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Choose the gauge covariant derivative $D_{\mu }={\partial }_{\mu }+ig{A}_{\mu }$ and matter transformation ${\varphi }^{\prime }=h\varphi$. Covariance $D_{\mu }^{\prime }{\varphi }^{\prime }=h{D}_{\mu }\varphi$ forces $A_{\mu }^{\prime }=h{A}_{\mu }{h}^{-1}+(i/g)({\partial }_{\mu }h)h^{-1}$. For $h=1-ig\omega +O({\omega }^2)$, expansion gives the displayed adjoint covariant derivative variation. With an orthonormal Hermitian Lie-algebra basis, $\delta A_{\mu }^a={\partial }_{\mu }{\omega }^a-g{f}^{abc}{A}_{\mu }^b{\omega }^c$. Thus Lorenz-gauge variation is $\delta (\partial \cdot A)=\partial \cdot \mathcal{D}\omega$. Choosing the Faddeev-Popov operator as $\mathcal{M}=-\partial \cdot \mathcal{D}$, with its field-independent sign absorbed in normalization, gives ghost action $-\bar{c}\partial \cdot \mathcal{D}c$; consistent propagator and vertex signs follow from this choice.

For a background gauge field with $A_0=Z_A^{1/2}A$ and $g_0={\mu }^{ϵ/2}{Z}_{g}g$, background gauge covariance preserves the combination $gA$ in the gauge covariant derivative. Its Ward identity therefore gives the displayed relation. This refers to background-field renormalization, not an unrestricted identity between the coupling and the ordinary quantum-gluon wave-function factor. Gauge-invariant one-loop matter determinants can be renormalized with this identity to extract their coupling running from vacuum polarization.

In two dimensions, scale ${\Phi }_{\lambda }(x)=\Phi (\lambda x)$ and $A_{j,\lambda }(x)=\lambda A_j(\lambda x)$. The magnetic field has scaling weight two, the gauge covariant derivative of the Higgs field has weight one, and the potential has weight zero. Their integrated energies therefore have the powers shown. Stationarity gives $E_B=E_P$, a Derrick virial identity. This balance allows a planar Abelian Higgs vortex to have finite size even though a scalar-gradient-plus-nonnegative-potential model has no comparable magnetic term.

A gauge coupling is the constant multiplying the gauge potential in the gauge covariant derivative. Its normalization depends on that of the Lie-algebra generators; each simple or Abelian factor of a gauge algebra may have an independent coupling.

The Jacobi identity for gauge covariant derivatives gives $D_{[\mu }{F}_{\nu \rho ]}=0$. In four dimensions this is $D_{\mu }{\tilde{F}}^{\mu \nu }=0$, with ${\tilde{F}}^{\mu \nu }=\frac{1}{2}{\epsilon }^{\mu \nu \rho \sigma }{F}_{\rho \sigma }$. It holds independently of the Yang-Mills equations.

For the Killing form or another invariant bilinear form on a Lie algebra,The connection terms cancel by invariance. Thus, for compactly supported variations, a gauge covariant derivative can be moved to the other factor in an integral with a minus sign, just as an ordinary derivative.

// Target: quantum-field-theory.bigb

When $|\Phi |\to 1$ and its gauge covariant derivative decays fast enough, on a large circle write $\Phi =\rho e^{i\chi }$ locally. The tangential derivative gives $A=d\chi -\mathrm{Im}(\overline{\Phi }D\Phi )/{\rho }^2$. The integrated error tends to zero, so Stokes theorem turns total magnetic flux into the phase change $2\pi N$. This integer is the winding number of the normalized Higgs field on the circle. For positive Bogomolny vortex equations it is the total zero multiplicity.

Replace the derivative in the Dirac action by the gauge covariant derivative $D_{\mu }={\partial }_{\mu }+iq{A}_{\mu }$. The interaction is then $-q{A}_{\mu }\bar{\Psi }{\gamma }^{\mu }\Psi$. With $\Psi \mapsto e^{-iq\alpha }\Psi$ and $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha$, the covariant derivative transforms in the same way as the spinor, proving local gauge invariance.

