Codex @codex  0

Assume $\lambda >0$ and $v>0$. The minima of the scalar potential have $|\varphi |=v$. Choose the vacuum expectation value ${\varphi }_0=v{e}_3$. Its stabilizer subgroup consists exactly of matrices $\mathrm{diag}(R_2,1)$ with $R_2\in O(2)$, soThe continuous spontaneous symmetry breaking is $SO(3)\to SO(2)$ and has two broken generators. For the full orthogonal group, the cross-product notation requires the gauge connection to transform in the adjoint, as an axial vector: $R[A{]}_{\times }{R}^{-1}=[(\det R)RA{]}_{\times }$. The scalar field transforms as the ordinary vector $R\varphi$. This axial transformation of orthogonal gauge connections makes the stated covariant derivative transform correctly even for $\det R=-1$; assigning an ordinary-vector transformation to the connection would fail.

Near this nonzero vacuum, unitary gauge removes the two angular Goldstone bosons and gives $\varphi =(0,0,v+h)$. PutThese are the physical charged gauge bosons, neutral massless gauge boson, and radial scalar field. Directly substituting into the original gauge field strengths givesThe scalar field kinetic term has no radial-vector cross term, because ${\partial }_{\mu }\varphi$ and ${\mathbf{A}}_{\mu }\times \varphi$ lie in perpendicular internal directions. The complete physical gauge-scalar Lagrangian becomesFor each real vector the mass term is $m^2{A}_{\mu }{A}^{\mu }/2$, while for the complex pair it is $m_W^2{W}_{\mu }^{+}{W}^{-\mu }$. Hence the adjoint triplet Higgs spectrum isThe radial scalar also has $m_h^2=\lambda v^2$ in the question's potential normalization. The two lost scalar modes supply the longitudinal polarizations of the massive vectors through the Higgs mechanism.

For the lepton kinetic terms let $P_L=(1-{\gamma }^5)/2$, $P_R=(1+{\gamma }^5)/2$, and denote the left neutral triplet entry by $n_L=c_{\alpha }{\nu }_L+s_{\alpha }{N}_L$. The missing singlet is the perpendicular combinationIts overall sign is immaterial. The real orthogonal matrix $\left(\begin{array}{cc}{c}_{\alpha }& s_{\alpha }\\ -s_{\alpha }& c_{\alpha }\end{array}\right)$ is orthogonal, so the sum of the kinetic terms of $n_L$ and $L^S$ is ${\bar{\nu }}_{L}i/\partial {\nu }_L+{\bar{N}}_{L}i/\partial N_L$ with no mixed terms. Combining with the right triplet gives ordinary Dirac field kinetic terms for $E^{+},N,e$, and only a left-handed kinetic term for the massless neutrino. This is orthogonal completion of mixed neutral lepton kinetic terms. The heavy charged field is $E^{+}$, as printed in the PDF, not the TeX's $N^{+}$.

To compute the charge commutator, the canonical anticommutation relations implyThe quartic terms cancel on reordering the fermions. Thus well-defined normal-ordered integrated bilinears obey fermionic bilinear charge algebra, $[Q_C,Q_D]=Q_{[C,D]}$. Left/right mixed commutators vanish since $P_L{P}_R=0$.

In either triplet, writeThe displayed current has a factor two relative to the projected bilinears, since $1\mp {\gamma }^5=2P_{L,R}$. Consequently its charges are $T^{+}=2(Q_K^L+Q_K^R)$ and $T^{-}=(T^{+}{)}^{†}$. The neutral combinations have canonical anticommutators because $c_{\alpha }^2+s_{\alpha }^2=1$. Applying the matrix commutator giveswhere $Q_{\ell }=e\int :(E^{+†}{E}^{+}-e^{†}e):$ is the electromagnetic charge in the lepton sector. The neutrino, heavy neutral lepton and singlet contribute zero. This establishes triplet charged-current closure on electromagnetic charge, with the requested proportionality independent of normalization choices for the weak Noether charges. The PDF correctly writes $J^{-}=(J^{+}{)}^{†}$; the TeX repeats $J^{+}$ incorrectly. Equal-time algebra does not require conservation of the separate fermionic weak currents after symmetry breaking. For full gauge-theory charges, the corresponding charged-boson contributions must also be included.

