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\hrule 
\leftline{}
\leftline{Chiba Univ. Preprint
          \hfill   \hbox{\bf CHIBA-EP-128}}
\leftline{\hfill   \hbox{hep-th/0105268}}
\leftline{\hfill   \hbox{May 2001}}
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\centerline{\large\bf
  The Most General and Renormalizable Maximal Abelian Gauge
} 

   
\vskip 1cm

\centerline{{\bf 
Toru Shinohara,${}^{1,\dagger}$
Takahito Imai${}^{1,\ddagger}$ and
Kei-Ichi Kondo${}^{1,2,\ast}$
}}
\vskip 1cm
\begin{description}
\item[]{\it 
$^1$ Graduate School of Science and Technology,
  Chiba University, Chiba 263-8522, Japan
  %$^\ddagger$
  }
\item[]{\it \centerline{ %$^1$ 
$^2$ Department of Physics, Faculty of Science, 
Chiba University,  Chiba 263-8522, Japan}
  }
%\item[]{\centerline{$^\dagger$ 
%  E-mail:  kondo@cuphd.nd.chiba-u.ac.jp }
%  }
%\item[]{$^\ddagger$ 
%  E-mail:   
%  }
\end{description}
%\vskip 0.5cm

\centerline{{\bf Abstract}}

We have constructed the most general gauge fixing term for the $SU(2)$ Yang-Mills theory, which leaves the global $U(1)$ gauge symmetry intact (i.e., the most general Maximal Abelian gauge).
From the viewpoint of renormalizability, we point out the importance of an additional gauge fixing parameter that is not included in the ordinary gauge fixing term with the global $SU(2)$ gauge symmetry.
By requiring a number of symmetries without spoiling the renormalizability, we restrict the most general gauge to a minimum gauge which is considered to be the essential part of the maximal Abelian gauge.  Then we demonstrate that a new parameter $\kappa$ introduced in this gauge is very efficient in order to examine the renormalizability.  In fact, we confirm the renormalizability of this gauge together with
a new parameter $\kappa$ by calculating explicitly the anomalous
dimensions of all the parameters  at one loop order.



\vskip 0.5cm
Key words: maximal Abelian gauge, Abelian dominance,
           renormalizability, non-Abelian gauge theory

PACS: 12.38.Aw, 12.38.Lg 
\vskip 0.2cm
\hrule  
${}^\dagger$ 
  E-mail: {\tt sinohara@cuphd.nd.chiba-u.ac.jp}
\par 
${}^\ddagger$ 
  E-mail: {\tt takahito@physics.s.chiba-u.ac.jp}
\par 
${}^\ast$ 
  E-mail: {\tt kondo@cuphd.nd.chiba-u.ac.jp}

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%\begin{description}
%\item[]{
%$^\ddagger$
%Address from March 1996 to December 1996.
%  On leave of absence from: \\
%  Department of Physics, Faculty of Science,
%  Chiba University, Chiba 263, Japan.
%  }
%\item[]{
%$^*$ To be published in Phys. Rev. Lett.
% Submitted to .
% }  
%\end{description}

\end{titlepage}


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\section{Introduction} %
%%%%%%%%%%%%%%%%%%%%%%%%
\par
The gauge fixing is an indispensable procedure in quantizing the continuum gauge theory.
It is believed that the physically meaningful results do not
depend on the gauge fixing condition.
Therefore we can adopt any favorite gauge fixing condition
for our purpose.
The maximal Abelian (MA) gauge is one of the gauge fixing conditions which seems to be very useful to investigate the physics of the low energy sector of quantum chromodynamics (QCD).  
The reason is as follows. 
In the low energy region of QCD, the Abelian projection
procedure\cite{tHooft81} or a hypothesis of Abelian dominance\cite{EI82}
has been justified by the recent research mainly based on numerical
simulations\cite{SY90}.
Here the Abelian dominance has played an important role in the low energy phenomenon of QCD, for instance, quark confinement and chiral symmetry breaking.
Especially, quark confinement can be explained
by the dual superconductor picture\cite{dsc} at least qualitatively.
Dual superconductivity of QCD is expected to be described by the dual Ginzburg-Landau (DGL) theory.
However, the DGL theory is an Abelian gauge theory, while QCD is
an $SU(3)$ non-Abelian gauge theory.
Therefore, in order to adopt the dual superconductor picture responsible
for the low energy physics in QCD such as quark confinement, the Abelian 
projection procedure is necessary.
Thus we expect that the maximal Abelian gauge \cite{QR98} is the most useful gauge
for describing the low energy region of QCD.
\par
In a series of papers\cite{KondoI,KondoII,KondoIII,KS00a,KS00b,Shinohara01a}, we have attempted to give an analytical framework which is able to explain the  Abelian dominance in QCD in the MA gauge
from the viewpoint of renormalizability.
The MA gauge  is a nonlinear gauge fixing condition, in sharp contrast with the conventional gauge fixing of the Lorentz type which is a linear gauge.
Due to this non-linearity, we must introduce the quartic ghost--anti-ghost
self-interaction to maintain the renormalizability.
The {\it modified} MA gauge fixing term\cite{KondoII,KS00a} was devised to incorporate such a self-interaction term in a natural way.
We have pointed out a possibility of dynamical
mass generation of off-diagonal gluons and off-diagonal ghosts
due to the ghost--anti-ghost condensation.
The fact that the off-diagonal fields become massive while the diagonal fields remain massless gives an analytical explanation of Abelian dominance
in the low energy region.

\par
In this paper, we investigate the $SU(N)$ Yang-Mills theory in the maximal Abelian gauge from a viewpoint of renormalizability.
Various phenomena supporting the Abelian dominance
in the low-energy sector of QCD have been reported
for the maximal Abelian gauge.
The MA gauge partially fixes the non-Abelian
gauge symmetry leaving the residual $U(1)^{N-1}$ gauge symmetry.
In previous papers\cite{KondoII,KS00a}, we have proposed
a  modified MA gauge fixing term that includes a
quartic off-diagonal ghost self interaction term from the viewpoint
of renormalizability.
%However, it is not yet clear if we should adopt the gauge fixing
%condition with respect to residual $U(1)^{N-1}$ gauge fixing.

\par
After the MA gauge fixing, there remains the residual $U(1)^{N-1}$ gauge symmetry.
Then we must fix the residual symmetry to completely fix the gauge degrees of freedom.
The most naive choice of the gauge fixing condition for the residual
$U(1)^{N-1}$ gauge symmetry is the Lorentz gauge just as in the ordinary
Abelian gauge theory.
However, it is not clear whether this choice is available or not in our context.  
This is because we must consider more general gauge fixing term than that in the conventional Lorentz gauge,
since the MA gauge fixes partially not only the 
local $SU(N)$ gauge symmetry but also the global $SU(N)$ gauge symmetry.
In the most general MA gauge, therefore, we can not require
the global $SU(N)$ gauge symmetry for the gauge fixing term,
but require only the  global $U(1)^{N-1}$ gauge symmetry.
A detailed consideration of such a gauge fixing term in the case
of $SU(2)$ has already been attempted by Min, Lee and Pac\cite{MLP85}
or Hata and Niigata\cite{HN93}.
However, in this paper, we point out that an introduction of one more parameter is more efficient for discussing the total renormalizability of the theory by keeping various symmetries which we want to require for the {\it renormalized} theory.
In fact, we demonstrate its usefulness by a perturbative calculation at one-loop level by making use of the dimensional regularization.
\par
This paper is organized as follows. 
In section 2, we give a general consideration on the renormalizable gauge fixing term respecting the global $U(1)$ gauge symmetry in the $SU(2)$ non-Abelian gauge theory.
By  taking account of the symmetries, we can fix some of the 
parameters without spoiling the renormalizability.
Then we restrict our consideration to a fixed parameter subspace. 
It is possible to choose a minimum set of the maximal Abelian gauge 
by restricting the parameter space to three independent parameters.
In section 3, we determine all the remaining parameter in the minimum choice of the most general MA gauge,
although some of the anomalous dimensions
have already been obtained in the previous papers\cite{KS00b,Shinohara01a}.
By a thorough treatment of a new parameter $\kappa$,
the renormalizability of the  modified  MA gauge is confirmed.
We give the conclusion and discussion in the final section.
In Appendix, we discuss the rescaling of the fields
preserving BRST transformation and its connection to the renormalization.


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\section{The most general gauge fixing terms} %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:The most general gauge fixing terms}
In this section, we construct the most general gauge fixing term
for the Maximal Abelian gauge in the $SU(2)$ Yang-Mills theory.
Note that we require only the global $U(1)$ symmetry to the gauge fixing term, not the global $SU(2)$ gauge symmetry.
The most general gauge fixing term is obtained in the BRST exact form,
\begin{equation}
S_{\rm GF+FP}
 =-i\int d^4x\mbox{\boldmath$\delta$}_{\rm B} G ,
\label{eq:GF+FP}
\end{equation}
where $G$
%$G^{(a)}$, $G^{(i)}$ and $G^{\rm ex}$ 
is a  functional of
gluons ${\cal A}_{\mu}^A=(A_\mu^a,a_\mu)$,
ghosts ${\cal C}^A=(C^a,C^3)$,
anti-ghosts $\bar{\cal C}^A=(\bar C^a,\bar C^3)$ 
and Nakanishi-Lautrup fields ${\cal B}^A=(B^a,B^3)$ with $a=1,2$.
\par
First, the functional $G$ must satisfy the following requirements.
\begin{enumerate}
\item[1)]
 The functional $G$ must be of mass dimension 3.
Since ${\cal A}_{\mu}^A=(A_\mu^a,a_\mu)$,
${\cal C}^A=(C^a,C^3)$,
$\bar{\cal C}^A=(\bar C^a,\bar C^3)$ and
${\cal B}^A=(B^a,B^3)$ has respectively the mass dimension  1, 1, 1 and 2,  a monomial in the functional $G$ consists of at most three fields.

\item[2)]
The functional $G$ must has the global $U(1)$ symmetry.  

\item[3)]
The functional $G$ must has the ghost number $-1$.
Note that ${\cal A}_{\mu}^A=(A_\mu^a,a_\mu)$,
${\cal C}^A=(C^a,C^3)$,
$\bar{\cal C}^A=(\bar C^a,\bar C^3)$ and
${\cal B}^A=(B^a,B^3)$ has the ghost number 0, 1, $-1$ and 0, respectively.

