# Revisiting Gribov’s copies inside the horizon

## Abstract

In this work, we revisit the problem of legitimate topologically trivial Gribov copies inside the Gribov horizon. We avoid the reducibility problem which hampered the standard construction of van Baal, and then we are able to build a valid example with spherical symmetry. We also apply the same technique in the presence of a background of a Polyakov instanton in a Euclidian 3D spacetime, in order to study the effect of a non-trivial environment in the generation of multiple copies inside the horizon.

## Keywords

Gauge Transformation Zero Mode Configuration Space Morse Theory Topological Sector## 1 Introduction

It is important here to mention the fundamental result of Dell’Antonio and Zwanziger proving that every gauge orbit has at least one copy in \(C_{0}\) [5, 6]. This showed that the restriction of the configuration space to this region would not cut out any physical configuration from the action functionals.

After this reasoning, van Baal started to look for an example of these copies. One important point that should be mentioned is that he was mainly interested in gauge copies with vanishing winding number. In fact, several criticisms to Gribov’s work had been previously based on the idea that Gribov copies were an exclusive feature of improper large gauge transformations, carrying a non-vanishing winding number (see for instance [13]). Following this idea, all the construction on the restriction of the configuration space could be irrelevant, since such ambiguities of the gauge potential could be allowed so as to accommodate field configurations with non-null Pontryagin index [14, 15]. In order to prove that copies could be found even among homotopically trivial gauge configurations, Henyey constructed an explicit example [16]. The strategy designed by van Baal was then to start with Henyey’s example and show that it fit the deformation picture that he apprehended from Morse. The conclusion then was that two copies with vanishing winding number would be generated inside Gribov’s horizon [7].

We should also mention that presently the existence of Gribov copies inside the first region has also strong evidence coming from lattice simulations. These studies started even before van Baal’s work on Morse theory [21, 22, 23, 24], but even nowadays it is systematically pointed out that some of these copies are just lattice artefacts [25]. It is unlikely that all lattice copies would disappear in the continuum theory, although no formal proof that this is the case has been shown up to now (specially for the zero winding number configurations inside the first region). But it must be highlighted the recent result that numerical simulations indicate that the lattice theory shows different deep infrared behaviours for expectation values calculated over configurations in the Fundamental Modular Region, free of Gribov copies, from those obtained in the first Gribov region [26], showing once more the relevance of the analytical identification of Gribov copies inside the first region in the continuum.

Anyway, the fake copies shown in [7] are very useful to reveal that the argument on Morse theory is not enough to ensure the presence of copies with vanishing winding number inside the Gribov horizon. The question of rigid gauge copies, or the asymptotic demands on gauge fields and transformations, which we shall address in the following, are issues among the physical restrictions which need to be imposed on the results coming from the basic appliance of Morse theory. Then this is the first goal of our work: to provide a simple example of such gauge copies inside \(C_0\) for an Euclidian \(R^3\) spacetime, in the spirit of Henyey’s work, but avoiding the reducibility problem. This will be the topic of Sect. 2. In Sect. 3, based on the analysis developed, we spend a few words in an alternative interpretation of the fact that the perturbative region is insensitive to the presence of Gribov copies. In Sect. 4, we study the same problem in the presence of an instantonic background, but still in a 3D Euclidian spacetime. The analogy with the null background is straightforward, but the change in the asymptotic conditions imposed by the instanton alters the final conclusion. Section 5 is devoted to the analysis of these results, with an interesting possibility of absence of Gribov copies inside the horizon for gauge configurations with non-vanishing winding number in \(R^3\) spacetime.

