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A bare Dirac mass couples left and right chirality, so the distinct electroweak representations prevent writing it as a gauge-invariant term before symmetry breaking. Instead the Yukawa interactions areWeak indices are contracted to singlets. Substituting the vacuum value gives mass matrices $M_f=Y_{f}v/\sqrt{2}$ and a physical Higgs coupling $-{\bar{f}}_L{M}_f{f}_R\sigma /v$ plus its conjugate. Biunitary diagonalization defines mass eigenstates; a mismatch between the up- and down-quark left rotations appears in the CKM matrix. Gauge-boson masses arise from the Higgs kinetic term, whereas these fermion masses arise from the Yukawa term.

With only the minimal field content there is no right-handed neutrino, so no renormalizable neutrino Yukawa term. Adding a gauge-singlet $N_R$ permits both a Dirac scale $m_D=y_{\nu }v/\sqrt{2}$ and a Majorana scale $M_R$. For one family the symmetric mass matrix in a left-chiral basis isIts signed eigenvalues are $(M_R\pm \sqrt{M_R^2+4m_D^2})/2$. Rephasing gives positive physical masses, and for $|M_R|\gg |m_D|$ the seesaw mechanism yieldsFor $m_D\sim 100\mathrm{GeV}$ and $M_R\sim 10^{14}\mathrm{GeV}$, the light scale is about $0.1\mathrm{eV}$. Integrating out the heavy state generates the dimension-five Weinberg operator, whose Higgs vacuum insertion gives the same suppression. Alternatively a Dirac neutrino can have a very small Yukawa coupling with lepton number conserved. A tiny observed neutrino mass does not by itself distinguish these mechanisms.

The left- and right-chiral components of a charged Standard Model fermion transform in different electroweak group representations. A bare Dirac mass term $-m{\bar{f}}_L{f}_R+h.c.$ is therefore not invariant under the full electroweak gauge group. The Higgs doublet supplies the missing transformation properties. With $\tilde{H}=i{\sigma }_2{H}^{\ast }$, the gauge-invariant Yukawa interactions arewhere each $Y_f$ is a flavor-space Yukawa matrix. For example, hypercharges in the down-quark term add to $-1/6+1/2-1/3=0$, while those in the up-quark term add to $-1/6-1/2+2/3=0$. Thus the Higgs and conjugate Higgs are required in different terms.

After electroweak symmetry breaking, choose unitary gauge with $H=(0,(v+h)/\sqrt{2}{)}^T$. The fermion mass matrices becomeA singular value decomposition $U_{fL}^{†}{M}_f{U}_{fR}=\mathrm{diag}(m_{f,1},m_{f,2},m_{f,3})$ defines nonnegative physical tree-level masses. The mismatch $U_{uL}^{†}{U}_{dL}$ is the Cabibbo-Kobayashi-Maskawa matrix in the weak charged current. The corresponding physical Higgs boson couples diagonally as $-(m_f/v)h\bar{f}f$ in the minimal one-doublet theory. This relates its coupling strength to each mass, but the measured Yukawa couplings remain independent input parameters: the minimal theory does not predict their hierarchy.

The minimal renormalizable Standard Model contains no right-handed neutrino, so it has neither a neutrino Dirac Yukawa term nor a gauge-invariant renormalizable left-handed Majorana mass term. Its neutrinos are massless. Observed neutrino oscillations require going beyond that minimal field content or operator set: adding right-handed singlets permits Dirac masses, while the dimension-five Weinberg operator gives a Majorana neutrino mass of order $v^2/\Lambda$ after symmetry breaking. These alternatives should be distinguished from the charged-fermion Yukawa mechanism.