\end{enumerate}
From the above requirements 1) and 2), the possible form of the monomials in $G$ can be classified into seven groups:
\begin{equation}
  \delta^{ab}X^a Y^b Z^3,
\quad
  \epsilon^{ab3}X^a Y^b Z^3, 
\quad
  X^3 Y^3 Z^3 ,
\end{equation}
\begin{equation}
  \delta^{ab} X^a Y^a,
\quad 
  \epsilon^{ab} X^a Y^b,
\quad
  X^3 Y^3 ,
\end{equation}
\begin{equation}
  X^3 .
\end{equation}
Taking account of the fact that the functional $G$ is of the form, 
\begin{equation}
 G=\bar{C} \Phi ,
\end{equation}
apart from the index, we find that one of $X$, $Y$ and $Z$ must be an anti-ghost $\bar{\cal C}^A=(\bar C^a,\bar C^3)$ and that $\Phi$ must be of dimension 2 and of ghost number zero from the requirement 3).  

\par
Second, we consider the global $SU(2)$ symmetry which is broken by the MA gauge fixing. 
The invariants under the global $SU(2)$ rotation are 
\begin{equation}
  \epsilon^{ABC} X^A Y^B Z^C, \quad \delta^{AB} X^A X^B .
\end{equation}
Therefore, the three groups of the seven groups belong to this type:
\begin{equation}
  \epsilon^{ab3}X^a Y^b Z^3, 
\quad
   \delta^{ab} X^a Y^b,
\quad
  X^3 Y^3 .
  \label{eq:1stgroup}
\end{equation}
The remaining four groups,
\begin{equation}
  \delta^{ab}X^a Y^b Z^3,
\quad
  X^3 Y^3 Z^3 ,
\quad
  \epsilon^{ab} X^a Y^b,
\quad
  X^3 ,
  \label{eq:2ndgroup}
\end{equation}
 are incompatible with the global $SU(2)$ symmetry if they exist in the functional $G$.
They are called the exceptional terms.  
Thus, the possible form of the functional is rewritten as
\begin{equation}
S_{\rm GF+FP}
 =-i\int d^4x\mbox{\boldmath$\delta$}_{\rm B}
   \left(G^{(a)}+G^{(i)}+G^{\rm ex}\right),
\label{eq:S_GF+FP}
\end{equation}
where
we have decomposed the terms belonging to the first group (\ref{eq:1stgroup}) into two
 functionals~$G^{(a)}$ and $G^{(i)}$ according to their forms,
\begin{equation}
  G^{(a)} = \bar{C}^a \Phi^a ,
\quad
  G^{(i)} = \bar{C}^3 \Psi^3 ,
\end{equation}
and
$G^{\rm ex}$ denotes the exceptional terms of the form,
\begin{equation} 
  G^{\rm ex} = \bar{C}^a \Phi'^a + \bar{C}^3 \Psi'^3 .
\end{equation}
\par
The first functional~$G^{(a)}$ plays the role of partially fixing
the $SU(2)$ gauge symmetry to $U(1)$.
The possible form of monomials in $G^{(a)}$ is either 
$\epsilon^{ab3}\bar{C}^aY^b Z^3$
or
$\delta^{ab}\bar{C}^aY^b$.
It is easy to see that the possible choices are given as
$
 \epsilon^{ab3}\bar{C}^aY^b Z^3
 \sim \epsilon^{ab3}\bar{C}^a (C^b \bar{C}^3, \bar{C}^b C^3, A_\mu^b a_\mu) 
$
and
$
 \delta^{ab} \bar{C}^aY^b 
\sim \delta^{ab} \bar{C}^a (B^b, \partial^\mu A_\mu^b)  .
$
Thus the most general form of $G^{(a)}$ is given by
\begin{equation}
G^{(a)}
 :=\bar C^a\left[
    (\partial^\mu\delta^{ab}-\xi g\epsilon^{ab}a^\mu)A_\mu^b
     +\frac\alpha2B^a
     -i\frac\zeta2g\epsilon^{ab}\bar C^bC^3
     +i\eta g\epsilon^{ab}\bar C^3C^b
    \right].
\label{eq:G^a}
\end{equation}
It turns out that the off-diagonal component of the Nakanishi-Lautrup
field $B^a$ is generated from this functional after performing 
the BRST transformation explicitly.
\par
By making use of the anti-BRST transformation, this functional is recast  into
\begin{eqnarray}
G^{(a)}
 &\equiv&
   -\bar{\mbox{\boldmath$\delta$}}_{\rm B}
    \left[\frac12A^{\mu a}A_\mu^a
          -\frac\zeta2iC^a\bar C^a\right]
   \nonumber\\
 & &+\bar C^a\left[
     i(1-\xi)g\epsilon^{ab}a^\mu A_\mu^a
     +\frac{\alpha-\zeta}2B^a
     +i\eta g^2\epsilon^{ab}\bar C^3C^b
    \right] .
\label{eq:G'}
\end{eqnarray}
%where we have introduced a new parameter
%\begin{equation}
%\lambda:=\alpha-\zeta.
%\end{equation}
The first term of the right hand side of (\ref{eq:G'})
is both BRST and anti-BRST exact, and give rise to
the  modified  MA gauge fixing term
proposed in the previous papers\cite{KondoII,KS00a}.
After performing the BRST transformation, we obtain
\begin{eqnarray}
S^{(a)}
 &:=&
    -i\int d^4x\mbox{\boldmath$\delta$}_{\rm B}G^{(a)}
    \nonumber\\
 &=&\int d^4x\biggl\{
    %%%%%%%%%% $(B^a)$ %%%%%%%%%
    \frac{\alpha}{2}B^aB^a
    +
    B^aD^{\xi\mu} A_\mu^a
    -
    ig\zeta\epsilon^{ab}
    B^a\bar{C}^bC^3
    +
    ig\eta\epsilon^{ab}B^a\bar{C}^3C^b
    \nonumber \\
    %%%%%%%%%% $(B^3)$ %%%%%%%%%
    &&
    -
    ig\eta\epsilon^{ab}B^3\bar{C}^aC^b
    %%%%%%%%%% $(Cs and As)$ %%%%%%%%%%
    +
    i\bar{C}^aD^{\xi\mu} D_{\mu} C^a
    \nonumber \\
    &&
    +
    ig\epsilon^{ab}\bar{C}^aD^{\xi\mu}A_\mu^bC^3
    +ig(1-\xi)\epsilon^{ab}\bar C^aA_\mu^b\partial^\mu C^3
    -ig\xi\epsilon^{ad}\epsilon^{cb}
    \bar{C}^aC^bA_\mu^cA^{\mu d}
    \nonumber \\
    %%%%%%%%%%% $(CCCC)$ %%%%%%%%%%%%
    &&
    +
    g^2\frac{\zeta}{4}\epsilon^{ab}\epsilon^{cd}\bar{C}^a\bar{C}^bC^cC^d
    -
    g^2\eta\bar{C}^3C^3\bar{C}^aC^a
    \biggr\},
\label{eq:S^a}
\end{eqnarray}
where we have defined a covariant derivative $D_\mu^\xi$
in terms of the Abelian gluon $a_\mu$ as
\begin{equation}
D_\mu^\xi{\mit\Phi}^a
  :=(\partial_\mu\delta^{ab}
     -\xi g\epsilon^{ab}a_\mu){\mit\Phi}^b,
\label{eq:Abelian covariant derivative with xi}
\end{equation}
which is abbreviated in the special case of $\xi=1$ as
\begin{equation}
D_\mu{\mit\Phi}^a
 :=D_\mu^{\xi=1}{\mit\Phi}^a
  =(\partial_\mu\delta^{ab}
    -g\epsilon^{ab}a_\mu){\mit\Phi}^b.
\label{eq:Abelian covariant derivative}
\end{equation}

