QED and Yang–Mills theory

Quantum electrodynamics and Yang–Mills theory use the same central idea—replace an ordinary derivative by a connection—but the consequences differ sharply. In QED the gauge algebra is Abelian, so the photon carries no electric charge and the covariant ghost determinant is field independent. In Yang–Mills theory the gauge bosons carry the non-Abelian charge, the curvature itself is nonlinear, and gauge, matter, and ghost interactions are tied to one coupling. This lesson develops that comparison far enough to check a perturbative calculation without attempting a survey of the Standard Model.

Required background. Vector fields and gauge redundancy supplies covariant gauge fixing and physical polarizations. Perturbative expansion and Feynman rules supplies the rule-extraction and fermion-sign conventions. Loops and regularization and renormalization and the renormalization group supply the regulator, counterterm, and running-coupling language used below.

Helpful background. Fermions, spin, and anticommutation fixes the Dirac propagator and closed-fermion-loop sign. Symmetry, currents, and Ward identities explains contact terms, while LSZ and tree amplitudes explains why an on-shell polarization-replacement test is stronger than a check on one Green function.

Connections and curvature distinguish the theories

For QED, let $q$ be the signed $U(1)$ transformation weight of a Dirac field. The inherited charge convention is

D_\mu^{\mathrm{QED}}=\partial_\mu-iqA_\mu, \qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.

The local transformation

\psi\longmapsto e^{+iq\lambda(x)}\psi, \qquad A_\mu\longmapsto A_\mu+\partial_\mu\lambda

makes $D_\mu\psi$ transform like $\psi$ and leaves $F_{\mu\nu}$ invariant. Moreover,

[D_\mu^{\mathrm{QED}},D_\nu^{\mathrm{QED}}] =-iqF_{\mu\nu}.

With the $(+---)$ metric, a gauge-fixed QED action is

S_{\mathrm{QED}}=\int\mathrm d^4x\left[ -\frac14F_{\mu\nu}F^{\mu\nu} +\bar\psi(i\gamma^\mu D_\mu^{\mathrm{QED}}-m)\psi -\frac1{2\xi}(\partial_\mu A^\mu)^2 \right].

Because $i\gamma^\mu D_\mu^{\mathrm{QED}} =i\gamma^\mu\partial_\mu+q\gamma^\mu A_\mu$ , the interaction is $+q\bar\psi\gamma^\mu\psi A_\mu$ . Its vertex is $+iq\gamma^\mu$ ; for the electron label $q=-e$ , $e>0$ , this is $-ie\gamma^\mu$ . The resulting electron interaction and covariant gauge fixing agree with Schwartz 2014, §§ 13.1–13.3, pp. 224–236.

Keep the charge label distinct from a potential convention. Write $j_q^\mu=q\bar\psi\gamma^\mu\psi$ . With the potential $A$ above, Maxwell’s equation is $\partial_\nu F^{\nu\mu}=-j_q^\mu$ . In the common convention $D=\partial+iqA^{\rm em}$ , use $A^{\rm em}=-A$ , $F^{\rm em}=-F$ , and the opposite gauge parameter; then $\partial_\nu F^{{\rm em}\,\nu\mu}=j_q^\mu$ and the same label $q=-e$ is the physical electron charge. The QED action and static-source calculation derive this translation and the unchanged Coulomb force.

For Yang–Mills theory, let $G$ be a compact gauge group and write $A_\mu=A_\mu^aT_R^a$ in a unitary representation $R$ . We use Hermitian generators and the site convention

[T_R^a,T_R^b]=if^{abc}T_R^c, \qquad D_\mu^{\mathrm{YM}}=\partial_\mu-igA_\mu, \qquad g>0.

If $\psi\mapsto U\psi$ , covariance requires

A_\mu\longmapsto UA_\mu U^{-1}-\frac{i}{g}(\partial_\mu U)U^{-1}.

