The Gribov Horizon and Ghost Interactions in Euclidean Gauge Theories

TL;DR

This paper investigates how the Gribov horizon influences ghost interactions in Euclidean SU(2) gauge theory, revealing ghost condensation and gauge behavior changes in the infrared limit.

Contribution

It demonstrates that ghost zero modes induced by the Gribov horizon lead to additional ghost interactions and a nonlinear gauge, with recovery of Landau gauge in the infrared.

Findings

Ghost zero modes produce additional interactions.

Ghost condensation occurs in nonlinear gauges.

Landau gauge is recovered in the infrared limit.

Abstract

The effect of the Gribov horizon in Euclidean $SU(2)$ gauge theory is studied. Gauge fields on the Gribov horizon yield zero modes of ghosts and anti-ghosts. We show these zero modes can produce additional ghost interactions, and the Landau gauge changes to a nonlinear gauge effectively. In the infrared limit, however, the Landau gauge is recovered, and ghost zero modes may appear again. We show ghost condensation happens in the nonlinear gauge, and the zero mode repetition is avoided.

Equations

$${\partial }_{\mu }{A}_{\mu }=0.$$

$$F\cdot G=F^A{G}^A,(F\times {)}^{AB}=f^{ACB}{F}^C,(F\times G{)}^A=f^{ABC}{F}^B{G}^C,A=1,2,3,$$

$$Z_{\alpha }=\int D\mu e^{-\int dx({\mathcal{L}}_{\mathrm{inv}}+{\mathcal{L}}_{\alpha })},D\mu =D{A}_{\mu }DBDcD\bar{c},$$

$${\mathcal{L}}_{\mathrm{inv}}=\frac{1}{4}{F}_{\mu \nu }^2,{\mathcal{L}}_{\alpha }=B\cdot {\partial }_{\mu }{A}_{\mu }-\frac{\alpha }{2}{B}^2+i\bar{c}\cdot {\partial }_{\mu }{D}_{\mu }c,$$

$${\partial }_{\mu }{D}_{\mu }=D_{\mu }{\partial }_{\mu },\int dxi\bar{c}\cdot {\partial }_{\mu }{D}_{\mu }c=\int dxi({\partial }_{\mu }{D}_{\mu }\bar{c})\cdot c.$$

$$-{\partial }_{\mu }{D}_{\mu }{u}_n(x)={\lambda }_n{u}_n(x).$$

$$-{\partial }_{\mu }{D}_{\mu }{u}_n^{\ast }(x)={\lambda }_n{u}_n^{\ast }(x),$$

$$c(x)=\xi u_0(x)+{\xi }^{†}{u}_0^{\ast }(x)+\cdots ,$$

$$\bar{c}(x)=\bar{\xi }{u}_0(x)+{\bar{\xi }}^{†}{u}_0^{\ast }(x)+\cdots .$$

$$\int d\zeta\,\zeta^{n}=\left\{\begin{array}[]{@{\,}ll}1&(n=1)\\ 0&(n=0,2,3,\cdots)\end{array}\right.,$$

$$\int DcD\bar{c}{e}^{-\int dx{\mathcal{L}}_{\alpha }}=0.$$

$$\int DcD\bar{c}\xi {\xi }^{†}\bar{\xi }{\bar{\xi }}^{†}{e}^{-\int dx{\mathcal{L}}_{\alpha }}\ne 0.$$

$$c^A{c}^B{\bar{c}}^C{\bar{c}}^D={\Psi }^{ABCD}\xi {\xi }^{†}\bar{\xi }{\bar{\xi }}^{†}+\cdots ,$$

$$\int DcD\bar{c}{\sigma }^{[AB][CD]}{\Psi }^{ABCD}\xi {\xi }^{†}\bar{\xi }{\bar{\xi }}^{†}{e}^{-\int dx{\mathcal{L}}_{\alpha }}=\int DcD\bar{c}{\sigma }^{[AB][CD]}{c}^A{c}^B{\bar{c}}^C{\bar{c}}^D{e}^{-\int dx{\mathcal{L}}_{\alpha }},$$

$${\sigma }^{[AB][CD]}{c}^A{c}^B{\bar{c}}^C{\bar{c}}^D=(\bar{c}\times \bar{c})\cdot (c\times c)=-2(\bar{c}\times c)\cdot (\bar{c}\times c),$$

$$\int DcD\bar{c}(\bar{c}\times c)\cdot (\bar{c}\times c)e^{-\int dx{\mathcal{L}}_{\alpha }}.$$

$$\int d\zeta e^{\zeta }=1$$

$$\int DcD\bar{c}{e}^{-\int dx\frac{K_1}{4}(i\bar{c}\times c{)}^2}{e}^{-\int dx{\mathcal{L}}_{\alpha }},$$

$${\sigma }^{[AB][CD]}{c}^A{c}^B{\bar{c}}^C{\bar{c}}^D=-2[B\cdot (c\times \bar{c})][B\cdot (c\times \bar{c})],$$