\par
The second functional~$G^{(i)}$ is used to fix the residual
$U(1)$ gauge symmetry.
The possible monomials are of two types:
$\epsilon^{ab3}\bar{C}^3 Y^a Z^b$ and $\delta^{33}\bar{C}^3Y^3$.
Therefore, we obtain
$\epsilon^{ab3}\bar{C}^3 Y^a Z^b
\sim \epsilon^{ab3}\bar{C}^3(\bar{C}^a C^b, A_\mu^a A^\mu{}^b=0)$ 
and 
$
 \delta^{33}\bar{C}^3Y^3
\sim \delta^{33}\bar{C}^3(B^3, \partial^\mu a_\mu) .
$ 
It should be remarked that
the term proportional to $\bar C^3\epsilon^{ab}\bar C^aC^b$
is a candidate for the terms in this functional.
However, such a term has already been included
in Eq.~(\ref{eq:G^a}) as the last term.
Thus the general form of $G^{(i)}$ is given by
\begin{equation}
G^{(i)}
 :=\bar C^3\left[
    \kappa\partial^\mu a_\mu
    +\frac\beta2B^3
   \right].
\label{eq:G^i}
\end{equation}
After performing the BRST transformation, we obtain
\begin{eqnarray}
S^{(i)}
 &:=&
    -i\int d^4x\mbox{\boldmath$\delta$}_{\rm B}G^{(i)}
    \nonumber\\
 &=&\int d^4x\biggl\{
    %%%%%%%%%%% $(B^i)$ %%%%%%%%%%
    \frac{\beta}{2}B^3B^3
    +
    \kappa B^3\partial^\mu a_\mu
%    \nonumber \\
    %%%%%%%%%% $(Cs and As)$ %%%%%%%%%
%    &&
    +
    i\kappa\bar{C}^3\partial^2C^3
    +
    ig\kappa\bar C^3\epsilon^{ab}\partial^\mu(A_\mu^aC^b)
    \biggr\}.
\end{eqnarray}
\par
The last functional $G^{\rm ex}$ in Eq.~(\ref{eq:S_GF+FP}) includes
exceptional terms.
The possible forms are classified into
$\delta^{ab}X^a Y^b Z^3$, $X^3 Y^3 Z^3$, $\epsilon^{ab}X^a Y^b$ and $X^3$.
The trilinear monomials are 
$
 \delta^{ab}\bar{C}^a Y^b Z^3
\sim \delta^{ab}\bar{C}^a (A_\mu^b a^\mu, \bar{C}^bC^3=0,C^b \bar{C}^3)$
and
$
 \delta^{ab}\bar{C}^3 X^a Y^b
 \sim \bar{C}^3 \delta^{ab}(A_\mu^a A^\mu{}^b, \bar{C}^a C^b) .
$
Moreover, 
$
  X^3 Y^3 Z^3 \sim \bar{C}^3 (a_\mu a^\mu, \bar{C}^3 C^3=0) .
$
The bilinear terms are 
$\epsilon^{ab}X^a Y^b \sim \epsilon^{ab}\bar{C}^a (B^b, \partial^\mu A_\mu^b)$.
The linear term is 
$X^3 \sim \bar{C}^3 (\Lambda^2, \partial^2)$ with a parameter $\Lambda$ of mass dimension one.
Thus $G^{\rm ex}$ is given by
\begin{eqnarray}
G^{\rm ex}
 &:=&g\bar C^3\left[
     \frac\chi2a^\mu a_\mu
     +\frac\varrho2A^{\mu a}A_\mu^a
     +i\varsigma\bar C^aC^a
     \right]
     +g\omega(\Lambda^2+\partial^2)\bar C^3
     \nonumber\\
  & &+\bar C^a\epsilon^{ab}
      (\vartheta\partial^\mu\delta^{bc}
       -\varpi g\epsilon^{bc}a^\mu)A_\mu^c ,
\label{eq:G^ex}
\end{eqnarray}
where we have omitted the bilinear term $\epsilon^{ab}\bar{C}^a B^b$, since it gives a vanishing contribution after the BRST transformation,
$\mbox{\boldmath$\delta$}_{\rm B}(\epsilon^{ab}\bar{C}^a B^b)=0$.
Now we require only the global $U(1)$ gauge symmetry
for the gauge fixing terms
so that the terms included in Eq.~(\ref{eq:G^ex}) are not forbidden in spite of the fact that the diagonal index is not contracted.
The second term in the right hand side of (\ref{eq:G^ex})
becomes a linear term in $B^3$ after carrying out the BRST transformation.
We make use of the dimensional regularization in this paper so that
the divergence coming from the tadpole of $B^3$ does not appear
as a result of perturbative loop expansions.
Therefore we can set the parameter $\omega=0$ without spoiling
the renormalizability.
After performing the BRST transformation, we obtain
\begin{eqnarray}
S^{\rm ex}
 &:=&
    -i\int d^4x\mbox{\boldmath$\delta$}_{\rm B}G^{\rm ex}
    \nonumber\\
 &=&\int d^4x\biggl\{
%%%%%%%%(B^a)%%%%%%%%
-
ig\varsigma B^a\bar{C}^3C^a
+
\vartheta\epsilon^{ab}B^aD^\mu A_\mu^b
+
g(\varpi-\vartheta)B^aa_\mu A^{\mu a}
\nonumber \\
%%%%%%%%% $(B^3)$ %%%%%%%%
&&
+
g\frac{\varrho}{2}B^3A_\mu^a A^{\mu a}
+
g\frac{\chi}{2}B^3a_\mu a^\mu
+
ig\varsigma B^3\bar{C}^aC^a
\nonumber \\
%%%%%%%%% $(C-propagator?)$ %%%%%%%%
&&
+
i\vartheta\epsilon^{ab}
\bar{C}^aD^\mu D_\mu C^b
+
ig(\varpi-\vartheta)\bar{C}^aa_\mu D^\mu C^a
\nonumber \\
%%%%%%%%% $(C^3)$ %%%%%%%%%
&&
-
ig\vartheta\bar{C}^aD^\mu A_\mu^aC^3
+
ig(\varpi-\vartheta)\bar{C}^a\partial^\mu C^3A_\mu^a
+
ig^2(\varpi-\vartheta)\epsilon^{ab}\bar{C}^aC^3a_\mu A^{\mu b}
\nonumber \\
%%%%%%%%% $(\bar{C}^3)$ %%%%%%%%
&&
%ig\varrho\bar{C}^3A_\mu^aD^\mu C^a
+ig\varrho\bar C^3A_\mu^aD^\mu C^a
+
ig\chi\epsilon^{ba}\bar{C}^3a_\mu A^{\mu b}C^a
\nonumber \\
%%%%%%%%% $(others)$ %%%%%%%%
&&
+
ig\chi\bar{C}^3a_\mu\partial^\mu C^3
+
ig^2\varpi\epsilon^{bc}\bar{C}^aA_\mu^bC^cA^{\mu a}
-
g^2\varsigma\epsilon^{ab}\bar{C}^3\bar{C}^aC^bC^3
\biggr\}.
\end{eqnarray}



Summing up three functionals and integrating out
the Nakanishi-Lautrup fields, we obtain the most general
form of the gauge fixing term with global $U(1)$ symmetry as
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\begin{eqnarray}
S_{\rm GF+FP}
 &=&\int d^4x\bigg\{
%%%%%%%%%%%(C propagator)%%%%%%%%%%
+
i\bar{C}^a D^{\mu\xi}D_\mu C^a
+
i\vartheta\epsilon^{ab}\bar{C}^aD^\mu D_\mu C^b
+
i\kappa\bar{C}^3\partial^2C^3
\nonumber \\
%%%%%%%%%%%%(\bar{C}^aA^3C^c)%%%%%%%%%%%%
&&
+
i\frac{g\kappa}{\beta}(\eta\epsilon^{ab}-\varsigma\delta^{ab})
\bar{C}^aC^b\partial^\mu a_\mu
+
i(\varpi-\vartheta)\bar{C}^aa_\mu(D^\mu C)^a
\nonumber \\
%%%%%%%%%%%(\bar{C}^aA^3A^3C^c)%%%%%%%%%%%
&&
+
i\frac{g^2\chi}{\beta}(\eta\epsilon^{ab}-\varsigma\delta^{ab})
\bar{C}^aC^ba_\mu a^\mu
\nonumber \\
%%%%%%%%%%%(\bar{C}^aA^bC^3)%%%%%%%%%%%%
&&
+
ig\frac{\alpha-\zeta}{\alpha}\epsilon^{ab}\bar{C}^aD^{\xi\mu}A_\mu^bC^3
+
ig(1-\xi)\epsilon^{ab}\bar{C}^aA_\mu^b\partial^\mu C^3
-
ig\frac{\alpha-\zeta}{\alpha}\vartheta\bar{C}^aD^\mu A^{\mu a}C^3
\nonumber \\
%
&&
+
ig(\varpi-\vartheta)\bar{C}^a\partial^\mu C^3A_\mu^a
\nonumber \\
%
&&
-
ig^2\frac{\alpha-\zeta}{\alpha}(\varpi-\vartheta)\epsilon^{ab}
\bar{C}^aC^3a_\mu A^{\mu b}
\nonumber \\
%%%%%%%%%%%(\bar{C}^3A^bC^c)%%%%%%%%%%%%
&&
-
ig\kappa\epsilon^{ab}\partial^\mu\bar{C}^3A_\mu^aC^b
+
ig\varrho\bar{C}^3A_\mu^aD^\mu C^a
\nonumber \\
%
&&
-
i\frac{g}{\alpha}(\eta\epsilon^{ab}-\varsigma\delta^{ab})
\bar{C}^3C^bD^{\xi\mu} A_\mu^b
-
i\frac{g\vartheta}{\alpha}(\eta\delta^{ab}-\varsigma\epsilon^{ab})
\bar{C}^3C^bD^\mu A_\mu^a
\nonumber \\
%
&&
+
ig\chi\epsilon^{ab}\bar{C}^3a_\mu A^{\mu a}C^b
-
i\frac{g^2}{\alpha}(\varpi-\vartheta)(\eta\epsilon^{ab}-\varsigma\delta^{ab})
\bar{C}^3C^ba_\mu A^{\mu a}
\nonumber \\
%%%%%%%%%%%(\bar{C}^3A^3C^3)%%%%%%%%%%%
&&
+
ig\chi\bar{C}^3a_\mu\partial^\mu C^3
\nonumber \\
%%%%%%%%%%%(\bar{C}^aC^bA^cA^d)%%%%%%%%%%
&&
+
ig^2\left((-\xi\epsilon^{ad}+\varpi\delta^{ad})\epsilon^{cb}
          +
          \frac{\varrho}{\beta}(\eta\epsilon^{ab}-\varsigma\delta^{ab})\delta^{cd}
\right)
\bar{C}^aC^bA_\mu^cA^{\mu d}
\nonumber \\
%%%%%%%%%%%(C^aC^bC^cC^d)%%%%%%%%%%%%
&&
+
g^2\frac{1}{2}\left(\zeta-\frac{\varsigma^2+\eta^2}{\beta}\right)
\delta^{ac}\delta^{bd}
\bar{C}^a\bar{C}^bC^cC^d
\nonumber \\
%%%%%%%%%%%(\bar{C}^a\bar{C}^3\C^bC^3)%%%%%%%%%%%%
&&
-
g^2\frac{\alpha-\zeta}{\alpha}(\eta\delta^{ab}-\varsigma\epsilon^{ab})
\bar{C}^3\bar{C}^aC^aC^3
\nonumber \\
%
&&
-
\frac{1}{2\alpha}
\left(D^{\xi\mu} A_\mu^a
      +
      \vartheta\epsilon^{ab}D^\mu A_\mu^b
      +
      g(\varpi-\vartheta)a_\mu A^{\mu a}\right)^2
\nonumber \\
%
&&
-
\frac{1}{2\beta}
\left(\partial^\mu a_\mu
      +
      g\varrho A_\mu^aA^{\mu a}
      +
      g\chi a_\mu a^\mu\right)^2
\bigg\}.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
The GF term $S_{\rm GF}$ has eleven independent
(gauge fixing) parameters $\xi$, $\alpha$,
$\zeta$, $\eta$, $\kappa$, $\beta$, $\chi$, $\varrho$, $\varsigma$,
$\vartheta$ and $\varpi$.
We can investigate the fixed points through the renormalization group
flow in the parameter space.
However, we expect some fixed points to exist from the viewpoint of
symmetries.
For instance, there is a fixed subspace in the parameter space
protected by the following symmetries.%
  \footnote{%
    Some of these symmetries were first pointed out
    by Hata and Niigata in Ref.~\cite{HN93}.
  }
%\begin{enumerate}
\begin{description}
%--------------------------------
\item[Charge conjugation:]
  The exceptional part~(\ref{eq:G^ex}) breaks
  the ``charge conjugation'' symmetry\cite{HN93}
  under the discrete transformation:
  \begin{equation}
  {\mit\Phi}^1\rightarrow{\mit\Phi}^1,\quad
  {\mit\Phi}^2\rightarrow-{\mit\Phi}^2,\quad
  {\mit\Phi}^3\rightarrow-{\mit\Phi}^3,
  \end{equation}
  where ${\mit\Phi}^A$ denotes all fields.
  Any term belonging to the group~(\ref{eq:2ndgroup}) is not invariant
  under this charge conjugation, while any term belonging to
  the group~(\ref{eq:1stgroup})
  is invariant under the ``charge conjugation''.
  Therefore, by setting the parameter
  $\chi=\varrho=\varsigma=\omega=\vartheta=\varpi=0$,
  the charge conjugation symmetry is recovered.
  However, once we consider the non-perturbative effect, for instance
  ghost--anti-ghost condensation proposed in the previous paper\cite{KS00a},
  the ``charge conjugation'' invariance%
  \footnote{%
    There are two types ghost--anti-ghost condensation,
    $C^a\bar C^a=C^1\bar C^1+C^2\bar C^2$ and
    $\epsilon^{ab}\bar C^aC^b=C^1\bar C^2-C^2\bar C^1$.
    The ``charge conjugation'' invariance is broken by the latter one.
    (See Ref.~\cite{KS00a} for more details.) 
  }
  is not expected to hold.
%--------------------------------
\item
  [Translational invariance for $\bar C^3$:]
  By setting the parameter to
  $\eta=0$ and $\chi=\varrho=\varsigma=\omega=0$,
  the GF term respects a global symmetry under the translation
  of the diagonal anti-ghost $\bar C^3(x)$ as
  $\bar C^3(x)\rightarrow\bar C^3(x)+\bar\theta^3$
  where $\bar\theta^3$ is a constant Grassmann variable.
  This is because the diagonal anti-ghost $\bar C^3$
  appears only in the differentiated form  $\partial_\mu\bar C^3$
  for this choice of the parameters.
  Then the translational symmetry of $\bar C^3$ exist in the theory.