The commutator now defines a matrix-valued curvature,

\begin{aligned} [D_\mu^{\mathrm{YM}},D_\nu^{\mathrm{YM}}]&=-igF_{\mu\nu},\\ F_{\mu\nu} &=\partial_\mu A_\nu-\partial_\nu A_\mu-ig[A_\mu,A_\nu],\\ F_{\mu\nu}^a &=\partial_\mu A_\nu^a-\partial_\nu A_\mu^a +g f^{abc}A_\mu^bA_\nu^c. \end{aligned}

Thus $F_{\mu\nu}\mapsto UF_{\mu\nu}U^{-1}$ , and a minimal action with Dirac matter is

S_{\mathrm{inv}}=\int\mathrm d^4x\left[ -\frac14F_{\mu\nu}^aF^{a\mu\nu} +\bar\psi(i\gamma^\mu D_\mu^{\mathrm{YM}}-m)\psi \right].

The commutator term in $F$ is the decisive difference: squaring the curvature forces cubic and quartic gauge interactions. Yang and Mills introduced this nonlinear gauge curvature in Yang and Mills 1954, pp. 191–195; a modern action-level derivation appears in Schwartz 2014, §§ 25.1–25.3, pp. 481–495.

When the generators commute, the identification $gT_R\leftrightarrow q$ turns $D_\mu^{\mathrm{YM}}$ into $D_\mu^{\mathrm{QED}}$ . The Abelian representation label carries either sign even though $g>0$ . Transport this identification through the field transformation, curvature, action, and vertices; changing only one line would be inconsistent.

Gauge fixing introduces ghosts only when the orbit derivative depends on the field

The invariant gauge-field kinetic operator has null directions. In a linear covariant gauge, both theories add

\mathcal L_{\mathrm{gf}} =-\frac1{2\xi}(\partial^\mu A_\mu^a)^2,

where the adjoint index is omitted in QED. The free gauge propagator is

D_{\mu\nu}^{ab}(p) =\frac{-i\delta^{ab}}{p^2+i0} \left[ \eta_{\mu\nu} -(1-\xi)\frac{p_\mu p_\nu}{p^2+i0} \right].

This propagator acts in an enlarged gauge-fixed state space; its numerator is not a sum over physical polarizations.

The Faddeev–Popov determinant measures how the gauge function $G^a[A]=\partial^\mu A_\mu^a$ changes along a gauge orbit. In QED, $\delta A_\mu=\partial_\mu\lambda$ , so $\delta G=\Box\lambda$ . The determinant is independent of $A_\mu$ and can be absorbed into the overall normalization.

For Yang–Mills theory,

\delta A_\mu^a=(D_\mu\alpha)^a, \qquad (D_\mu\alpha)^a =\partial_\mu\alpha^a+g f^{abc}A_\mu^b\alpha^c,

so $\delta G^a=\partial^\mu(D_\mu\alpha)^a$ depends on the gauge field. Grassmann fields $c^a,\bar c^a$ exponentiate the determinant. With the same adjoint derivative, the gauge-fixing and ghost terms are

\mathcal L_{\mathrm{gf+gh}} =-\frac1{2\xi}(\partial^\mu A_\mu^a)^2 -\bar c^a\partial^\mu(D_\mu c)^a.

After integrating by parts,

\mathcal L_{\mathrm{gh}} =(\partial^\mu\bar c^a)\partial_\mu c^a +g f^{abc}(\partial^\mu\bar c^a)A_\mu^b c^c.

Ghosts therefore interact in Yang–Mills loops even though they are never physical external particles. The determinant construction is due to Faddeev and Popov 1967, pp. 29–30.

An auxiliary field $B^a$ makes the organizing BRST symmetry nilpotent off shell. One consistent convention is

\begin{aligned} sA_\mu^a&=(D_\mu c)^a, &sc^a&=-\frac g2f^{abc}c^bc^c,\\ s\bar c^a&=B^a, &sB^a&=0,\\ s\psi&=igc^aT_R^a\psi. \end{aligned}

Then $\mathcal L_{\mathrm{gf+gh}}$ follows by applying $s$ to $\bar c^a(\partial\cdot A^a+\xi B^a/2)$ and eliminating $B^a$ . The signs in $sc$ and $s\psi$ are fixed by $s^2=0$ , using both the anticommutation of the ghosts and the Jacobi identity. This is the perturbative BRST construction, not a claim that a linear gauge condition intersects every nonperturbative gauge orbit exactly once Becchi, Rouet, and Stora 1976, pp. 287–321.

