= Solution
<Chirality> distinguishes the two inequivalent <Weyl spinor> representations of the proper <Lorentz group>. For a <Dirac spinor>, define $\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3$ and the <chiral projectors>
$$
P_L=\frac{1-\gamma^5}{2},\qquad P_R=\frac{1+\gamma^5}{2},\qquad\psi=\psi_L+\psi_R.
$$
They satisfy $P_LP_R=0$ and $P_L+P_R=I$. Proper <Lorentz transformations> preserve each chiral representation, whereas <parity> exchanges them. <Helicity> instead measures the projection of <spin angular momentum> along <momentum>. For a massive particle it can change under a boost that overtakes the particle, so it must not be identified with <chirality>. For massless positive-energy particles the left-chiral field annihilates negative-helicity particles and creates positive-helicity antiparticles.
The <Standard Model> is a <chiral gauge theory>: left- and right-chiral fields have different <gauge group representations>. In the convention $Q=T_3+Y$, one generation contains
$$
\begin{array}{c|c}
\text{field}&SU(3)_c\times SU(2)_L\text{ representation and }Y\\\hline
Q_L=(u_L,d_L)^T&(\mathbf3,\mathbf2)_{1/6}\\
u_R&(\mathbf3,\mathbf1)_{2/3}\\
d_R&(\mathbf3,\mathbf1)_{-1/3}\\
L_L=(\nu_L,e_L)^T&(\mathbf1,\mathbf2)_{-1/2}\\
e_R&(\mathbf1,\mathbf1)_{-1}
\end{array}
$$
The minimal field content has no <right-handed neutrino>. A sterile right-handed field is a possible extension, not a required member of that minimal list. The table displays <weak isospin>, <hypercharge> and <color charge> assignments, not assignments of massive-particle helicity.
The <weak charged current> contains $\bar\nu\gamma^\mu P_Le\,W_\mu^+$ and its <Hermitian conjugate>, with coupling $g/\sqrt2$. Only the left-chiral lepton doublet participates, giving <parity violation>. Both chiralities can couple to the <Z boson>, but their coefficients differ:
$$
\mathcal L_Z=-\frac g{\cos\theta_W}Z_\mu\bar f\gamma^\mu\left[(T_3-Q\sin^2\theta_W)P_L-Q\sin^2\theta_WP_R\right]f.
$$
Here $T_3$ is that of the left-chiral component. For the electron, it is $-1/2$. The <photon> and <gluon> couplings of a charged massive <fermion> are vectorlike; this does not make the full <electroweak interaction> vectorlike.
A bare <Dirac mass term> $m\bar\psi\psi=m(\bar\psi_L\psi_R+\bar\psi_R\psi_L)$ joins the two chiral representations. Their different electroweak quantum numbers forbid that term before <electroweak symmetry breaking>. A <Yukawa interaction>, such as $-y_e\bar L_LHe_R+\mathrm{h.c.}$, is gauge invariant; the <Higgs doublet> expectation value then gives $m_e=y_ev/\sqrt2$. Up-type quarks use the conjugate doublet. Thus <chirality> explains the weak charge assignments, their <parity violation>, and why masses arise through the <Higgs mechanism>. The consistency of these chiral assignments also requires <Standard Model anomaly cancellation>.
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