%--------------------------------
\item
  [Translational invariance for $C^3$:]
  By setting the parameter to $\alpha=\zeta$,
  the action has a global symmetry
  under the translation of the diagonal ghost as
  $C^3(x)\rightarrow C^3(x)+\theta^3$
  where $\theta^3$ is a constant Grassmann variable.
  In the similar manner to the previous case,
  we can confirm that the translational
  symmetry of $C^3$ exists in this case.

%--------------------------------
\item
  [Implicit residual $U(1)$ invariance:]
  By setting the parameter to
  $\xi=1$, $\chi=\varrho=\varsigma=\omega=0$ and $\varpi=\vartheta$,
  the action has the residual $U(1)$ gauge symmetry mentioned in the previous
  paper\cite{KS00b}, although the gauge fixing
  for the residual $U(1)$ gauge symmetry has already been accomplished.
  As we have mentioned in the previous paper\cite{KS00b},
  there is the $U(1)$ gauge symmetry if the diagonal gluon does not
  appear in the action after replacing
  all the derivatives with the Abelian covariant derivative defined by
  (\ref{eq:Abelian covariant derivative})
  except for a quadratic term as $(\partial^\mu a_\mu)^2$.
  In the view of the background field method\cite{Abbott82}, there is
  a gauge symmetry with respect to the background diagonal field.


%--------------------------------
\item
  [FP conjugation invariance:]
  After setting the parameter to
  $\chi=\varrho=\varsigma=\omega=0$, $\vartheta=\varpi=0$,
  $\xi=0$, $\kappa=1$, $\beta=-2\eta$ and $\alpha=\zeta+\eta$,
  the action has the invariance under the FP ghost conjugation:
\begin{equation}
 C^A \rightarrow \pm \bar C^A, \quad
 \bar C^A \rightarrow \mp C^A, \quad
 B^A \rightarrow - \bar B^A, \quad
 \bar B^A \rightarrow - B^A, \quad
 {\cal A}_\mu^A \rightarrow {\cal A}_\mu^A .
\end{equation}


%--------------------------------
\item
  [Anti-BRST symmetry:]
  By setting the parameter to
  $\chi=\varrho=\varsigma=\omega=0$, $\vartheta=\varpi=0$,
  $1-\xi-\kappa=0$ and
  $\alpha-\beta+\eta-\zeta=0$,
  the action has the anti-BRST invariance.
  Then the action is given by
\begin{eqnarray}
S_{\rm GF}
 &\equiv&
   \int d^4x\biggl\{
   i\mbox{\boldmath$\delta$}_{\rm B}
    \bar{\mbox{\boldmath$\delta$}}_{\rm B}
    \biggl[\frac12A^{\mu a}A_\mu^a
           -\frac i2(\eta+\zeta)C^a\bar C^a
           +\frac\kappa2a^\mu a_\mu
           -\eta iC^3\bar C^3
           \biggr]
    \nonumber\\
 & &+\frac12(\alpha-\eta-\zeta)B^AB^A
    \biggr\}.
\label{eq:BRST--Anti-BRST}
\end{eqnarray}
Here, the second term in the integrand of the right hand side of the
Eq.~(\ref{eq:BRST--Anti-BRST}) is not exact in the combined BRST and
anti-BRST transformations,
$
\mbox{\boldmath$\delta$}_{\rm B}
\bar{\mbox{\boldmath$\delta$}}_{\rm B}
$,
differently from the first term.
However, the second term is both BRST and anti-BRST invariant since
$B^AB^A
  =-i\mbox{\boldmath$\delta$}_{\rm B}
    (\bar C^AB^A)
  =i\bar{\mbox{\boldmath$\delta$}}_{\rm B}
    (C^AB^A)
$.


%--------------------------------
\item
  [Global $SU(2)$ invariance:]
  After setting the parameter to
  $\chi=\varrho=\varsigma=\omega=0$, $\vartheta=\varpi=0$,
  $\xi=0$, $\kappa=1$, $\alpha=\beta$ and $\zeta=\eta$,
  the action has the global $SU(2)$ invariance.
  Then the action is given by
\begin{equation}
S_{\rm GF}
 \equiv
  -\int d^4x
   i\mbox{\boldmath$\delta$}_{\rm B}
    \biggl\{\bar C^A
    \biggl[\partial^\mu A_\mu^A
           +\frac\alpha2B^A
           -\frac\zeta2\epsilon^{ABC}\bar C^BC^C
           \biggr]
    \biggr\}.
\label{eq:SU2 action}
\end{equation}
It is easy to see that this choice of parameters is
a spacial case of the anti-BRST symmetric case.
As a result, the action~(\ref{eq:SU2 action}) is recast into 
\begin{equation}
S_{\rm GF}
 =\int d^4x
  \left[
   i\mbox{\boldmath$\delta$}_{\rm B}
    \bar{\mbox{\boldmath$\delta$}}_{\rm B}
    \left(\frac12{\cal A}_\mu^A{\cal A}^{\mu A}
          -\frac\xi2i{\cal C}^A\bar{\cal C}^A\right)
    +\frac{\xi^\prime}2{\cal B}^A{\cal B}^A
    \right],
\end{equation}
by introducing $\xi$ and $\xi^\prime$ as $\alpha=\xi+\xi^\prime$,
$\zeta=\frac{-i}2\xi g$.
This form agrees with the global $SU(2)$ invariant action which is invariant under the BRST and anti-BRST transformation obtained by Baulieu and Thierry-Mieg\cite{BT82}.

\end{description}
%\end{enumerate}
There might be the other fixed subspace of parameters protected by the other
symmetries, for example, $Sp(2)$ symmetry for the multiplet of ghost and
  anti-ghost $(C,\bar C)$, see e.g. Ref.~\cite{KS00a}.