The action fixes every interaction vertex

Define the linear part of the Yang–Mills curvature by

F_{\mu\nu}^{a(0)} =\partial_\mu A_\nu^a-\partial_\nu A_\mu^a.

Expanding the gauge action gives

\begin{aligned} \mathcal L_{\mathrm{YM}} ={}&-\frac14F_{\mu\nu}^{a(0)}F^{a(0)\mu\nu}\\ &-\frac g2 f^{abc}F_{\mu\nu}^{a(0)}A^{b\mu}A^{c\nu}\\ &-\frac{g^2}{4}f^{abc}f^{ade} A_\mu^bA_\nu^cA^{d\mu}A^{e\nu}. \end{aligned}

The last two lines are the three- and four-gauge interactions. They are not optional additions, and their relative coefficient cannot be adjusted independently. The matter terms and their full vertex factors are

\begin{array}{c|c|c} &\mathcal L_{\mathrm{int}}&\text{vertex factor}\\ \hline \mathrm{QED}&+q\bar\psi\gamma^\mu\psi A_\mu&+iq\gamma^\mu\\ \mathrm{Yang\text{-}Mills} &+g\bar\psi\gamma^\mu T_R^a\psi A_\mu^a &+ig\gamma^\mu T_R^a \end{array}

Use the Fourier transform $\widetilde f(p)=\int\mathrm d^4x\,e^{+ip\cdot x}f(x)$ and take all momenta incoming. For $A_\mu^a(p)$ , $A_\nu^b(q)$ , and $A_\rho^c(r)$ with $p+q+r=0$ , the three-gauge vertex factor is

\begin{aligned} \mathcal V_{\mu\nu\rho}^{abc}(p,q,r) =g f^{abc}\big[{} &(q-r)_\mu\eta_{\nu\rho} +(r-p)_\nu\eta_{\rho\mu}\\ &+(p-q)_\rho\eta_{\mu\nu} \big]. \end{aligned}

The derivative supplies the factor that makes this rule real in the stated Fourier convention; the four-gauge rule instead carries $-ig^2$ times a sum of three $ff$ color tensors and metric products. Order the ghost-vertex legs as antighost $\bar c^a$ , gauge field $A_\mu^b$ , and ghost $c^c$ . For incoming antighost momentum $q$ , the displayed interaction gives

\mathcal V_\mu^{\bar a b c} =i\,g f^{abc}(-iq_\mu)=g f^{abc}q_\mu.

If the labels are instead ordered as gauge $a$ , antighost $b$ , ghost $c$ , the tensor is $g f^{bac}q_\mu=-g f^{abc}q_\mu$ . Relabeling legs does not supply an extra Grassmann permutation. A closed ghost loop has the usual additional minus sign of a closed Grassmann loop. The full vertex construction appears in Srednicki 2006 draft, § 72, pp. 424–426, PDF and Yang–Mills color algebra and perturbative vertices. The draft uses the opposite metric and labels its ghost vertex in gauge–antighost–ghost order. Its tensor must be translated with both its momentum components and its leg labels; the derivative calculation above fixes the sign directly in our conventions.

The comparison is now structural:

Feature	QED	Yang–Mills theory
Gauge algebra	Commuting	$[T^a,T^b]=if^{abc}T^c$
Gauge self-interaction	None	Three- and four-gauge vertices
Covariant ghost	Field-independent determinant; decouples	Interacting adjoint Grassmann field
Matter vertex	Charge $q$	Matrix $T_R^a$ and coupling $g$
Quantum identity	Ward–Takahashi	Slavnov–Taylor

Color algebra reduces before momentum integration

For each compact simple factor, define

\operatorname{tr}_R(T_R^aT_R^b)=T(R)\delta^{ab}, \qquad T_R^aT_R^a=C_2(R)\mathbf1_R,

and

f^{acd}f^{bcd}=C_A\delta^{ab}.

