A few remarks on the zero modes of the Faddeev-Popov operator in the Landau and maximal Abelian gauges
aa r X i v : . [ h e p - t h ] S e p A few remarks on the zero modes of the Faddeev-Popovoperator in the Landau and maximal Abelian gauges
M. S. Guimaraes a ∗ , S. P. Sorella a †‡ , a UERJ − Universidade do Estado do Rio de JaneiroInstituto de F´ısica − Departamento de F´ısica Te´oricaRua S˜ao Francisco Xavier 524, 20550-013 Maracan˜a, Rio de Janeiro, RJ, Brasil.
June 6, 2018
Abstract
The construction outlined by Henyey [1] is employed to provide examples of normalizable zero modesof the Faddeev-Popov operator in the Landau and maximal Abelian gauges in SU (2) Euclidean Yang-Mills theories in d = 3 dimensions. The corresponding gauge configurations have all finite norm || A || < ∞ . In particular, in the case of the Landau gauge, the explicit construction of an infinite classof normalizable zero modes with finite norm || A || is provided. ∗ [email protected] † [email protected] ‡ Work supported by FAPERJ, Funda¸c˜ao de Amparo `a Pesquisa do Estado do Rio de Janeiro, under the program
Cientistado Nosso Estado , E-26/101.578/2010. Introduction
The characterization of the zero modes of the Faddeev-Popov operator in various gauges is relevant forthe issue of the Gribov copies [2], which affect in a non-perturbative way the quantization of Yang-Millstheories. Zero modes of the Faddeev-Popov operator give rise to gauge field configurations which are infact Gribov copies.The issue of the zero modes of the Faddeev-Popov operator is usually addressed by focusing on a spe-cific gauge configuration as, for example, an instanton configuration [3]. A different strategy to face theexplicit construction of Gribov copies in the Coulomb gauge has been outlined by Henyey [1], see also[4] for a pedagogical review of the method and [5] for a recent application to Abelian gauge theories incurved backgrounds. Within Henyey’s construction, both the zero mode and the corresponding gaugefield configuration are determined in a self-consistent way. More precisely, the differential equation forthe zero mode is employed as an algebraic equation allowing us to express the gauge field in terms of thefunction which parametrizes the zero mode itself. In this way, Henyey has been able to provide examplesof gauge copies with the desired asymptotic behavior for the resulting gauge field configuration.The aim of this work is twofold. First, we revise Henyey’s construction in the case of SU (2) Landau gaugein d = 3 dimensions. Here, we shall be able to provide the construction of an infinite class of normalizablezero modes whose corresponding gauge configurations have all finite norm || A || = T r R d x |−→ A | < ∞ .Second, we point out that Henyey’s strategy can be employed to obtain examples of zero modes in othergauges as, for example, the maximal Abelian gauge, which proves to be useful in order to investigate thedual superconductivity mechanism for confinement. In this case, despite the complexity of the Faddeev-Popov operator, Henyey’s set up turns out to be helpful in order to obtain in a simple way an exampleof a normalizable zero mode whose corresponding gauge field has finite norm || A || < ∞ .Let us remind here that the condition of finite norm || A || plays an important role in the understandingof the Gribov issue which has been achieved so far. It turns out in fact that the requirement that thefunctional f A [ U ] = T r R d x |−→ A U | , with A Ui = U † ∂ i U + U † A i U , is well defined and differentiable alongthe gauge orbit of A i is at the basis of well established results concerning the properties of the Gribovand fundamental modular regions in the Landau gauge, see [6, 7, 8]. The Gribov region correspondsin fact to the set of all relative minima of the functional f A [ U ] in the space of the gauge orbits, whilethe fundamental modular region is defined as the set of all absolute minima of f A [ U ]. In principle, thefundamental modular region is known to be free of Gribov copies. As such, the restriction of the domainof integration in the Feynman path integral to that region would provide the correct way of quantizingYang-Mills theories. Though, till now, a practical way of implementing such a restriction is still lacking.Nevertheless, it turns out that the restriction of the domain of integration to the Gribov region can beimplemented within a local and renormalizable framework, although this region is still plagued by thepresence of residual copies. This has resulted in the so called Gribov-Zwanziger action [9, 10] and its morerecent refined version [11, 12]. We underline that the functional f A [ U ] can also be introduced in the maxi-mal Abelian gauge, where a few properties of the corresponding Gribov region have been established [13].The present work is organized as follows. In Sect.2, Henyey’s construction is employed in the caseof the Landau gauge, where an infinite class of normalizable zero modes exhibiting finite norm || A || ispresented. Sect.3 is devoted to discuss Henyey’s set up in the case of the maximal Abelian gauge.2 Henyey’s construction of the zero modes in the Landau gauge