Use the metric $g^{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$ and $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2g^{\mu \nu }$. Transposing a product reverses its order but does not conjugate the factor $i$. Thus the charge-conjugation matrix identity giveswhere reversing the four distinct gamma matrices requires six exchanges. Similarly,

To fix the transpose sign, work in the Weyl representation of the gamma matrices:The Pauli identity checks $C_0{\gamma }^{\mu T}{C}_0^{-1}=-{\gamma }^{\mu }$ directly, and ${\epsilon }^T=-\epsilon$ gives $C_0^T=-C_0$, with $C_0^{†}{C}_0=I$. Any other intertwiner differs from $C_0$ by a scalar: its ratio with $C_0$ commutes with all gamma matrices, whose products span the full four-by-four matrix algebra. A matrix commuting with all matrix units is scalar. Under a spinor basis change $S$, the intertwiner becomes $S{C}_0{S}^T$, whose transpose is still its negative. Therefore in every such four-component representationThis proves the antisymmetry of the four-dimensional charge-conjugation matrix, rather than leaving the sign undetermined by the general transpose argument.

Hermitian conjugating the charged Dirac equation, using real $A_{\mu },e,m$ and ${\gamma }^0{\gamma }^{\mu †}{\gamma }^0={\gamma }^{\mu }$, gives the adjoint Dirac equationTranspose it and multiply by $C$. Since $C{\gamma }^{\mu T}=-{\gamma }^{\mu }C$, the field ${\psi }^c=C{\bar{\psi }}^T$ obtained by charge conjugation obeysThus algebraic charge conjugation reverses the electromagnetic coupling.

If ${\gamma }_5\psi =\eta \psi$, $\eta =\pm 1$, Hermiticity of ${\gamma }_5$ and its anticommutation with ${\gamma }^0$ implyUsing the first transpose identity,Hence $\boxed{\bar{\psi }{\gamma }_5=-\eta \bar{\psi },{\gamma }_5{\psi }^c=-\eta {\psi }^c}$: algebraic charge conjugation reverses chirality. This concerns conjugating the already projected spinor, not the distinct operation of keeping a chiral projector outside an operator conjugation.

For the Grassmann variation of a chiral Majorana mass term, variations of odd fields must be moved past odd fields with a minus sign. If $A^T=-A$, componentwise reordering givesBoth $C$ and $C^{-1}$ are antisymmetric. Treating $\psi ,\bar{\psi }$ independently, integrating the kinetic variation by parts, and placing varied Grassmann fields on the left givesThe factors two from variation cancel the mass-term factors $1/2$. The first coefficient gives one equation; multiplying the second by $C$ gives the other:The variations can be restricted to their chiral subspaces: the two terms in each equation have the same opposite output chirality, so no inactive component equation has been imposed.

Applying $i/\partial$ to the first equation and substituting the second, using $(i/\partial {)}^2=-\square$, givesA phase rotation removes the phase of $m$, and combining the chiral field with its conjugate then gives a Majorana fermion of this mass, not two independent chiral species.

Let $\chi ={\varphi }^{\prime †}\Psi$ and $\bar{\chi }=\bar{\Psi }{\varphi }^{\prime }$. Question 1 shows they are gauge singlets, so both terms in ${\mathcal{L}}_G=G{\chi }^T{C}^{-1}\chi -G^{\ast }\bar{\chi }C{\bar{\chi }}^T$ preserve electroweak gauge invariance. The original PDF has $C^{-1}$ in the first term; the TeX aid incorrectly replaces it by $G^{-1}$. At the Higgs field vacuum,soComparing with the just-derived chiral mass equations yields the Majorana neutrino mass from the Weinberg operator:For nonzero $G$ this generates a mass without introducing a right-handed neutrino. Under lepton number rotation $\Psi \mapsto e^{i\alpha }\Psi$, the first operator acquires $e^{2i\alpha }$ and its conjugate acquires $e^{-2i\alpha }$. Thus the interaction violates lepton number by $\boxed{\Delta L=2}$. Each composite contains one scalar of mass dimension one and one fermion of dimension $3/2$, so the interaction is dimension five and $G$ has dimension $-1$. This is the Weinberg operator, an effective extension of the minimal renormalizable Standard Model, consistent with the absence of bare quadratic masses in Question 1.