Taking a trace of the Casimir relation gives the dimension check

d_R C_2(R)=d_A T(R).

For the fundamental representation of $SU(N)$ with $T(F)=1/2$ ,

d_F=N, \qquad d_A=N^2-1, \qquad C_F=\frac{N^2-1}{2N}, \qquad C_A=N.

Two useful reductions are

T_R^bT_R^aT_R^b =\left(C_2(R)-\frac12C_A\right)T_R^a, \qquad f^{abe}f^{cde}+f^{bce}f^{ade}+f^{cae}f^{bde}=0.

The second equation is Jacobi. For a color tensor with all external legs treated as incoming, gauge invariance also implies color conservation,

\left(\sum_i\mathbf T_i^a\right)\mathcal C=0,

where an incoming antifundamental carries $-(T^a)^T$ . Reduce these tensors before substituting numerical $SU(N)$ values; $T(R)$ , $C_2(R)$ , and $C_A$ answer different contractions.

The Abelian limit means that all generators commute, so $f^{abc}\to0$ and $C_A\to0$ . It is not the substitution $N=1$ in formulas for $SU(N)$ . In this limit the pure-gauge and ghost vertices vanish, the representation matrices become ordinary charges, and the nonlinear BRST and Slavnov–Taylor structures reduce to their Abelian forms.

Ward and Slavnov–Taylor identities test the complete result

Distinguish the full Feynman fermion correlator $S_F$ from the reduced propagator $S$ by

S_F(p)=iS(p),\qquad S^{-1}(p)=p\!\!\!/-m-\Sigma(p).

Define the proper photon-vertex rule to be $+iq\Gamma^\mu(p+k,p)$ , so $\Gamma^\mu$ has both $i$ and the signed charge removed. The Ward–Takahashi identity is

k_\mu\Gamma^\mu(p+k,p) =S^{-1}(p+k)-S^{-1}(p).

At tree level, $\Gamma^\mu=\gamma^\mu$ and $S_0^{-1}(p)=p\!\!\!/-m$ , so both sides equal $k\!\!\!/$ . Using $S_F^{-1}$ instead would require an additional factor $i$ on the right-hand side. Between equal-mass on-shell spinors the reduced inverse terms vanish, with the same external-state and infrared prescription used throughout. Consequently a complete amplitude with an external photon must obey

\mathcal M=\varepsilon_\mu(k)\mathcal M^\mu, \qquad k_\mu\mathcal M^\mu=0.

This makes the amplitude invariant under $\varepsilon_\mu\mapsto\varepsilon_\mu+\alpha k_\mu$ and removes the longitudinal part of an internal photon propagator between conserved currents. Off shell, however, the contraction is a difference of inverse propagators, not zero. The original on-shell identity and its off-shell extension are Ward 1950, p. 182 and Takahashi 1957, pp. 371–375.

Apply this to the complete two-diagram Compton amplitude. That calculation derives the incoming-photon cancellation, recoil relation, Klein–Nishina rate, and Thomson limit in the same convention. Neither graph passes the Ward test alone. Exercise 3 below transfers the cancellation to the outgoing photon, without spin or polarization averaging.

In Yang–Mills theory, the nonlinear gauge transformation and the interacting determinant prevent one simple linear identity from controlling every vertex. A BRST change of variables gives the functional Slavnov–Taylor equation, schematically

\mathcal S(\Gamma)=0,

where $\Gamma$ includes sources for the nonlinear BRST variations. Differentiating this equation relates gauge, ghost, and matter propagators and vertices. It implies the same on-shell polarization-replacement test for a complete perturbative amplitude,

\varepsilon_\mu^a(k)\mathcal M^{a\mu} \longrightarrow k_\mu\mathcal M^{a\mu}=0,

but it does not imply that an arbitrary off-shell gauge-vertex contraction vanishes. The necessary ghost functions and inverse propagators appear in the off-shell identity. The generalized identities were derived independently in Taylor 1971, pp. 436–444 and Slavnov 1972, pp. 99–104.

