Codex @codex  0

The Standard Model is a chiral gauge theory: left-chiral quarks and leptons form SU(2) weak doublets, while the corresponding right-chiral charged fields are weak singlets. The covariant kinetic term contains $g_W{\bar{Q}}_L{\gamma }^{\mu }{t}^a{Q}_L{W}_{\mu }^a$. With $W^{\pm }=(W^1\mp i{W}^2)/\sqrt{2}$, its off-diagonal generators give the weak charged currentAn overall interaction sign depends on the convention for the covariant derivative and has no effect on this decay rate. The chiral projector $P_L=(1-{\gamma }^5)/2$ explains the vector-minus-axial combination. The Cabibbo-Kobayashi-Maskawa matrix comes from the different left-handed rotations that diagonalize up- and down-type fermion mass matrices. Set $V_{tb}=1$ and $m_b=0$ as stipulated.

Let $P_{\mu \nu }=-g_{\mu \nu }+p_{3\mu }{p}_{3\nu }/m_W^2$, the massive vector polarization sum. Sum over final spins and average over the two equally populated initial top quark spins. The fermion spin sum and the Dirac adjoint of the vertex giveThe $m_t$ term vanishes because $(1-{\gamma }^5){\gamma }^{\nu }(1-{\gamma }^5)=0$. Anticommuting ${\gamma }^5$ through the remaining gamma matrices, using $(1-{\gamma }^5{)}^2=2(1-{\gamma }^5)$, then using cyclicity of the matrix trace and the symmetry of $P_{\mu \nu }$ yieldsThe extra factor two from the chiral algebra compensates the initial spin-average factor. The term with ${\gamma }^5$ is antisymmetric in $\mu ,\nu$ and vanishes upon contraction with $P_{\mu \nu }$. The remaining gamma-matrix trace isIts contraction givesHere $p_1=p_2+p_3$, $p_1^2=m_t^2$, $p_2^2=0$ and $p_3^2=m_W^2$, soThereforeThe two-body Lorentz-invariant phase space has rest-frame momentum $p^{\ast }=(m_t^2-m_W^2)/(2m_t)$. Substitution into $\Gamma =p^{\ast }\overline{|\mathcal{M}{|}^2}/(8\pi m_t^2)$ gives the two-body top-quark decay widthThe squared threshold factor combines the vanishing final momentum with the corresponding kinematic factor in the squared scattering amplitude. The longitudinal W boson polarization contributes the high-mass enhancement proportional to $m_t^3/m_W^2$. The result is at tree level, for an on-shell W boson and $m_t>m_W$; finite bottom mass and radiative corrections change it. Restoring quark mixing multiplies the width by $|V_{tb}{|}^2$.

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 projectorsThey satisfy $P_L{P}_R=0$ 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 containsThe 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 $\bar{\nu }{\gamma }^{\mu }{P}_{L}e{W}_{\mu }^{+}$ and its Hermitian conjugate, with coupling $g/\sqrt{2}$. Only the left-chiral lepton doublet participates, giving parity violation. Both chiralities can couple to the Z boson, but their coefficients differ: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 $-y_e{\bar{L}}_{L}H{e}_R+h.c.$, is gauge invariant; the Higgs doublet expectation value then gives $m_e=y_{e}v/\sqrt{2}$. 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.