Let us start by recalling the relationship between zero modes of the Faddeev-Popov operator and Gribovcopies. Let A i denotes a SU (2) Lie-algebra valued field in d = 3 Euclidean space, i = 1 , ,
3, obeying theLandau gauge condition ∂ i A i = 0. Under a gauge transformation, the field A i transforms as e A i = U † ∂ i U + U † A i U . (1)For an infinitesimal gauge transformation, U = 1 + α , eq.(1) reduces to e A i = A i + ( ∂ i α + [ A i , α ]) . (2)The gauge transformed field e A i is a Gribov copy of A i if it fulfills the Landau gauge condition ∂ i e A i = 0.We see thus, from eq.(2), that this is the case if there exists a SU (2) Lie-algebra valued function α = ω such that: ∇ ω + [ A i , ∂ i ω ] = 0 . (3)The differential operator M = ( ∇ · +[ A i , ∂ i · ]) appearing in eq.(3) is nothing but the Faddeev-Popovoperator of the Landau gauge, and ω is a zero mode of M .In order to construct normalizable solutions of eq.(3), we shall follow the strategy outlined by Henyey [1]and consider the following gauge configuration: −→ A = i −→ a σ , −→∇ · −→ A = 0 , (4)that is, we are considering a particular transverse gauge field configuration along the abelian directionof the gauge group SU (2) defined by the Pauli matrix σ . Adopting spherical coordinates ( r, θ, ϕ ), theLandau gauge condition is fulfilled by setting −→ a = a ( r, θ )ˆ e ϕ . (5)We next consider the following ansatz [1] for the zero mode ωω ( r, θ, ϕ ) = rb ( r ) sin θ ( σ cos ϕ + σ sin ϕ ) , (6)where the radial function b ( r ) is left free. Using eqs.(5), (6), the differential equation (3) becomes(4 b ′ + rb ′′ ) sin θ + ab = 0 . (7)It is thus apparent that this equation can be seen as an algebraic equation enabling us to express thefield configuration a in eq.(5) in terms of the function b parametrizing the zero mode ω , namely a ( r, θ ) = − rb sin θ (cid:18) b ′′ + 4 b ′ r (cid:19) . (8)We suffice now to find a function b ( r ) providing a normalizable zero mode ω and a gauge field A i withfinite norm || A || < ∞ .To that purpose we choose the following form for the function b ( r ): b ( r ) = kr + αr γ + β , (9)3here k , α , β and γ are real constants, and we initially consider 0 < γ <
3. Expression (9) can be seenas a generalization of the original Henyey’s proposal [1]. Note that this function is regular at the origin r = 0 if β = 0 and thus, by eq.(6), the zero mode ω is also regular. The function a ( r, θ ) defining thegauge field configuration can be easily evaluated by means of eq.(8), and reads a ( r, θ ) = sin θ ( r + αr γ + β ) (cid:2) α ( γ − γ + 18) r γ +2 − α ( γ − γ ) r γ − + αβγ ( γ + 3) r γ − + 18 βr (cid:3) . (10)We see thus that the gauge field is also regular at the origin provided γ ≥ ω is normalizable, i.e. || ω || = T r Z d x ω † ω < ∞ . (11)In the present case we have T rω † ω = 2 r b sin θ , so that T r Z d x ω † ω = 4 πk Z π dθ sin θ Z ∞ dr r ( r + αr γ + β ) < ∞ . (12)Since the radial integrand goes as 1 /r for r → ∞ (for 1 ≤ γ < || A || is also required to be finite, ı.e. ||−→ A || = T r Z d x |−→ A | < ∞ . (13)From eq.(10) we have: ||−→ A || = 4 π Z π dθ sin θ Z ∞ dr (cid:2) α ( γ − γ + 18) r γ +3 − α ( γ − γ ) r γ + αβγ ( γ + 3) r γ + 18 βr (cid:3) ( r + αr γ + β ) . (14)Note that the requirement γ < r . Thus, for r → ∞ , the radial integrand behaves as r γ − and the integral will be finite if2 γ − < − ⇒ γ < . (15)Therefore, choosing the radial function b ( r ) as b ( r ) = kr + αr γ + β , ≤ γ < , (16)will lead to an infinite class of normalizable zero modes, corresponding to the infinite set of values of theexponent γ . The related gauge field configurations have all finite norm || A || < ∞ . The aim of this section is that of showing that Henyey’s strategy can be successfully employed also inthe case of the maximal Abelian gauge. Here, the gauge field is split into diagonal and off-diagonal4omponents. In the case of SU (2), the diagonal component corresponds to the diagonal generator of theCartan subgroup, i.e. σ , and is identified with A i , while the off-diagonal components A ai , with a = 1 , i.e. σ a . In this case, different gauge conditions areimposed on the diagonal and off-diagonal components, namely D abi A bi = 0 , ∂ i A i = 0 , (17)where D abi = δ ab ∂ i − ε ab A i , (18)is the covariant derivative with respect to the diagonal component A i . For the Faddeev-Popov operatorwe have now [13] M ab = − (cid:16) D aci D cbi + ε ac ε bd A ci A di (cid:17) , (19)so that the differential equation for a zero mode reads D aci D cbi α b + ε ac ε bd A ci A di α b = 0 . (20)In order to construct examples of zero modes we shall consider the following gauge field configuration A i = 0 , A ci = δ c a i , (21)The maximal Abelian gauge fixing conditions, eqs.