Preprint number: XXXX-XXXX The Gribov Horizon and Ghost Interactions in Euclidean Gauge Theories Hirohumi Sawayanagi1 7 1 ................................................................................ 0 The effect of the Gribov horizon in Euclidean SU(2) gauge theory is studied. Gauge fields 2 on the Gribov horizon yield zero modes of ghosts and anti-ghosts. We show these zero modes n can produce additional ghostinteractions,and the Landau gauge changes to a nonlinear gauge a effectively.Inthe infraredlimit, however,the Landaugaugeis recovered,andghostzeromodes J may appear again. We show ghost condensation happens in the nonlinear gauge, and the zero 2 mode repetition is avoided. 1 ..................................................................................................... Subject Index B05,B06 ] h t - p e h [ 1 v 9 3 2 3 0 . 1 0 7 1 : v i X r a 1 typeset using TEX.cls PTP 1 Introduction A perturbative calculation in gauge theories requires gauge fixing. However, in non- Abelian gauge theories, there is a problem of gauge copies [1]. Gribov showed that gauge- equivalent copies exist in the Landau gauge ∂ A = 0. (1.1) µ µ In the Coulomb gauge, it was shown that almost all gauge transformations are responsible for gauge fixing degeneracies [2]. If gauge copies are connected by an infinitesimal gauge transformationwitha gaugeparameter ε(x), (1.1)gives ∂ D ε(x) = 0. That is, theFaddeev- µ µ Popov (FP) operator ∂ D has zero eigenvalues. The boundary that the lowest eigenvalue µ µ − of the FP operator equals zero is called the (first) Gribov horizon ∂Ω. The region inside ∂Ω, where eigenvalues of ∂ D are positive, is called the Gribov region Ω. In general, gauge µ µ − copies may exist outside of Ω [1] and on the horizon [3]. There are some ideas to solve the problem. One of them is to restrict a functional integral in the Gribov region Ω [1, 4]. (Strictly speaking, there may be some copies in Ω. Hence more restricted region in Ω, that is called a fundamental modular region (FMR) Λ, is considered [5].) Another idea is to sum over all gauge copies [6, 7]. For a solvable gauge model, it was shown that correct results are obtained by collecting all gauge copies [8, 9]. The Gribov horizon yield some effects. In the first approach, the horizon perturbs gluons into shadow particles [4, 10]. Even if the region is restricted to the FMR Λ, there are points that the boundary of Λ touches the horizon ∂Ω [5]. These points give the singularity of the operator 1/∂ D . As a result, the color Coulomb potential is enhanced and the confinement µ µ might be shown [11]. In the second approach, gauge configurations on the Gribov horizon contribute in general, and the FP operator has zero modes. These zero modes can cause a trouble in proving the gauge equivalence [12]. Thus physical effects of the horizon ∂Ω are worth studying. In this paper, we study the effect of these zero modes. In the next section, we show that a pair of zero modes in the Landau gauge can yield additional ghost interactions. If we require the BRS invariance, an effective Lagrangian becomes a Lagrangian in a nonlinear gauge. In 3, the gauge ∂ A = 0 is considered. If there is a pair of zero modes, the nonlinear gauge µ µ § 6 is realized as well. We also show that the partition function does not vanish even if the FP operator yields a single zero mode. In 4, the effect of a single zero mode is discussed in the § Landau gauge. In the low energy region, ghost condensation appears in the nonlinear gauge. The effect of the zero modes under the condensation is discussed in 5. 