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The unitary charge-conjugation operator acts on field operators and commutes with numerical chiral projectors, so $\hat{C}(P_L\psi ){\hat{C}}^{-1}=P_L{\psi }^c$. This component is left-chiral. The algebraic conjugate of an already projected spinor instead satisfies $(P_L\psi {)}^c=P_R{\psi }^c$, because its adjoint carries the opposite projector. Keeping the two operations separate avoids a false chirality conclusion.

Take $D_{\mu }={\partial }_{\mu }+i{K}_{\mu }$, with the Pauli matrices acting on the Higgs doublet. For electroweak hypercharge $Y=1/2$,This specifies every component of the gauge covariant derivative for the electroweak interaction. Expanding the gauge-covariant kinetic term makes its derivative, trilinear and quartic interactions explicit:where the Pauli matrix multiplication law givesThe antisymmetric Pauli contribution vanishes because $W_{\mu }^a{W}^{b\mu }$ is symmetric in $a,b$. Reversing the sign convention for $D_{\mu }$ reverses the linear gauge interactions consistently, without changing the masses.

A nonzero vacuum expectation value requires ${\mu }^2<0$. Minimizing the Higgs potential gives $v^2=-{\mu }^2/\lambda$. By an gauge transformation choosein unitary gauge. The electroweak doublet gauge-boson mass matrix follows by inserting the vacuum expectation value in the gauge-covariant kinetic term:Define the charged electroweak gauge bosons and the neutral rotation through the Weinberg angle bywith inverse $W_{\mu }^3=c_W{Z}_{\mu }+s_W{A}_{\mu }$ and $B_{\mu }=-s_W{Z}_{\mu }+c_W{A}_{\mu }$. ThenThe factors differ because $W^{+}$ and $W^{-}$ are conjugate fields whereas $Z$ is real. The massless photon corresponds to the unbroken Lie algebra generator $Q=T_3+Y$, which annihilates $⟨\varphi ⟩$. Thus three of the four real gauge bosons acquire mass, with $m_W=m_Z{c}_W$. The three would-be Goldstone bosons provide their longitudinal polarizations; the remaining scalar $h$ is the Higgs boson.

Introduce the left-handed lepton doublet $L=({\nu }_{eL},e_L{)}^T$, with $Y_L=-1/2$, and the right-handed singlet $e_R$, with $Y_R=-1$. Use the chiral projectors $P_L=(1-{\gamma }^5)/2$ and $P_R=(1+{\gamma }^5)/2$ in the course's convention. The minimal Standard Model has no right-handed neutrino. The gauge-invariant fermion terms areThe doublet contraction in the Yukawa interaction is a singlet, and its total hypercharge is $+1/2+1/2-1=0$. A bare term ${\bar{e}}_L{e}_R$ would fail electroweak gauge invariance. The gauge covariant derivatives above already give all the requested fermion-gauge couplings. In terms of mass eigenstates, $e=g{s}_W=g^{\prime }{c}_W$ and they becomeThus the weak charged current is chiral and the neutrino has zero electric charge. The gauge-invariant electron Yukawa mass follows from the Yukawa interaction after electroweak symmetry breaking:Rephase $e_R$ to make $y_e$ real and positive. It givesThe electron Dirac mass is therefore compatible with the original gauge symmetry through the Higgs mechanism. The neutrino remains massless in this minimal renormalizable lepton sector.

The strong-interaction matrix element between two spin-zero pseudoscalar mesons has only $p^{\mu }$ and $k^{\mu }$ available. The product of the two intrinsic parities is positive. An axial current would require a pseudovector constructed from these momenta, but an expression involving the Levi-Civita symbol needs three independent four-vectors and therefore vanishes. This is a consequence of parity conservation in the hadronic matrix element, not of parity conservation in the weak interaction. The vector current can have the two independent structures $p+k$ and $p-k$. Its coefficients are Lorentz scalars; with $p^2=m_K^2$ and $k^2=m_{\pi }^2$ fixed, their only varying invariant is $s=q^2$. HenceThese are the pseudoscalar-to-pseudoscalar form factors. With relativistically normalized states they are dimensionless. The vanishing axial matrix element and this decomposition explain the two equalities separately.

Write $q_1$ and $q_2$ for the outgoing electron and antineutrino momenta. From the Fermi interaction, an invariant scattering amplitude, up to an irrelevant overall sign or phase, isThe CKM matrix element $V_{us}$ multiplies the quark current in the convention of this interaction. Let ${\ell }_{\mu }=\bar{u}(q_1){\gamma }_{\mu }(1-{\gamma }^5)v(q_2)$. For massless leptons, the massless Dirac equation and chirality matrix anticommutation giveIn the second term move $/q_2$ through the chiral projector before applying $/q_{2}v=0$. Since $p+k=2p-q$, this transverse massless leptonic current givesThe disappearance of $f_{-}$ uses the massless approximation; for a massive charged lepton its contraction is proportional to the lepton mass.

