The Electroweak Theory

The charged weak interaction acts only on left-handed fields and violates parity maximally; the electromagnetic interaction acts on left and right alike and conserves parity. Two interactions with such different symmetries look unrelated. Glashow, Weinberg, and Salam showed they are not: both descend from one gauge theory whose symmetry group is $SU(2{)}_L\times U(1{)}_Y$, broken down to the electromagnetic $U(1{)}_{\text{EM}}$ by the Higgs field of the next lessons. This lesson builds the unbroken gauge structure — the fermion representations, the four gauge fields, and the rotation that separates the photon from the $Z$ — and reads off the relations the single mixing angle imposes. It assumes the $V-A$ structure of the charged current and the QED Feynman rules.

Throughout, $\hslash =c=1$ and ${\eta }^{\mu \nu }=\mathrm{diag}(+,-,-,-)$. The chiral projectors are $P_{L,R}=\frac{1}{2}(1\mp {\gamma }^5)$, and ${\psi }_L=P_L\psi$, ${\psi }_R=P_R\psi$.

The gauge group and its charges

The gauge group of the electroweak sector is a product of two factors, each with its own coupling constant.

• Weak isospin $SU(2{)}_L$. A non-abelian factor with coupling $g$ and three generators $T^a=\frac{1}{2}{\sigma }^a$ ($a=1,2,3$), the Pauli matrices halved. The subscript $L$ marks that only left-handed fields transform under it; right-handed fields are singlets. The eigenvalue of $T_3$ is the third component of weak isospin.
• Weak hypercharge $U(1{)}_Y$. An abelian factor with coupling $g^{\prime }$ and a single generator $Y$, the weak hypercharge. Every field, left- or right-handed, carries a hypercharge.

The two abelian-looking charges $T_3$ and $Y$ are not the electric charge. The electric charge is the specific combination that survives symmetry breaking, fixed by the Gell-Mann–Nishijima relation

This is the same relation that organizes hadrons by isospin and hypercharge in the flavor-$SU(3)$ story, carried over to the electroweak charges. It is an input that fixes $Y$ once the electric charges and the isospin assignments are known.¹

Fermion representations

Left-handed fermions of one generation form two $SU(2{)}_L$ doublets, the leptons and the quarks; every right-handed fermion is an $SU(2{)}_L$ singlet. Writing the first generation,

The upper member of each doublet has $T_3=+\frac{1}{2}$, the lower $T_3=-\frac{1}{2}$; the singlets have $T_3=0$. The hypercharges follow from $Y=2(Q-T_3)$:

Both members of a left-handed doublet share the same hypercharge because $Y$ is an $SU(2)$ singlet quantum number: it cannot distinguish the two isospin partners. The second and third generations repeat the pattern with $({\nu }_{\mu },\mu )$, $(c,s)$ and $({\nu }_{\tau },\tau )$, $(t,b)$. A right-handed neutrino, if it exists, is neutral under the entire group ($T_3=Q=Y=0$) and so is invisible to the electroweak interaction; the original Standard Model omits it.

Gauge fields and the covariant derivative

Gauge invariance under $SU(2{)}_L\times U(1{)}_Y$ requires one gauge field per generator: a triplet $W_{\mu }^1,W_{\mu }^2,W_{\mu }^3$ for the three $SU(2)$ generators and a single $B_{\mu }$ for hypercharge. The interaction with matter is dictated by the covariant derivative, which replaces the ordinary derivative in the fermion kinetic term. For a field in a representation with isospin generators $T^a$ and hypercharge $Y$,

On a right-handed singlet the $T^a$ term drops out ($T^a=0$), leaving only the hypercharge coupling; on a left-handed doublet both terms act. Expanding $W_{\mu }^a{T}^a=\frac{1}{2}(W^1{\sigma }^1+W^2{\sigma }^2+W^3{\sigma }^3)$ and defining the charged combinations

the off-diagonal part of the doublet covariant derivative becomes the charged current: a $W^{+}$ turns a lower doublet member into the upper one (raising $T_3$ by one), a $W^{-}$ does the reverse. The $\sqrt{2}$ and the sign convention are fixed so that $W^{+}$ carries electric charge $+1$. This is the interaction responsible for beta decay and muon decay: the charged $W$ couples exclusively to the left-handed doublets, which is why the charged weak current is purely $V-A$.

