Gauge-Fixed Yang–Mills Action and Ghost Sector

Covariant Yang–Mills perturbation theory requires more than adding $-(\partial\cdot A)^2/(2\xi)$. The gauge condition has a field-dependent Jacobian, represented by interacting Grassmann ghosts, and the combined gauge-fixing and ghost sector is organized by a nilpotent BRST differential. Ghosts and the auxiliary field are not physical particles; they are the fields that make covariance and the physical-state cancellation compatible.

Required background. Yang–Mills equations, constraints, and observables supplies the gauge orbits and Gauss constraint being fixed. The BRST differential and gauge-fixed complex supplies the general cohomological construction specialized here.

Helpful background. Gribov copies and the limits of local gauge fixing explains why this perturbative gauge slice is not a global nonperturbative section of configuration space.

The Faddeev–Popov operator

For an infinitesimal transformation with parameter $\alpha^a$,

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

Choose the linear covariant gauge function $G^a[A]=\partial^\mu A_\mu^a$. Its derivative along a gauge orbit is

$$\delta_\alpha G^a =\partial^\mu(D_\mu\alpha)^a, \qquad \mathcal M^{ab}[A] =-\partial^\mu D_\mu^{ab}[A].$$

The determinant $\det\mathcal M[A]$ is field dependent because $D_\mu$ contains $A_\mu$. Independent Grassmann fields $c^a$ and $\bar c^a$ exponentiate it. After the gauge average is represented with parameter $\xi$,

$$\mathcal L_{\rm gf+gh} =-\frac{1}{2\xi}(\partial^\mu A_\mu^a)^2 -\bar c^a\partial^\mu(D_\mu c)^a.$$

After an integration by parts, the ghost term is

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

The ghost–gauge interaction is absent in Abelian theory but unavoidable in non-Abelian loops. Faddeev and Popov derived the determinant prescription for arbitrary Yang–Mills diagrams in Faddeev and Popov 1967, pp. 29–30.

Gauge fermion and nilpotent BRST symmetry

Introduce a commuting Nakanishi–Lautrup field $B^a$ and the odd differential

$$\begin{aligned} sA_\mu^a&=(D_\mu c)^a,\\ sc^a&=-\frac g2 f^{abc}c^bc^c,\\ s\bar c^a&=B^a,\\ sB^a&=0,\\ s\psi&=igc^aT_R^a\psi. \end{aligned}$$

With gauge fermion density

$$\Psi=\bar c^a\left(\partial^\mu A_\mu^a+\frac\xi2B^a\right),$$

the BRST-exact term is

$$s\Psi =\frac\xi2B^aB^a+B^a\partial^\mu A_\mu^a -\bar c^a\partial^\mu(D_\mu c)^a,$$

The algebraic equation $B^a=-\partial\cdot A^a/\xi$ returns the gauge-fixing term above. Integrating the ghost term by parts gives the displayed positive-kinetic convention. This construction makes $s(\mathcal L_{\rm YM}+s\Psi)=0$ immediate.

Nilpotency is a nontrivial sign check. On matter, antisymmetry of $c^bc^c$ turns $T^bT^c$ into $[T^b,T^c]/2$, and the stated $sc$ cancels the second variation. On $A_\mu$, the remaining cubic ghost term vanishes by

$$f^{abe}f^{ecd}+f^{bce}f^{ead}+f^{cae}f^{ebd}=0.$$

Hence $s^2=0$ on every field off shell; keeping $B$ is what makes $s^2\bar c=0$ immediate. The explicit derivation is given in Srednicki 2007, § 74, pp. 435–442.