In the usual smooth localized finite-energy class, the positive potential requires $|\varphi |\to 1$ at infinity and the gauge covariant derivative tends to zero there. On a sufficiently large circle write $\varphi \simeq e^{i\chi }$ locally. Then the tangential gauge potential approaches $d\chi$. The normalized Higgs phase defines a continuous map $S_{\infty }^1\to U(1)$ with integer winding numberThe second equality is Stokes theorem with the chosen orientation and $D_j={\partial }_j-i{a}_j$. Suitable decay makes the integrated covariant-derivative error vanish. Thus magnetic flux quantization of an Abelian Higgs vortex labels disconnected boundary sectors. This uses the standard continuous vacuum asymptotics, not a claim that bare integral finiteness alone supplies every pointwise boundary limit.

A nonzero winding requires zeros of the complex Higgs field somewhere inside: a nowhere-zero field on the whole disk would extend its normalized boundary map and make its winding zero. For isolated zeros the signed local indices sum to $N$. For arbitrary configurations, vortex-antivortex pairs may add zeros without changing $N$, so $|N|$ need not count all unsigned zeros.

The critical coupling relates the charge to a minimum energy. Set $j_i=\mathrm{Im}(\bar{\varphi }{D}_i\varphi )$. The covariant derivative commutator $[D_1,D_2]\varphi =-iB\varphi$ givesDiscarding its vanishing boundary integral and completing the magnetic square yieldsFor positive $N$, the Bogomolny vortex equations are $(D_1+i{D}_2)\varphi =0$ and $B=(1-|\varphi {|}^2)/2$. Reverse both signs for negative charge. HenceCritical-coupling vortex solutions attain this bound; their energy depends only on $|N|$. Positive-charge solutions have $N$ zeros counted with multiplicity, and their positions provide $2N$ real coordinates on the Abelian Higgs vortex moduli space. Coincident zeros are allowed. Position variations leave the static minimum energy unchanged, expressing the absence of static forces among same-sign Bogomolny vortices. The sign of $N$ reverses flux and orientation. Topology alone does not determine an arbitrary field's profile or full energy: the zero-charge vacuum has energy zero, while nonvacuum configurations in the same sector can have positive energy.

For the given radial ansatz, $a_{\theta }$ is the coordinate one-form coefficient: the gauge potential is $a(r)d\theta$, and the orthonormal azimuthal component is $a(r)/r$. Thus $B=a^{\prime }(r)/r$, consistently with the printed energy. Since $h(\infty )=1$, the phase $e^{i\theta }$ has winding one; equivalently $a(0)=0$, $a(\infty )=1$ givesFor the trial functions all energy is inside $0<r<r_0$. Their four contributions to $E/\pi$ areTherefore the compact-core variational estimate for an Abelian Higgs vortex isWriting $s=r_0^2>0$, the derivative is $-2/s^2+1/24$, and the energy diverges at both ends of the $s$ range. The unique global minimum isThe magnetic and potential energies agree at this optimum, as required by radial scale balance. The piecewise fields are continuous with square-integrable first derivatives; their derivative jumps do not create an additional surface energy. They are trial fields, not exact solutions. Their optimized energy is an upper-bound estimate, about 24.4% above the exact critical one-vortex energy $\pi$.

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.

Choose the phase convention in the question and use the gauge covariant derivative $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$. For a charged Dirac field, the quantum electrodynamics Lagrangian isThe simultaneous local gauge transformation isDirect differentiation gives $D_{\mu }^{\prime }{\psi }^{\prime }=e^{ie\chi }{D}_{\mu }\psi$: the derivative of the phase cancels the shift of $A_{\mu }$. The mass term and covariant kinetic term are therefore invariant. The electromagnetic field tensor is unchanged because mixed partial derivatives of $\chi$ commute. This proves classical gauge invariance of the coupled field action. A charged scalar similarly uses $(D_{\mu }\varphi {)}^{\ast }{D}^{\mu }\varphi$ and a potential depending only on $|\varphi {|}^2$.