\par

A remarkable difference between our gauge fixing procedure 
and that of the previous works (Min, Lee and Pac \cite{MLP85} and Hata and Niigata\cite{HN93})
is the existence of a new parameter $\kappa$. 
If we do not require the recovery of global $SU(2)$ gauge symmetry
in our gauge fixing,  then there is no need to set the parameter
$\kappa$ to 1 against Ref.~\cite{HN93}.
Of course, if the parameter $\kappa$ is not affected by renormalization
or there is a fixed point at $\kappa=1$, the special choice of $\kappa=1$
is allowed.
However, as we shall see later, neither the anomalous dimension of $\kappa$
is a non-renormalized parameter nor there is a fixed point at $\kappa=1$.
In the work \cite{MLP85}, the introduction of an extra parameter was avoided by introducing different renormalization factors for the ghost and anti-ghost.  
  It may seem that the parameter $\kappa$ can be absorbed by rescaling
  the diagonal ghost $C^3$ and diagonal anti-ghost $\bar C^3$.
  However, such a rescaling varies the BRST transformation
  and hence the renormalized BRST transformation fails to absorb the divergences
  due to the absence of renormalization of $\kappa$.
This is a reason why we have introduced an extra parameter by keeping the same renormalization factors for the ghost and anti-ghost.
See Appendix, for more details.
Another advantage of this procedure is that the symmetry of the {\it renormalized} theory under the FP conjugation (i.e., the symmetry of the renormalized theory under the exchange of the ghost and anti-ghost) is easily examined for the renormalized theory, since the renormalized ghost and anti-ghost fields are defined through the same renormalization factor.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The minimum choice of the gauge fixing terms} %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
By requiring some of the symmetries listed in the previous section,
that is, charge conjugation,
translational invariance of the diagonal ghost $C^3$
or anti-ghost $\bar C^3$ and implicit $U(1)$ gauge symmetry,
we obtain the minimum choice of the renormalizable MA gauge.
Setting parameters as
\begin{equation}
\alpha=\zeta,\quad
\xi=1,\quad
\mbox{and}\quad
\eta=\chi=\varrho=\varsigma=\omega=\vartheta=\varpi=0,
\end{equation}
we arrive at the gauge fixing terms with three parameters
$\alpha$, $\beta$ and $\kappa$.
\begin{eqnarray}
S_{\rm GF}
 &:=&
    i\int d^4x
     \mbox{\boldmath$\delta$}_{\rm B}
     \bar{\mbox{\boldmath$\delta$}}_{\rm B}
     \left[\frac12A^{\mu a}A_\mu^a
           -\frac\alpha2iC^a\bar C^a\right]
    -i\int d^4x
     \mbox{\boldmath$\delta$}_{\rm B}
     \left\{
     \bar C^3\left[
     \kappa\partial^\mu a_\mu
     +\frac\beta2B^3
    \right]
    \right\}
    \nonumber\\
 &=&\int d^4x
    \biggl\{
    B^aD^\mu A_\mu^b
    +\frac\alpha2B^aB^a
    +i\bar C^aD^2C^b
    -ig^2\epsilon^{ad}\epsilon^{cb}\bar C^aC^bA^{\mu c}A_\mu^d
    \nonumber\\
 & &+ig\epsilon^{ab}\bar C^a(D^\mu A_\mu^b)C^3
    -i\alpha g\epsilon^{ab}B^a\bar C^bC^3
    +\frac\alpha4g^2\epsilon^{ab}\epsilon^{cd}\bar C^a\bar C^bC^cC^d
    \nonumber\\
 & &+\kappa B^3\partial^\mu a_\mu
    +\frac\beta2B^3B^3
    +i\kappa\bar C^3\partial^2C^3
    +i\kappa\bar C^3\partial^\mu(g\epsilon^{bc}A_\mu^bC^c)
    \biggr\}.
\label{eq:S_kappa1}
\end{eqnarray}
By integrating out $B^3$ and $B^a$, we obtain
\begin{eqnarray}
S_{\rm GF}
 &=&\int d^4x
    \biggl\{
    -\frac1{2\alpha}(D^\mu A_\mu^a)^2
    -\frac{1}{2\hat\beta}(\partial^\mu a_\mu)^2
    \nonumber\\
 & &+i\bar C^aD^2C^b
    -ig^2\epsilon^{ad}\epsilon^{cb}\bar C^aC^bA^{\mu c}A_\mu^d
    +\frac\alpha4g^2\epsilon^{ab}\epsilon^{cd}\bar C^a\bar C^bC^cC^d
    \nonumber\\
 & &+i\kappa\bar C^3\partial^2C^3
    +i\kappa\bar C^3\partial^\mu(g\epsilon^{bc}A_\mu^bC^c)
    \biggr\},
\label{eq:S_kappa2}
\end{eqnarray}
where we have defined
\begin{equation}
\hat\beta:=\beta/(\kappa^2),
\end{equation}
for later convenience.
We notice that the diagonal ghost $C^3$ does not appear
in the interaction terms in (\ref{eq:S_kappa2}).
%there does not appear diagonal ghost $C^3$ in the
%gauge fixing term~(\ref{eq:S_kappa2}) except for kinetic term
%due to translational symmetry of $C^3$.
Therefore we do not need to take account of the internal diagonal ghost
in the calculation of perturbative loop expansions.
The anomalous dimensions of the diagonal gluon $a_\mu$, off-diagonal gluon
$A_\mu^a$, coupling constant $g$ and gauge fixing parameters $\alpha$ and
$\hat\beta$ have already been obtained
in previous papers\cite{KS00b,Shinohara01a}.
In this paper, we determine the anomalous dimension
of a remaining parameter $\kappa$,
the ghost field $C$ and anti-ghost field $\bar C$ by making use of
the dimensional regularization at the one-loop level.

From the total action:
\begin{equation}
S:=S_{\rm YM}
   +S_{\rm GF},
\label{eq:S}
\end{equation}
with the Yang-Mills action
\begin{equation}
S_{\rm YM}
 =-\int d^4x\frac14F_{\mu\nu}^AF^{\mu\nu A},
\label{eq:S_YM}
\end{equation}
we obtain the following Feynman rules.

%==========================%
\subsection{Feynman rules} %
%==========================%
%The Feynman rules are given as follows (see Fig.~\ref{fig:Feynman Rules}).
%We give only those rules that are necessary for the renormalization
%at the one-loop level.
%The two-loop result will be given in a subsequent paper.\cite{KS00c}

\unitlength=0.001in
%%%%%%% BEGIN FIGURE (Feynman Rules) %%%%%%%%%%
\begin{figure}[tbp]
\begin{center}
\begin{picture}(5600,2400)%(0,-3000)
%\put(0,2400){\tframe[500][100](5600,2400)}%
%===========%
% Graph (a) %
%===========%
\put(0,2200){\mbox{(a)}}%
\put(550,2100){%
   \put(-150,0){\epsfysize=5mm \epsfbox{apropa.eps}}%
   \put(-300,110){\mbox{$\mu$}}%
   \put(550,110){\mbox{$\nu$}}%
   \put(80,-60){\mbox{$p$}}%
   }%
%===========%
% Graph (b) %
%===========%
\put(1400,2200){\mbox{(b)}}%
\put(1950,2100){%
   \put(0,0){\epsfysize=5mm \epsfbox{apropa.eps}}%
   \put(-280,110){\mbox{$a,\mu$}}%
   \put(700,110){\mbox{$b,\nu$}}%
   \put(230,-50){\mbox{$p$}}%
   }%
%===========%
% Graph (c) %
%===========%
\put(3050,2200){\mbox{(c)}}%
\put(3450,2200){%
   \put(-50,0){\epsfysize=2mm \epsfbox{ghost.eps}}%
%   \put(-150,0){\mbox{$i$}}%
%   \put(630,0){\mbox{$j$}}%
   \put(200,-100){\mbox{$p$}}%
   }%
%===========%
% Graph (d) %
%===========%
\put(4350,2200){\mbox{(d)}}%
\put(4800,2200){%
   \put(-50,0){\epsfysize=2mm \epsfbox{ghost.eps}}%
   \put(-180,0){\mbox{$a$}}%
   \put(630,0){\mbox{$b$}}%
   \put(200,-100){\mbox{$p$}}%
   }%
%===========%
% Graph (e) %
%===========%
\put(0,1700){\mbox{(e)}}%
\put(200,1000){%
   \put(0,0){\epsfysize=20mm \epsfbox{aaa.eps}}%
   \put(150,200){\mbox{$p$}}%
   \put(580,750){\mbox{$q$}}%
   \put(880,150){\mbox{$r$}}%
   \put(100,500){\mbox{$3,\mu$}}%
   \put(860,600){\mbox{$a,\rho$}}%
   \put(470,30){\mbox{$b,\sigma$}}%
   }%
%===========%
% Graph (f) %
%===========%
\put(1450,1700){\mbox{(f)}}%
\put(1700,1000){%
   \put(0,0){\epsfysize=20mm \epsfbox{acc.eps}}%
   \put(570,700){\mbox{$p$}}%
   \put(700,150){\mbox{$q$}}%
   \put(100,500){\mbox{$3,\mu$}}%
   \put(700,600){\mbox{$a$}}%
   \put(570,30){\mbox{$b$}}%
   }%
%===========%
% Graph (g) %
%===========%
\put(2700,1700){\mbox{(g)}}%
\put(3000,1000){%
   \put(0,0){\epsfysize=20mm \epsfbox{acc.eps}}%
   \put(570,700){\mbox{$p$}}%
   \put(700,150){\mbox{$q$}}%
   \put(100,500){\mbox{$c,\mu$}}%
   \put(720,600){\mbox{$3$}}%
   \put(570,30){\mbox{$b$}}%
   }%
%===========%
% Graph (h) %
%===========%
\put(4020,1700){\mbox{(h)}}%
\put(4500,1000){%
   \put(0,0){\epsfysize=20mm \epsfbox{aaaa.eps}}%
   \put(-250,600){\mbox{$3,\mu$}}%
   \put(-250,100){\mbox{$3,\nu$}}%
   \put(730,600){\mbox{$a,\rho$}}%
   \put(730,100){\mbox{$b,\sigma$}}%
   }%
%===========%
% Graph (i) %
%===========%
\put(0,700){\mbox{(i)}}%
\put(400,0){%
   \put(0,0){\epsfysize=20mm \epsfbox{aaaa.eps}}%
   \put(-250,600){\mbox{$a,\mu$}}%
   \put(-250,100){\mbox{$b,\nu$}}%
   \put(730,600){\mbox{$c,\rho$}}%
   \put(730,100){\mbox{$d,\sigma$}}%
   }%
%===========%
% Graph (j) %
%===========%
\put(1500,700){\mbox{(j)}}%
\put(1950,0){%
   \put(0,0){\epsfysize=20mm \epsfbox{aacc.eps}}%
   \put(-250,600){\mbox{$3,\mu$}}%
   \put(-250,100){\mbox{$3,\nu$}}%
   \put(650,600){\mbox{$a$}}%
   \put(650,100){\mbox{$b$}}%
   }%
%===========%
% Graph (k) %
%===========%
\put(2850,700){\mbox{(k)}}%
\put(3300,0){%
   \put(0,0){\epsfysize=20mm \epsfbox{aacc.eps}}%
   \put(-250,600){\mbox{$c,\mu$}}%
   \put(-250,100){\mbox{$d,\nu$}}%
   \put(650,600){\mbox{$a$}}%
   \put(650,100){\mbox{$b$}}%
   }%
%===========%
% Graph (l) %
%===========%
\put(4300,700){\mbox{(l)}}%
\put(4550,0){%
   \put(0,0){\epsfysize=20mm \epsfbox{cccc.eps}}%
   \put(0,580){\mbox{$a$}}%
   \put(0,130){\mbox{$c$}}%
   \put(750,600){\mbox{$b$}}%
   \put(750,100){\mbox{$d$}}%
   }%
\end{picture}
\caption[]{%
    The wavy line corresponds to the gluon,
    and the broken line corresponds to the ghost or anti-ghost.
    The graphs in (a) and (b) represent respectively diagonal and
    off-diagonal gluons while the graphs in (c) and (d) represent
    respectively diagonal and off-diagonal ghosts.
    The graphs in (e), (f) and (g) represent the three-point vertex
    and the graphs in (h), (i), (j), (k) and (l)
    represent the four-point vertex.
}
\label{fig:Feynman Rules}
\end{center}
\end{figure}
%%%%%%% END FIGURE (Feynman Rules) %%%%%%%%%%


%---------------------------%
\subsubsection{Propagators} %
%---------------------------%
\begin{enumerate}
\item[(a)] diagonal gluon propagator:
\begin{equation}
iD_{\mu\nu}
  =-\frac i{p^2}
    \left[g_{\mu\nu}
          -\left(1-\hat\beta\right)
           \frac{p_\mu p_\nu}{p^2}\right].
\label{eq:propagator of diagonal gluon}
\end{equation}

\item[(b)] off-diagonal gluon propagator:
\begin{equation}
iD_{\mu\nu}^{ab}
  =-\frac i{p^2}
    \left[g_{\mu\nu}-(1-\alpha)\frac{p_\mu p_\nu}{p^2}\right]\delta^{ab} .
\label{eq:propagator of off-diagonal gluon}
\end{equation}