## 2 Gribov copies inside the horizon

When presenting the Morse theory point of view of the Gribov problem, we mentioned the study of the constant copies coming from Henyey’s configuration done in [7]. But, actually, the first example of copies in the literature which can be associated to Morse theory is the original example given by Gribov [1, 4]. This example showed that corresponding to any given field in \(C_{n-1}\) next to the boundary \(l_{n}\) there is a Gribov copy in \(C_{n}\) close to the same boundary. The demonstration of this starts from a generic boundary configuration \(C\) in \(l_{n}\) where a continuous first order small displacement is applied. Then the same displacement is applied to the (infinitesimal) copy of \(C\) in \(l_{n}\), let us call it \({\tilde{C}}\). Now, demanding that the fields so obtained, \(A\) and \({\tilde{A}}\), respectively, satisfy the gauge condition, one can show that they are gauge copies of each other, and that they belong to the different regions, internal and external to the boundary \(l_{n}\). So, they are Gribov copies in \(C_{n-1}\) and \(C_{n}\) [1, 4]. The displacement used in this demonstration is just the kind of continuous deformation which is encompassed by the analogy with Morse theory. It meets a maximum and a minimum along the direction described by the gauge transformation connecting \(A\) and \({\tilde{A}}\) (in the other directions in the gauge orbit they both have \(n-1\) negative eigenvalues in order to be localised in such regions). The difference between this case, where only one copy is generated inside the boundary, and the example of the two constant copies of [7] is that now the deformation is of first order in the gauge parameter used to expand the configurations around the boundary [4] while in [7] the first order term is null, and the expansion begins in second order [20]. This vanishing of the first order is in fact a necessary condition in the deformation of a minimum configuration, in accordance to the Morse description. In this process we certainly cross the boundary in the way to generate a local maximum, such that the boundary configuration that needs to be described must still be a local minimum. This is only possible if the first order term vanishes. We will use this fact to start our analysis from the boundary configuration.

## 3 Brief commentary on the perturbative region

The question of why the QCD perturbative region is insensitive to the Gribov copies has received several different answers since the presentation of Gribov’s problem. We mentioned in the Introduction, for example, the belief that such copies would always be associated to large gauge transformations, and in this way effects of the corrections demanded by this problem would not affect the perturbative calculations. As we know, this explanation does not stand anymore.

We can also find the argument that the zero modes of the Faddeev–Popov operator do not couple to the physical spectrum [29]. This can be accepted as part of the explanation, but the fact is that Gribov copies are not restricted to infinitesimal boundary copies, which are associated to the zero modes. The existence of the finite copies shown by Henyey [16], not included in this subspace, show that this argument is incomplete.

We may cite the point of view that the corrections coming from the implementation of Gribov’s ideas in the action functional display the property of becoming negligible in the UV part of the spectrum, where we expect the perturbative approximation to hold, by the asymptotic freedom of QCD’s gauge coupling (details on the field theory implementing Gribov can be found in [30, 31, 32, 33] and references therein). Certainly this is an important feature of such a theory, but this is a conclusion obtained a posteriori. It can be seen as a guidance along its construction, rather than an inevitable effect.

There is then an improvement of the first argument, based on the fact that the gauge copies of the perturbative vacuum belong to different topological sectors [34]. Then they cannot be accessed perturbatively. But, again, the existence of Gribov copies with vanishing winding number, [16], compromise the functional integrals around this vacuum (another criticism can be found in [35], where some examples of trivial copies of the vacuum are built for curved spaces; see also [36]). However, a further development, showing that the wave functionals are localised around \(A=0\) for weak coupling, and that their spread in configuration space is proportional to the gauge coupling [37], gave new support to this point of view.

Our intention here is just to give an alternative point of view, and in a certain sense, glue together these results. In the process of calculating the copies inside the boundary, we made use of a perturbative expansion around the zero mode configurations, (29) and (35). In fact, we just followed Gribov’s original example of copies around the horizon, where the concept of a perturbative expansion is essential. In all these cases, an expansion parameter needs to be explicitly introduced to make this possible. Certainly we have such a parameter already available in QCD: it is the gauge coupling itself. In [38] gauge transformations with this form were used in a context very similar to ours, to study spherical Gribov copies (although some conventions used differ from ours). There, the existence of these copies was interestingly employed to give a possible explanation for the confinement of physical colour charges predefined in a non-perturbative way. If we stick to this idea, we understand that the scope of the theory to see different configurations as gauge copies is associated to the gauge coupling level. For an extremely low value of the coupling, the gauge freedom would be restricted to infinitesimal gauge transformations. Only boundary configurations, related by infinitesimal transformations, would be understood as gauge copies among those satisfying the gauge fixing condition. As the coupling increases, the theory begins to correlate more distant gauge configurations, reaching then the copies around the boundary, which are related by gauge transformations linear in the expansion parameter. One step further, the multiple copies of the kind we have described, of second order as can be seen in (29) and (35), are reached. This interpretation is in accordance with the result described in [37], and the general view exposed in the last paragraph. At the extreme perturbative level, one should only worry with the zero mode configurations of the boundaries. Then the argument of [29] fits nicely, showing that at this level of coupling the perturbative treatment will be precise, without the need to any restriction prescribed by Gribov to the configuration space. With the asymptotic freedom of QCD’s coupling, we know that this happens for the UV limit. The imprecision will only appear as the energy drops, when the copies will gain relevance in any calculation.