With the Standard Model field content, every gauge-invariant, Lorentz-invariant operator of dimension at most four automatically preserves baryon number and total lepton number. They are therefore accidental symmetries, rather than symmetries imposed in constructing the renormalizable Lagrangian. This explains perturbative conservation in ordinary reactions. They are not exact principles: the electroweak chiral anomaly violates $B+L$, the dimension-five Weinberg operator violates lepton number, and dimension-six proton-decay operators can violate baryon number. The combination $B-L$ is anomaly free when a right-handed neutrino is included in each generation.

The unique dimension-five operator built solely from Standard Model fields is the Weinberg operatorwhich gives a Majorana mass term $m_{\nu }\sim c_5{v}^2/\Lambda$ after the Higgs condenses. A dimension-six example that mediates proton decay iswith Lorentz and weak indices contracted appropriately; $u^c{u}^c{d}^c{e}^c/{\Lambda }^2$ is another. Their dimensions explain why neutrino masses are tiny and proton decay is rare: both probe a high scale, while the stronger proton-lifetime bound places especially severe constraints on baryon-number violation.

In the convention $(SU(3{)}_c,SU(2{)}_L{)}_Y$, one Standard Model fermion generation including a right-handed neutrino isA bare Dirac mass term pairs left- and right-handed fields in the same gauge representation, but every charged left-handed Standard Model fermion is an $SU(2{)}_L$ doublet while its right-handed partner is a singlet. The Higgs field $H:(1,2{)}_{1/2}$ and $\tilde{H}=i{\sigma }^2{H}^{\ast }:(1,2{)}_{-1/2}$ permit the gauge-invariant Yukawa interactionsThe neutral Higgs vacuum expectation value turns them into masses after electroweak symmetry breaking.

Because ${\nu }_R$ is a gauge singlet, it may also have a large Majorana mass term $M$. Together with its Dirac mass $m_D$, the seesaw mechanism gives a light neutrino mass $m_{\nu }\simeq m_D^2/M$. If no right-handed neutrino is retained, the same low-energy physics is encoded by the dimension-five Weinberg operator $(L_{L}H)(L_{L}H)/\Lambda$.

Now let $\varphi$ be an electroweak scalar triplet with $Y=1$. Its $T^3$ weights are $1,0,-1$, and electric charge is $Q=T^3+Y$, so its components have chargesIn the given matrix convention,Thus the electromagnetic-neutral direction satisfies ${\varphi }_3=0$ and ${\varphi }_1-i{\varphi }_2=0$.

The gauge-invariant type-II seesaw mechanism Yukawa interaction isThe two lepton doublets have total hypercharge $-1$, which is cancelled by $Y(\varphi )=1$, and the displayed $SU(2)$ contraction is a singlet. Expanding it containsTherefore a neutral condensate $⟨{\varphi }^0⟩\ne 0$ preserves electromagnetism and generates a left-handed-neutrino Majorana mass term proportional to $y_{\varphi }⟨{\varphi }^0⟩$.

The Standard Model is the quantum field theory of quarks, leptons and their strong, weak and electromagnetic interactions, organized by the gauge group $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$. Quantum chromodynamics describes the color sector; the electroweak interaction combines the remaining gauge fields with the Higgs field.

Electroweak symmetry breaking leaves the electromagnetic $U(1{)}_Q$ unbroken. The W bosons and Z boson become massive, while the photon stays massless; Yukawa interactions generate charged-fermion masses. This structure relates scattering, decay and particle-production processes using a common set of couplings. Predictions require specifying parameters and the approximation used, not only listing the group.

The minimal renormalizable field content has massless neutrinos. Observed neutrino oscillations require masses and mixing, motivating extensions such as the Weinberg operator or additional neutrino fields. The Standard Model also does not supply a quantum description of gravity.

The Weinberg operator is the dimension-five Standard Model operator $(LH)(LH)/\Lambda$. After the Higgs field condenses, it gives a left-handed neutrino Majorana mass of order $v^2/\Lambda$.