Background-Field Yang–Mills Effective Action

The background-field method splits the gauge potential into a prescribed background and an integrated quantum fluctuation, then fixes only the fluctuation with a background-covariant condition. Ordinary gauge redundancy is still fixed, but a separate background gauge invariance remains manifest. Its Ward identity forces the background effective action to use gauge-invariant counterterms and yields the especially useful relation $Z_gZ_{\bar A}^{1/2}=1$.

Required background. Gauge-fixed Yang–Mills action and ghost sector supplies the Faddeev–Popov and BRST construction. Coupling to background gauge fields and bundles supplies the general background Ward-identity viewpoint.

Helpful background. The 1PI effective action and mean-field equations supplies the Legendre transform and loop expansion used below.

Background and quantum transformations

Write the full integration variable as

$$A_\mu=\bar A_\mu+a_\mu.$$

There are two useful transformation laws. A quantum gauge transformation leaves $\bar A$ fixed and transforms the full field in the usual way:

$$\delta_{\rm q}\bar A_\mu=0, \qquad \delta_{\rm q}a_\mu=D_\mu[\bar A+a]\alpha.$$

A background transformation acts on the two pieces so that the background is a connection and the fluctuation is covariant:

$$\bar A_\mu' =U\bar A_\mu U^{-1} -\frac{i}{g}(\partial_\mu U)U^{-1}, \qquad a_\mu'=Ua_\mu U^{-1}.$$

The total $A=\bar A+a$ therefore transforms as a connection in either description, but the roles differ: the quantum transformation is the redundancy to be fixed, while the background transformation will remain as a manifest symmetry.

Choose

$$G^a[a;\bar A]=(\bar D^\mu a_\mu)^a, \qquad \mathcal L_{\rm gf} =-\frac{1}{2\xi}(\bar D^\mu a_\mu)^a(\bar D^\nu a_\nu)^a.$$

Differentiating $G$ along a quantum orbit gives

$$\mathcal M^{ab}[a;\bar A] =-\bigl(\bar D^\mu D_\mu[\bar A+a]\bigr)^{ab},$$

so the ghost term is

$$\mathcal L_{\rm gh} =\bar c^a\mathcal M^{ab}c^b +(\bar D^\mu\bar c)^a(D_\mu[\bar A+a]c)^a.$$

Under a background transformation, $G$, $c$, and $\bar c$ all transform in the adjoint. Contracting their adjoint indices makes $\mathcal L_{\rm gf}+\mathcal L_{\rm gh}$ background invariant. Abbott’s all-orders formulation makes this preserved symmetry and its renormalization consequences explicit in Abbott 1981, pp. 189–203.

Curvature expansion and quadratic operators

The full curvature separates as

$$F_{\mu\nu}[\bar A+a] =\bar F_{\mu\nu} +\bar D_\mu a_\nu-\bar D_\nu a_\mu -ig[a_\mu,a_\nu].$$

Terms quadratic in $a$ from the Yang–Mills action are

$$\begin{aligned} \mathcal L_{\rm YM}^{(2)}={}& -\frac12(\bar D_\mu a_\nu)^a(\bar D^\mu a^\nu)^a +\frac12(\bar D_\mu a_\nu)^a(\bar D^\nu a^\mu)^a\\ &-\frac g2f^{abc}\bar F^a_{\mu\nu}a^{b\mu}a^{c\nu}. \end{aligned}$$

Commuting two background derivatives in the second term produces another curvature coupling. After adding the gauge-fixing term, the quadratic kernel is

$$\mathcal L_a^{(2)} =\frac12a_\mu^b\left[ (\bar D^2)^{bc}\eta^{\mu\nu} +\left(\frac1\xi-1\right)(\bar D^\mu\bar D^\nu)^{bc} -2g f^{abc}\bar F^{a\mu\nu} \right]a_\nu^c.$$

In background Feynman gauge, $\xi=1$, this reduces to a minimal covariant Laplacian plus the spin-one curvature term:

$$\mathcal L_{a,\xi=1}^{(2)} =-\frac12(\bar D_\mu a_\nu)^a(\bar D^\mu a^\nu)^a -g f^{abc}\bar F^a_{\mu\nu}a^{b\mu}a^{c\nu}.$$