6 is devoted to § § summary. In Appendix A, examples of zero modes in the Coulomb gauge are given in three 2 dimensional space-time. In Appendix B, the effective Lagrangian in 2 is derived by the use § of a source term. The nonlinear gauge has two gauge parameters. Renormalization group equations for these parameters are presented in Appendix C. In Appendix D, symmetries in the nonlinear gauge are discussed. 2 Effect of ghost zero modes in the Landau gauge We consider the SU(2) gauge theory with structure constants fABC. Using the notations F G = FAGA, (F )AB = fACBFC, (F G)A = fABCFBGC, A = 1,2,3, · × × a partition function in the Landau gauge is Z = Z with L α=0 Z = Dµe−R dx(Linv+Lα), Dµ = DA DBDcDc¯, (2.1) α µ Z 1 α = F2 , = B ∂ A B2 +ic¯ ∂ D c, (2.2) inv µν α µ µ µ µ L 4 L · − 2 · where ic¯ ∂ D c = ic¯A∂ (∂ +gA )ABcB. The gauge condition (1.1) leads to the relations µ µ µ µ µ · × ∂ D = D ∂ , dxic¯ ∂ D c = dxi(∂ D c¯) c. (2.3) µ µ µ µ µ µ µ µ · · Z Z Namely, ∂ D is hermitian, and its eigenvalues are real. µ µ The eigenfunction u with the eigenvalue λ satisfies n n ∂ D u (x) = λ u (x). (2.4) µ µ n n n − When A is on the first Gribov horizon, the lowest eigenvalue is λ = 0 and u (x) is a zero µ 0 0 mode. If we can make u (x) complex, as (2.4) leads to 0 ∂ D u∗(x) = λ u∗(x), (2.5) µ µ n n n − u∗(x) is also a zero mode. We assume a pair of zero modes (u (x),u∗(x)) exists. Some 0 0 0 examples of a zero-mode pair (u (x),u∗(x)) are presented in Appendix A. If u is real, it 0 0 0 may be a single zero mode. An example of such a zero mode is given in Appendix A, and its effect is discussed in 4. § Now we expand the ghost c as 1 c(x) = ξu (x)+ξ†u∗(x)+ , (2.6) 0 0 ··· where ξ and ξ† are independent Grassmann variables. Other modes, i.e. nonzero modes and a single zero mode, are not written explicitly. In the same way, the property (2.3) implies 1We assume that eigenfunctions of the FP operatorforman orthonormalcomplete set. Strictly speaking, to ensure it, spacesand/orconfigurationsofA mustbe restricted.We emphasize whatis importanthere is µ that c contains ξu0,ξ†u∗0 and c¯contains ξ¯u0,ξ¯†u∗0. 3 that the expansion c¯(x) = ξ¯u (x)+ξ¯†u∗(x)+ . (2.7) 0 0 ··· j j∗ holds. We note, if there are some pairs of zero modes (u (x),u (x)) (j = 1,2, ), ξu (x)+ 0 0 ··· 0 ξ†u∗(x) and ξ¯u (x)+ξ¯†u∗(x) are replaced by [ξ uj(x)+ξ†uj∗(x)] and [ξ¯uj(x)+ 0 0 0 j j 0 j 0 j j 0 ¯† j∗ ξ u (x)], respectively. However the discussion below is also applicable. j 0 P P Eqs.(2.4) and(2.5) imply that theLagrangian dxic¯ ∂ D c doesnot containthe Grass- µ µ · mann variables ξ,ξ†,ξ¯and ξ¯†. However the measures Dc and Dc¯ contain dξdξ† and dξ¯dξ¯†, R respectively. Since a Grassmann variable ζ satisfies 1 (n = 1) dζζn = , (2.8) (0 (n = 0,2,3, ) Z ··· the partition function vanishes: DcDc¯e−R dxLα = 0. Z We know that fermions in an instanton background have zero modes. These zero modes yield the additional interaction of fermions [13, 14]. Likewise, the above ghost zero modes may produce additional ghost interactions, because DcDc¯ξξ†ξ¯ξ¯†e−R dxLα = 0. (2.9) 6 Z From (2.6) and (2.7), we obtain cAcBc¯Cc¯D = ΨABCDξξ†ξ¯ξ¯† + , ··· where ΨABCD = uAuBu∗Cu∗D, and terms denoted by lack some or all of ξ,ξ†,ξ¯and ξ¯†. 0 0 0 0 ··· Therefore (2.9) leads to DcDc¯σ[AB][CD]ΨABCDξξ†ξ¯ξ¯†e−R dxLα = DcDc¯σ[AB][CD]cAcBc¯Cc¯De−R dxLα, (2.10) Z Z where σ[AB][CD] is antisymmetric with respect to A and B, and C and D as well. Thus ghost zero modes produce effective ghost interactions. 