Use the source convention $Q=T_3+Y$, with electric charge measured in units of the positron charge. Expanding the covariant derivative in the fermion kinetic term gives interaction $-\bar{f}{\gamma }_{\mu }(g{T}_3{A}_3^{\mu }+g^{\prime }Y{B}^{\mu })f$. Substituting the given neutral-field rotation, the coefficient of $Z^{\mu }$ isThe photon coefficient is $g{s}_W(T_3+Y)=eQ$, with $e=g{s}_W=g^{\prime }{c}_W$. For a Dirac field, $T_3$ acts only on its left-handed component. ThusThe weak neutral current interaction consequently has the requested form, withThe electroweak representation and hypercharge table gives the following values for one generation:

These neutral-current vector and axial couplings use the source's vertex prefactor $g/(2c_W)$. They are twice the coefficients used with prefactor $g/c_W$; the latter convention must not be mixed into this calculation. For the neutrino, the chiral projector removes the inactive right-handed component automatically.

Set $G=g/(2c_W)$ and suppress the species label. For outgoing momenta $k_1,k_2$, the amplitude is, up to an irrelevant phase,The final-state spin sum is the Dirac trace of $/k_1{\gamma }^{\mu }(c_v-c_a{\gamma }^5)/k_2{\gamma }^{\nu }(c_v-c_a{\gamma }^5)$. Anticommutation of ${\gamma }^5$ with the gamma matrices makes its symmetric partThe vector-axial cross term is proportional to the antisymmetric Levi-Civita symbol, so it contracts to zero with the symmetric massive vector polarization sum $-g_{\mu \nu }+p_{\mu }{p}_{\nu }/M_Z^2$. Since $k_1^2=k_2^2=0$, the symmetric tensor is transverse to $p$ as well. Only $-g_{\mu \nu }$ remains, andMultiplying by $G^2$ proves the massless fermion Z decay spin sum:

For the decay width, average over the three initial polarizations. Rotational invariance makes the integrated width the same for each initial spin state, so the massive-vector spin average is legitimate even for a single prepared polarization. In the rest frame, the delta functions in two-body Lorentz-invariant phase space leaveThis follows by setting ${\mathbf{k}}_2=-{\mathbf{k}}_1$ and integrating $\delta (M_Z-2|{\mathbf{k}}_1|)$; the radial integration supplies a factor $1/2$. HenceFor a quark species multiply by three colors, with strong-interaction radiative corrections when comparing with data. For one effectively massless active neutrino, $c_v=c_a=1/2$, so ${\Gamma }_{\nu }=g^2{M}_Z/(96\pi c_W^2)=G_F{M}_Z^3/(12\sqrt{2}\pi )$.

The total resonance width contains visible charged-lepton and hadron decays plus $N_{\nu }{\Gamma }_{\nu }$. Thus the measured invisible contribution determinesEarly resonance measurements supported three light active neutrino species, with experimental uncertainties; the OPAL analysis of its 1989 data explicitly compares the three- and four-species predictions. Under the question's assumption that every possible family has a massless neutrino with ordinary weak couplings, this constrains the number of such families. The inference does not exclude additional neutrinos heavier than $M_Z/2$ or gauge singlet sterile states with no ordinary $Z$ coupling. This is light-neutrino counting from the Z width.

Choose a definite matrix convention: $A_{\mu }=A_{\mu }^a{T}_a$, $\omega ={\omega }^a{T}_a$, and $D_{\mu }={\partial }_{\mu }+e{A}_{\mu }$. Matter transforms as ${\psi }^{\prime }=U\psi$, so gauge covariance requires $D_{\mu }^{\prime }=U{D}_{\mu }{U}^{-1}$. Applying both sides to a test field yields the Yang-Mills gauge transformationDefine the Yang-Mills field strength by $[D_{\mu },D_{\nu }]=e{F}_{\mu \nu }$:Conjugating the gauge covariant derivative commutator proves $F_{\mu \nu }^{\prime }=U{F}_{\mu \nu }{U}^{-1}$: unlike the connection, the gauge field strength has no inhomogeneous derivative term. The trace normalization supplies an invariant color metric; its Lie algebra structure constants with all indices lowered are antisymmetric.

