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Work with canonically normalized real components of the scalar field; a complex representation can first be written in terms of its real and imaginary components. At a vacuum expectation value ${\phi }_0$, expansion of the scalar potential givesThe linear term vanishes by stationarity. Thus the scalar mass matrix is the Hessian matrix of the scalar potential. Invariance under each Lie algebra generator gives the identity ${\partial }_{r}V(\phi )(i{T}_a\phi {)}_r=0$. Differentiating this identity with respect to ${\phi }_s$ and evaluating at the stationary point removes the term containing ${\partial }_{r}V$, leavingThe kernel of the map $T\mapsto iT{\phi }_0$ is precisely the Lie algebra of the stabilizer subgroup H. Consequently its restriction to a complementary space of broken Lie algebra generators is injective: a linear combination that annihilates ${\phi }_0$ would belong both to the stabilizer algebra and its complement, and hence would vanish. There are therefore $\dim G-\dim H$ independent Goldstone directions in the scalar mass matrix, tangent to the vacuum manifold. There are $\dim G-\dim H$ symmetry-protected massless scalars.

This Goldstone theorem guarantees at least that many zero quadratic masses. The claim that these are all the massless scalars additionally requires nonzero curvature transverse to the vacuum orbit. For example, $V=\kappa ({\phi }^T\phi -v^2{)}^4$, with $\kappa >0$, has an ordinary spherical vacuum manifold but its entire Hessian matrix vanishes there. These accidental massless radial modes beyond Goldstone modes are not excluded by the hypotheses. For the indicated orthogonal group breaking, the protected number is

For the Higgs doublet, put $r=\sqrt{2{\varphi }^{†}\varphi }=v+H$ and $z=\sqrt{2}\varphi /r$ wherever $r\ne 0$. If $z=(z_1,z_2{)}^T$, thenThis explicitly exhibits the three angular coordinates as an SU(2) transformation of a real radial field. In particular, the polar coordinates for the electroweak Higgs doublet can locally be chosen in the formAt the vacuum the three variations generated by $-i{\sigma }_1$, $-i{\sigma }_2$, and $i(I-{\sigma }_3)$ acting on $(0,1{)}^T$ are independent over the reals. They span the three broken directions; the remaining generator $Q=({\sigma }_3+I)/2$ annihilates the vacuum. This proves that the exponential supplies local angular coordinates. An inverse local gauge transformation, applied also to the gauge fields, removes these three fields in unitary gauge. They supply the longitudinal polarizations of the massive W bosons and Z boson through the Higgs mechanism. The parameterization is local about a nonzero vacuum, rather than a coordinate system through every possible zero of the Higgs doublet.

Use the metric $(+---)$ and define the charged W bosons and neutral gauge fields bywhere $s_W=\sin {\theta }_W$, $c_W=\cos {\theta }_W$, and $\tan {\theta }_W=g^{\prime }/g$. Direct substitution into the gauge-covariant kinetic term givesand henceThe electroweak doublet gauge-boson mass matrix in the $(A^3,B)$ basis isIts normalized zero direction is $(s_W,c_W)$ and its massive direction is $(c_W,-s_W)$. Thus the scalar-vector terms in physical fields arewith the tree-level electroweak gauge-boson massesThese are also charge eigenstates. The unbroken U(1) gauge symmetry has generator $Q=T_3+Y$; on the Higgs doublet, $Y=I/2$. Since $[Q,T_{\pm }]=\pm T_{\pm }$, the combinations $W^{\pm }$ have charges $\pm 1$, while Z, the photon, and H are neutral. The photon disappears from the displayed scalar-vector terms because $g{A}^3-g^{\prime }B=\sqrt{g^2+g^{\prime 2}}Z$. The photon is massless and has no direct tree-level coupling to the remaining radial scalar in this sector. This does not remove its interactions with charged W bosons, or exclude a scalar-to-photon interaction generated by loops.

Finally, retain the printed normalization of the Higgs field potential:The kinetic term is $\frac{1}{2}(\partial H{)}^2$, so the Higgs radial mass for a half-normalized quartic potential is $\boxed{m_H^2=\lambda v^2}$. The extra factor of two familiar from a potential with prefactor $\lambda$, rather than $\lambda /2$, does not apply here.