\item[(c)] diagonal ghost propagator:
\begin{equation}
i\Delta
  =-\frac1{\kappa p^2}.
\end{equation}

\item[(d)] off-diagonal ghost propagator:
\begin{equation}
i\Delta^{ab}
  =-\frac1{p^2}\delta^{ab}.
\end{equation}
\end{enumerate}

%------------------------------------%
\subsubsection{Three-point vertices} %
%------------------------------------%
\begin{enumerate}
\item[(e)] One diagonal and two off-diagonal gluons:
\begin{eqnarray}
&&i\left<a_\mu(p)A_\rho^a(q)A_\sigma^b(r)\right>_{\rm bare}
  \nonumber\\
&&\textstyle
 =g\epsilon^{ab}
  \left[(q-r)_\mu g_{\rho\sigma}
        +\left\{r-p+\frac q\alpha\right\}_\rho g_{\sigma\mu}
        +\left\{p-q-\frac r\alpha\right\}_\sigma g_{\mu\rho}
  \right].
\end{eqnarray}

\item[(f)] One diagonal gluon, one off-diagonal ghost and one anti-ghost:
\begin{equation}
i\left<\bar C^a(p)C^b(q)a_\mu\right>_{\rm bare}
  =-i(p+q)_\mu g\epsilon^{ab}.
\end{equation}

\item[(g)] One off-diagonal gluon, one off-diagonal ghost and one anti-ghost:
\begin{equation}
i\left<\bar C^3(p)C^b(q)A_\mu^c\right>_{\rm bare}
  =-i\kappa g\epsilon^{cb}p_\mu.
\end{equation}

%\item[(h)] One diagonal gluon, one off-diagonal ghost and one anti-ghost:
%\begin{equation}
%i\left<\bar C^a(p)C^3(q)A_\mu^c\right>_{\rm bare}
%  =0.
%\end{equation}

\end{enumerate}

%-----------------------------------%
\subsubsection{Four-point vertices} %
%-----------------------------------%
\begin{enumerate}
\item[(h)] Two diagonal gluons and two off-diagonal gluons:
\begin{equation}
i\left<A_\mu^3A_\nu^3
       A_\rho^aA_\sigma^b\right>_{\rm bare}
  =-ig^2\delta^{ab}
    \left[2g_{\mu\nu}g_{\rho\sigma}
          -\left(1-\frac1\alpha\right)
          (g_{\mu\rho}g_{\nu\sigma}+g_{\mu\sigma}g_{\nu\rho})\right].
\end{equation}

\item[(i)] Four off-diagonal gluons:
%\begin{eqnarray}
\begin{equation}
%&&\textstyle
  i\left<A_\mu^aA_\nu^bA_\rho^cA_\sigma^d\right>_{\rm bare}
%     \nonumber\\
%&&\textstyle
    =-i2g^2\bigl[\epsilon^{ab}\epsilon^{cd}I_{\mu\nu,\rho\sigma}
            +\epsilon^{ac}\epsilon^{bd}I_{\mu\rho,\nu\sigma}
            +\epsilon^{ad}\epsilon^{bc}I_{\mu\sigma,\nu\rho}],
%&&\textstyle
%    =-i2g^2\bigl[f^{Eab}f^{Ecd}I_{\mu\nu,\rho\sigma}
%            +f^{Eac}f^{Ebd}I_{\mu\rho,\nu\sigma}
%            +f^{Ead}f^{Ebc}I_{\mu\sigma,\nu\rho}],
\end{equation}
%\end{eqnarray}
where
$I_{\mu\nu,\rho\sigma}
  :=(g_{\mu\rho}g_{\nu\sigma}
     -g_{\mu\sigma}g_{\nu\rho})/2
$
.
%and the capital $E$ runs over the diagonal and off-diagonal indices.


\item[(j)] Two diagonal gluons, one off-diagonal ghost and one anti-ghost:
\begin{equation}
i\left<\bar C^aC^bA_\mu^3A_\nu^3\right>_{\rm bare}
  =2g^2\delta^{ab}g_{\mu\nu}.
\end{equation}

\item[(k)] Two diagonal gluons, one off-diagonal ghost and one anti-ghost:
\begin{equation}
i\left<\bar C^aC^bA_\mu^cA_\nu^d\right>_{\rm bare}
  =g^2\left[\epsilon^{ad}\epsilon^{cb}
            +\epsilon^{ac}\epsilon^{db}\right]g_{\mu\nu}.
\end{equation}

\item[(l)] Two diagonal gluons, one off-diagonal ghost and one anti-ghost:
\begin{equation}
i\left<\bar C^a\bar C^bC^cC^d\right>_{\rm bare}
  =ig^2\zeta\epsilon^{ab}\epsilon^{cd}.
\end{equation}

\end{enumerate}



%=========================%
\subsubsection{Counterterms} %
%=========================%
In order to construct the renormalized theory, we define the following
renormalized fields%
\footnote{%
  More generally, the renormalization factors $Z_C$ and $Z_{\bar{C}}$ of $\bar C$ and $C$
  are not necessarily identical to each other, as adopted as in Ref.~\cite{MLP85}.
  However, in our case, it is possible to take $Z_C=Z_{\bar{C}}$, since we have introduced an extra parameter $\kappa$.
  }
and parameters:
\begin{equation}
\begin{array}{c}
\begin{array}{ccc}
% 1st line
a_\mu=Z_a^{1/2}a_{{\rm R}\mu}, &
C^3=Z_c^{1/2}C_{{\rm R}}^3, &
\bar C^3=Z_c^{1/2}\bar C_{{\rm R}}^3, \\[2mm]
% 2nd line
A_\mu^a=Z_A^{1/2}A_{{\rm R}\mu}^a, &
C^a=Z_C^{1/2}C_{{\rm R}}^a, &
\bar C^a=Z_C^{1/2}\bar C_{{\rm R}}^a,
\end{array}\\[6mm]
% 3rd line
g=Z_gg_{\rm R},
\quad
\alpha=Z_\alpha\alpha_{\rm R},
\quad
\hat\beta=Z_{\hat\beta}\hat\beta_{\rm R},
\quad
\kappa=Z_\kappa\kappa_{\rm R}.
\end{array}
\label{eq:renormalization}
\end{equation}
By substituting
the above renormalization relations~(\ref{eq:renormalization})
into the action~(\ref{eq:S}), we obtain
\begin{equation}
S=S_{\rm R}
  +\Delta S_{\rm gauge}
  +\Delta S_{\rm ghost}.
\end{equation}
Here $S_{\rm R}$ is the renormalized action obtained
from the bare action~(\ref{eq:S})
by replacing all the fields and parameters with the renormalized ones,
while $\Delta S_{\rm ghost}$ and $\Delta S_{\rm gauge}$ are
counterterms with and without ghost fields respectively.
In this paper we focus on the renormalizability of the terms
with ghost fields $\Delta S_{\rm ghost}$.
This is explicitly given by
\begin{eqnarray}
\Delta S_{\rm ghost}
 &=&\int d^4x\biggl\{
    i\delta_a
     \bar C_{\rm R}^aD_{\rm R}^2C_{\rm R}^a
    +i\delta_b
     \kappa_{\rm R}\bar C_{\rm R}^3\partial^2C_{\rm R}^3
    \nonumber\\
 & &-i\delta_c
     g_{\rm R}^2\epsilon^{ad}\epsilon^{cb}
     \bar C_{\rm R}^aC_{\rm R}^bA_{\rm R}^{\mu c}A_{{\rm R}\mu}^d
    +\delta_d
     \frac{\alpha_{\rm R}}4g_{\rm R}^2\epsilon^{ab}\epsilon^{cd}
     \bar C_{\rm R}^a\bar C_{\rm R}^bC_{\rm R}^cC_{\rm R}^d
    \nonumber\\
 & &+i\delta_e
     \kappa_{\rm R}\bar C_{\rm R}^3\partial^\mu
     (g_{\rm R}\epsilon^{bc}A_{{\rm R}\mu}^bC_{\rm R}^c)
    \biggr\},
\label{eq:dS_ghost}
\end{eqnarray}
where we have defined the renormalized Abelian covariant derivative
$D_{\rm R}$ by
\begin{equation}
D_{{\rm R}\mu}{\mit\Phi}^a
  :=\left(\partial_\mu\delta^{ab}
         -g_{\rm R}\epsilon^{ab}a_{{\rm R}\mu}\right){\mit\Phi}^b.
\end{equation}
Indeed, substituting the renormalized relations~(\ref{eq:renormalization})
into the definition of the bare Abelian covariant
derivative~(\ref{eq:Abelian covariant derivative}) and using the
relation $Z_g=Z_a^{-1/2}$ due to the implicit residual $U(1)$ gauge
symmetry pointed out in the previous paper\cite{KS00b},
we obtain
\begin{eqnarray}
D_\mu{\mit\Phi}^a
 &=&\left(\partial_\mu\delta^{ab}
          -g\epsilon^{ab}a_\mu\right){\mit\Phi}^b
    \nonumber\\
 &=&\left(\partial_\mu\delta^{AB}
          -Z_gZ_a^{1/2}g_{\rm R}\epsilon^{ab}a_{{\rm R}\mu}\right)
    {\mit\Phi}^b
    \nonumber\\
 &=&\left(\partial_\mu\delta^{AB}
          -g_{\rm R}\epsilon^{ab}a_{{\rm R}\mu}\right){\mit\Phi}^b
    \nonumber\\
 &=&D_{{\rm R}\mu}{\mit\Phi}^a.
\end{eqnarray}
Thus Abelian covariant derivative itself does not change
under the renormalization.