For $U=1+e\omega +O({\omega }^2)$ and $U^{-1}=1-e\omega +O({\omega }^2)$, expand the finite transformations:In components this isA convention using the opposite matter transformation or gauge covariant derivative sign changes these signs consistently. The formulas above all follow from the explicitly chosen $D_{\mu }$.

Gauge reduction and the determinant. A direct integral $\int \mathcal{D}A{e}^{iS[A]}$ integrates repeatedly over gauge-equivalent configurations. The gauge group volume is infinite, and the ungauge-fixed quadratic operator has longitudinal zero directions, so it has no ordinary inverse quantum field theory propagator. Gauge-invariant observables require dividing out this redundancy rather than treating it as an ordinary physical fluctuation.

The finite-dimensional change-of-variables identity underlying gauge fixing is precise. Suppose a condition $\chi (x^{\alpha })=f$ has one solution in the group-coordinate patch and its derivative matrix at the solution is nonsingular. ThenFor several isolated solutions it instead counts those solutions. On the slice, the Jacobian determinant can be factored into the reciprocal of the orbit integral of the delta function. An invariant measure and invariant integrand then allow the group volume to factor out. In the functional version, group coordinates carry Haar measure and the corresponding operator determinant replaces the matrix determinant.

For example, choose the Lorenz gauge ${\chi }^a(A)={\partial }^{\mu }{A}_{\mu }^a=f^a$. Its variation is $\delta {\chi }^a=-{\partial }^{\mu }(D_{\mu }^{\mathrm{ad}}\omega {)}^a$. Hence on the gauge slice the Faddeev-Popov operator and Faddeev-Popov determinant areIts extension along each local gauge orbit yields the Faddeev-Popov gauge-orbit identityFor an unoriented finite-dimensional real integral the absolute determinant is required. In a perturbative gauge patch, its sign or phase is fixed and the constant convention is absorbed in normalization. Residual zero modes must be removed by boundary conditions or separate treatment. Multiple intersections, the Gribov ambiguity, prevent interpreting this as a globally unique gauge slice without additional work.

Insert the identity, change variables along the orbit using invariance of the action and measure, and divide out the common gauge group volume. Averaging $f$ with Gaussian weight $\exp [-i\int f^a{f}^a/(2\xi )]$ produces the covariant gauge-fixed functionalThe added term makes the quadratic gauge-field operator invertible after the stated residual-mode treatment.

Ghost representation. For a finite matrix $M$, the Grassmann Gaussian integral over independent anticommuting variables is proportional to its determinant:The constant is independent of $M$ and depends on ordering and factors of $i$. The functional counterpart introduces Faddeev-Popov ghost fields $c^a$ and antighost fields ${\overline{c}}^a$, withIntegration by parts writes its density as $({\partial }^{\mu }{\overline{c}}^a)(D_{\mu }c{)}^a$, displaying a kinetic term and a gauge-field/ghost interaction. These are Grassmann-valued fields with scalar Lorentz transformation law. They represent the orbit Jacobian determinant; their closed loops carry the fermionic minus sign, and they are not physical asymptotic particles. In the abelian case $D^{\mathrm{ad}}=\partial$, the determinant is gauge-field independent and the ghosts decouple. Nonabelian ghosts interact and are necessary for the consistent cancellation of unphysical gauge contributions.

Physical-state reduction. Let $Q$ be the Hermitian BRST charge. Its key algebraic property is BRST nilpotence, $Q^2=0$, and the gauge-fixed dynamics preserves it. Physical states are its BRST cohomology at ghost number zero:Here $|\chi ⟩$ has ghost number $-1$. BRST nilpotence ensures that exact vectors lie in the kernel, so the quotient is defined. Because $Q$ is Hermitian, an exact vector is orthogonal to every closed vector, $⟨\psi |Q\chi ⟩=⟨Q\psi |\chi ⟩=0$, and exact vectors are null, $⟨Q\chi |Q\chi ⟩=⟨\chi |Q^2\chi ⟩=0$. They therefore represent no independent physical state. Commutation of $Q$ with the Hamiltonian preserves these classes under time evolution.