Covariant quantization uses an auxiliary state space with an indefinite inner product $[\cdot,\cdot]_K$ . Candidate physical states are ghost-number-zero BRST cohomology classes: closed states satisfy $Q|\mathrm{phys}\rangle=0$ , and states differing by an exact state $Q|\chi\rangle$ are identified. Assume that $Q$ is conserved, nilpotent and self-adjoint with respect to this indefinite product on a common $Q$ -invariant domain. For a closed state $\psi$ , $[\psi,Q\chi]_K=[Q\psi,\chi]_K=0$ . Exact states are therefore orthogonal to closed states, so the product descends to cohomology.

Physical unitarity requires more. Assume that the ghost-number-zero quotient has a positive-definite induced product and a Hilbert completion. If asymptotic scattering exists and both $S$ and $S^{-1}$ preserve the domain and ghost number, commute with $Q$ , and preserve the auxiliary form, they induce mutually inverse isometries on the quotient: commutation preserves closed states and sends $Q|\chi\rangle$ to $Q(S|\chi\rangle)$ . Physical scattering thus extends unitarily to the completion. Positivity and a compatible asymptotic and infrared construction are additional assumptions; anomaly freedom of the Slavnov–Taylor identity alone does not establish them. The BRST state-space discussion explains the positivity conditions. Individual gauge-fixed propagators, diagrams and cuts can still depend on the gauge choice.

Renormalization must preserve the constrained interaction pattern

Write the Yang–Mills bare quantities as

A_0=Z_A^{1/2}A, \qquad c_0=Z_c^{1/2}c, \qquad \psi_0=Z_\psi^{1/2}\psi, \qquad g_0=\mu^\epsilon Z_g g.

The Slavnov–Taylor identity requires the complete multiplicative factors of the interaction terms to be compatible with one $Z_g$ :

\begin{aligned} Z_{3A}&=Z_gZ_A^{3/2}, &Z_{4A}&=Z_g^2Z_A^2,\\ Z_{\bar c cA}&=Z_gZ_cZ_A^{1/2}, &Z_{\bar\psi\psi A}&=Z_gZ_\psi Z_A^{1/2}. \end{aligned}

Thus extracting $Z_g$ from two different vertices must agree after the appropriate wave-function factors are divided out. A symmetry-breaking regulator may require finite restoring counterterms; an uncanceled gauge anomaly prevents restoration. The corresponding Abelian statement is the QED relation $Z_1=Z_2$ , which follows from the Ward–Takahashi identity and leaves charge renormalization tied to photon vacuum polarization. These constraints and their perturbative use are developed in Srednicki 2006 draft, § 74, pp. 438–439, PDF.

One-loop running provides a useful normalization and sign check. For Dirac QED species with $q_f=eQ_f$ ,

\beta(e) =\mu\frac{\mathrm de}{\mathrm d\mu} =\frac{e^3}{12\pi^2}\sum_f Q_f^2+O(e^5).

For Yang–Mills theory with Dirac fermions in representations $R_f$ ,

\beta(g) =-\frac{g^3}{16\pi^2} \left[ \frac{11}{3}C_A -\frac{4}{3}\sum_f T(R_f) \right] +O(g^5).

When the algebra is Abelian, $C_A=0$ and $T(R_f)=Q_f^2$ , so the second formula reproduces the positive QED coefficient. This cross-check requires the ghost contribution in the non-Abelian gauge-sector sum. The coefficients and their one-loop interpretation are derived in Srednicki 2006 draft, § 73, Eqs. (73.31)–(73.41), pp. 433–434, PDF and Schwartz 2014, §§ 26.4–26.6, pp. 517–528. In the draft, $T(A)=C_A$ and its $d=4-\epsilon_{\rm draft}$ means $\epsilon_{\rm draft}=2\epsilon$ when comparing poles with our $d=4-2\epsilon$ convention; the four-dimensional beta coefficient agrees.