Then the point we want to reach is that, following this line of reasoning, the restriction to the Gribov horizon is sufficient to characterise an intermediate energy level in the way to the deep IR. This restriction gets rid of the copies linear in the gauge coupling of the kind reported originally by Gribov and at the same time already describes a new behaviour of the theory in the IR. This is the range where the results coming from [30, 31, 32, 33] will be relevant. But as the energy decreases even more, the restriction to a fundamental modular region [7], free from the copies depending exclusively on higher orders of the coupling, probably becomes imperative. Actually, this is also the current vision coming from the lattice [26], where new effects in a deeper IR scale are arising.

## 4 Gribov copies in the presence of an instanton

In this section we will repeat the same exercise now in the presence of a non-trivial background. We still remain in a 3D Euclidian space. Our intention is to describe the behaviour of the horizon in a region of the configuration space with non-vanishing Pontryagin number.

In [39], it was found that gauge orbits in non-trivial topological sectors contribute with a different multiplicity factor due to Gribov copies in relation to what happens in the trivial sector. This is also associated to the fact that large gauge copies can be located in different Gribov regions, which is already known for a long time in the case of the perturbative vacuum [14]. This implies that conclusions derived for a trivial topological sector cannot be naively extrapolated to a non-trivial one.

The study of Gribov copies in non-trivial sectors was also the subject of [40], where zero modes of a single \(SU(2)\) 4D instanton in a maximally abelian gauge were found. In this work, this horizon configuration was determined with the use of the norm functional (1) adapted to this gauge fixing, but the study did not concluded if such a configuration would bifurcate and allow for the presence of Gribov copies inside the horizon.

This environment also allowed the construction of sphalerons in the superposition of the Gribov horizon with that of the fundamental modular region. The conclusion was that such configurations gathered the conditions to generate the kind of bifurcation predicted by the Morse approach to Gribov’s problem [18, 19, 20]. But these three dimensional sphalerons are associated to the non-trivial mappings among three spheres, \(\pi _{3}(S^{3})\). This demands a non-trivial topology for the three dimensional space, which must be that of a \(S^{3}\).

We will focus on different configurations, also based on a \(SU(2)\) gauge group, but of the kind of the spherical ones described by (8) for a \(R^3\) Euclidian space. Non trivial configurations cannot be implemented in this case for a pure Yang–Mills theory. This is achieved only in the presence of a scalar field, minimally coupled to the \(SU(2)\) potential. This is the Polyakov instanton, first described in [41].

## 5 Conclusion

In the first part of this work, we addressed the problem of actually constructing examples of Gribov’s copies inside the horizon. We showed how the standard example of [7] cannot be considered as legitimate, as the copies become rigid global copies inside the horizon due to the reducibility of the starting Henyey configuration [17]. We succeeded in presenting such an example for a topologically trivial field, which confirms the assumption that copies inside the horizon would not be restricted to those coming from large gauge transformations, or topologically non-trivial spacetimes. There is also an interesting question that could be raised here. In our present case, and also in that shown in [7], multiple copies are generated only *inside* the horizon. We never see the possibility of a bifurcation allowing copies on the outside. One could associate this effect to the symmetries of the initial gauge configurations of these examples, and imagine that in general the bifurcation process would work in both ways from any boundary. But this is not so. In order to see this, we just need to retrace the origin of the bifurcation from the analogy with Morse theory. As reasoned in [7], this process begins with the deformation of a critical point of the topographic functional (1), such that its third order contribution in the gauge parameter expansion vanishes. Then, in this gauge orbit, an expansion of the functional (1) around the boundary configuration will only start in a fourth order. If this critical configuration represents a maximum in its gauge orbit, Morse theory indicates that a bifurcation would generate copies outside the boundary, from the conservation of the Euler characteristic (3). But the fact is that for the \(SU(2)\) group, this fourth order element is always positive definite, as proved in the appendix of [43]. This corresponds to a minimum condition. Then any bifurcation starting from an \(SU(2)\) horizon configuration (with a non-null contribution in its gauge orbit appearing only in the fourth order) will only possibly generate double copies inside the horizon.