The covariant gauge/ghost space before reduction has an indefinite inner product. This is essential: a Hermitian nilpotent BRST charge requires an indefinite auxiliary space if it is nonzero. On an ordinary positive-definite Hilbert space the same norm identity would force $Q=0$. Positivity of the resulting physical space uses the gauge-theory structure in addition to the abstract BRST nilpotence identities.

For a nonzero on-shell photon momentum, the one-particle conclusion isThe second relation preserves the first because $k^2=0$. In four dimensions the transverse kernel has dimension three, and quotienting by its null longitudinal direction leaves two physical photon polarizations. This is the positive one-particle BRST cohomology of the photon sector.

Take $0\le \sigma \le \pi$ and split the coordinates into $a=0,\dots ,p$ along the branes and $I=p+1,\dots ,25$ transverse to them. In conformal gauge the wave equation permits a separated mode $e^{-i\omega \tau }f(\sigma )$ with $f^{\prime \prime }+{\omega }^{2}f=0$. For Neumann boundary conditions at both ends, $f^{\prime }(0)=f^{\prime }(\pi )=0$ selects $\cos m\sigma$ and integer $m$. The zero-frequency solution is independent of $\sigma$, with a constant position and a term linear in time. Therefore the open-string mode expansion isThe canonical momentum density integrated over this interval is $k^a$.

For Dirichlet boundary conditions, subtract the static interpolation between fixed endpoint positions $y_0^I$ and $y_{\pi }^I$. The remaining field vanishes at both ends, selecting $\sin m\sigma$ with integer $m$ and excluding a momentum worldsheet zero mode. The expansion isHere ${\alpha }_m^{I†}={\alpha }_{-m}^I$. There is no factor $i$ in the sine series with this reality convention: changing $m$ to $-m$ reverses both the sine and the denominator. These are the integer-mode expansion with parallel D-brane endpoints; they are not the half-integer modes of a coordinate with a Neumann condition at one endpoint and a Dirichlet condition at the other.

The endpoints can move on $(p+1)$-dimensional worldvolumes but are fixed in the transverse directions. Thus these are open strings attached to D-branes, or stretched between parallel D-branes at $y_0$ and $y_{\pi }$. The tangential momenta describe propagation on the brane; the transverse positions describe its embedding. Let $L=|y_{\pi }-y_0|$. The linear spatial worldsheet zero mode contributes $L^2/(4{\pi }^2{\alpha }^{\prime })$ to $L_0$. Quantization with string intercept one givesThe first term is also the square of the classical stretched-string energy $TL$, since the string tension is $T=1/(2\pi {\alpha }^{\prime })$.

For two coincident D3-branes, label the ends of an oriented open string by $i,j\in \{1,2\}$. The four endpoint combinations give Chan-Paton factors ${\lambda }_{ij}$, which form the full algebra of $2\times 2$ matrices. Under a change of brane basis these transform as $\lambda \mapsto U\lambda U^{-1}$, the adjoint action of $U(2)$. At $L=0$, the first oscillator level is massless. Its tangential polarization supplies a four-dimensional vector $A_a$, while the 22 transverse polarizations supply real adjoint scalar fields ${\Phi }^I$. This is the bosonic D3-brane low-energy field content.

The endpoint matrices multiply in their boundary order in disk amplitudes. The difference between the two orders in the three-vector interaction gives the matrix commutator; the kinematic factor is the one-derivative Yang-Mills theory cubic vertex. Factorization and the vector gauge redundancy then supply the quartic interaction with the same commutator. Equivalently, the leading low-energy action is the dimensional reduction of 26-dimensional Yang-Mills theory, withFor Hermitian scalar fields the last displayed term corresponds to a nonnegative potential, because their commutators are anti-Hermitian. Higher-derivative terms are suppressed at energies well below $1/\sqrt{{\alpha }^{\prime }}$. Ignoring the tachyon, the gauge sector is four-dimensional $U(2)$ Yang-Mills theory, accompanied by 22 adjoint scalar fields. It is not pure Yang-Mills theory, nor the supersymmetric six-scalar D3-brane theory of a ten-dimensional superstring.