These beta functions assume four-dimensional perturbation theory and a mass-independent description above the relevant thresholds. Thresholds require matching, higher coefficients depend more strongly on the renormalization scheme, and a running coupling is not itself an observable. A negative leading Yang–Mills beta function establishes weak ultraviolet coupling when the bracket is positive; it does not by itself prove confinement or a mass gap.

Before trusting a gauge-theory calculation

Use the following checks on the complete result, not on a conveniently chosen diagram:

Theory data: state the gauge group, global representation content needed for the calculation, generator normalization, signed charges, and Fourier convention.
Action signs: recompute $[D_\mu,D_\nu]$ and expand $\bar\psi i\gamma^\mu D_\mu\psi$ . The curvature and matter vertices must follow from the same $D_\mu$ .
Gauge fixing: record $\xi$ , the gauge propagator, the Faddeev–Popov operator, and whether ghosts interact.
Diagram content: include every graph at the stated order, including closed fermion and ghost loops with their Grassmann signs.
Algebra: reduce Lorentz and color contractions symbolically; test Jacobi, Casimir identities, dimensions, and the Abelian limit.
Physical states and identities: apply the Ward or Slavnov–Taylor contraction to the complete amplitude and keep ghosts and auxiliary polarizations out of the external state set.
Renormalization: state the regulator and scheme, compare coupling renormalization from independent vertices, remove the regulator at fixed input, and label the perturbative and infrared limits of the conclusion.

A disagreement should be traced back to one of these declared ingredients. Adjusting an isolated sign until one identity happens to vanish can conceal a second inconsistency elsewhere.

Scope and limitations

This lesson treats unbroken, perturbative QED and Yang–Mills theory in linear covariant gauges around the usual vacuum. It does not derive the Higgs mechanism, electroweak mixing, chiral gauge-anomaly cancellation, the Standard Model matter content, or QCD factorization. Those require additional fields and assumptions rather than more terms appended to the formulas above.

Several limitations are already present inside this scope. Linear covariant gauge fixing is local in field space and does not resolve Gribov copies globally. Perturbative colored gauge bosons are useful short-distance external states, but confinement prevents interpreting them as nonperturbative asymptotic particles in QCD. In massless QED, ordinary charged-particle LSZ amplitudes have infrared difficulties and must ultimately be replaced by suitable inclusive or dressed observables. BRST cohomology and Slavnov–Taylor identities also assume compatible boundary conditions and no gauge anomaly. Ultraviolet renormalizability settles none of these infrared or global questions.

Exercises

Starting from the two covariant derivatives used on this page, derive both commutators and both matter-vertex signs. Then show how the commuting limit of the Yang–Mills convention becomes the QED convention.

Solution

In QED, multiplication by $A_\mu$ commutes, so acting on a test field gives
$[\partial_\mu-iqA_\mu,\partial_\nu-iqA_\nu] =-iq(\partial_\mu A_\nu-\partial_\nu A_\mu) =-iqF_{\mu\nu}.$
The matter kinetic term expands as
$\bar\psi i\gamma^\mu(\partial_\mu-iqA_\mu)\psi =\bar\psi i\gamma^\mu\partial_\mu\psi +q\bar\psi\gamma^\mu A_\mu\psi.$
Multiplication by $i$ in the perturbative expansion $e^{iS}$ therefore gives the vertex $+iq\gamma^\mu$ .

