Charged and Neutral Weak Currents

Electroweak mixing reorganizes the gauge interaction into a purely left-handed charged current, a universal vector electromagnetic current, and a chiral neutral $Z$ current. For a fermion of charge $Q_f$ and weak isospin $T_3^f$,

$$\mathcal L_Z= \frac{g}{c_W}Z_\mu\, \bar f\gamma^\mu\left(T_3^fP_L-s_W^2Q_f\right)f ,$$

which exposes both the chiral weak charge and the electromagnetic subtraction.

Required background. Gauge-boson masses and electroweak mixing supplies the $W^\pm,Z,A$ rotation and $e=gs_W=g'c_W$. Quantum currents, improvements, and conservation supplies the operator meaning of the currents.

Helpful background. Cross sections and decay rates explains how these vertices enter observable rates.

Currents in the physical gauge-field basis

For one generation,

$$\begin{aligned} Q_L&\sim(\mathbf3,\mathbf2)_{1/6},& u_R&\sim(\mathbf3,\mathbf1)_{2/3},& d_R&\sim(\mathbf3,\mathbf1)_{-1/3},\\ L_L&\sim(\mathbf1,\mathbf2)_{-1/2},& e_R&\sim(\mathbf1,\mathbf1)_{-1}. \end{aligned}$$

With $Q=T_3+Y$, the gauge part of $i\bar f\gamma^\mu D_\mu f$ becomes

$$\mathcal L_{\rm int} =\frac{g}{\sqrt2}\left(W_\mu^+J_+^\mu+W_\mu^-J_-^\mu\right) +eA_\mu J_{\rm em}^\mu +\frac{g}{c_W}Z_\mu J_Z^\mu .$$

The one-generation currents before flavor mixing are

$$\begin{aligned} J_+^\mu&=\bar\nu_L\gamma^\mu e_L+\bar u_L\gamma^\mu d_L, \qquad J_-^\mu=(J_+^\mu)^\dagger,\\ J_{\rm em}^\mu&=\sum_fQ_f\bar f\gamma^\mu f,\\ J_Z^\mu&=\sum_f \bar f\gamma^\mu\left(T_3^fP_L-s_W^2Q_f\right)f . \end{aligned}$$

Here $T_3^f$ is $+\tfrac12$ or $-\tfrac12$ for the corresponding left-handed doublet component and zero for a right-handed singlet. The derivation from the chiral representations and the $W^3$-$B$ rotation is given in Schwartz 2014, §29.3.1, pp. 592–595.

The photon coupling is vectorlike because $Q_f$ is the same for the two chiralities of each charged Dirac fermion. By contrast, only $P_L=(1-\gamma^5)/2$ appears in $J_\pm$. In the mass basis the quark term becomes $\bar u_L\gamma^\mu V_{\rm CKM}d_L$; that matrix belongs to the Yukawa-sector rotation, not to the gauge representation itself.

Vector and axial neutral couplings

Writing the same $Z$ interaction as

$$\mathcal L_Z= \frac{g}{2c_W}Z_\mu\sum_f \bar f\gamma^\mu \left(g_V^f-g_A^f\gamma^5\right)f$$

gives

$$g_V^f=T_3^f-2Q_fs_W^2, \qquad g_A^f=T_3^f .$$

For the fermions of one generation,

Fermion	$Q_f$	$T_3^f$	$g_V^f$	$g_A^f$
$\nu$	$0$	$+\tfrac12$	$+\tfrac12$	$+\tfrac12$
$e$	$-1$	$-\tfrac12$	$-\tfrac12+2s_W^2$	$-\tfrac12$
$u$	$+\tfrac23$	$+\tfrac12$	$+\tfrac12-\tfrac43s_W^2$	$+\tfrac12$
$d$	$-\tfrac13$	$-\tfrac12$	$-\tfrac12+\tfrac23s_W^2$	$-\tfrac12$

The factors in this table depend on the displayed normalization of $\mathcal L_Z$. A source that puts $g/c_W$, rather than $g/(2c_W)$, in front will tabulate couplings smaller by a factor of two.

As one low-energy illustration, integrating out the $Z$ produces parity-violating electron interactions. In the common normalization where the tree electron weak charge is $Q_W^e=2g_V^e$,

$$Q_W^e=-1+4s_W^2 .$$

This is a tree-level identity in the declared convention, not a claim about a measured value: radiative corrections replace the single angle by process- and scheme-dependent effective form factors.

Algebraic checks and physical limits

Electromagnetic check. The coefficient of $A_\mu$ must be $e(T_3+Y)=eQ$ for both chiralities. Any axial photon coupling signals an incorrect rotation or charge assignment.

Charged-current check. Since $T^\pm$ act only within $SU(2)_L$ doublets, there is no minimal right-handed $W$ current. A term $\bar u_R\gamma^\mu d_RW_\mu^+$ requires new representations or higher-dimensional interactions.

Neutral-current check. Before Yukawa rotations, the $Z$ generator is diagonal and generation universal. Unitary rotations therefore leave the neutral current flavor diagonal, whereas the relative up- and down-sector rotation remains in the charged current.

Neutrino check. For a minimal left-handed neutrino with $Q=0$, $g_V^\nu=g_A^\nu=\tfrac12$, so the displayed vector–axial form equals a purely left-handed coupling.

Weak-angle check. At $s_W^2=0$, the $Z$ couples only to $T_3$. This limit is an algebraic diagnostic, not the physical electroweak point.

Common pitfalls

Using $T_3$ for a right-handed singlet. A right-handed field has $T_3=0$, even though its Dirac partner shares the same electric charge. Its $Z$ coupling comes entirely from the $-s_W^2Q$ term.

Adding flavor mixing by hand to the neutral current. Universal gauge generators commute with generation rotations. Tree-level flavor-changing neutral currents do not arise from the minimal gauge kinetic terms.

Comparing tables without their prefactors. Vector and axial couplings are conventional until the overall $Z$-vertex normalization is included. Compare the full vertex, not isolated $g_V$ symbols.

Handoff

Low-energy matching and flavor rotations receive

$$\left\{J_\pm^\mu,\ J_{\rm em}^\mu,\ J_Z^\mu;\ \frac{g}{\sqrt2},e,\frac{g}{c_W};\ T_3^f,Q_f;\ \text{current normalization}\right\}.$$

Use these data in the Fermi limit and combine them with the mass rotations on Yukawa couplings and fermion masses. Observable precision corrections require a declared electroweak input scheme.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, §29.3.1, pp. 592–595. DOI.