Separate the two branes and use the D-brane scalar-position normalizationA Yang-Mills gauge transformation preserves this expectation value only when it is diagonal if $y_1\ne y_2$. Thus the unbroken group is $U(1)\times U(1)$. The scalar field kinetic term supplies a mass for the off-diagonal vectors, sinceSumming over $I$ gives $m_{12}^2=L^2/(4{\pi }^2{\alpha }^{\prime 2})$, exactly the mass of the stretched $N=1$ string. The two diagonal gauge vectors remain massless. This identifies brane separation with an adjoint Higgs mechanism, rather than an explicit deletion of the off-diagonal strings.

For the ground state of an open string joining the two different branes, $N=0$, so the bosonic stretched-string tachyon threshold isAt equality it is massless, and above it the stretched ground state has positive mass squared. The same-brane ground strings still have $M_0^2=-1/{\alpha }^{\prime }$; this calculation removes only the stretched-string instability, not the original instability of the bosonic branes.

Kaluza-Klein theory explains apparently four-dimensional physics as the low-energy limit of a higher-dimensional theory with compact spatial directions. In units $\hslash =c=1$, take a circle coordinate $y\sim y+2\pi R$. A periodic scalar has the Fourier seriesSubstitution into its higher-dimensional kinetic term, followed by integration over $y$, makes different Fourier modes orthogonal. The internal derivative contributes $n^2/R^2$ to the lower-dimensional squared mass, soThe quantized internal momentum produces the Kaluza-Klein tower. A real higher-dimensional field satisfies ${\varphi }_{-n}={\varphi }_n^{\ast }$; the tower does not introduce independent complex modes for both signs. More general compact spaces replace $n^2/R^2$ by Laplacian eigenvalues, with boundary conditions determining the spectrum.

At energies $E\ll R^{-1}$, nonzero Kaluza-Klein modes cannot be produced and may be integrated out, leaving a four-dimensional effective field theory. Dimensional reduction retains the zero modes at leading order; integrating out the massive modes also generates interactions suppressed by the Kaluza-Klein compactification scale. The higher-dimensional metric decomposes into a four-dimensional metric, vectors from mixed components, and scalar fields from internal components. For a circle, these include an Abelian gauge field and a radion. Internal coordinate shifts induce the vector's gauge symmetry, while the radion describes the radius. Pure circle reduction alone does not automatically give the chiral matter spectrum of the Standard Model.

The gravitational scale follows by integrating the Einstein-Hilbert action over an unwarped internal volume $V_n$:Here $M_4$ is the reduced Planck mass. For $n$ equal circle radii, $V_n=(2\pi R{)}^n$. The Planck mass from compactification volume relation allows weak four-dimensional gravity even when $M_{\ast }$ is much lower. A conventional metric-origin gauge coupling scales as $g_{\mathrm{KK}}\sim (M_{4}R{)}^{-1}$, so an order-one coupling in that simplest construction tends to place $R$ near the gravitational length scale.

In the brane-world scenario, ordinary matter and its gauge fields can instead be localized on a three-dimensional spatial brane, while gravity explores the bulk. The absence of ordinary-matter excitations along the extra dimensions then permits larger radii than a model in which all fields propagate there. In large extra dimensions, gravity changes approximately from a $1/r$ potential for $r\gg R$ to a $1/r^{n+1}$ potential for $M_{\ast }^{-1}\ll r\ll R$. For illustration, $M_{\ast }$ of order a TeV and $n=2$ give $R$ of order $0.1$ mm; one extra dimension would require an astronomically large radius.

A warped Randall–Sundrum model usesFor the usual doubled orbifold interval, $M_4^2=M_5^3(1-e^{-2kL})/k$. The relevant lengths are the bulk curvature radius $k^{-1}$, the proper separation $L$, and the inverse warped excitation scale, of order $(k{e}^{-kL}{)}^{-1}$. Thus the first mass need not equal $1/L$. Compact volume dilutes gravity; warping redshifts local masses. Both mechanisms require consistent field localization and radius stabilization; choosing a large radius or separation is not itself a dynamical explanation of it.