$$\int DcD\bar{c}[B\cdot (c\times \bar{c})][B\cdot (c\times \bar{c})]e^{-\int dx{\mathcal{L}}_{\alpha }}.$$

$$\int DcD\bar{c}{e}^{-\int dx{K}_{2}B\cdot (\bar{c}\times c)}{e}^{-\int dx{\mathcal{L}}_{\alpha }},$$

$${\delta }_B{A}_{\mu }=D_{\mu }c,{\delta }_{B}c=-\frac{g}{2}c\times c,{\delta }_B\bar{c}=iB,$$

$${\delta }_B\left\{\frac{K_1}{2}(i\bar{c}\times c{)}^2+K_2[B\cdot (\bar{c}\times c)]\right\}=(-i{K}_1-g{K}_2)(B\times c)\cdot (\bar{c}\times c).$$

$${\mathcal{L}}_{\mathrm{eff}}=-\frac{{\alpha }_2}{2}(ig\bar{c}\times c{)}^2+{\alpha }_{2}B\cdot (ig\bar{c}\times c)=\frac{{\alpha }_2}{2}{B}^2-\frac{{\alpha }_2}{2}{\bar{B}}^2,$$

$$Z$$

$$Z_{\alpha }^{NL}$$

$${\mathcal{L}}_{NL}$$

$$\int dxi\bar{c}\cdot {\partial }_{\mu }{D}_{\mu }c=\int dxi(D_{\mu }{\partial }_{\mu }\bar{c})\cdot c,{\partial }_{\mu }{D}_{\mu }\ne D_{\mu }{\partial }_{\mu },$$

$$c(x)=\xi u_0(x)+{\xi }^{†}{u}_0^{\ast }(x)+\zeta v_0+\cdots ,$$

$$\int DcD\bar{c}{e}^{-\int dx{\mathcal{L}}_{\alpha }}=0,$$

$$\int DcD\bar{c}{c}^A{c}^B{c}^C{e}^{-\int dx{\mathcal{L}}_{\alpha }}\ne 0.$$

$$\bar{c}(x)=\bar{\xi }{\bar{u}}_0+{\bar{\xi }}^{†}{\bar{u}}_0^{\ast }(x)+\bar{\zeta }{\bar{v}}_0(x)+\cdots .$$

Full text

\subjectindex

B05,B06

1 Introduction

A perturbative calculation in gauge theories requires gauge fixing. However, in non-Abelian gauge theories, there is a problem of gauge copies gri . Gribov showed that gauge-equivalent copies exist in the Landau gauge

[TABLE]

In the Coulomb gauge, it was shown that almost all gauge transformations are responsible for gauge fixing degeneracies masnak . If gauge copies are connected by an infinitesimal gauge transformation with a gauge parameter $\varepsilon(x)$, (1.1) gives $\partial_{\mu}D_{\mu}\varepsilon(x)=0$. That is, the Faddeev-Popov (FP) operator $-\partial_{\mu}D_{\mu}$ has zero eigenvalues. The boundary that the lowest eigenvalue of the FP operator equals zero is called the (first) Gribov horizon $\partial\Omega$. The region inside $\partial\Omega$, where eigenvalues of $-\partial_{\mu}D_{\mu}$ are positive, is called the Gribov region $\Omega$. In general, gauge copies may exist outside of $\Omega$ gri and on the horizon bal .

There are some ideas to solve the problem. One of them is to restrict a functional integral in the Gribov region $\Omega$ gri ; zw . (Strictly speaking, there may be some copies in $\Omega$. Hence more restricted region in $\Omega$, that is called a fundamental modular region (FMR) $\Lambda$, is considered vb .) Another idea is to sum over all gauge copies fuj ; hir . For a solvable gauge model, it was shown that correct results are obtained by collecting all gauge copies flpr ; fuj2 .

The Gribov horizon yield some effects. In the first approach, the horizon perturbs gluons into shadow particles zw ; ms . Even if the region is restricted to the FMR $\Lambda$, there are points that the boundary of $\Lambda$ touches the horizon $\partial\Omega$ vb . These points give the singularity of the operator $1/\partial_{\mu}D_{\mu}$. As a result, the color Coulomb potential is enhanced and the confinement might be shown zw2 . 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 fuj3 . Thus physical effects of the horizon $\partial\Omega$ 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 $\partial_{\mu}A_{\mu}\neq 0$ is considered. If there is a pair of zero modes, the nonlinear gauge 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 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 $f^{ABC}$. Using the notations

[TABLE]

a partition function in the Landau gauge is $Z_{L}=Z_{\alpha=0}$ with

[TABLE]

where $i\bar{c}\cdot\partial_{\mu}D_{\mu}c=i\bar{c}^{A}\partial_{\mu}(\partial_{\mu}+gA_{\mu}\times)^{AB}c^{B}$. The gauge condition (1.1) leads to the relations

[TABLE]

Namely, $\partial_{\mu}D_{\mu}$ is hermitian, and its eigenvalues are real.