The coefficients
$\delta=(\delta_a,\delta_b,\delta_c,\delta_d,\delta_e)$
in the counter terms~(\ref{eq:dS_ghost}) are related
to the renormalization factors~$Z_X=(Z_c,Z_C,Z_\kappa)$ as
\begin{equation}
\begin{array}{ll}
% 1st line
\delta_a
 =Z_C-1,&
\delta_b
 =Z_cZ_\kappa-1,\\[2mm]
% 2nd line
\delta_c
 =Z_g^2Z_CZ_A-1,&
\delta_d
 =Z_\alpha Z_g^2Z_C^2-1,\\[2mm]
% 3rd line
\delta_e
 =Z_\kappa Z_c^{1/2}Z_g
  Z_A^{1/2}Z_C^{1/2}-1.&
\label{eq:deltas}
\end{array}
\end{equation}
Therefore we can determine the renormalization factors $Z$s
by calculating $\delta$s.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Anomalous dimensions} %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this subsection, we determine the renormalization factors and
anomalous dimensions of the fields and parameters.
The renormalization factor $Z_X$ is expanded
order by order of the loop expansion as
\begin{equation}
Z_X=1+Z_X^{(1)}+Z_X^{(2)}+\cdots,
\end{equation}
where $Z_X^{(n)}$ is the $n$th order contribution.
The anomalous dimension of the respective field $X=Z_XX_{\rm R}$
is defined by
\begin{equation}
\gamma_X
 :=\frac12\mu\frac{\partial}{\partial\mu}\ln Z_X
 :=\frac12\mu\frac{\partial}{\partial\mu}Z_X^{(1)}+\cdots,
\label{eq:def. of anomalous dimension 1}
\end{equation}
and the anomalous dimension of the respective parameter
$Y=Z_YY_{\rm R}$ is defined by
\begin{equation}
\gamma_Y
 :=\mu\frac{\partial}{\partial\mu}Y_{\rm R}
 :=-Y_{\rm R}\mu\frac{\partial}{\partial\mu}Z_Y^{(1)}+\cdots.
\label{eq:def. of anomalous dimension 2}
\end{equation}

The anomalous dimension of the diagonal gluon $a_\mu$
can be determined by requiring the renormalizability for the transverse
part of the propagator of the diagonal gluon.
On the other hand, the anomalous dimension of the Abelian gauge
fixing parameter $\hat\beta$ can be determined by requiring the
renormalizability for the longitudinal part of the propagator
of the diagonal gluon.
Similarly, the anomalous dimensions of the off-diagonal gluons $A_\mu^a$
and the gauge fixing parameter $\alpha$ can be respectively determined
by considering the transverse and longitudinal part of
the propagators of off-diagonal gluons.
Then we can obtain the renormalization factors $Z_a$, $Z_{\hat\beta}$,
$Z_A$ and $Z_\alpha$ by calculating the counterterms $\Delta S_{\rm gauge}$.
Moreover, from the counterterms $\Delta S_{\rm gauge}$
we can calculate also the anomalous dimension of the QCD coupling
constant $g$, that is, the $\beta$-function.
These renormalization parameters have already been calculated
in Ref.~\cite{KS00b,Shinohara01a}.
According to the results of Ref.~\cite{KS00b,Shinohara01a},
the renormalization factors are given as
\begin{equation}
Z_a^{(1)}
 =Z_{\hat\beta}^{(1)}
 =\frac{22}3\frac{(\mu^{-\epsilon}g_{\rm R})^2}{(4\pi)^2\epsilon},
\label{eq:Z_a}
\end{equation}
\begin{equation}
Z_A^{(1)}
 = \frac{(g_{\rm R}\mu^{-\epsilon})^2}{(4\pi)^2\epsilon}
   \left[\frac{17}6-\frac{\alpha_{\rm R}}2-\hat\beta_{\rm R}
   \right],
\label{eq:Z_A}
\end{equation}
\begin{equation}
Z_\alpha^{(1)}
 =\frac{(g_{\rm R}\mu^{-\epsilon})^2}{(4\pi)^2\epsilon}
  \left[\frac43-\alpha_{\rm R}-\frac3{\alpha_{\rm R}}
  \right],
\label{eq:Z_alpha}
\end{equation}
\begin{equation}
Z_g^{(1)}
 =-\frac12Z_A^{(1)}
 =-\frac{11}3\frac{(\mu^{-\epsilon}g_{\rm R})^2}{(4\pi)^2\epsilon}.
\label{eq:Z_g}
\end{equation}

In this paper, we determine the remaining renormalization factors,
$Z_c$, $Z_C$ and $Z_\kappa$,
by making use of the dimensional regularization.
In order to determine these three factors,
we must calculate three independent coefficients
$\delta$s in Eqs.~(\ref{eq:deltas}).
For instance, $Z_C$ is obtained by calculating $\delta_a$
in Eq.~(\ref{eq:deltas}).
By calculating $\delta_b$, we obtain a relation of $Z_c$ and $Z_\kappa$.
One more relation is obtained by calculating $\delta_e$.
In the actual calculations,
it is useful to remember the fact that the diagonal ghost
does not appear in the internal line.

%======================================================%
%\subsubsection{Quantum contribution to ghost propagators} %
%======================================================%
First, we consider the quantum correction to the propagator of
the diagonal ghost.
There is no divergent graph for the diagonal ghost propagator
in the dimensional regularization,
so that we immediately obtain a relation
between $Z_c^{(1)}$ and $Z_\kappa^{(1)}$:
\begin{equation}
\delta_b
 =Z_c^{(1)}+Z_\kappa^{(1)}
 =0.
\label{eq:relation1}
\end{equation}

Next, we consider the quantum correction to propagators of
the off-diagonal ghosts.
The divergent graphs for the propagator of the off-diagonal ghost
are enumerated in Fig.~{\ref{fig:off-diagonal ghost propagators}}.
%%%%%%% BEGIN FIGURE (Feynman Rules) %%%%%%%%%%
\begin{figure}[tbp]
\begin{center}
\begin{picture}(5400,800)%(0,-3000)
%\put(0,800){\tframe[500][100](5400,800)}%
%============%
% Graph (a1) %
%============%
\put(0,650){\mbox{(a1)}}%
\put(0,0){%
   \put(0,0){\epsfxsize=30mm \epsfbox{cc1.eps}}%
   }%
%============%
% Graph (a2) %
%============%
\put(1400,650){\mbox{(a2)}}%
\put(1400,0){%
   \put(0,0){\epsfxsize=30mm \epsfbox{cc2.eps}}%
   \put(550,470){\mbox{3}}%
   }%
%============%
% Graph (a3) %
%============%
\put(2800,650){\mbox{(a3)}}%
\put(2800,0){%
   \put(0,0){\epsfxsize=30mm \epsfbox{cc2.eps}}%
   }%
%============%
% Graph (a4) %
%============%
\put(4200,650){\mbox{(a4)}}%
\put(4200,0){%
   \put(0,0){\epsfxsize=30mm \epsfbox{cc3.eps}}%
   }%
\end{picture}
\caption[]{%
   The graphs corresponding to one-loop radiative corrections for the propagator of the off-diagonal ghost.
   The wavy line labeled by 3 represents the diagonal gluon
   while the wavy line without any label represents the off-diagonal gluon.
   Similarly the broken line with no label represents the
   off-diagonal ghost or anti-ghost.
}
\label{fig:off-diagonal ghost propagators}
\end{center}
\end{figure}
%%%%%%% END FIGURE (Feynman Rules) %%%%%%%%%%
Non-trivial contribution is given by only one graph~(a1).
Thus, by making use of the dimensional regularization,
$\delta_a$ or $Z_C^{(1)}$ is obtained as
\begin{equation}
\delta_a=Z_C^{(1)}
 =\frac{(g_{\rm R}\mu^{-\epsilon})^2}{(4\pi)^2\epsilon}
  (3-\hat\beta_{\rm R}),
\label{eq:Z_C}
\end{equation}
where $\epsilon$ is defined as $\epsilon:=(4-d)/2$.

%===================================================================%
%\subsection{Quantum contribution to $\bar C^3$-$C^a$-$A_\mu^b$ vertex} %
%===================================================================%
In order to determine $\delta_e$ we calculate the quantum correction
to the three point vertex of one diagonal anti-ghost,
one off-diagonal ghost and one off-diagonal gluon.
The divergent graphs for this vertex are collected
in Fig.~{\ref{fig:3 point vertex}}.
%%%%%%% BEGIN FIGURE (Feynman Rules) %%%%%%%%%%
\begin{figure}[tbp]
\begin{center}
\begin{picture}(5300,1000)%(0,-3000)
%\put(0,1000){\tframe[500][100](5300,1000)}%
%============%
% Graph (b1) %
%============%
\put(0,850){\mbox{(b1)}}%
\put(0,0){%
   \put(0,0){\epsfxsize=30mm \epsfbox{acc3.eps}}%
   \put(970,870){\mbox{3}}%
   }%
%============%
% Graph (b2) %
%============%
\put(1400,850){\mbox{(b2)}}%
\put(1400,0){%
   \put(0,0){\epsfxsize=30mm \epsfbox{acc2.eps}}%
   \put(900,870){\mbox{3}}%
   }%
%============%
% Graph (b3) %
%============%
\put(2800,850){\mbox{(b3)}}%
\put(2800,0){%
   \put(0,0){\epsfxsize=30mm \epsfbox{acc1.eps}}%
   \put(500,870){\mbox{3}}%
   \put(970,870){\mbox{3}}%
   }%
%============%
% Graph (b4) %
%============%
\put(4200,850){\mbox{(b4)}}%
\put(4200,0){%
   \put(0,0){\epsfysize=1in \epsfbox{acc4.eps}}%
   \put(970,770){\mbox{3}}%
   \put(370,170){\mbox{3}}%
   }%
\end{picture}
\caption[]{%
   The  graphs corresponding to one-loop radiative corrections for the three point vertex of
   one diagonal anti-ghost, one off-diagonal ghost
   and one off-diagonal gluon.
   The wavy line and the broken line labeled by 3
   represent the diagonal gluon and diagonal ghost respectively,
   while the line without any label represents the off-diagonal gluon
   or off-diagonal ghost.
}
\label{fig:3 point vertex}
\end{center}
\end{figure}
%%%%%%% END FIGURE (Feynman Rules) %%%%%%%%%%
Then we obtain
\begin{eqnarray}
\delta_e
 &=&Z_\kappa^{(1)}
    +\frac12Z_c^{(1)}
    +Z_g^{(1)}
    +\frac12Z_A^{(1)}
    +\frac12Z_C^{(1)}
    \nonumber\\
 &=&\frac{(g_{\rm R}\mu^{-\epsilon})^2}{(4\pi)^2\epsilon}
    \left[\hat\beta_{\rm R}+\frac94+\frac34\alpha_{\rm R}\right],
\end{eqnarray}
or, by solving with respect to $Z_\kappa^{(1)}$, we also obtain
\begin{equation}
Z_\kappa^{(1)}
 =-Z_c^{(1)}
 =-(3+\alpha_{\rm R})
   \frac{(g_{\rm R}\mu^{-\epsilon})^2}{(4\pi)^2\epsilon},
\label{eq:Z_kappa}
\end{equation}
where we have made use of Eqs.~(\ref{eq:relation1}), (\ref{eq:Z_A}),
(\ref{eq:Z_g}) and (\ref{eq:Z_C}).