Free propagators and field content

Use $\widetilde f(k)=\int\mathrm d^4x\,e^{+ik\cdot x}f(x)$. Inverting the quadratic action gives

$$D_{\mu\nu}^{ab}(k) =\frac{-i\delta^{ab}}{k^2+i0} \left[ \eta_{\mu\nu} -(1-\xi)\frac{k_\mu k_\nu}{k^2+i0} \right],$$

and the oriented ghost propagator

$$\langle c^a(k)\bar c^b(-k)\rangle_0 =\frac{i\delta^{ab}}{k^2+i0}.$$

If $B$ is retained rather than integrated out, the first-order free rules are

$$\langle A_\mu^a(k)B^b(-k)\rangle_0 =-\frac{\delta^{ab}k_\mu}{k^2+i0}, \qquad \langle B^a(k)B^b(-k)\rangle_0=0,$$

for the stated Fourier and $s\Psi$ convention. The algebraic relation $B=-\partial\cdot A/\xi$ reproduces the longitudinal part of the $A$ propagator; contact terms depend on whether $B$ is eliminated before or after forming time-ordered products. There is no independent $B$ pole or asymptotic particle.

The mass dimensions are

$$[A_\mu]=[c]=[\bar c]=1, \qquad [B]=2, \qquad [\xi]=0.$$

Every term therefore has dimension four. In Feynman gauge $\xi=1$ the gauge propagator is proportional to $\eta_{\mu\nu}$; in Landau gauge $\xi=0$ it is transverse. Neither choice changes a complete physical amplitude.

BRST handoff to physical states

The conserved BRST charge $Q$ satisfies $Q^2=0$. The perturbative physical state space is not the full indefinite-metric Fock space but the cohomology

$$\mathcal H_{\rm phys}=\frac{\ker Q}{\operatorname{im}Q}.$$

Longitudinal and timelike gauge modes, ghosts, antighosts, and the auxiliary sector assemble into BRST-trivial combinations; the free one-particle cohomology contains the two transverse gauge polarizations. A BRST-invariant time evolution preserves $\ker Q$ and descends to equivalence classes. The symmetry’s renormalized gauge-theory formulation is developed in Becchi, Rouet, and Stora 1976, pp. 287–321. Establishing positive physical unitarity additionally requires the Slavnov identity, absence of gauge anomalies, and suitable asymptotic states; that proof architecture is completed on Consistency of perturbative Yang–Mills theory.

This construction is perturbative around a gauge slice near a chosen background. Gribov copies can obstruct a global gauge condition, and BRST cohomology in a confining nonperturbative theory need not be represented by colored one-gauge-boson states. Covariant ghosts are indispensable internal fields, but they never appear as physical external particles.

Independent checks and limits

1. Determinant check: differentiating $G^a[A^\alpha]$ must give $\partial^\mu D_\mu^{ab}$ with the same $g f^{abc}$ sign as the ghost vertex.
2. Nilpotency check: calculate $s^2c$; the three-ghost coefficient must vanish by Jacobi, not by treating the ghosts as commuting.
3. Free inverse check: multiplying the gauge kinetic operator by $D_{\mu\nu}$ returns transverse plus longitudinal identity projectors with weights $1$ and $\xi$.
4. Abelian limit: $f^{abc}\to0$ makes $\mathcal M=-\partial^2$ field independent, so the ghost determinant is an irrelevant constant and ghost interactions disappear.

Common pitfalls

Dropping ghosts because they are unphysical. “Unphysical” means absent from external cohomology classes, not absent from loops. Ghost loops are required by the Slavnov identities and the one-loop beta function.

Checking BRST invariance only after eliminating $B$. Nilpotency on $\bar c$ is off shell with $B$ present. After elimination it generally closes only modulo the auxiliary equation of motion.

Interpreting $\xi$ as a coupling. It changes the representation of off-shell Green functions. A residual $\xi$ dependence in a complete physical amplitude is a failed gauge-consistency check.

References

• C. Becchi, A. Rouet, and R. Stora, “Renormalization of Gauge Theories,” Annals of Physics 98 (1976), 287–321, DOI.
• L. D. Faddeev and V. N. Popov, “Feynman Diagrams for the Yang–Mills Field,” Physics Letters B 25 (1967), 29–30, DOI.
• Mark Srednicki, Quantum Field Theory, Cambridge University Press (2007), §§ 71–74, DOI.