The physical relevance of the existence of Gribov copies is unquestionable. Their existence inside Gribov’s horizon is again a physical puzzle of undeniable relevance. Not only for the main fact that their presence can change the physical behaviour of particles described by gauge theories, but also because no one has actually any idea of how to implement a restriction on field space in order to get rid of them, and so define the Fundamental Modular Region. When we began our search, our first idea was that if we could not find these copies among spherical trivial configurations (which gather the mathematical conditions for not allowing global copies among them, as we explained in the text), then this could enable us to even conjecture that in the trivial sector the FMR would be all the first Gribov region, at least in a continuum Euclidian spacetime. The fact that we could not find any physical argument to avoid the construction of our example in Sect. 2 made this conjecture false.

In the second part, after spending a few words on the repercussion of this development on the effects of Gribov’s copies at the perturbative level, we analysed the same question of copies inside the horizon in the presence of a Polyakov instanton background. In the end, as we have seen, the same approach which showed the copies among trivial fields is obstructed by the special asymptotic behaviour demanded by this non-trivial configuration. Obviously, this does not mean that such copies are absent in general for any non-trivial sector of gauge orbits, but it induces the idea that we may have a coincidence between Gribov’s first region and the FMR for some special conditions of the spacetime, necessary for the development of instantonic configurations. In such sectors, the confirmation of this hypothesis would allow the study of a gauge theory actually free of Gribov copies by implementing at the action level the restriction of the configuration space to the region up to the horizon [30, 31, 32, 33]. As already indicated in lattice simulations [26], we believe that this restriction to a FMR will probably unveil new phenomena in the deep infrared of Yang–Mills theory not described up to now.