The eigenfunction $u_{n}$ with the eigenvalue $\lambda_{n}$ satisfies

[TABLE]

When $A_{\mu}$ is on the first Gribov horizon, the lowest eigenvalue is $\lambda_{0}=0$ and $u_{0}(x)$ is a zero mode. If we can make $u_{0}(x)$ complex, as (2.4) leads to

[TABLE]

$u_{0}^{*}(x)$ is also a zero mode. We assume a pair of zero modes $(u_{0}(x),u_{0}^{*}(x))$ exists. Some examples of a zero-mode pair $(u_{0}(x),u_{0}^{*}(x))$ are presented in Appendix A. If $u_{0}$ is real, it 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 111We assume that eigenfunctions of the FP operator form an orthonormal complete set. Strictly speaking, to ensure it, spaces and/or configurations of $A_{\mu}$ must be restricted. We emphasize what is important here is that $c$ contains $\xi u_{0},\xi^{\dagger}u_{0}^{*}$ and $\bar{c}$ contains $\bar{\xi}u_{0},\bar{\xi}^{\dagger}u_{0}^{*}$.

[TABLE]

where $\xi$ and $\xi^{\dagger}$ 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 that the expansion

[TABLE]

holds. We note, if there are some pairs of zero modes $(u_{0}^{j}(x),u_{0}^{j*}(x))\ (j=1,2,\cdots)$, $\xi u_{0}(x)+\xi^{\dagger}u_{0}^{*}(x)$ and $\bar{\xi}u_{0}(x)+\bar{\xi}^{\dagger}u_{0}^{*}(x)$ are replaced by $\sum_{j}[\xi_{j}u_{0}^{j}(x)+\xi_{j}^{\dagger}u_{0}^{j*}(x)]$ and $\sum_{j}[\bar{\xi_{j}}u_{0}^{j}(x)+\bar{\xi_{j}}^{\dagger}u_{0}^{j*}(x)]$, respectively. However the discussion below is also applicable.

Eqs.(2.4) and (2.5) imply that the Lagrangian $\int dx\,i\bar{c}\cdot\partial_{\mu}D_{\mu}c$ does not contain the Grassmann variables $\xi,\xi^{\dagger},\bar{\xi}$ and $\bar{\xi}^{\dagger}$. However the measures $Dc$ and $D\bar{c}$ contain $d\xi d\xi^{\dagger}$ and $d\bar{\xi}d\bar{\xi}^{\dagger}$, respectively. Since a Grassmann variable $\zeta$ satisfies

[TABLE]

the partition function vanishes:

[TABLE]

We know that fermions in an instanton background have zero modes. These zero modes yield the additional interaction of fermions tH ; tH2 . Likewise, the above ghost zero modes may produce additional ghost interactions, because

[TABLE]

From (2.6) and (2.7), we obtain

[TABLE]

where $\Psi^{ABCD}=u_{0}^{A}u_{0}^{B}u_{0}^{*C}u_{0}^{*D}$, and terms denoted by $\cdots$ lack some or all of $\xi,\xi^{\dagger},\bar{\xi}$ and $\bar{\xi}^{\dagger}$. Therefore (2.9) leads to

[TABLE]

where $\sigma^{[AB][CD]}$ is antisymmetric with respect to $A$ and $B$, and $C$ and $D$ as well. Thus ghost zero modes produce effective ghost interactions.

Now we determine $\sigma^{[AB][CD]}$, and construct effective Lagrangians. The first candidate is $\sigma^{[AB][CD]}=f^{EAB}f^{ECD}(=\delta^{AC}\delta^{BD}-\delta^{AD}\delta^{BC})$. This choice gives the term

[TABLE]

and (2.10) becomes

[TABLE]

From (2.8), the equality

[TABLE]

holds. Therefore, as in the instanton case cr , (2.11) is derived from the nonvanishing partition function

[TABLE]

where $K_{1}$ is a dimensionless constant.

Interaction with other fields is also possible. If we use $\sigma^{[AB][CD]}=B^{E}B^{F}(f^{EAC}f^{FBD}-f^{EBC}f^{FAD})$, 222Instead of $B$, we can use $A_{\mu}$. Examples are $F_{\mu\nu}$ and $\partial_{\mu}A_{\mu}$. However, using them, we cannot construct a Lagrangian which has mass dimension four (or lower than four) and has the off-shell BRS invariance. we obtain the term

[TABLE]

and (2.10) becomes

[TABLE]

Taking account of (2.12), we find (2.14) is derived from

[TABLE]

where $K_{2}$ is a dimensionless constant.