Neutral fields and the Weinberg rotation

The diagonal part of the covariant derivative involves $W_{\mu }^3$ (from $SU(2)$) and $B_{\mu }$ (from hypercharge). Neither is the photon. Both couple to matter, and the physical neutral bosons — the massless photon $A_{\mu }$ and the massive $Z_{\mu }$ — are orthogonal linear combinations of the two. The mixing is a rotation by the Weinberg angle (or weak mixing angle) ${\theta }_W$:

The angle is not free once the couplings are chosen. The combination that stays massless after symmetry breaking must couple to matter exactly as the photon does — proportional to the electric charge $Q=T_3+Y/2$ — and this fixes

Substituting the rotation into the covariant derivative and demanding that the $A_{\mu }$ coupling reproduce $-ieQ{A}_{\mu }$ gives the electromagnetic coupling as a product of the two electroweak couplings and the mixing angle,

The electric charge is therefore smaller than either weak coupling: unification does not make the electromagnetic force the strongest but distributes one underlying strength between the two neutral bosons. The measured value at the $Z$ scale is ${\sin }^2{\theta }_W\approx 0.231$, so $g\approx 0.64$ and $g^{\prime }\approx 0.34$, and $e\approx 0.31$, consistent with $\alpha =e^2/4\pi \approx 1/128$ at that scale.²

Neutral currents and their couplings

The $Z$ boson couples to a neutral current that, unlike the photon's, treats left- and right-handed fields differently and is not simply proportional to charge. In Dirac notation the $Z$ interaction of a fermion $f$ is

with vector and axial-vector couplings set by the isospin and charge of the fermion,

The axial coupling is just the weak isospin; the vector coupling carries the whole ${\sin }^2{\theta }_W$ dependence. Because ${\sin }^2{\theta }_W\approx \frac{1}{4}$, the vector coupling of the charged leptons, $g_V^e=-\frac{1}{2}+2{\sin }^2{\theta }_W\approx -0.04$, is anomalously small — a near-cancellation that makes the leptonic $Z$ coupling almost purely axial and provides one of the most sensitive measurements of the mixing angle. Neutral currents were the decisive early prediction of the theory: the reaction ${\nu }_{\mu }+e^{-}\to {\nu }_{\mu }+e^{-}$, which has no charged-current diagram, was observed in the Gargamelle bubble chamber in 1973, before the $W$ and $Z$ themselves were seen.³

Predicted masses and the $\rho$ parameter

The unbroken theory has four massless gauge fields. Symmetry breaking, treated in the Higgs mechanism, gives three of them mass while leaving the photon massless, and the same mixing angle that separated the photon from the $Z$ also relates their masses. The results, derived there, are

where $v\approx 246\text{GeV}$ is the Higgs vacuum expectation value. The second equality is a genuine prediction with no free parameter: the ratio of the two boson masses is fixed by the mixing angle already measured in neutral-current data,

The $\rho$ parameter equals one in the minimal theory with a single Higgs doublet; its measured value $\rho \approx 1.0004$ is a stringent test of the doublet structure. Using ${\sin }^2{\theta }_W=0.231$ and the measured $M_Z=91.19\text{GeV}$, the theory predicts $M_W=M_Z\cos {\theta }_W\approx 80.0\text{GeV}$, against the measured $80.38\text{GeV}$ — the small difference coming from radiative corrections dominated by the heavy top quark.⁴

Charged and neutral currents together

The full electroweak interaction of the fermions, written after the rotation to physical fields, is a sum of three currents coupled to the three physical gauge bosons (plus the photon):

The three currents have distinct chiral content, summarized below. The charged current is purely left-handed; the electromagnetic current is vector-like (left and right equally); the neutral current is a definite mixture set by ${\sin }^2{\theta }_W$.

That the neutral current changes no flavor is a structural fact: the $Z$ is a diagonal combination of $W^3$ and $B$, both of which are diagonal in flavor, so there are no tree-level flavor-changing neutral currents. Their absence in the data — the strong suppression of $K^0\to {\mu }^{+}{\mu }^{-}$, for instance — was a central clue that pointed to the theory and, through the GIM mechanism, to the charm quark.

Summary

The electroweak interaction is a gauge theory of $SU(2{)}_L\times U(1{)}_Y$ with couplings $g$ and $g^{\prime }$. Left-handed fermions sit in weak-isospin doublets, right- handed fermions in singlets, and each carries a hypercharge fixed by $Q=T_3+Y/2$. The four gauge fields are the $SU(2)$ triplet $W^{1,2,3}$ and the hypercharge singlet $B$. The charged combinations $W^{\pm }=(W^1\mp i{W}^2)/\sqrt{2}$ mediate the purely left-handed charged current; the neutral $W^3$ and $B$ rotate through the Weinberg angle ${\theta }_W$, with $\tan {\theta }_W=g^{\prime }/g$, into the massless photon and the massive $Z$. The single angle ties everything together: $e=g\sin {\theta }_W$, the neutral-current couplings $g_V=T_3-2Q{\sin }^2{\theta }_W$ and $g_A=T_3$, and the mass relation $M_W=M_Z\cos {\theta }_W$. What the unbroken theory cannot supply is the masses themselves — gauge invariance forbids explicit mass terms — which is the problem the Higgs mechanism solves, after the spontaneous symmetry breaking of the next lesson.