The ghost quadratic operator at $a=0$ is the adjoint scalar Laplacian $-\bar D^2$. Thus the one-loop contribution has the schematic determinant form

$$\Gamma^{(1)}[\bar A] =\frac i2\operatorname{Tr}\ln\Delta_1 -i\operatorname{Tr}\ln\Delta_0 +\Gamma_{\rm matter}^{(1)},$$

where $\Delta_1$ is the vector kernel and $\Delta_0=-\bar D^2$ is the ghost kernel. The relative minus sign is the closed Grassmann loop. The explicit expansion and one-loop evaluation are given in Srednicki 2007, § 78, pp. 465–471, with the effective-action organization summarized in Schwartz 2014, § 34.3, pp. 752–758.

Background Ward identity

After integrating over $a,c,\bar c$ and setting external quantum fields to zero, background invariance gives

$$\bar D_\mu^{ab} \frac{\delta\Gamma_{\rm bg}[\bar A]}{\delta\bar A_\mu^b}=0.$$

Consequently the divergent background-only effective action must be a local background-gauge invariant. In four-dimensional pure Yang–Mills theory, the CP-even dimension-four term is proportional to

$$-\frac14\int\mathrm d^4x\, \bar F^a_{\mu\nu}\bar F^{a\mu\nu},$$

apart from BRST-exact, total-derivative, and field-independent terms. The two-point divergence therefore determines the same counterterm that renormalizes all background three- and four-point vertices.

Let

$$\bar A_{0\mu}=Z_{\bar A}^{1/2}\bar A_\mu, \qquad g_0=\mu^\epsilon Z_g g.$$

Because the background covariant derivative contains the single combination $g\bar A$, the Ward identity requires

$$Z_g Z_{\bar A}^{1/2}=1.$$

This is the background-field analogue of the QED charge identity. It means that the coupling beta function can be extracted from the background two-point function alone. Evaluating the vector, ghost, and matter determinants returns

$$b_0 =\frac{11}{3}C_A -\frac43\sum_fT(R_f) -\frac13\sum_sT(R_s),$$

in agreement with Non-Abelian screening and asymptotic freedom. The method simplifies the organization; it does not change the beta function.

Dependence that remains

Background gauge invariance does not make $\Gamma_{\rm bg}$ an observable. At generic off-shell $\bar A$, the effective action can depend on the quantum gauge parameter $\xi$. The split $A=\bar A+a$ is also an auxiliary organization, and the gauge-fixing term distinguishes the two pieces. Physical on-shell quantities agree with ordinary gauges after the appropriate BRST identities are imposed.

Nor is $\bar A$ necessarily a solution of the quantum equations. If the expansion is performed away from a stationary background, terms linear in $a$ occur and the interpretation of the loop expansion must retain them or the associated sources. Background covariance alone cannot replace an on-shell or gauge-independence check.

Independent checks and limits

1. Transformation check: $\bar D^\mu a_\mu$ must transform homogeneously under the combined background transformation; otherwise its square is not invariant.
2. Operator check: setting $\bar F=0$ and $\bar A=0$ reduces the vector kernel to the ordinary covariant-gauge inverse propagator and the ghost kernel to $-\partial^2$.
3. Ward check: every local background counterterm must be expressible through $\bar F$, background-covariant matter fields, and covariant derivatives; a standalone $\bar A^2$ term fails.
4. Renormalization check: the same $Z_g$ inferred from $Z_{\bar A}$ must reproduce the one-loop $b_0$ obtained in an ordinary covariant gauge.

Common pitfalls

Transforming only the background. The fluctuation and ghosts must rotate covariantly for $G^aG^a$ and the ghost action to be background invariant.

Confusing background and quantum gauge invariance. Quantum redundancy is fixed and governed by BRST; background invariance is a manifest Ward symmetry of the organized effective action.

Treating off-shell background invariance as gauge-parameter independence. A functional can be background gauge invariant and still depend on $\xi$ and on the split away from physical observables.

References

• Laurence F. Abbott, “The Background Field Method Beyond One Loop,” Nuclear Physics B 185 (1981), 189–203, DOI.
• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2014), § 34.3, DOI.
• Mark Srednicki, Quantum Field Theory, Cambridge University Press (2007), § 78, DOI.