## References

- 1.V.N. Gribov, Nucl. Phys. B
**139**, 1 (1978)CrossRefADSMathSciNetGoogle Scholar - 2.S. Sciuto, Phys. Rep.
**49**, 181 (1979)CrossRefADSGoogle Scholar - 3.M. Semenov-Tyan-Shanskii, V. Franke, SLOM Instituta, AN SSSR
**120**, 159 (1982) [English: Plenum Press, New York (1986)]Google Scholar - 4.R.F. Sobreiro, S.P. Sorella, hep-th/0504095
- 5.G. Dell’Antonio, D. Zwanziger, Commun. Math. Phys.
**138**, 291 (1991)CrossRefADSzbMATHMathSciNetGoogle Scholar - 6.G. Dell’Antonio, D. Zwanziger, in
*Proceedings, Probabilistic Methods in Quantum Field Theory and Quantum Gravity*(Cargese 1989), pp. 107–130 [see HIGH ENERGY PHYSICS INDEX 29, No. 10571 (1991)]Google Scholar - 7.P. van Baal, Nucl. Phys. B
**369**, 259 (1992)CrossRefADSGoogle Scholar - 8.E. Witten, J. Differ. Geom.
**17**, 661 (1982)zbMATHMathSciNetGoogle Scholar - 9.J.M.F. Labastida, Commun. Math. Phys.
**123**, 641 (1989)CrossRefADSzbMATHMathSciNetGoogle Scholar - 10.J. Milnor,
*Annals of Mathematics Studies*, vol. 51. (Princeton University Press, Princeton, 1973)Google Scholar - 11.Y. Matsumoto,
*An Introduction to Morse Theory*. Iwanami Series in Modern Mathematics, vol. 208. (AMS Bookstore, Boston, 2002)Google Scholar - 12.M.F. Atiyah, R. Bott, Phil. Trans. R. Soc. Lond. A
**308**, 523 (1982)CrossRefADSMathSciNetGoogle Scholar - 13.P. Ramond,
*Field Theory. A Modern Primer*, 2nd edn. Frontiers of Physics, vol. 74. (Addison-Wesley Publishing Company, Upper Saddle River, 1989), p 236Google Scholar - 14.R. Jackiw, I. Muzinich, C. Rebbi, Phys. Rev. D
**17**, 1576 (1978)CrossRefADSGoogle Scholar - 15.R. Benguria, P. Cordero, C. Teitelboim, Nucl. Phys. B
**122**, 61 (1977)CrossRefADSGoogle Scholar - 16.F.S. Henyey, Phys. Rev. D
**20**, 1460 (1979)CrossRefADSGoogle Scholar - 17.P. Van Baal, in
*Proceedings of the International Symposium on Advanced Topics of Quantum Physics*, ed. by J.Q. Liang (Science Press, Beijing, 1993), pp. 133–136. hep-lat/9207029 - 18.B. van den Heuvel, P. van Baal, Nucl. Phys. Proc. Suppl.
**42**, 823 (1995). hep-lat/9411046 - 19.P. van Baal, Trento QCD Workshop 1995:0004-23. hep-th/9511119
- 20.P. van Baal, in
*Confinement, Duality, and Nonperturbative Aspects of QCD*(Cambridge, 1997), pp. 161–178. hep-th/9711070 - 21.C. Parrinello, S. Petrarca, A. Vladikas, Phys. Lett. B
**268**, 236 (1991)CrossRefADSGoogle Scholar - 22.E. Marinari, C. Parrinello, R. Ricci, Nucl. Phys. B
**362**, 487 (1991)CrossRefADSMathSciNetGoogle Scholar - 23.M.L. Paciello, C. Parrinello, S. Petrarca, B. Taglienti, A. Vladikas, Phys. Lett. B
**276**, 163 (1992) [Erratum-ibid. B**281**, 417 (1992)]Google Scholar - 24.J.E. Hetrick, P. de Forcrand, A. Nakamura, R. Sinclair, Nucl. Phys. Proc. Suppl.
**26**, 432 (1992)CrossRefADSGoogle Scholar - 25.A. Cucchieri, T. Mendes, Phys. Rev. D
**88**, 114501 (2013). arXiv:1308.1283 [hep-lat] - 26.V.G. Bornyakov, V.K. Mitrjushkin, R.N. Rogalyov, Phys. Rev. D
**89**, 054504 (2014). arXiv:1304.8130 [hep-lat] - 27.R. Rajaraman,
*Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory*(North Holland, Amsterdam, 1982)zbMATHGoogle Scholar - 28.R.R. Landim, L.C.Q. Vilar, O.S. Ventura, V.E.R. Lemes, J. Math. Phys.
**55**, 022901 (2014). arXiv:1212.4098 [hep-th] - 29.M. Kaku,
*Quantum Field Theory: A Modern Introduction*(Oxford University Press, New York, 1993), p 313Google Scholar - 30.N. Vandersickel, D. Zwanziger, Phys. Rep.
**520**, 175 (2012). arXiv:1202.1491 [hep-th] - 31.J. Greensite, S. Olejnik, D. Zwanziger, Phys. Rev. D
**69**, 074506 (2004). hep-lat/0401003 - 32.D. Dudal, N. Vandersickel, H. Verschelde, S.P. Sorella, PoS QCD-TNT
**09**, 012 (2009). arXiv:0911.0082 [hep-th] - 33.M.A.L. Capri, A.J. Gomez, M.S. Guimaraes, V.E.R. Lemes, S.P. Sorella, D.G. Tedesco, Phys. Rev. D
**83**, 105001 (2011). arXiv:1102.5695 [hep-th] - 34.D. Amati, A. Rouet, Phys. Lett. B
**73**, 39 (1978)CrossRefADSGoogle Scholar - 35.M. Astorino, F. Canfora, J. Zanelli, arXiv:1205.5579 [hep-th]
- 36.F. Canfora, P. Salgado-Rebolledo, Phys. Rev. D
**87**, 045023 (2013). arXiv:1302.1264 [hep-th] - 37.M. Luscher, Nucl. Phys. B
**219**, 233 (1983)CrossRefADSMathSciNetGoogle Scholar - 38.A. Ilderton, M. Lavelle, D. McMullan, JHEP
**0703**, 044 (2007). hep-th/0701168 - 39.L. Baulieu, A. Rozenberg, M. Schaden, Phys. Rev. D
**54**, 7825 (1996). hep-th/9607147 - 40.F. Bruckmann, T. Heinzl, A. Wipf, T. Tok, Nucl. Phys. B
**584**, 589 (2000). hep-th/0001175 CrossRefADSzbMATHMathSciNetGoogle Scholar - 41.A.M. Polyakov, Nucl. Phys. B
**120**, 429 (1977)CrossRefADSMathSciNetGoogle Scholar - 42.S. Weinberg,
*The Quantum Theory of Fields*. Modern Applications, vol. 2 (Cambridge University Press, Cambridge, 1996)Google Scholar - 43.D. Zwanziger, Phys. Rev. D
**69**, 016002 (2004). hep-ph/0303028

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