4 Now we determine σ[AB][CD], and construct effective Lagrangians. The first candidate is σ[AB][CD] = fEABfECD(= δACδBD δADδBC). This choice gives the term − σ[AB][CD]cAcBc¯Cc¯D = (c¯ c¯) (c c) = 2(c¯ c) (c¯ c), × · × − × · × and (2.10) becomes DcDc¯(c¯ c) (c¯ c)e−R dxLα. (2.11) × · × Z From (2.8), the equality dζeζ = 1 (2.12) Z holds. Therefore, as in the instanton case [15], (2.11) is derived from the nonvanishing partition function DcDc¯e−R dxK41(ic¯×c)2e−R dxLα, (2.13) Z where K is a dimensionless constant. 1 Interaction with other fields is also possible. If we use σ[AB][CD] = BEBF(fEACfFBD − fEBCfFAD), 2 we obtain the term σ[AB][CD]cAcBc¯Cc¯D = 2[B (c c¯)][B (c c¯)], − · × · × and (2.10) becomes DcDc¯[B (c c¯)][B (c c¯)]e−R dxLα. (2.14) · × · × Z Taking account of (2.12), we find (2.14) is derived from DcDc¯e−R dxK2B·(c¯×c)e−R dxLα, (2.15) Z where K is a dimensionless constant. 2 We can combine (2.13) and (2.15) in a BRS invariant form. Carrying out the BRS transformation g δ A = D c, δ c = c c, δ c¯= iB, (2.16) B µ µ B B −2 × we obtain K δ 1(ic¯ c)2 +K [B (c¯ c)] = ( iK gK )(B c) (c¯ c). B 2 1 2 2 × · × − − × · × (cid:26) (cid:27) If we set K = iK = igα , we get the BRS invariant effective Lagrangian 2 −g 1 2 α α α = 2(igc¯ c)2 +α B (igc¯ c) = 2B2 2B¯2, (2.17) eff 2 L − 2 × · × 2 − 2 where B¯ = B +igc¯ c, and α is a new dimensionless constant. 2 − × 2Instead of B, we can use A . Examples are F and ∂ A . However,using them, we cannot construct a µ µν µ µ Lagrangianwhich has mass dimension four (or lower than four) and has the off-shell BRS invariance. 5 Here we used the property (2.8) to derive the effective Lagrangian (2.17). In Appendix B, we derive it by using a source term. Now we summarize the result. In the Landau gauge, when the configuration A on the µ Gribov horizon contribute to the partition function, the FP operator has zero modes. If a pair of zero modes (u (x),u∗(x)) exists, the effective Lagrangian (2.17) is produced. From 0 0 (2.1) and (2.17), we obtain the partition function Z = ZNL α=0 ZNL = Dµe−R dx(Linv+Lα+Leff) = Dµe−R dx(Linv+LNL), (2.18) α Z Z α α = B ∂ A +ic¯ ∂ D c 1B2 2B¯2, (2.19) NL µ µ µ µ L · · − 2 − 2 where α = α α . Thus the Gribov horizon yields the Lagrangian in the nonlinear gauge 1 2 − [16–18]. NL L 3 α = 0 gauge 6 In the α = 0 gauge, as ∂ A = 0 and µ µ 6 6 dxic¯ ∂ D c = dxi(D ∂ c¯) c, ∂ D = D ∂ , µ µ µ µ µ µ µ µ · · 6 Z Z the operator ∂ D is not hermitian. We assume that the operator ∂ D has a pair of zero µ µ µ µ modes (u ,u∗) and a real single zero mode v . Then c is expanded as 0 0 0 c(x) = ξu (x)+ξ†u∗(x)+ζv + , (3.1) 0 0 0 ··· where ξ,ξ† and ζ are independent Grassmann variables. Although the Lagrangian (2.2) does not contain ξ,ξ† and ζ, the measure Dc contains dξdξ†dζ. Thus we find DcDc¯e−R dxLα = 0, Z DcDc¯cAcBcC e−R dxLα = 0. (3.2) 6 Z However (3.2) contradicts with the ghost number conservation. To avoid this problem, a pair of zero modes (u¯ ,u¯∗) and a real single zero mode v¯ of the operator D ∂ must exist, 3 and 0 0 0 µ µ 3Let us consider a square matrix , which is not necessarily hermitian. There are eigenvectors V which k D satisfy V =λ V . Since det( λE)=det(t λE), t has the same eigenvalues as . Thus we have k k k D D− D− D D t U =λ U .AstU satisfiestU =λ tU ,theseeigenvectorssatisfytU V =0ifλ =λ [19].Inthepresent l l l l l l l l k l k D D 6 case, we assign =∂ D , t =D ∂ , V =(u ,u∗,v ) and U =(u¯ ,u¯∗,v¯). D µ µ D µ µ k k k k l l l l 6 c¯is expanded as c¯(x) = ξ¯u¯ +ξ¯†u¯∗(x)+ζ¯v¯ (x)+ . (3.3) 0 0 0 ··· Since ∂ D = D ∂ , a zero-mode pair (u¯ , u¯∗) is different from (u , u∗), and v¯ = v . µ µ 6 µ µ 0 0 0 0 0 6 0 Now we consider the effect of the zero-mode pairs (u , u∗) and (u¯ , u¯∗). Since the 0 0 0 0 Lagrangian does not contain ξ,ξ†,ξ¯and ξ¯†, and the measure contains dξdξ†dξ¯dξ¯†, to obtain a non-zero partition function, we must repeat the consideration in 2. Namely the zero-mode § pairs give rise to the effective Lagrangian , and the nonlinear gauge is realized. eff L ¯ Next we study tha terms ζv in (3.1) and ζv¯ in (3.3). The Lagrangian has the term 0 0 eff L igα B (c¯ c). Although this term is necessary to ensure the BRS symmetry, as 2 · × ¯ B (c¯ c) = B ζζv¯ (x) v (x)+ , (3.4) 0 0 · × ·{ × ···} ¯ the partition function does not vanish even if DcDc¯contains dζdζ. Thus, when α = 0, the partition function changes from (2.1) to (2.18), if the FP operator 6 ∂ D has a pair of zero modes. This result is unchanged even if this operator has a single µ µ zero mode. 4 Renormalization group flow of α We return to the gauge α = 0, and assume ∂ D has a single zero mode v . Now ∂ D = µ µ 0 µ µ D ∂ holds, we must set v¯ (x) = v (x) in (3.3), i.e. µ µ 0 0 ¯ c = ζv (x)+ , c¯= ζv (x)+ . 0 0 ··· ··· ¯ Since v (x) v (x) = 0, c¯ c and (3.4) do not contain ζζ. Namely we cannot say that 0 0 × × ZNL = 0 is guaranteed. α=0 6 To evade this difficulty, we first construct the partition function ZNL = 0, and then take α 6 the limit α 0, i.e. lim ZNL. α→0 α → From the Lagrangian , the equation of motion for B is NL L ∂ A αB = igα (c¯ c). µ µ 2 − − × So, when α 0, the term igα (c¯ c) must be taken into account. In this section, treating 2 → − × the interactions perturbatively at the one-loop level, we study the behavior of α. 7 In Appendix C, we derive the renormalization group (RG) equations ∂α g2C (G) 13 ∂α g2C (G) 13 1 2 2 2 µ = α α , µ = α α , (4.1) ∂µ 16π2 1 3 − 1 ∂µ 16π2 2 3 − 2 (cid:18) (cid:19) (cid:18) (cid:19) which coincide with the results in Refs. [20] and [21].4 We emphasize that the equation for α does not contain α , and vice versa. From (4.1), α = α +α satisfies 1 2 1 2 ∂α g2C (G) 13 µ = 2 α α2 +2(α α )α . (4.2) 2 2 ∂µ 16π2 3 − − (cid:26) (cid:27) When α 1, (4.2) becomes | | ≪ ∂α g2C (G) µ 2 α2. (4.3) ∂µ ≃ − 8π2 2 Therefore, when α = 0, α increases as µ decreases. The quartic ghost interaction makes 2 6 α = 0, and the situation in 3 realizes. Even if a single zero mode v exists, the partition 0 6 § function does not vanish. Eq.(4.1) shows that (α ,α ) = (0,0) is an infrared fixed point. Does this fact imply that 1 2 the Landau gauge (1.1) is retrieved as µ 0? Does the process in 2 repeat again? In the → § next section, we show such a trouble does not happen. 5 Ghost condensation In Appendix B, we present the Lagrangian [18, 22] α ϕ2 = 1B2 +B(∂ A +ϕ w)+ic¯ (∂ D +gϕ )c+ . (5.1) ϕ µ µ µ µ L − 2 − · × 2α 2 This Lagrangian has the BRS invariance, if ϕ transforms as δ ϕ = gϕ c. Setting the con- B × stant w = 0, and performing the ϕ integration, we find yields . Namely, ϕ is an ϕ NL L L ¯ auxiliary field which represents α B. 2 However, in a low energy region, ϕ is not an auxiliary field. In Ref. [22], we derived another RG equation for α given by 2 ∂ g2C (G) 2 µ α = (β 2α )α , (5.2) ∂µ 2 (4π)2 0 − 2 2 which is different from (4.1). Eq.(5.2) was derived by making the Wilsonian effective action for ϕ.5 We also showed that ϕ acquires the vacuum expectation value ϕ = ϕ under the 0 h i 4Theparametersα1 andα2 inthisarticlearerelatedtotheparametersinRefs.