We can combine (2.13) and (2.15) in a BRS invariant form. Carrying out the BRS transformation

[TABLE]

we obtain

[TABLE]

If we set $K_{2}=-\frac{i}{g}K_{1}=ig\alpha_{2}$, we get the BRS invariant effective Lagrangian

[TABLE]

where $\bar{B}=-B+ig\bar{c}\times c$, and $\alpha_{2}$ is a new dimensionless constant.

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 $A_{\mu}$ on the Gribov horizon contribute to the partition function, the FP operator has zero modes. If a pair of zero modes $(u_{0}(x),u_{0}^{*}(x))$ exists, the effective Lagrangian (2.17) is produced. From (2.1) and (2.17), we obtain the partition function

[TABLE]

where $\alpha_{1}=\alpha-\alpha_{2}$. Thus the Gribov horizon yields the Lagrangian in the nonlinear gauge $\mathcal{L}_{NL}$ btm ; zj ; hs1 .

3 $\alpha\neq 0$ gauge

In the $\alpha\neq 0$ gauge, as $\partial_{\mu}A_{\mu}\neq 0$ and

[TABLE]

the operator $\partial_{\mu}D_{\mu}$ is not hermitian. We assume that the operator $\partial_{\mu}D_{\mu}$ has a pair of zero modes $(u_{0},u_{0}^{*})$ and a real single zero mode $v_{0}$. Then $c$ is expanded as

[TABLE]

where $\xi,\xi^{\dagger}$ and $\zeta$ are independent Grassmann variables. Although the Lagrangian (2.2) does not contain $\xi,\xi^{\dagger}$ and $\zeta$, the measure $Dc$ contains $d\xi d\xi^{\dagger}d\zeta$. Thus we find

[TABLE]

However (3.2) contradicts with the ghost number conservation. To avoid this problem, a pair of zero modes $(\bar{u}_{0},\bar{u}_{0}^{*})$ and a real single zero mode $\bar{v}_{0}$ of the operator $D_{\mu}\partial_{\mu}$ must exist, 333Let us consider a square matrix $\mathcal{D}$, which is not necessarily hermitian. There are eigenvectors $V_{k}$ which satisfy $\mathcal{D}V_{k}=\lambda_{k}V_{k}$. Since $\det(\mathcal{D}-\lambda E)=\det(^{t}\mathcal{D}-\lambda E)$, ${}^{t}\mathcal{D}$ has the same eigenvalues as $\mathcal{D}$. Thus we have ${}^{t}\mathcal{D}U_{l}=\lambda_{l}U_{l}$. As ${}^{t}U_{l}$ satisfies ${}^{t}U_{l}\mathcal{D}=\lambda_{l}\,^{t}U_{l}$, these eigenvectors satisfy ${}^{t}U_{l}V_{k}=0$ if $\lambda_{l}\neq\lambda_{k}$ yk . In the present case, we assign $\mathcal{D}=\partial_{\mu}D_{\mu}$, ${}^{t}\mathcal{D}=D_{\mu}\partial_{\mu}$, $V_{k}=(u_{k},u_{k}^{*},v_{k})$ and $U_{l}=(\bar{u}_{l},\bar{u}_{l}^{*},\bar{v}_{l})$. and $\bar{c}$ is expanded as

[TABLE]

Since $\partial_{\mu}D_{\mu}\neq D_{\mu}\partial_{\mu}$, a zero-mode pair ($\bar{u}_{0}$, $\bar{u}_{0}^{*}$) is different from ($u_{0}$, $u_{0}^{*}$), and $\bar{v}_{0}\neq v_{0}$.

Now we consider the effect of the zero-mode pairs ($u_{0}$, $u_{0}^{*}$) and ($\bar{u}_{0}$, $\bar{u}_{0}^{*}$). Since the Lagrangian does not contain $\xi,\xi^{\dagger},\bar{\xi}$ and $\bar{\xi}^{\dagger}$, and the measure contains $d\xi d\xi^{\dagger}d\bar{\xi}d\bar{\xi}^{\dagger}$, 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 $\mathcal{L}_{eff}$, and the nonlinear gauge is realized.

Next we study tha terms $\zeta v_{0}$ in (3.1) and $\bar{\zeta}\bar{v}_{0}$ in (3.3). The Lagrangian $\mathcal{L}_{eff}$ has the term $ig\alpha_{2}B\cdot(\bar{c}\times c)$. Although this term is necessary to ensure the BRS symmetry, as

[TABLE]

the partition function does not vanish even if $DcD\bar{c}$ contains $d\zeta d\bar{\zeta}$.

Thus, when $\alpha\neq 0$, the partition function changes from (2.1) to (2.18), if the FP operator $\partial_{\mu}D_{\mu}$ has a pair of zero modes. This result is unchanged even if this operator has a single zero mode.

4 Renormalization group flow of $\alpha$

We return to the gauge $\alpha=0$, and assume $\partial_{\mu}D_{\mu}$ has a single zero mode $v_{0}$. Now $\partial_{\mu}D_{\mu}=D_{\mu}\partial_{\mu}$ holds, we must set $\bar{v}_{0}(x)=v_{0}(x)$ in (3.3), i.e.