In Yang–Mills theory, matrix multiplication matters:
$\begin{aligned} [\partial_\mu-igA_\mu,\partial_\nu-igA_\nu] ={}&-ig(\partial_\mu A_\nu-\partial_\nu A_\mu)\\ &-g^2[A_\mu,A_\nu]\\ ={}&-igF_{\mu\nu}. \end{aligned}$
Since $[A_\mu,A_\nu]=i f^{abc}A_\mu^aA_\nu^bT^c$ , this gives $F_{\mu\nu}^a=\partial_\mu A_\nu^a-\partial_\nu A_\mu^a +g f^{abc}A_\mu^bA_\nu^c$ . The matter term is
$\bar\psi i\gamma^\mu(\partial_\mu-igA_\mu)\psi =\bar\psi i\gamma^\mu\partial_\mu\psi +g\bar\psi\gamma^\mu A_\mu^aT_R^a\psi,$
hence the vertex is $+ig\gamma^\mu T_R^a$ . If the generators commute, set $gT_R=q$ . Then $\partial_\mu-igA_\mu T_R=\partial_\mu-iqA_\mu$ , and the vertex becomes $+iq\gamma^\mu$ . The action, curvature, and vertex use the same identification.
Derive the Faddeev–Popov operator in Lorenz gauge for QED and Yang–Mills theory. Explain precisely why only the non-Abelian ghost interacts.

Solution

With $G[A]=\partial\cdot A$ , the QED variation is
$\delta_\lambda G =\partial^\mu\partial_\mu\lambda =\Box\lambda.$
Thus one may take $\mathcal M=-\Box$ . It contains no $A_\mu$ , so $\det\mathcal M$ is the same for every field configuration and cancels in normalized correlators. Introducing Abelian ghosts would only produce a free determinant with no coupling to photons.

For Yang–Mills theory,
$\delta_\alpha G^a =\partial^\mu(D_\mu\alpha)^a, \qquad \mathcal M^{ab}[A] =-\partial^\mu D_\mu^{ab}[A].$
Because $D_\mu^{ab}=\delta^{ab}\partial_\mu+g f^{acb}A_\mu^c$ in the declared index ordering, $\mathcal M[A]$ depends on $A$ . Exponentiating its determinant gives
$-\bar c^a\partial^\mu(D_\mu c)^a \doteq (\partial^\mu\bar c^a)\partial_\mu c^a +g f^{abc}(\partial^\mu\bar c^a)A_\mu^b c^c,$
where $\doteq$ denotes equality after discarding the boundary term. The second term is the ghost–gauge interaction. Setting $f^{abc}=0$ removes it and recovers the QED result.
Let $k=p'-p$ and let $u(p)$ and $u(p')$ be on-shell spinors of the same mass. Prove the tree-level QED Ward contraction and use it to remove the gauge-parameter-dependent part of a photon exchange. Then use the Compton amplitude, whose kinematics instead obeys $p+k=p'+k'$ , to prove the outgoing-photon replacement test $\varepsilon'^*\to k'$ . Explain why one diagram fails.

Solution

The Dirac equations are
$(p\!\!\!/-m)u(p)=0, \qquad \bar u(p')(p'\!\!\!/-m)=0.$
Therefore
$\begin{aligned} k_\mu\bar u(p')\gamma^\mu u(p) &=\bar u(p')(p'\!\!\!/-p\!\!\!/)u(p)\\ &=m\bar u(p')u(p)-m\bar u(p')u(p)=0. \end{aligned}$
This is the tree-level, on-shell limit of $k_\mu\Gamma^\mu=S^{-1}(p')-S^{-1}(p)$ . If two conserved currents exchange a photon, the gauge-dependent propagator term is proportional to
$J^\mu(k)k_\mu k_\nu J'^\nu(-k).$
Each factor vanishes by current conservation, so the exchange is independent of $\xi$ . This conclusion applies after the complete conserved currents have been assembled; an individual diagram or an off-shell vertex need not vanish under the same contraction.

For Compton scattering, define the reduced tree propagator $R(r)=(r\!\!\!/+m)/(r^2-m^2)$ away from its poles. Momentum conservation and the two external Dirac equations give
$\begin{aligned} \bar u(p')k'\!\!\!/\,R(p+k) &=\bar u(p')\bigl[(p\!\!\!/+k\!\!\!/-m) -(p'\!\!\!/-m)\bigr]R(p+k) =\bar u(p'),\\ R(p-k')k'\!\!\!/\,u(p) &=R(p-k')\bigl[(p\!\!\!/-m) -(p\!\!\!/-k'\!\!\!/-m)\bigr]u(p) =-u(p). \end{aligned}$
Thus the two terms in the amplitude reduce to $-e^2\bar u(p')\varepsilon\!\!\!/u(p)$ and $+e^2\bar u(p')\varepsilon\!\!\!/u(p)$ , respectively. They cancel for each spin and incoming polarization. Keeping either graph alone, or changing their relative plus sign, leaves a generally nonzero result.
For the $SU(N)$ fundamental representation with $\operatorname{tr}(T^aT^b)=\delta^{ab}/2$ , derive $C_F$ and reduce $T^bT^aT^b$ .