Use the fermion spin sum and the supplied gamma matrix trace identities. The symmetric part of the leptonic tensor isThe Levi-Civita symbol term is antisymmetric and drops out when contracted with $p^{\mu }{p}^{\nu }$. ThusThere is no initial-spin average because the kaon is spinless. For the integrated massless leptonic tensor, keep every factor of $2\pi$ explicit and define the unnormalized two-lepton Lorentz-invariant phase spaceThe leptons are massless, so $q_i^0=|{\mathbf{q}}_i|$. Its trace is $g^{\mu \nu }{I}_{\mu \nu }=\pi s$. ThereforeThe three Lorentz-invariant phase-space measures and their momentum delta function contribute $1/[8(2\pi {)}^5]$, in addition to $1/(2m_K)$ in the decay rate. Combining them with the spin sum givesThis massless semileptonic pseudoscalar decay rate uses a two-lepton integral over future-timelike $q$ and the pion integral is restricted to the physically allowed region. The null endpoint follows by continuity. The coefficient has mass dimension $-5$, so the complete expression has mass dimension one, as a decay rate must in natural units.

In the kaon centre-of-momentum frame, put $E=k^0$ and $\kappa =|\mathbf{k}|$. ThenThe dimensionally consistent Källén function isThe pion-only mass term must have fourth power: the second power printed in the PDF is dimensionally inconsistent. This repair also follows directly from squaring $E=(m_K^2+m_{\pi }^2-s)/(2m_K)$. Angular integration and the change of variable givewhere the negative sign reverses the endpoints. Combining this with the bracket $\lambda /4$ yieldsThus $\boxed{B=G_F^2|V_{us}{|}^2/(192{\pi }^3{m}_K^3),a=0,b=(m_K-m_{\pi }{)}^2}$. The lower limit is the minimum invariant mass of two massless leptons; at the upper limit the pion is at rest. The coefficient has mass dimension $-7$, while $ds{\lambda }^{3/2}$ has dimension eight. The Källén function also shows why the differential decay rate vanishes at zero pion momentum.

Use the metric $g_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$ and the usual Dirac adjoint $\bar{\psi }={\psi }^{†}{\gamma }^0$.

Field transformation. The mode expansion of a Dirac field isCharge conjugation exchanges particle and antiparticle operators. Choose their phases so thatThe conjugate relations for creation and annihilation operators preserve their canonical anticommutation relations. Thus the transformed expansion is ${\eta }_C$ times the same integral with $a_s{u}^s$ replaced by $b_s{u}^s$ and $b_s^{†}{v}^s$ replaced by $a_s^{†}{v}^s$. On the other hand, transposing the adjoint expansion gives $b_{s}C{\bar{v}}^{sT}{e}^{-ipx}+a_s^{†}C{\bar{u}}^{sT}{e}^{ipx}$. The given spinor identities identify these expressions, proving

Symmetry of the spin matrix. Transposing $C{\gamma }^{\mu T}=-{\gamma }^{\mu }C$ gives ${\gamma }^{\mu }{C}^T=-C^T{\gamma }^{\mu T}$. Substitute ${\gamma }^{\mu T}=-C^{-1}{\gamma }^{\mu }C$ and multiply on the right by $C^{-1}$:The complex four-dimensional Clifford algebra representation generated by the gamma matrices is irreducible; equivalently its sixteen independent gamma products span the full matrix algebra. The Schur lemma therefore gives $C^T{C}^{-1}=zI$. Hence $C^T=zC$. Transposing once more gives $C=z^{2}C$, so $z^2=1$ andThis is the transpose symmetry of a charge-conjugation matrix. The given antisymmetric choice is consistent with this conclusion.

Equation of motion. Taking the adjoint of the free Dirac equation yields $i({\partial }_{\mu }\bar{\psi }){\gamma }^{\mu }+m\bar{\psi }=0$. Its transpose, multiplied by $C$, isThe defining matrix relation converts this intoThus the charge conjugation of a Dirac field maps free solutions to free solutions. With an external electromagnetic field, its charge would also reverse; no such background is present here.

Chirality. Define the chiral projectors $P_L=(1-{\gamma }^5)/2$ and $P_R=(1+{\gamma }^5)/2$. The operator $\hat{C}$ acts on the field operators and commutes with these numerical matrices:The transformed component is therefore left-chiral, as requested. There is a useful distinction: the algebraic charge conjugate of a left-chiral spinor is right-chiral. Indeed $C{\gamma }^{5T}{C}^{-1}={\gamma }^5$ and $\overline{P_L\psi }=\bar{\psi }{P}_R$ giveThis is the operator conjugation of a chiral field component; it avoids confusing projection after operator conjugation with charge conjugation of the already projected spinor.

Baryogenesis. Consider equal initial populations of a heavy particle $X$ and its antiparticle, with two baryon-number-violating decay channels carrying different final baryon numbers $b_1,b_2$. Let $r$ and $\bar{r}$ be the branching fractions into a channel and its antiparticle channel. The net baryon number produced per initial particle-antiparticle pair isExact charge conjugation symmetry equates the conjugate branching fractions and makes this vanish. Exact CP symmetry also equates the fully summed conjugate decay rates, because parity reverses momenta and helicities without changing baryon number. Thus both violation of charge conjugation and CP violation are necessary for generating an asymmetry from a symmetric initial state.

Violation of charge conjugation alone can generate a helicity asymmetry while leaving total baryon number zero. For example, with $L,R$ denoting helicities, CP symmetry may enforce $\Gamma (X\to q_L{q}_L)=\Gamma (\bar{X}\to {\bar{q}}_R{\bar{q}}_R)$ and the analogous relation with $L,R$ exchanged, even when same-helicity charge-conjugate rates differ. The CP rate cancellation in baryogenesis means that summing over helicities cancels the baryon asymmetry. These are the symmetry requirements in the Sakharov conditions; baryon-number violation and departure from thermal equilibrium are also needed in the conventional decay mechanism.