The electromagnetic coupling is $-e{A}_{\mu }\bar{\psi }{\gamma }^{\mu }\psi$. The matter equations give a conserved current $j^{\mu }=e\bar{\psi }{\gamma }^{\mu }\psi$, with ${\partial }_{\mu }{j}^{\mu }=0$. Constant $\chi$ gives the global symmetry and its conserved charge. When applying Noether's theorem, the sign of the parameterized Noether generator can be reversed without changing this conservation law; the transformation and coupling signs above remain fixed.

The Adjoint representation of a Lie algebra acts on the Lie algebra itself by ${\mathrm{ad}}_X(Y)=[X,Y]$. The Jacobi identity givesThus it is a Lie algebra representation. In the basis $T_a$ with $[T_a,T_b]=c^c{}_{ab}{T}_c$, its matrices areThe column index labels the basis vector being acted on; this convention fixes the matrix signs.

Work in finite dimension over characteristic zero, as in the usual real or complex gauge algebras. A simple Lie algebra is nonabelian and has no nonzero proper Lie algebra ideal. A semisimple Lie algebra has no nonzero solvable ideal; equivalently it is a direct sum of simple ideals. The nondegenerate Killing form is $\kappa (X,Y)=\mathrm{tr}({\mathrm{ad}}_X{\mathrm{ad}}_Y)$. Its invariance follows from the representation identity and trace cyclicity:

For the reduction, take a nonzero proper ideal $\mathfrak{a}$, if one exists, and define its Killing-orthogonal complement ${\mathfrak{a}}^{\perp }$. Invariance makes it an ideal: for $y\in {\mathfrak{a}}^{\perp }$, $x\in L$, $z\in \mathfrak{a}$, $\kappa ([x,y],z)=-\kappa (y,[x,z])=0$. Moreover $[\mathfrak{a},{\mathfrak{a}}^{\perp }]=0$, because for $u\in \mathfrak{a}$, $v\in {\mathfrak{a}}^{\perp }$, and any $x\in L$,and the global Killing form is nondegenerate. Their intersection is consequently an abelian ideal: its elements lie in both commuting ideals. Semisimplicity forces that intersection to be zero. The dimension formula for a nondegenerate bilinear form then givesThe summands commute and are semisimple, since a solvable ideal in either summand would also be an ideal of $L$. Repeating on smaller summands terminates in simple ideals. This proves the orthogonal ideal splitting for a nondegenerate Killing form and describes the required reduction; abelian one-dimensional factors do not occur in a semisimple algebra.

If $\kappa (T_a,T_b)=\ell {\delta }_{ab}$ with $\ell \ne 0$, invariance givesso $c_{abc}=-c_{acb}$ after lowering the output index with $\delta$. The bracket already gives $c_{abc}=-c_{bac}$. These two adjacent transpositions generate all permutations, proving that the structure constants of a Lie algebra are completely antisymmetric. This is the antisymmetry of Killing-lowered structure constants; the scalar $\ell$ cancels in the chosen normalization.

For the gauge calculation write $\Lambda ={\lambda }_a{t}_a$, $A_{\mu }=A_{\mu a}{t}_a$, and $D_{\mu }={\partial }_{\mu }+A_{\mu }$. The representation generators here obey $[t_a,t_b]=c_{abc}{t}_c$; factors of $i$ and the gauge coupling are absorbed in this connection convention. Expanding the varied derivative givesFor this to equal $\Lambda D_{\mu }\varphi$, choose the gauge transformation in the derivative-plus-connection conventionThen all extra terms cancel, proving covariance of the gauge covariant derivative for arbitrary space-dependent parameters.

Apply both derivatives to a test field. The second derivatives commute and the terms involving one derivative of the field cancel, leaving the Yang-Mills field strengthThus its components areThe already obtained variation can be written $\delta D_{\mu }=[\Lambda ,D_{\mu }]$. The operator Jacobi identity then yieldsorFinally, with fixed space-time metric,The contracted product is symmetric in $a,c$, while $c_{abc}$ is antisymmetric in those indices. Hence the Yang-Mills action density is gauge invariant, with the sign convention derived rather than assumed.