Solution

By definition, $T^aT^a=C_F\mathbf1_N$ , with the adjoint index summed. Taking the trace gives
$NC_F =\sum_{a=1}^{N^2-1}\operatorname{tr}(T^aT^a) =\frac{N^2-1}{2},$
hence
$C_F=\frac{N^2-1}{2N}.$
Next use the adjoint Casimir through the double commutator,
$[T^b,[T^b,T^a]]=C_A T^a.$
Expanding the left-hand side and using $T^bT^b=C_F\mathbf1$ gives
$2C_FT^a-2T^bT^aT^b=C_AT^a.$
Therefore
$T^bT^aT^b =\left(C_F-\frac12C_A\right)T^a.$
For $SU(N)$ , $C_A=N$ , so the coefficient is
$\frac{N^2-1}{2N}-\frac N2=-\frac1{2N}.$
The negative result is a useful test that $C_F$ and $C_A$ have not been confused.
Suppose a renormalized Yang–Mills calculation determines the three-gauge and ghost–gauge vertex factors $Z_{3A}$ and $Z_{\bar c cA}$ , together with $Z_A$ and $Z_c$ . Give two extractions of $Z_g$ and state what a disagreement means.

Solution

The Slavnov–Taylor-compatible factorizations are
$Z_{3A}=Z_gZ_A^{3/2}, \qquad Z_{\bar c cA}=Z_gZ_cZ_A^{1/2}.$
Hence
$Z_g^{(3A)}=Z_{3A}Z_A^{-3/2}, \qquad Z_g^{(\bar c cA)} =Z_{\bar c cA}Z_c^{-1}Z_A^{-1/2}.$
They must agree through the calculated order after all diagrams, subdivergence counterterms, and finite symmetry-restoring terms required by the chosen regulator have been included. A disagreement is not evidence for two independent gauge couplings. It indicates missing diagrams—often a ghost contribution—an inconsistent color or sign convention, an incomplete subtraction, or an anomalous symmetry breaking. Only the last possibility can remain after every allowed local counterterm is exhausted.

Continue from gauge structure to controlled predictions

You are ready to continue when you can derive both curvatures from their covariant derivatives, explain why the non-Abelian ghost interacts, reconstruct the vertex signs from the action, reduce the basic color factors, apply the correct on-shell or off-shell identity, and compare coupling renormalization across independent vertices without overextending the perturbative conclusion.

Continue to effective field theory and matching to organize physics across separated scales, then to infrared-safe observables and synthesis to turn massless gauge-theory amplitudes into defensible predictions. For a deeper theory-specific treatment, enter the QED chapter or the Yang–Mills chapter.

References

Becchi, C., A. Rouet, and R. Stora. “Renormalization of Gauge Theories.” Annals of Physics 98, no. 2 (1976): 287–321. DOI.
Faddeev, L. D., and V. N. Popov. “Feynman Diagrams for the Yang–Mills Field.” Physics Letters B 25, no. 1 (1967): 29–30. DOI.
Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
Slavnov, A. A. “Ward Identities in Gauge Theories.” Theoretical and Mathematical Physics 10, no. 2 (1972): 99–104. DOI.
Srednicki, Mark. Quantum Field Theory. Prepublication draft, 2006. Author’s draft and published-edition errata. Open PDF.
Takahashi, Yasushi. “On the Generalized Ward Identity.” Il Nuovo Cimento 6 (1957): 371–375. DOI.
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