# Absence of the Gribov ambiguity in a quadratic gauge

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## Abstract

The Gribov ambiguity exists in various gauges. Algebraic gauges are likely to be ambiguity free. However, algebraic gauges are not Lorentz invariant, which is their fundamental flaw. In addition, they are not generally compatible with the boundary conditions on the gauge fields, which are needed to compactify the space i.e., the ambiguity continues to exist on a compact manifold. Here we discuss a quadratic gauge fixing, which is Lorentz invariant. We consider an example of a spherically symmetric gauge field configuration in which we prove that this Lorentz invariant gauge removes the ambiguity on a compact manifold \(\mathbb {S}^3\), when a proper boundary condition on the gauge configuration is taken into account. Thus, we provide one example where the ambiguity is absent on a compact manifold in the algebraic gauge. We also show that the *BRST* invariance is preserved in this gauge.

## Keywords

Ghost Compact Manifold Landau Gauge Auxiliary Field Proper Boundary Condition## 1 Introduction

Defining the path integral in gauge theories is a major issue of infinite redundant functional integrations. The fact that the Yang–Mills action is invariant under the gauge transformation is the cause of the issue. The issue is addressed by invoking a gauge condition such as the Landau gauge \(\partial ^\mu A_\mu =f\). However, it is shown in Ref. [1] that even after the Landau gauge fixing, there still exist equivalent configurations, which contribute to the measure of the path integral. This implies that the Landau gauge does not uniquely choose a configuration, which is the problem known as the Gribov ambiguity. We need only inequivalent configurations in the measure in order to properly quantize the theory. The inequivalent configurations can be extracted out by restricting the space of integration to the fundamental modular region \(C^0\), where the Faddeev–Popov operator has positive eigenvalues [1]. However, the region \(C^0\) still contains Gribov copies [1]. The restriction on the space of integration is achieved by adding suitable terms to the effective action \(S_{{\text {eff}}}\) resulting from the Landau gauge fixing [2, 3]. This modified action is known as the Gribov–Zwanziger action. The GZ action is not *BRST* invariant [4]. So, in an attempt to eliminate the Gribov copies, we lose the *BRST* invariance of the theory. The same ambiguity is claimed to exist in all gauges [5].

An essential reason why some gauges have the ambiguity is the differential operator involved in the gauge. Algebraic gauges are likely to be ambiguity free since they do not have a differential operator, but they have one disadvantage. In general, they violate the Lorentz invariance, which is a basic requirement for any theory, whereas the gauge under consideration in this paper is Lorentz invariant. It also turns out that the theory is *BRST* invariant. Alternative formulations addressing the Gribov ambiguity are suggested in Refs. [6, 7]. The former reference particularly is an approach using Lorentz invariant algebraic gauge conditions.

The contents of this paper are arranged as follows: in the next section, we discuss a particular quadratic gauge and its consequences at the infrared scale. In Sect. 3, we examine the case of a spherically symmetric gauge configuration. We prove that when a proper boundary condition on the gauge configuration at \(\infty \) is taken into account, the quadratic gauge uniquely chooses the configuration on a compact manifold \(\mathbb {S}^3\).

## 2 A quadratic gauge and effective Lagrangian

*x*. This gauge condition results in an effective Lagrangian of the form [14]

## 3 Spherically symmetric gauge potential and the quadratic gauge

*n*is an integer. The gauge transformation \(A_\mu \longrightarrow {{\tilde{A}}}_\mu = UA_\mu U^{-1}+i(\partial _\mu U)U^{-1}\) results in transformations of \(f_1, f_2\), and \(f_3\) as follows:

*r*. Now, the

*a*th component of \(A_i\) can be derived using the following formula:

*r*only, the first term and the coefficient of \(\sin ^2\theta \cos ^2\phi \) in the second term of Eq. (18) must individually vanish, giving us two different copy equations,

- 1.
\(f_2+\frac{1}{2} \ne -f_1\cot \frac{\alpha }{2}\),

- 2.
\(f_2+\frac{1}{2} = -f_1 \cot \frac{\alpha }{2}\).

The result is true under stronger general boundary conditions, such as \(\frac{1}{r^2}\), \(e^{-r}\), and all cases where \(\cot \frac{\alpha }{2}\rightarrow \infty \) faster than \(f_1\) decays. Similarly, it can be shown that the condition for the other two components, \( {\tilde{A}}_{i}^2 {\tilde{A}}^{i 2}=A_i^2A^{i 2}\) and \({\tilde{A}}_{i}^3 {\tilde{A}}^{i 3}=A_i^3A^{i 3}\), produce the same two equations for the copy.

## 4 *BRST* symmetry in quadratic gauge

*BRST*invariant. We begin by writing the

*BRST*transformations in the quadratic gauge:

*BRST*invariant.

## 5 Conclusion

We discussed a particular quadratic gauge, which is a Lorentz invariant algebraic gauge. We worked out an example of the spherically symmetric configuration in the quadratic gauge and proved that the configuration with a proper boundary condition does not have any copy on \(\mathbb {S}^3\). Thus, we provided one example where an algebraic gauge is compatible with the boundary condition on the fields and the compactification of the space is possible in an algebraic gauge. We also proved that the theory is *BRST* invariant.

## Notes

### Acknowledgments

Haresh is sincerely thankful to Professor Urjit A. Yajnik for useful comments on the subject.

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