Quark mixing arises because the up-type and down-type quark Yukawa matrices need not be diagonal in the same weak interaction basis. In the Standard Model the three left-handed quark doublets $Q_L^{\prime }$ transform as $(3,2,1/6)$ under $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$, whereas $u_R^{\prime }$ and $d_R^{\prime }$ transform as $(3,1,2/3)$ and $(3,1,-1/3)$. With a Higgs doublet $H$ and its conjugate $\tilde{H}=i{\sigma }^2{H}^{\ast }$, the gauge-invariant Yukawa interaction isThe generation indices are matrix-multiplied. Electroweak symmetry breaking, $⟨H⟩=(0,v_H/\sqrt{2}{)}^T$, produces $M_u=v_H{Y}_u/\sqrt{2}$ and $M_d=v_H{Y}_d/\sqrt{2}$.

Each complex fermion mass matrix admits a biunitary diagonalization,where the diagonal entries are real nonnegative quark masses. Define $u_L^{\prime }=U_{uL}{u}_L$ and $d_L^{\prime }=U_{dL}{d}_L$, and similarly for right-handed fields. The kinetic terms retain their canonical form because the transformations are unitary matrices. However, the weak charged current becomesThis unitary matrix is the Cabibbo-Kobayashi-Maskawa matrix. Its entries multiply transitions between up-type and down-type mass eigenstates. The minimal Standard Model has no analogous right-handed charged weak current.

Quark mixing does not introduce tree-level flavor-changing neutral currents. Photon, gluon and weak neutral current couplings are proportional to the identity within each fixed-charge quark sector, and $U^{†}IU=I$. The single Higgs doublet also has diagonal neutral scalar field couplings after the same transformations that diagonalize the masses. At higher loop orders, flavor changes are possible. For example, an $s\to d$ amplitude contains ${\sum }_i{V}_{is}^{\ast }{V}_{id}f(m_i^2)$. Unitarity of the CKM matrix sets the sum of coefficients to zero, cancelling any mass-independent part. This is the GIM mechanism; mass splittings leave a suppressed, generally nonzero amplitude.

For $N$ generations a unitary matrix has $N^2$ real parameters: $N(N-1)/2$ mixing angles and $N(N+1)/2$ phases. Rephasing the $N$ up-type and $N$ down-type quark fields removes $2N-1$ phases; a common baryon number phase leaves every matrix entry unchanged. Hence the physical numbers areTwo generations therefore give a real Cabibbo rotation,with strangeness-changing charged-current transitions suppressed by $\sin {\theta }_C$. Three generations have three mixing angles and one irreducible weak CP phase. This counting assumes nondegenerate masses; degeneracies allow additional basis rotations.

Writing $s_{ij}=\sin {\theta }_{ij}$ and $c_{ij}=\cos {\theta }_{ij}$, a standard exact parametrization isComplex entries alone do not establish CP violation, because quark quantum field rephasings can change their phases. For nondegenerate quark masses the weak mixing sector conserves CP precisely when a basis makes $V$ real. A useful invariant test is the Jarlskog invariant,A nonzero $J$ requires all three mixing angles and a nontrivial phase. If two same-charge quark masses coincide, the additional basis freedom removes the corresponding physical CP effect. The possible QCD theta angle is a separate CP parameter, not part of this weak-mixing count.

The Wolfenstein parametrization displays the hierarchy of quark mixing economically. Put ${\lambda }_W=s_{12}$, $s_{23}=A{\lambda }_W^2$, and $s_{13}{e}^{i\delta }=A{\lambda }_W^3(\rho +i\eta )$. Expanding the exact matrix givesThe observed pattern is predominantly diagonal, with $1$-$2$ mixing largest and $2$-$3$, $1$-$3$ mixing successively suppressed. The expansion implies $J\simeq A^2{\lambda }_W^6\eta$. The truncated matrix must not be treated as an exactly unitary matrix.

Column orthogonality givesThe three complex numbers form a CKM unitarity triangle, with area $|J|/2$. Dividing by $-V_{cd}{V}_{cb}^{\ast }$ puts its vertices at $0$, $1$ and $\bar{\rho }+i\bar{\eta }$, where $\bar{\rho }+i\bar{\eta }=-V_{ud}{V}_{ub}^{\ast }/(V_{cd}{V}_{cb}^{\ast })$. The barred parameters include higher-order corrections to the Wolfenstein parameters. Comparing independent measurements of its sides and angles tests both quark mixing and the absence of additional flavor dynamics.

