Gribov Copies

Definition

In a non-abelian gauge theory (e.g., Yang-Mills with gauge group $G=SU(N)$), two gauge field configurations $A_{\mu }^a$ and $A_{\mu }^{\prime a}$ are Gribov copies if:

1. They are related by a gauge transformation: $A_{\mu }^{\prime a}=A_{\mu }^g\equiv g^{-1}({\partial }_{\mu }+A_{\mu })g$ for some $g:{\mathbb{R}}^d\to G$.
2. Both satisfy the same gauge-fixing condition, e.g., Landau gauge ${\partial }^{\mu }{A}_{\mu }^a=0$.

The existence of Gribov copies means the gauge-fixing condition does not uniquely pick one representative from each gauge orbit — the Faddeev-Popov procedure over-counts field configurations in the path integral. This was discovered by Gribov in 1978, who showed it occurs inevitably for non-abelian groups in Landau and Coulomb gauges.

Quantitatively, a Gribov copy $A^{\prime }$ of a configuration $A$ satisfying ${\partial }^{\mu }{A}_{\mu }=0$ satisfies:

${\partial }^{\mu }{A}_{\mu }^{\prime a}=0\text{with}{A}^{\prime }=A+D_{\mu }^{ab}(A){ϵ}^b+O({ϵ}^2)$

for some infinitesimal gauge parameter ${ϵ}^a$. The condition ${\partial }^{\mu }{D}_{\mu }^{ab}(A){ϵ}^b=0$ shows that copies are associated with zero modes of the Faddeev-Popov operator $M^{ab}=-{\partial }^{\mu }{D}_{\mu }^{ab}$.

The Faddeev-Popov determinant $\det M$ changes sign across the Gribov horizon $\partial \Omega$ (where $\det M=0$), invalidating the perturbative expansion that assumes $\det M>0$.

Role in Research

Gribov copies are the central problem of the gribov-zwanziger research program, which dominated the research portfolio from 2011 to 2022. Their resolution via restriction to the Gribov region $\Omega$ is the foundation of the entire Gribov-Zwanziger (GZ) and Refined Gribov-Zwanziger (RGZ) framework.

Key results depending on the Gribov copy analysis:

• No-pole condition (Gribov’s original approach): requiring the ghost propagator to have no unphysical poles forces the path integral to remain within $\Omega$.
• All-order equivalence of no-pole and horizon conditions (Phys. Lett. B 719, 2013; 43 citations): proved that Gribov’s no-pole condition and Zwanziger’s horizon function approach are equivalent to all perturbative orders.
• BRST consistency: the discovery of an exact nilpotent BRST symmetry for the GZ action (Phys. Rev. D 92, 2015; 92 citations) showed that the restriction to $\Omega$ is physically consistent — Gribov copies, being gauge artifacts, do not contribute to physical observables when BRST symmetry is preserved.

An open question (listed in gribov-zwanziger) is the extent to which Gribov copies affect gauge-invariant, physical observables. While gauge-dependent quantities (propagators, vertices) are modified by the restriction, the physical spectrum should be copy-independent.

Relations

• gribov-horizon — the boundary $\partial \Omega$ where the Faddeev-Popov operator $M$ first develops zero eigenvalues; copies are most problematic near this boundary
• refined-gribov-zwanziger — the RGZ theory is built on the Gribov copy resolution plus dimension-two condensates; Gribov copies motivate the entire RGZ framework
• brst-symmetry — BRST invariance is the key consistency check for the Gribov copy restriction; the discovery of exact BRST in GZ (Phys. Rev. D 92, 2015) resolved the long-standing concern that restricting the path integral breaks gauge invariance in an uncontrolled way
• confinement — the Gribov copy problem is intimately linked to confinement: the modified propagators in the restricted path integral exhibit spectral positivity violation, signaling gluon confinement
• spectral-positivity-violation — direct consequence of the Gribov restriction on the gluon propagator

Papers

See gribov-zwanziger for full paper list.

Notes

• Gribov copies exist in all non-abelian gauge theories in Landau gauge and Coulomb gauge; they are absent only in abelian (e.g., QED) theories where the gauge group is $U(1)$.
• The first Gribov region $\Omega$ (where $M>0$) still contains Gribov copies — complete gauge fixation would require restricting to the fundamental modular region $\Lambda \subset \Omega$, the true global minimum of $\Vert A{\Vert }^2$ on each gauge orbit. The GZ approach uses $\Omega$ (more tractable) rather than $\Lambda$.
• Common confusion: “Gribov copies” is sometimes used loosely to mean any gauge ambiguity. Precisely, they are gauge-equivalent configurations that both satisfy the gauge-fixing condition — not just any two gauge-related configurations.
• The lattice gauge theory community has extensively studied Gribov copies numerically; the effect of different copy-selection algorithms on the gluon and ghost propagators is well-documented and provides non-perturbative evidence for the GZ picture.