(17), reduce to −→∇ · −→ a = 0 . (22)Proceeding as in the previous section, we adopt spherical coordinates ( r, θ, ϕ ), so that condition (22) isfulfilled by setting −→ a = a ( r, θ )ˆ e ϕ . (23)The differential equation for the zero mode, eq.(20), becomes thus ∇ α (1) = 0 , ∇ α (2) + a α (2) = 0 . (24)Setting α (1) = 0 and α (2) = ω , one immediately realizes that the gauge field configuration a can beexpressed in terms of ω , according to a = − ω ∇ ω . (25)Therefore, we are in a situation very similar to that of the Landau gauge, and Henyey’s set up can beeasily adapted to the present case. Adopting in fact the following ansatz for ωω ( r, θ, ϕ ) = rb ( r ) sin θ cos ϕ , (26)it follows that the gauge configuration is fully determined by the radial function b ( r ), namely a = − rb (cid:0) rb ′′ + 4 b ′ (cid:1) . (27)It suffices thus to find a suitable function b ( r ) which gives rise to a normalizable zero mode while providinga finite norm for the gauge configuration. It is worth to notice here the interesting feature that equation(27) directly provides the norm of the gauge field, namely || A || = 2 Z d x a . (28)5n the present case, it turns out that a simple choice for the function b ( r ) is provided by b ( r ) = kr m + β , (29)where ( k, β ) are non-vanishing constants and the exponent m > in order to ensure that the zero mode ω has finite norm, i.e. Z d x ω ∝ Z ∞ dr r ( r m + β ) < ∞ for m > . (30)From expression (29) it turns out that a = 1( r m + β ) (cid:0) (3 m − m ) r m − + β ( m + 3 m ) r m − (cid:1) , (31)from which it follows that the gauge configuration a is regular at the origin r = 0, while a r →∞ ∼ r , (32)unless m = 3, in which case a = 18 βr ( r + β ) , and a r →∞ ∼ r , (33)giving rise to a finite norm || A || .As such, expression b ( r ) = kr + β , (34)gives rise to a normalizable zero mode whose corresponding gauge field a , eq.(33), has finite norm, showingthus that Henyey’s set up can be successfully employed in the maximal Abelian gauge. Let us conclude this section with a remark on the expression (31). For a generic value of the exponent < m < < m <
3, is needed in order to ensure that a in expression (31) does notbecome negative for large r ), the resulting gauge configuration a does not provide a finite norm || A || ,due to the asymptotic behavior (32). However, it is interesting to observe that such a configuration willgive rise to a finite Yang-Mills action. In fact, a simple calculation shows that S Y M = 14 g Z d x (cid:0) F ij F ij + F aij F aij (cid:1) = 2 πg Z ∞ dr r ∂a∂r ∂a∂r , (35)which is finite since, according to expression (32), the gauge configuration a behaves as 1 /r for r → ∞ .In summary, although not exhibiting a finite norm || A || for a generic value of the exponent < m <
3, expressions (29) and (31) provide an example of an infinite class (labelled by the exponent m ) ofnormalizable zero modes with finite Yang-Mills action. An analogous situation is easily found also in thecase of the Landau gauge. 6 Conclusion
In this work, Henyey’s set up [1] has been employed to construct examples of normalizable zero modesof the Faddeev-Popov operators corresponding to the Landau and maximal Abelian gauges in SU (2)Yang-Mills theories in d = 3 dimensions. In particular, in the Landau gauge, an infinite class of nor-malizable zero modes whose corresponding gauge configurations have finite norm || A || has been provided.Let us conclude with a few general remarks on the issue of the Gribov copies: • needless to say, the examples of the zero modes and related gauge field configurations which wehave been able to construct in a rather easy way give a simple confirmation of the pivotal relevanceand, at the same time, of the complexity of the issue of the Gribov copies for a correct quantizationof Yang-Mills theories. • although a full resolution of the Gribov issue is still lacking, we would have a tendency to believe thatthe thermodynamical/statistical treatment of this issue, as started by Gribov [2] and systematized byZwanziger [9, 10], seems very suitable. In a rough pictorial way, one could figure out that the infinitedimensional space of the Gribov copies could be associated to a kind of thermodynamical reservoir towhich Yang-Mills theories are intrinsically connected. In this sense, the emergence of the dynamicalmassive Gribov parameter γ [2, 9, 10] looks quite natural. To some extent, this parameter wouldencode the information of the existence of the infinite dimensional reservoir of Gribov copies, asmuch as the temperature T carries the information of the presence of a thermodynamical bath.It is remarkable that the Gribov-Zwanziger treatment of the Gribov issue, amounting to cut offthe domain of integration in the functional integral to the Gribov region, has