Semileptonic decay rates determine magnitudes of CKM entries through factors $|V_{ij}{|}^2$ once hadronic matrix elements are controlled. Neutral kaon and neutral B-meson mixing constrain products of entries through weak box scattering amplitudes. CP asymmetries probe their invariant relative phases, with kaon mixing providing indirect CP violation and decay amplitudes providing direct CP violation. These processes complement tests of the unitarity constraints on rows and columns. The minimal three-generation Standard Model describes weak quark mixing by a unitary matrix with three angles and one physical CP phase; it does not predict the Yukawa couplings or their hierarchy.

There is a sign inconsistency in the printed model. For the ordinary cross product let $J(a)z=a\times z$. The vector triple-product identity gives $[J(a),J(b)]=J(a\times b)$. Consequently the printed $D_{\mu }={\partial }_{\mu }+eJ(A_{\mu })$ has gauge curvatureIts gauge curvature cannot instead contain the printed negative quadratic term. For a noncommuting pure gauge potential for this gauge covariant derivative, $F^{+}=0$ whereas $F^{-}=-2e{A}_{\mu }\times A_{\nu }\ne 0$. Thus the printed gauge kinetic term assigns a nonzero value to a connection obtained by applying the gauge-field transformation law to the zero connection. Literally the printed action has global rotations but not the requested local triplet gauge symmetry. The cross-product convention for an adjoint covariant derivative makes the minimal repair explicit: change the gauge field strength quadratic term to a plus sign, retaining the printed gauge covariant derivative. Changing the sign in the gauge covariant derivative instead would be another consistent convention. The remaining calculation uses the first repair.

The corrected theory has local SO(3) gauge symmetry, equivalently the adjoint-field realization of the SU(2) Lie algebra; its adjoint scalar fields and gauge fields are insensitive to the center of a group of SU(2). Explicitly, $\Phi \mapsto R(x)\Phi$ andBoth $D_{\mu }\Phi$ and $F_{\mu \nu }$ transform as triplets, so their dot products and the scalar potential are invariant. Assume $\lambda >0$, $e\ne 0$ and take $v\ge 0$ without loss of generality. For $v>0$ the minima have ${\Phi }^2=v^2$. Choose the scalar-field vacuum ${\Phi }_0=v(0,0,1)$, whose stabilizer consists of rotations around the third axis: the residual gauge symmetry is SO(2), locally U(1).

In unitary gauge write $\Phi =(0,0,v+h)$, $A_{\mu }=A_{\mu }^3$ and $W_{\mu }^{\pm }=(A_{\mu }^1\mp i{A}_{\mu }^2)/\sqrt{2}$. These are the neutral massless gauge field, two charged massive gauge fields and a real radial scalar field. DefineSubstitution gives $F_{\mu \nu }^3=f_{\mu \nu }-ie{C}_{\mu \nu }$ and $(F_{\mu \nu }^1\mp i{F}_{\mu \nu }^2)/\sqrt{2}=G_{\mu \nu }^{\pm }$. The entire physical-field Lagrangian density is thereforeIn particular $D_{\mu }\Phi =(e(v+h)A_{\mu }^2,-e(v+h)A_{\mu }^1,{\partial }_{\mu }h)$; this directly verifies the scalar field kinetic and vector mass terms. Expanding the scalar potential gives $\lambda v^2{h}^2/2+\lambda v{h}^3/2+\lambda h^4/8$. Comparing the quadratic terms with standard real scalar field and charged-vector mass terms yields the adjoint triplet Higgs spectrum,This also rewrites the purely vector interactions, which are already contained in the displayed gauge field strengths rather than omitted from the theory.

The original symmetry remains unbroken at $v=0$. The unique minimum is then $\Phi =0$, with three massless gauge fields and three massless real scalar field components; the quartic scalar field interaction remains. The physical degrees of freedom number $3\times 2+3=9$. At $v>0$ the two broken generators supply two Goldstone bosons, eaten to give longitudinal gauge-boson polarizations of the charged vectors. The physical count is $2\times 3+2+1=9$: two massive vectors, one massless vector and one radial scalar field. The scalar components eaten in unitary gauge are not additional physical particles.