'B_\mu)e_R, \qquad D_\mu H=\left(\partial_\mu-i gW_\mu^a\frac{\tau^a}{2} -i g'\frac12B_\mu\right)H.$$

Substituting these derivatives into $i\bar\psi\gamma^\mu D_\mu\psi$ fixes every gauge vertex and its chirality. The hypercharge table and its conversion to electromagnetic charge are derived in Schwartz 2014, §29.3, pp. 592–94.

The table specifies representations of the local gauge algebra. It does not by itself choose the exact global quotient of $SU(3)_c\times SU(2)_L\times U(1)_Y$ or its line-operator spectrum; that is a separate global-form question.

Gauge-invariant interactions constrain hypercharge

The kinetic terms are

$$\mathcal L_{\mathrm{kin}}= -\frac14W^a_{\mu\nu}W^{a\mu\nu} -\frac14B_{\mu\nu}B^{\mu\nu} +\sum_\psi i\bar\psi\gamma^\mu D_\mu\psi +(D_\mu H)^\dagger D^\mu H.$$

A bare Dirac mass such as $\bar e_Le_R$ is not $SU(2)_L$ invariant. The scalar permits

$$\mathcal L_Y= -\bar Q_LY_dHd_R -\bar Q_LY_u\widetilde H u_R -\bar L_LY_eHe_R+\mathrm{h.c.}, \qquad \widetilde H=i\tau^2H^*.$$

The hypercharge sums, including the sign from a barred field, are

$$-\frac16+\frac12-\frac13=0, \qquad -\frac16-\frac12+\frac23=0, \qquad +\frac12+\frac12-1=0.$$

These are fast convention checks. Using the alternative convention $Q=T_3+Y/2$ without doubling every tabulated $Y$ would fail them and change the $B_\mu$ coupling.

The gauge interactions and scalar potential preserve baryon and individual lepton numbers at the classical renormalizable level, while the Yukawa matrices break much of the flavor symmetry. Quantum anomalies and nonperturbative electroweak effects refine those statements; they should not be inferred from the kinetic terms alone.

Quantum consistency preview

For anomaly calculations, rewrite every fermion as a left-handed Weyl field. Thus $u_R,d_R,e_R$ become $u^c,d^c,e^c$ with conjugate non-Abelian representations and opposite hypercharges. One generation then satisfies

$$[SU(2)_L]^2U(1)_Y: \qquad 3\left(\frac16\right)-\frac12=0,$$ $$[SU(3)_c]^2U(1)_Y: \qquad 2\left(\frac16\right)-\frac23+\frac13=0,$$

and

$$\begin{aligned} [U(1)_Y]^3: &\quad 6\left(\frac16\right)^3 +3\left(-\frac23\right)^3 +3\left(\frac13\right)^3\\ &\quad+2\left(-\frac12\right)^3+1^3=0. \end{aligned}$$

The mixed gravitational–hypercharge sum also vanishes. There are four left-handed weak doublets after color multiplicity—three quark doublets and one lepton doublet—so the number is even, as required by the global anomaly of an $SU(2)$ theory with an odd number of Weyl doublets Witten 1982, pp. 324–328. The complete perturbative anomaly calculation and its relation to charge quantization are given in Schwartz 2014, §30.4, pp. 631–634.

Anomaly cancellation is necessary quantum consistency; it does not select the global gauge group uniquely and it is not the same check as invariance of an individual Yukawa term.

Checks and failure modes

Charge check. Apply $T_3+Y$ to both entries of every doublet and to every singlet. The neutral Higgs component must have $Q=0$, otherwise the proposed vacuum direction would break electromagnetism.

Vertex check. Right-handed singlets have no $W^{1,2,3}$ coupling. A right-handed charged current signals an incorrect representation or an extension beyond this model.

Yukawa check. Verify both the $SU(2)$ contraction and the hypercharge sum. Up-type masses require $\widetilde H$, not $H$.

Anomaly check. Convert right-handed fields to left-handed conjugates before summing. Keeping the original hypercharge while conjugating the representation produces a false nonzero result.

Generation check. Repeating the table gives three generations, but replication alone does not define flavor mixing. Mixing begins when independent Yukawa matrices are diagonalized.

Common pitfalls

Mixing the two hypercharge conventions. State $Q=T_3+Y$ or $Q=T_3+Y/2$ once and translate the table and coupling together. Electric charge is the invariant checkpoint.

Calling the table an anomaly proof. Classical covariant derivatives can be written for an anomalous spectrum. The triangle and global anomaly conditions are additional quantum tests.

Ignoring color multiplicity. Color does not enter $Q=T_3+Y$, but it enters anomaly sums and the count of weak doublets.

Handoff

The Higgs and current derivations require the typed input

$$\left\{SU(2)_L\times U(1)_Y,\ Q=T_3+Y,\ g,g',\ \{R_i,Y_i,\text{chirality}\},\ H\sim(\mathbf2)_{1/2}, \mathcal L_Y\text{ invariants}\right\}.$$

The next operation is to identify the Higgs vacuum orbit and physical scalar content. Detailed flavor misalignment belongs to Quark Flavor and CP.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, §§29.3 and 30.4, pp. 592–602 and 631–634. DOI.
• Weinberg, Steven. “A Model of Leptons.” Physical Review Letters 19, no. 21 (1967): 1264–1266. DOI.
• Witten, Edward. “An $SU(2)$ Anomaly.” Physics Letters B 117, nos. 5–6 (1982): 324–328. DOI.