[TABLE]

Since $v_{0}(x)\times v_{0}(x)=0$, $\bar{c}\times c$ and (3.4) do not contain $\bar{\zeta}\zeta$. Namely we cannot say that $Z_{\alpha=0}^{NL}\neq 0$ is guaranteed.

To evade this difficulty, we first construct the partition function $Z_{\alpha}^{NL}\neq 0$, and then take the limit $\alpha\to 0$, i.e. $\lim_{\alpha\to 0}Z_{\alpha}^{NL}$.

From the Lagrangian $\mathcal{L}_{NL}$, the equation of motion for $B$ is

[TABLE]

So, when $\alpha\to 0$, the term $-ig\alpha_{2}(\bar{c}\times c)$ must be taken into account. In this section, treating the interactions perturbatively at the one-loop level, we study the behavior of $\alpha$.

In Appendix C, we derive the renormalization group (RG) equations

[TABLE]

which coincide with the results in Refs. hn and kmsi .444The parameters $\alpha_{1}$ and $\alpha_{2}$ in this article are related to the parameters in Refs. hn and kmsi as follows:

after setting $\xi=0,\zeta=\eta$ and $\alpha=\beta$, $\alpha_{1}=(1+\eta)\alpha$ and $\alpha_{2}=-\eta\alpha$ in Ref. hn ,

$\alpha_{1}=(1-\xi)\lambda=\alpha^{\prime}+\alpha/2$ and $\alpha_{2}=\xi\lambda=\alpha/2$ in Ref. kmsi .

We emphasize that the equation for $\alpha_{1}$ does not contain $\alpha_{2}$, and vice versa. From (4.1), $\alpha=\alpha_{1}+\alpha_{2}$ satisfies

[TABLE]

When $|\alpha|\ll 1$, (4.2) becomes

[TABLE]

Therefore, when $\alpha_{2}\neq 0$, $\alpha$ increases as $\mu$ decreases. The quartic ghost interaction makes $\alpha\neq 0$, and the situation in §3 realizes. Even if a single zero mode $v_{0}$ exists, the partition function does not vanish.

Eq.(4.1) shows that $(\alpha_{1},\alpha_{2})=(0,0)$ is an infrared fixed point. Does this fact imply that the Landau gauge (1.1) is retrieved as $\mu\to 0$? 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 hs1 ; hs2

[TABLE]

This Lagrangian has the BRS invariance, if $\varphi$ transforms as $\delta_{B}\varphi=g\varphi\times c.$ Setting the constant $w=0$, and performing the $\varphi$ integration, we find $\mathcal{L}_{\varphi}$ yields $\mathcal{L}_{NL}$. Namely, $\varphi$ is an auxiliary field which represents $\alpha_{2}\bar{B}$.

However, in a low energy region, $\varphi$ is not an auxiliary field. In Ref. hs2 , we derived another RG equation for $\alpha_{2}$ given by

[TABLE]

which is different from (4.1). Eq.(5.2) was derived by making the Wilsonian effective action for $\varphi$.555In Appendix C.2, we explain how to derive (5.2) from $\mathcal{L}_{NL}$. We also showed that $\varphi$ acquires the vacuum expectation value $\langle\varphi\rangle=\varphi_{0}$ under the energy scale

[TABLE]

where $\Lambda$ is a momentum cut-off. Ghost-antighost bound states and ghost condensation appear below $\mu_{0}$. We substitute $\varphi(x)=\varphi_{0}+\varphi^{\prime}(x)$ into (5.1), and choose the constant $w=\varphi_{0}$. This choice is necessary to maintain the BRS symmetry kug .666This point is explained in Appendix D. The anti-BRS symmetry and the global gauge symmetry are also discussed. Then (5.1) becomes

[TABLE]

Because of the dimensional transmutation gn , the parameter below $\mu_{0}$ is not $\alpha_{2}$ but $\varphi_{0}$.

Contrary to $\alpha_{2}$, the gauge parameter $\alpha_{1}$ remains in (5.4). As we explain in Appendix C.2, the RG equation (4.1) for $\alpha_{1}$ persists, and $\alpha_{1}=0$ is an infrared fixed point. So, when $\mu\to 0$, (5.4) gives the gauge condition

[TABLE]

and the ghost Lagrangian

[TABLE]

As (5.5) means $\partial_{\mu}D_{\mu}\neq D_{\mu}\partial_{\mu}$, we assume $\partial_{\mu}D_{\mu}$ has a pair of zero modes $(u_{0},u_{0}^{*})$ and a single zero mode $v_{0}$, and $D_{\mu}\partial_{\mu}$ has zero modes $(\bar{u}_{0},\bar{u}_{0}^{*})$ and $\bar{v}_{0}$. Even if the measure $DcD\bar{c}$ contains $d\xi d\xi^{\dagger}d\zeta d\bar{\xi}d\bar{\xi}^{\dagger}d\bar{\zeta}$, because the term $i\bar{c}\cdot(g\varphi_{0}\times c)$ in (5.4) has

[TABLE]

the partition function does not vanish.