[20]and[21]asfollows: (1) after setting ξ =0,ζ =η and α=β, α1 =(1+η)α and α2 = ηα in Ref. [20], − (2) α1 =(1 ξ)λ=α′+α/2 and α2 =ξλ=α/2 in Ref. [21]. − 5In Appendix C.2, we explain how to derive (5.2) from . NL L 8 energy scale 2 2 µ = Λe−4π /(α2g ), (5.3) 0 whereΛisamomentumcut-off.Ghost-antighostboundstatesandghostcondensationappear below µ . We substitute ϕ(x) = ϕ +ϕ′(x) into (5.1), and choose the constant w = ϕ . This 0 0 0 choice is necessary to maintain the BRS symmetry [23].6 Then (5.1) becomes α 1B2 +B(∂ A +ϕ′)+ic¯ (∂ D +gϕ′ +gϕ )c. (5.4) µ µ µ µ 0 − 2 · × × Because of the dimensional transmutation [24], the parameter below µ is not α but ϕ . 0 2 0 Contrary to α , the gauge parameter α remains in (5.4). As we explain in Appendix 2 1 C.2, the RG equation (4.1) for α persists, and α = 0 is an infrared fixed point. So, when 1 1 µ 0, (5.4) gives the gauge condition → ∂ A +ϕ′ 0 (5.5) µ µ ≈ and the ghost Lagrangian dxic¯ (∂ D +gϕ′ )c = dxic¯ (D ∂ )c µ µ µ µ · × · Z Z = dxi(∂ D c¯) c. µ µ · Z As (5.5) means ∂ D = D ∂ , we assume ∂ D has a pair of zero modes (u ,u∗) and a µ µ 6 µ µ µ µ 0 0 single zero mode v , and D ∂ has zero modes (u¯ ,u¯∗) and v¯ . Even if the measure DcDc¯ 0 µ µ 0 0 0 contains dξdξ†dζdξ¯dξ¯†dζ¯, because the term ic¯ (gϕ c) in (5.4) has 0 · × igϕ ξ¯ξu¯ (x) u (x)+ξ¯†ξ†u¯∗(x) u∗(x)+ζ¯ζv¯ (x) v (x)+ , (5.6) 0 0 0 0 0 0 0 − ·{ × × × ···} the partition function does not vanish. 6 Summary In the Landau gauge α = 0, the FP operator ∂ D has zero modes on the Gribov hori- µ µ − zon. As the ghost c and the anti-ghost c¯are Grassmann variables, it is natural to expect that these zero modes yield effective ghost interactions. We have shown the quartic ghost interac- tion is produced by a pair of zero modes. If we impose the BRS invariance, the Lagrangian in the nonlinear gauge is obtained. Thus the Landau gauge changes to the nonlinear gauge. In the α = 0 gauge, the same result is obtained as well. 6 6ThispointisexplainedinAppendix D.Theanti-BRSsymmetryandtheglobalgaugesymmetryarealso discussed. 9 The effect of a single zero mode was also studied. Although there is no trouble in the α = 0 gauge, the partition function Z may vanish in the α = 0 gauge. We can avoid this 6 problem by taking the limit α 0. → Usually, when det∂ D = 0 for some configuration A , we can evade the Z = 0 prob- µ µ µ lem by choosing another gauge (locally) [25]. In this paper, we have shown that such a configuration changes the gauge to the nonlinear gauge automatically. The partition functions in the Landau gauge and the nonlinear gauge are equivalent perturbatively. In the nonlinear gauge, (α ,α ) = (0,0) is an infrared fixed point at the one- 1 2 loop level. In this case, the Landau gauge is retrieved and the zero-mode problem appears again. However, this scenario is not true. The nonlinear gauge yield the ghost condensation below the energy scale µ , and the zero-mode problem no longer happens. 0 A Examples of zero modes in the Coulomb gauge In this appendix, choosing the gauge ∂ A = 0, we study the eigenvalue equation j j ∂ D u = ( +gA ∂ )u = λu (A1) j j j j − − △ × in three-dimensional space-time. A.1 A pair of zero modes If the eigenfunction has the form uA = eiswA with gA (∂ w) = 0, (A1) becomes j j × iHABeiswB = ( +λ)eiswA, HAB = gfACBAC(∂ s). (A2) j j − △ Since H is a real antisymmetric 3 3 matrix, its eigenvalues are pure imaginary or 0, i.e. × HABwB = ih(x)wA, HABwB = ih(x)wA, HABwB = 0. (A3) + + − − 0 − The last equation of (A3) means that the effect of AC disappears and w does not become j 0 a zero mode. From (A2) and (A3), we obtain h(x)e±iswA = ( +λ)e±iswA. ± ± △ Thus we find the two functions u = e±isw become a zero-mode pair, if ± ± h(x)uA = uA (A4) ± ± △ holds. To give concrete examples, let us choose the abelian configuration AA(x) = a (x)δA3, ∂ a = 0. (A5) i i i i 10