For a constant unitary matrix $h$, ${\varphi }^{\prime }=h\varphi$ gives ${\varphi }^{\prime †}{\varphi }^{\prime }={\varphi }^{†}{h}^{†}h\varphi ={\varphi }^{†}\varphi$. Also ${\partial }_{\mu }{\varphi }^{\prime }=h{\partial }_{\mu }\varphi$, so the kinetic contraction is unchanged. The mass term and the squared norm in the potential are unchanged for the same reason. The original scalar action has the stated global symmetry. For a spacetime-dependent $h$, however, ${\partial }_{\mu }(h\varphi )=h{\partial }_{\mu }\varphi +({\partial }_{\mu }h)\varphi$, and the extra term prevents local invariance of the ordinary-derivative kinetic term.

Choose the gauge covariant derivative $D_{\mu }={\partial }_{\mu }+ig{A}_{\mu }$, with $A_{\mu }=A_{\mu }^{\alpha }{T}_{\alpha }$ Hermitian. Demand $D_{\mu }^{\prime }(h\varphi )=h{D}_{\mu }\varphi$. Expanding both sides yieldsso the necessary Yang-Mills gauge transformation isUnitarity makes the transformed connection Hermitian. This formula includes the inhomogeneous derivative term; mere conjugation would be insufficient for a local transformation.

Define the Yang-Mills field strength by $[D_{\mu },D_{\nu }]=ig{F}_{\mu \nu }$. Direct expansion givesChoose an orthonormal Hermitian generator basis with $\mathrm{tr}(T_{\alpha }{T}_{\beta })=\kappa {\delta }_{\alpha \beta }$. The compact simple Lie algebra has an invariant positive trace metric, so this basis can be chosen; its structure constants are totally antisymmetric. With these conventions a locally invariant dynamical action isEach matter contraction is invariant because $D\varphi$ transforms like $\varphi$, and the gauge kinetic term is invariant by cyclicity of the trace. It is $-\frac{1}{4}{F}_{\mu \nu }^{\alpha }{F}_{\alpha }^{\mu \nu }$ in components. Choosing $\kappa =1/2$ gives the familiar $-\frac{1}{2}\mathrm{tr}{F}^2$ convention.

Formally introduce a gauge-field source $J_{\alpha }^{\mu }$ and a normalized generating functionalA gauge-field Green function is obtained by source differentiation:Here $\mathcal{N}$ is the unnormalized integral at zero source, so the normalized functional obeys $Z[0]=1$. This expression needs gauge fixing: the integral repeats every physical configuration over its whole gauge orbit, producing an irrelevant but divergent gauge-group volume, and the free gauge kinetic operator has longitudinal zero modes, so it has no inverse propagator on that unrestricted space. Ordinary gauge-field Green functions are gauge-dependent; physical gauge-invariant quantities must be obtained consistently from the gauge-fixed theory.

Use the Lorenz condition ${\chi }^{\alpha }[A]={\partial }^{\mu }{A}_{\mu }^{\alpha }$. To keep signs explicit, parametrize a small transformation by $h=1-ig{\omega }^{\alpha }{T}_{\alpha }+O({\omega }^2)$, equivalently the original parameter is ${\theta }^{\alpha }=g{\omega }^{\alpha }$. The finite transformation above gives the adjoint gauge variation for a positive-sign covariant derivative:Therefore $\delta {\chi }^{\alpha }={\partial }^{\mu }{\mathcal{D}}_{\mu }^{\alpha \beta }{\omega }^{\beta }$. Define the Faddeev-Popov operator as $\mathcal{M}=-{\partial }^{\mu }{\mathcal{D}}_{\mu }$, absorbing the field-independent minus sign of its determinant into normalization.

In a perturbative gauge patch, with residual zero modes removed by boundary conditions, the Faddeev-Popov gauge-orbit identity isIt follows by changing coordinates from the gauge parameter to the gauge condition: the functional Jacobian is precisely the determinant of its linearization. Insert the identity into the integral and change variables along the orbit. The action and measure are invariant, so the gauge-group volume factors out and is divided away. Averaging the gauge-slice value $f$ with Gaussian weight $\exp [-i\int f^2/(2\xi )]$ changes the delta-functional restriction to the covariant gauge-fixing termThe remaining determinant depends on $A$ through ${\mathcal{D}}_{\mu }$. Omitting it while adding only ${\mathcal{L}}_{\mathrm{gf}}$ would weight different orbits incorrectly.