6 Summary

In the Landau gauge $\alpha=0$, the FP operator $-\partial_{\mu}D_{\mu}$ has zero modes on the Gribov horizon. As the ghost $c$ and the anti-ghost $\bar{c}$ are Grassmann variables, it is natural to expect that these zero modes yield effective ghost interactions. We have shown the quartic ghost interaction 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 $\alpha\neq 0$ gauge, the same result is obtained as well.

The effect of a single zero mode was also studied. Although there is no trouble in the $\alpha\neq 0$ gauge, the partition function $Z$ may vanish in the $\alpha=0$ gauge. We can avoid this problem by taking the limit $\alpha\to 0$.

Usually, when $\det\partial_{\mu}D_{\mu}=0$ for some configuration $A_{\mu}$, we can evade the $Z=0$ problem by choosing another gauge (locally) na . 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, $(\alpha_{1},\alpha_{2})=(0,0)$ is an infrared fixed point at the one-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 $\mu_{0}$, and the zero-mode problem no longer happens.

Appendix A Examples of zero modes in the Coulomb gauge

In this appendix, choosing the gauge $\partial_{j}A_{j}=0$, we study the eigenvalue equation

[TABLE]

in three-dimensional space-time.

A.1 A pair of zero modes

If the eigenfunction has the form $u^{A}=e^{is}w^{A}$ with $gA_{j}\times(\partial_{j}w)=0$, (A.1) becomes

[TABLE]

Since $H$ is a real antisymmetric $3\times 3$ matrix, its eigenvalues are pure imaginary or [math], i.e.

[TABLE]

The last equation of (A.3) means that the effect of $A_{j}^{C}$ disappears and $w_{0}$ does not become a zero mode. From (A.2) and (A.3), we obtain

[TABLE]

Thus we find the two functions $u_{\pm}=e^{\pm is}w_{\pm}$ become a zero-mode pair, if

[TABLE]

holds.

To give concrete examples, let us choose the abelian configuration

[TABLE]

A.1.1 Three-torus $T^{3}$

Gribov copies in the three-torus $T^{3}$ are studied in Ref. vb . The constant configuration

[TABLE]

is on the first Gribov horizon, where $L$ is the size of the torus. Setting $s=2\pi x_{1}/L$, we find (A.4) is satisfied by a zero-mode pair

[TABLE]

A.1.2 Axially symmetric configuration in $\mathbb{R}^{3}$

Next we consider the configuration

[TABLE]

where $(r,\theta,\phi)$ are the spherical coordinates. Using the angular momentum operator $\hat{L}_{j}=-i\epsilon_{jkl}x_{k}\partial_{l}$, we find

[TABLE]

Then it is natural to set $e^{is}=e^{im\phi}$ and

[TABLE]

where $l$ and $m$ are integers, and

[TABLE]

We note $e^{im\phi}\Theta_{lm}(\theta)=Y_{lm}(\theta,\phi)$ is the spherical harmonics which satisfies $Y_{lm}^{*}=(-1)^{-m}Y_{l,-m}$. Then (A.4) becomes

[TABLE]

Now, following Henyey he , we substitute the functions

[TABLE]

into (A.7), where $K,r_{0},d,\rho,\kappa$ and $\sigma$ are constants. Eq.(A.7) is satisfied by

[TABLE]

Thus we obtain the abelian configuration and the corresponding zero-mode pairs as

[TABLE]

In Ref. he , the $l=1$ case is presented explicitly.

A.2 A single zero mode

In Ref. ma , a single zero mode was found in an instanton background. Here we give an example in $\mathbb{R}^{3}$. Generalizing (A.5) and (A.6), we choose the configuration

[TABLE]

Then (A.1) becomes

[TABLE]

First we solve the equation

[TABLE]

where $i\alpha$ is an eigenvalue of $\Xi$. We substitute the expansion

[TABLE]

and, for simplicity, choose $l=1$. Then we find that the eigenvalues are $\alpha=2,1$ and $-1$, and the numbers of eigenfunctions are $1,3$ and $5$, respectively. We choose the real eigenfunctions $u_{\alpha}^{A}=R_{\alpha}(r)w_{\alpha}^{A}(\theta,\phi)$, where $w_{\alpha}^{A}(\theta,\phi)$ are given by

[TABLE]

Next we determine $R_{\alpha}$. From (A.11) with $\lambda=0$ and (A.12), $R_{\alpha}$ satisfies

[TABLE]

As in the previous subsection, we substitute (A.8) into (A.13). Then we find

[TABLE]

Two real zero modes are replaced by a pair of zero modes. So one real zero mode remains for each value of $\alpha$.