resulted in a localand renormalizable action [9, 10]. More recently, a refined version of the Gribov-Zwanziger actionhas been worked out [11, 12]. It is worth mentioning that, till now, the predictions of the refinedaction on the infrared behavior of the gluon and ghost two point correlation functions are in verygood agreement with the most recent lattice data [14, 15, 16, 17]. Moreover, the refined action hasprovided rather good estimates for the lightest glueball states [18]. • Finally, it is worth to spend a few words on the possible questioning which our present results mightrise with respect to the frameworks which are available so far in order to face the issue of the Gribovcopies. Here, we shall focus on the Landau gauge, due to the relatively large amount of resultswhich have been already established on the issue of the Gribov copies [6, 7, 8]. Our first commentis devoted to the fundamental modular region, which is defined as the set of all absolute minimaof the functional f A [ U ] = T r R d x |−→ A U | along the gauge orbits. It is known that the fundamentalmodular region is free from Gribov copies [6, 7, 8]. Moreover, the very definition of this region relieson the existence of the norm || A || . On the other hand, we have seen that it is rather easy to con-struct copies which do not exhibit finite norm || A || . Rather, the corresponding field configurationshave finite action, which is believed to be a more physical requirement than that of finite norm.Till now, we are unaware of a physically motivated argumentation in order to demand that therelevant asymptotic behavior of the gauge fields at infinity should be that required by the finitenessof the norm || A || . Moreover, we should also remind that, at present, a practical implementation ofthe restriction of the domain of integration of the functional integral to the fundamental modularregion has not yet been achieved. Here, it seems safe to state that much work is needed to reacha satisfactory understanding of that region and of the physical consequences of its implementationin the functional integral.Our second remark concerns the so called Gribov-Zwanziger framework [2, 9, 10], which allows7ne to effectively restrict, within a local and renormalizable field theory framework, the functionalintegral to the Gribov region ΩΩ = { A aµ ; ∂ µ A aµ = 0 , − ∂ µ D abµ > } . (36)The boundary, ∂ Ω, of the region Ω is the first Gribov horizon. As it stands, expression (36) hasto be further characterized by specifying the asymptotic behavior at infinity of the gauge fieldconfigurations. This is usually done by stating that the region Ω corresponds to the set of allrealtive minima of the functional f A [ U ] = T r R d x |−→ A U | along the gauge orbits, a definition whichestablishes a relationship between Ω and the requirement of finite norm || A || . Once defined inthat way, it is possible to prove the important result that every gauge orbit crosses Ω at least once[6, 7, 8], a result which provides a well defined support for restricting the functional integral tothe region Ω. Though, in principle, one could argue that, contrary to the fundamental modularregion, the introduction of the region Ω could be done without any reference to the requirement offinite norm || A || < ∞ . It is apparent in fact from eq.(36) that the region Ω could be introducedwithout any reference to the norm || A || , which could be eventually replaced by the criterion offinite action. This will endow the Gribov region Ω with a different asymptotic behavior for thefields. However, as a consequence, the statement that every gauge orbit crosses the Gribov regionshould be completely re-investigated ab initio . Still, we might argue that the restriction to theregion Ω, endowed now with the criterion of finite action, would enable us to get rid at least ofall copies related to zero modes, as it follows from the positivity of the Faddeev-Popov operator.As such, the so called Gribov-Zwanziger action would still retain a relevant meaning, due to theabsence of zero modes. Though, important properties of the region Ω, derived by making use of thenorm || A || , should be completely re-examined, a task which does not look that easy. Evidently,we cannot be exhaustive here. Furthermore, we hope that the reader might get convinced of thecrucial role that the issue of the Gribov copies represents for Yang-Mills theories. At the same time,we hope to have given a rough idea of how far we are from a satisfactory resolution of this issue. Acknowledgments
The Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq-Brazil), the Faperj, Funda¸c˜aode Amparo `a Pesquisa do Estado do Rio de Janeiro, the Latin American Center for Physics (CLAF), theSR2-UERJ, the Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior (CAPES) are gratefullyacknowledged.
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