The local matrix $U(x)$ is a gauge transformation in the representation acting on the matter field. With Hermitian Lie algebra generators it can be written locally in spacetime, and pointwise for $SU(3)$, asThe real functions ${\omega }^a$ can instead include a factor of $g$ by redefining the transformation parameters. The gauge covariant derivative convention in this paper is $D_{\mu }={\partial }_{\mu }+ig{A}_{\mu }$. Inserting the stated transformed $A_{\mu }$ shows explicitly that the ${\partial }_{\mu }U$ term cancels, giving $D_{\mu }^{\prime }(U\psi )=U{D}_{\mu }\psi$.

In QCD, each quark field is in the colour fundamental representation $3$, with ${\psi }^i\mapsto U^i{}_j{\psi }^j$. Thus its colour generators are eight $3\times 3$ matrices, conventionally $T^a={\lambda }^a/2$ for the Gell-Mann matrices; their number is not their matrix dimension. The eight gluons themselves transform in the adjoint representation of SU(3).

With $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$, the pure Yang-Mills theory has Lagrangian ${\mathcal{L}}_{\mathrm{YM}}=-\frac{1}{4}{F}_{\mu \nu }^a{F}^{a\mu \nu }=-\frac{1}{2}\mathrm{tr}(F_{\mu \nu }{F}^{\mu \nu })$. Put $G_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a$ and $C_{\mu \nu }^a=f_{abc}{A}_{\mu }^b{A}_{\nu }^c$. Since $F=G-gC$,The cross term has two derivative contributions. Interchange $\mu ,\nu$ and then $b,c$ in the second; antisymmetry of $f_{abc}$ makes it equal to the first. The cubic and quartic Yang-Mills self-interactions are thereforeThe positive cubic sign follows from the negative $gfAA$ term in the given Yang-Mills field strength; changing the covariant-derivative convention consistently changes intermediate signs.

For the left-handed quark doublet $Q_L=(u_L,d_L{)}^T$, the hypercharge is $Y=1/6$ and the weak generators are ${\sigma }^{\alpha }/2$. Its electroweak term isThe Pauli matrices give diagonal entries $\pm {\mathcal{A}}^3$ and off-diagonal entries ${\mathcal{A}}^1\mp i{\mathcal{A}}^2$. Expanding the term givesIn particular the off-diagonal weak charged current coupling is $-g(W_{\mu }{\bar{u}}_L{\gamma }^{\mu }{d}_L+W_{\mu }^{\ast }{\bar{d}}_L{\gamma }^{\mu }{u}_L)/\sqrt{2}$.

Write $s_W=\sin {\theta }_W$, $c_W=\cos {\theta }_W$. Invert the neutral-field rotation: ${\mathcal{A}}_{\mu }^3=c_W{Z}_{\mu }+s_W{A}_{\mu }$, $B_{\mu }=c_W{A}_{\mu }-s_W{Z}_{\mu }$. The original PDF gives $\tan {\theta }_W=g^{\prime }/g$, soThe photon coefficients inside the derivative are $g{s}_W/2+g^{\prime }{c}_W/6=2e/3$ for $u_L$, and $-g{s}_W/2+g^{\prime }{c}_W/6=-e/3$ for $d_L$. Multiplication by the $i$ in the kinetic Lagrangian gives the requested electromagnetic coupling of a weak multiplet:Equivalently it is $-e{A}_{\mu }{\overline{Q}}_L{\gamma }^{\mu }(T_3+Y)Q_L$, making the electric charges explicit. The remaining neutral terms are $-(g/c_W)Z_{\mu }\{(1/2-2s_W^2/3){\bar{u}}_L{\gamma }^{\mu }{u}_L+(-1/2+s_W^2/3){\bar{d}}_L{\gamma }^{\mu }{d}_L\}$. The TeX's missing prime in the mixing-angle ratio and truncated first Pauli matrix are conversion errors, not physical assumptions.

Use the metric in the question and fix the Levi-Civita symbol convention ${ϵ}_{0123}=+1$. A consistent choice of rotation Lie algebra generators isSubstituting the indices into the printed Poincare algebra givesThe other cyclic brackets follow the same way. The negative spatial metric and antisymmetry of $M^{\rho \sigma }$ both enter this sign.