Appendix B Derivation of the Lagrangians (2.19) and (5.1) by the use of ”source”

In the instanton case, the fermion determinant does not vanish if fermion sources exist tH ; tH2 . Following this case, we introduce a field $\varphi(x)$, and replace $i\bar{c}\cdot\partial_{\mu}D_{\mu}c$ with

[TABLE]

The eigenvalue equation is

[TABLE]

We treat the term $g\varphi\times$ as pertubation, and perform the expansion

[TABLE]

where $\Lambda_{n}^{(0)}=\lambda_{n}$, and $w_{n}^{(0)}=u_{n}$ in (2.4) and $w_{n}^{(0)}=u_{n}^{*}$ in (2.5). Using the normalization $\int dx\,u_{n}^{*}\cdot u_{n}=1$ and $\int dx\,u_{n}\cdot u_{n}=0$, we obtain

[TABLE]

for $u_{n}$ and

[TABLE]

for $u_{n}^{*}$, where $f^{ABC}u_{n}^{A}\varphi^{B}u_{n}^{C}=0$ has been used. Therefore, if $\partial_{\mu}D_{\mu}$ has a pair of zero modes $(u_{0},u_{0}^{*})$, (B.1) gives rise to the determinant

[TABLE]

where $k_{n}$ is the number of eigenfunctions that have the eigenvalue $\Lambda_{n}$ or $\Lambda_{n}^{*}$. Thus, although $\Lambda_{0}^{(0)}=\lambda_{0}=0$, $\Lambda_{0}^{(1)}\neq 0$ makes the partition function non-zero.

Since

[TABLE]

we find

[TABLE]

gives the determinant (B.2). To derive (2.19), we multiply (B.3) by $\exp[-\int dx\,(\varphi+\alpha_{2}B)^{2}/(2\alpha_{2})]$, and integrate with respect to $\varphi$:

[TABLE]

After the $\varphi$ integration, we obtain (2.19).

We note, to derive (5.1), (B.3) must be multiplied by $\exp[-\int dx\,\left\{(\varphi-w+\alpha_{2}B)^{2}+2w\varphi-w^{2}\right\}/(2\alpha_{2})]$, where $w$ is a constant determined later.

Appendix C Derivation of the RG equations (4.1) and (5.2)

In subsection C.1, using $\mathcal{L}_{NL}$, we derive the RG equation (4.1). In subsection C.2, the RG equation (5.2) is derived. The RG equation for $\alpha_{1}$ under the scale $\mu_{0}$ is discussed.

C.1 The Lagrangian (2.19) and the RG equations (4.1)

C.1.1 Equation for $\alpha_{2}$

The Lagrangian $\mathcal{L}_{NL}$ contains the quartic ghost interaction

[TABLE]

We define the renormalization constant $Z_{4}$ by

[TABLE]

where $\bar{c}_{0}=\tilde{Z}_{3}^{1/2}\bar{c}$ and $c_{0}=\tilde{Z}_{3}^{1/2}c$. First we consider the ghost self-energy. Although $\mathcal{L}_{NL}$ gives additional one-loop diagrams, divergence of them cancels out. Thus we obtain, as usual, $\tilde{Z_{3}}=1+\tilde{Z_{3}}^{(1)}+\cdots$ with

[TABLE]

where $\varepsilon=(4-D)/2$, and $C_{2}(G)=2$ is inserted. We note the gauge parameter in $\mathcal{L}_{NL}$ is $\alpha=\alpha_{1}+\alpha_{2}$.

Next we study $Z_{4}$. Using the notation of Fig.C1, one-loop diagrams which contribute to $Z_{4}$ come from the diagrams in Figs.C2 and C3. However Fig.C2(b) does not yield divergence, and divergences of Figs.C2(c1)-(c3) cancel out. Furthermore some of the diagrams derived from Fig.C3 don’t diverge.

Thus divergent diagrams are depicted in Fig.C4, and they give the constant $Z_{4}=1+Z_{4a}^{(1)}+Z_{4b}^{(1)}+Z_{4c}^{(1)}+Z_{4d}^{(1)}+\cdots$, where

[TABLE]

Eq.(C.1) leads to

[TABLE]

Then performing the replacement $g\to g\mu^{-\varepsilon}$ or $1/\varepsilon\to 2\ln\Lambda/\mu$ in (C.2) and (C.3), and using the RG equation

[TABLE]

we obtain

[TABLE]

C.1.2 Equation for $\alpha_{1}$

Renormalization constants are defined as usual:

[TABLE]

Then $\mathcal{L}_{NL}$ gives the counter terms

[TABLE]

The first counter term cancels the divergence of Fig.C5(a), and we obtain

[TABLE]

As the gauge parameter in $\mathcal{L}_{NL}$ is $\alpha$, the constant $Z_{3}^{(1)}$ is

[TABLE]

as usual. Using these results, $Z_{B}^{(1)}$ becomes

[TABLE]

The divergence of Fig.C5(b) is canceled by the second counter term, i.e.