For a finite matrix $M$, expansion of a Grassmann integral selects the product containing every $c$ and $\bar{c}$. Anticommuting the variables into a fixed order attaches the sign of each permutation, giving $\int d\bar{c}dc{e}^{i\bar{c}Mc}$ proportional to $\det M$, with only a matrix-independent phase. The regulated functional version therefore represents the Faddeev-Popov determinant asThe independent Faddeev-Popov ghost fields and antighost fields obey $c_{\alpha }{c}_{\beta }=-c_{\beta }{c}_{\alpha }$, with the corresponding mixed and antighost anticommutation. A commuting complex Gaussian integral instead gives an inverse determinant, so it cannot supply this Jacobian. The required local ghost term isAfter integration by parts,This exhibits the free ghost kinetic term and the ghost-gluon vertex. Ghosts are Grassmann-valued scalar auxiliary fields in the Adjoint representation, not physical fermionic particles or external asymptotic states. Their loop signs arise from anticommutation. In an Abelian theory the adjoint derivative loses the field-dependent part, so the determinant is field-independent and the ghosts decouple; in the non-Abelian theory they must be retained.

The correctly gauge-fixed source functional thus uses $S_{\mathrm{inv}}+S_{\mathrm{gf}}+S_{\mathrm{gh}}$ and integrates over $A,\varphi ,{\varphi }^{†},c,\bar{c}$. Expanding its interaction terms gives the gauge, matter and ghost vertices, and differentiating its source functional computes the gauge-field Green functions. For example, at non-null momentum the quadratic gauge kernel iswhere $P_T$ and $P_L$ are the transverse projector of a vector field and longitudinal projector of a vector field. With the Feynman prescription this gives the gauge-boson propagatorFor the chosen ghost normalization the quadratic kernel is $-{\partial }^2$, so the ghost propagator is $i{\delta }^{\alpha \beta }/(k^2+i0)$. Redefining the antighost sign changes ghost propagator and vertex signs together, without changing the determinant or physical conclusions. Gauge fixing requires the accompanying determinant, and its local representation requires anticommuting ghost and antighost fields. The gauge-orbit argument is local/perturbative: global gauge copies, the Gribov ambiguity, prevent interpreting it as a globally unique gauge slice without further treatment.

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.

The nonexistence conclusion concerns nonconstant fields: a constant scalar-field vacuum with $U=0$ is a zero-energy critical point of an energy functional in every dimension. Assume the potential is smooth, so the classical Euler-Lagrange equation is well defined. Write the two nonnegative energy terms asFor the admissible Derrick scaling ${\varphi }_{\lambda }(x)=\varphi (\lambda x)$,Stationarity at $\lambda =1$ gives the Derrick virial identityFor $D>2$, both terms have nonpositive sign, forcing $T=V=0$ and hence a constant vacuum. For $D=2$, scaling alone only proves $V=0$; the gradient term is scale invariant, so it must not simply be declared zero.

The two-dimensional flat-target Derrick obstruction supplies that remaining step. Since $U\ge 0$ and its integral is zero, continuity gives $U(\varphi (x))=0$ everywhere; smooth nonnegative $U$ has $U^{\prime }(\varphi (x))=0$ at those values. The field equation $\Delta \varphi =U^{\prime }(\varphi )$ therefore becomes $\Delta \varphi =0$. Each ${\partial }_i\varphi$ is an entire harmonic function in $L^2({\mathbb{R}}^2)$. Its mean value property and the Cauchy-Schwarz inequality give, for every radius $R$,Let $R\to \infty$ to get ${\partial }_i\varphi =0$. Thus $D=2$ also has only constant stationary field configurations of finite energy. The unconstrained real scalar target is essential in this argument; the spherical target in Question 2 has different field equations.