The printed brackets are real Lie algebra brackets, without the factor $i$ used for ordinary commutators of Hermitian quantum observables. We use those brackets for the algebraic verifications. For physical eigenvalues below, angular momentum is Hermitian, its spin projection is the real number $j_3$, and $\hslash =1$. In that quantum convention the ordinary operator commutators are $i$ times the displayed brackets. This distinction prevents identifying a real spin projection with an anti-Hermitian matrix eigenvalue.

In the universal enveloping algebra, translations commute. The Pauli-Lubanski pseudovector therefore satisfiesFor fixed $\nu ,\rho$, the product of momenta is symmetric in $\mu ,\tau$ while the epsilon coefficient is antisymmetric in them. There is no need to commute the Lorentz generator through the momenta.

Using the commutator derivation identity and the mixed bracket,The two terms become equal after swapping $\nu ,\rho$, and the last expression vanishes by antisymmetry in $\nu ,\tau$. In the Hermitian observable convention, the calculation has one overall extra $i$ and still vanishes. Thus $W_{\mu }$ preserves each momentum eigenspace.

For the specified epsilon orientation, two useful component identities areThe order displayed matters: the momentum operator is on the right, so it can act first on the momentum eigenstate. At rest, $p^{\mu }=(m,0,0,0)$, and on a spin state with $J_3|j,j_3⟩=j_3|j,j_3⟩$ these giveThese are massive rest-frame Pauli-Lubanski eigenvalues. The raised component would be $W^3=+m{j}_3$; confusing $W_3$ with $W^3$ reverses the answer.

Helicity is the projection of spin angular momentum, or equivalently the rotation generator acting internally, along the momentum direction:For the momentum $p^{\mu }=(k,0,0,k)$ with $k>0$, the helicity operator is $J_3$. Hence a helicity-$j_3$ state hasThe result uses only the two longitudinal components and does not need a separate assumption about the transverse little-group generators. These massless longitudinal Pauli-Lubanski eigenvalues agree with $W^{\mu }=j_3{P}^{\mu }$ for ordinary finite-helicity representations.

Finally, at rest the contraction is $m{W}_0=0$. For the chosen null momentum it is $k(W_0+W_3)$, whose two eigenvalues cancel. Thus both results obey $W_{\mu }{P}^{\mu }=0$. Reversing the epsilon orientation reverses all $W$ eigenvalues together; it leaves both identities and both consistency checks intact. For a literal anti-Hermitian derived representation $\rho$ of the printed real algebra, write $H_X=i\rho (X)$ for the Hermitian observable. Since $W_{\mu }$ is bilinear in generators, its abstract enveloping-algebra image is $\rho (W_{\mu })=-W_{\mu ,\mathrm{phys}}$. Thus the corresponding formal-image eigenvalues, if that convention is intended, areHere $\rho (P^{\mu })$ has eigenvalue $-i{p}^{\mu }$, while $p^{\mu }$ and $j_3$ themselves remain real physical labels. This is the same result after the Hermitian quantum generator convention is applied to both factors, not an inconsistent choice of spin sign. Both the Hermitian-observable and literal anti-Hermitian interpretations are consequently specified.

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.

Let $V_i=\partial V/\partial {\varphi }_i$ and $M_{ij}^2={\partial }_i{\partial }_{j}V{|}_{{\varphi }_0}$ be the scalar mass matrix, which is the Hessian matrix of the scalar potential at the vacuum. Invariance under the infinitesimal Lie group action givesDifferentiate with respect to ${\varphi }_j$ and evaluate at the vacuum expectation value ${\varphi }_0$. Since $V_i({\varphi }_0)=0$ at a minimum,Thus every tangent vector $i{t}^a{\varphi }_0$ generated by a broken Lie algebra generator is a zero eigenvalue eigenvector of the scalar mass matrix. The unbroken generators are precisely those in the stabilizer Lie algebra $\mathfrak{h}$ and give the zero tangent vector; independent broken directions span $\mathfrak{g}/\mathfrak{h}$. Hence Goldstone theorem givesmassless scalar modes at the classical level.