[TABLE]

From (C.5) and (C.6),

[TABLE]

and

[TABLE]

is derived. Substituting (C.8) and (C.10) into (C.9), we obtain

[TABLE]

and

[TABLE]

C.2 RG equations near $\mu_{0}$ and under $\mu_{0}$

C.2.1 Eq.(5.2)

The RG equation (5.2) is derived from the Lagrangian $\mathcal{L}_{\varphi}$ hs2 . To derive it from the Lagrangian $\mathcal{L}_{NL}$, we must replace (C.4) with

[TABLE]

Namely, in the region $\mu_{0}<\mu<\Lambda$, the interaction between $\bar{c}$ and $c$ becomes strong, and Fig.C4(a) is the main contribution. In the limit $\mu\to\mu_{0}$, $\bar{c}$ and $c$ make bound states and ghost condensate.

C.2.2 RG equation for $\alpha_{1}$

Near $\mu_{0}$, as we stated above, the Lagrangian (5.1) should be used. Under $\mu_{0}$, we must use the Lagrangian (5.4). In these Lagrangians, the gauge parameter for $A_{\mu}$ is not $\alpha$ but $\alpha_{1}$. Then, instead of (C.7), we must use

[TABLE]

Since the self-energies for $BA_{\mu}$ and $BB$ don’t have divergence now, $Z_{\alpha_{1}}=Z_{B}^{-1}=Z_{3}$ holds. Thus we have

[TABLE]

That is, the RG equation for $\alpha_{1}$ is unchanged.

Appendix D Symmetries of the Lagrangian $\mathcal{L}_{\varphi}$ in (5.1)

D.1 BRS symmetry

It is easy to check that $\mathcal{L}_{\varphi}$ is invariant under the BRS transformation

[TABLE]

The constant $w$ is determined to conserve this symmetry. From the partition function

[TABLE]

we can derive the equation of motion for $B$ as

[TABLE]

where

[TABLE]

Since $D\mu$ and $\mathcal{L}_{inv}+\mathcal{L}_{\varphi}$ are invariant under the BRS transformation,

[TABLE]

holds. We substitute $B=-i\delta_{B}\bar{c}$ and $\varphi(x)=\varphi_{0}+\varphi^{\prime}(x)$ into (D.1), and use $\langle A_{\mu}\rangle=0$, $\langle\varphi^{\prime}\rangle=0$. Then (D.1) leads to $i\alpha_{1}\langle\delta_{B}\bar{c}\rangle=w-\varphi_{0}$. The consistency with (D.2) requires $w=\varphi_{0}$.

D.2 Anti-BRS symmetry

The anti-BRS transformation is given by

[TABLE]

When $\varphi_{0}\neq 0$, from the equation of motion for $\varphi$, $\langle\alpha_{2}\bar{B}\rangle=\langle\varphi\rangle\neq 0$ holds. Therefore the anti-BRS symmetry is broken spontaneously, because

[TABLE]

In addition, we must set $w=\varphi_{0}\neq 0$ to maintain the BRS symmetry. As $\bar{\delta}_{B}\mathcal{L}_{\varphi}=-g(B\times\bar{c})\cdot w$, the Lagrangian does not respect the anti-BRS symmetry.

D.3 Global gauge symmetry

Using the constant small parameter $\theta$, the global gauge transformation is defined by $\delta_{\theta}\Phi=\theta\times\Phi$, where $\Phi$ represents all the fields in $\mathcal{L}_{\varphi}$. This symmetry breaks down just like the anti-BRS symmetry. In fact, $\varphi_{0}\neq 0$ gives $\langle\delta_{\theta}\varphi\rangle=\theta\times\varphi_{0}$. and $w=\varphi_{0}$ brings $\delta_{\theta}\mathcal{L}_{\varphi}=-w\cdot(\theta\times B)$.

Next we study the partition function $Z_{\varphi}$. It transforms as $\delta_{\theta}Z_{\varphi}\propto\langle\delta_{\theta}\mathcal{L}_{\varphi}\rangle$. Using $B=-i\delta_{B}\bar{c}$ and (D.2), we find

[TABLE]

Namely, because of the BRS symmetry, $Z_{\varphi}$ remains invariant under this symmetry.

In the same way, we can show that the breaking by $w$ cannot be observed in any function $\langle\Psi(\Phi)\rangle$, if $\Psi(\Phi)$ is BRS-invariant. To show this, we consider the function

[TABLE]

which appears in $\delta_{\theta}\langle\Psi(\Phi)\rangle$. Using $\delta_{\theta}\mathcal{L}_{\varphi}=-i(w\times\theta)\cdot\delta_{B}\bar{c}$ and $\delta_{B}\Psi(\Phi)=0$, we find (D.3) vanishes. Thus BRS invariant Green functions aren’t broken by $\delta_{\theta}\mathcal{L}_{\varphi}$.