%=================================%
%\subsection{Anomalous dimensions} %
%=================================%
Substituting one loop renormalization factors $Z_X^{(1)}$
and $Z_Y^{(1)}$ into the definitions of
the anomalous dimension~(\ref{eq:def. of anomalous dimension 1})
or (\ref{eq:def. of anomalous dimension 2}), we obtain the following
anomalous dimensions:
\begin{equation}
\gamma_a(g_{\rm R})
 =-\frac{22}3\frac{g_{\rm R}^2}{(4\pi)^2},
\end{equation}
\begin{equation}
\gamma_A(g_{\rm R})
 =-\frac{g_{\rm R}^2}{(4\pi)^2}\frac12
   \left[\frac{17}6-\frac{\alpha_{\rm R}}2-\hat\beta_{\rm R}
   \right],
\end{equation}
\begin{equation}
\gamma_{\hat\beta}(g_{\rm R})
 =\frac{44}3\hat\beta_{\rm R}\frac{g_{\rm R}^2}{(4\pi)^2},
\end{equation}
\begin{equation}
\gamma_\alpha(g_{\rm R})
 =\frac{g_{\rm R}^2}{(4\pi)^2}
  \left[\frac83\alpha_{\rm R}-2\alpha_{\rm R}^2-6
  \right],
\end{equation}
\begin{equation}
\beta(g_{\rm R})
 =\gamma_g(g_{\rm R})
  =-\frac{22}3\frac{g_{\rm R}^3}{(4\pi)^2},
\end{equation}
\begin{equation}
\gamma_c(g_{\rm R})
 =-(3+\alpha_{\rm R})
   \frac{g_{\rm R}^2}{(4\pi)^2},
\end{equation}
\begin{equation}
\gamma_C(g_{\rm R})
 =-\frac{g_{\rm R}^2}{(4\pi)^2}
  (3-\hat\beta_{\rm R}),
\end{equation}
\begin{equation}
\gamma_\kappa(g_{\rm R})
 =-2(3+\alpha_{\rm R})\kappa_{\rm R}
   \frac{g_{\rm R}^2}{(4\pi)^2}.
\end{equation}
Thus we obtain  anomalous dimensions of all the fields and parameters
at the one-loop level of perturbative expansion based on the dimensional
regularization.
From the anomalous dimensions just calculated,
we find a remarkable feature
that the parameter $\kappa_{\rm R}$ is necessary,
although it does appear only in $\gamma_\kappa$.
By the explicit calculations given above,
it turns out that there is no fixed point of $\kappa$
except for at $\kappa=0$ where the gauge fixing term for residual
$U(1)$ gauge symmetry is not well-defined.
Therefore the necessity of the existence of the parameter $\kappa$
has been overlooked so far, except for the work \cite{MLP85} where the different renormalization factors are introduced for the ghost and anti-ghost as another option of avoiding this issue.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion and discussion} %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this paper, we have pointed out the necessity of a new parameter
  $\kappa$ in the MA gauge from the viewpoint of renormalizability.
By including this parameter $\kappa$,
the Yang-Mills theory in the  modified  MA gauge
  (proposed in the previous paper\cite{KondoII,KS00a})
  becomes renormalizable in the exact sense.
Making use of the dimensional regularization, we have calculated the
  anomalous dimension of $\kappa$ and the other parameters
  at the one-loop level in the minimum case.
Then, we have confirmed that the renormalizability of the theory
  is maintained.
By the reason that $\kappa_{\rm R}$ does not appear in
  the anomalous dimensions other than $\gamma_\kappa$,
  the investigations so far missed a possibility of introducing an extra parameter $\kappa$.

We have required several symmetries in order to
  restrict the parameter space and to make clear the efficiency of introducing $\kappa$.
One of purposes of this paper is the confirmation of the meaning
  of $\kappa$.
We can expect that the renormalizability is not spoiled
  in a subspace of the parameters protected by some symmetries.
Thus we have required some symmetries to simplify the theory.
However we do not know any symmetry which forces the value of a parameter
  $\kappa$ to be 1 without spoiling the renormalizability.
Indeed, by this restriction of the parameter space, we have found
  that at least three independent parameters $\alpha$, $\hat\beta$
  and $\kappa$ are necessary and sufficient
  to maintain the renormalizability.
It should be remarked that the partially gauge fixing part
  for off-diagonal gluons is identical to
  the  modified  MA gauge.

However we should not impose so many restrictions.
For instance, the ordinary Faddeev-Popov term
  $i\partial^\mu\bar C^A{\cal D}_\mu C^A$ is not invariant under
  the translation of the diagonal ghost,
  while it is invariant under
  the translation of the diagonal anti-ghost.
Thus, if we would like to compare the MA gauge
  with the ordinary Lorentz gauge,
  we should not require the translational symmetry for the diagonal ghost.

Similarly, in the low-energy region of QCD in the  modified MA gauge,
  we expect that the ghost--anti-ghost composite operators
  $\epsilon^{ab}C^a\bar C^b$ and $C^a\bar C^a$ have non-trivial
  expectation values due to the condensation and hence
  the charge conjugation symmetry breaks down.\cite{KS00a,Schaden}
Therefore we must not require the charge conjugation symmetry from this viewpoint.
The results of investigations to remedy these shortcomings
will be reported elsewhere.






%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements} %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The authors would like to thank Takeharu Murakami for helpful discussion.





\appendix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Rescaling of the fields preserving BRST transformation
         and its connection to the renormalization}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
First, we consider the gauge fixing term
with global $SU(2)$ gauge symmetry given by
\begin{equation}
{\cal L}_{\rm GF}
 =-i\gamma
   \mbox{\boldmath$\delta$}_{\rm B}
     \left[\bar{\cal C}^A
           \left(\partial^\mu{\cal A}_\mu^A
                 +\frac\alpha2{\cal B}^A
                 -\zeta gf^{ABC}\bar{\cal C}^B{\cal C}^C
                 \right)\right],
\label{eq:GF with SU2}
\end{equation}
where $\gamma$, $\alpha$ and $\zeta$ are arbitrary parameters and
$\mbox{\boldmath$\delta$}_{\rm B}$ is the BRST transformation:
\begin{equation}
\begin{array}{l}
% 1st line
\mbox{\boldmath$\delta$}_{\rm B}{\cal A}_\mu^A
 ={\cal D}_\mu{\cal C}^A
 =\partial_\mu{\cal C}^A
  +gf^{ABC}{\cal A}_\mu^B{\cal C}^C,\\[1mm]
% 2nd line
\displaystyle
\mbox{\boldmath$\delta$}_{\rm B}{\cal C}^A
 =-\frac g2f^{ABC}{\cal C}^B{\cal C}^C,\\[3mm]
% 3rd line
\mbox{\boldmath$\delta$}_{\rm B}\bar{\cal C}^A
 =i{\cal B}^A,\\[1mm]
% 4th line
\mbox{\boldmath$\delta$}_{\rm B}{\cal B}^A
 =0.
\end{array}
\label{eq:BRST}
\end{equation}
Because of the nilpotency of the BRST transformation,
it is trivial that (\ref{eq:GF with SU2}) is invariant
under the transformation:
\begin{equation}
\delta{\cal A}_\mu^A
 =\epsilon\mbox{\boldmath$\delta$}_{\rm B}
  {\cal A}_\mu^A,
\quad
\delta{\cal C}^A
 =\epsilon\mbox{\boldmath$\delta$}_{\rm B}
  {\cal C}^A,
\quad
\delta\bar{\cal C}^A
 =\epsilon\mbox{\boldmath$\delta$}_{\rm B}
  \bar{\cal C}^A,
\quad
\delta{\cal B}^A
 =\epsilon\mbox{\boldmath$\delta$}_{\rm B}
  {\cal B}^A,
\end{equation}
for arbitrary Grassmann parameter $\epsilon$.

Now we show that the parameter $\gamma$ can be set equal to 1
without losing generality.
It turns out that such a parameter can be absorbed by rescaling the fields
${\cal C}^A$ and $\bar{\cal C}^A$.
Indeed, by rescaling the fields as
\begin{equation}
{\cal A}_\mu^\prime
  =x{\cal A}_\mu,
\quad
{\cal B}_\mu^\prime
  =y{\cal B}_\mu,
\quad
{\cal C}^\prime
  =u{\cal C},
\quad
\bar{\cal C}^\prime
  =v\bar{\cal C},
\label{eq:rescaled fields}
\end{equation}
the BRST transformation (\ref{eq:BRST}) is rewritten as
\begin{equation}
\begin{array}{l}
% 1st line
\mbox{\boldmath$\delta$}_{\rm B}{\cal A}_\mu^{\prime A}
 =\left(\frac xu\right)\partial_\mu{\cal C}^{\prime A}
  +\left(\frac1u\right)
   gf^{ABC}{\cal A}_\mu^{\prime B}{\cal C}^{\prime C},\\[2mm]
% 2nd line
\mbox{\boldmath$\delta$}_{\rm B}{\cal C}^{\prime A}
 =-\left(\frac1u\right)
   \frac12gf^{ABC}{\cal C}^{\prime B}{\cal C}^{\prime C},\\[2mm]
% 3rd line
\mbox{\boldmath$\delta$}_{\rm B}\bar{\cal C}^{\prime A}
 =i\left(\frac vy\right){\cal B}^{\prime A},\\[2mm]
% 4th line
\mbox{\boldmath$\delta$}_{\rm B}{\cal B}^{\prime A}
 =0.
\end{array}
\end{equation}
If two conditions $x=1$ and $y=uv$ are satisfied,
the same form of the BRST transformation
as the original BRST transformation~(\ref{eq:BRST}) is obtained
for the rescaled field~(\ref{eq:rescaled fields})
by defining a new BRST transformation
$\mbox{\boldmath$\delta$}_{\rm B}^\prime
  :=u\mbox{\boldmath$\delta$}_{\rm B}$
as
\begin{equation}
\begin{array}{l}
% 1st line
\mbox{\boldmath$\delta$}_{\rm B}^\prime{\cal A}_\mu^{\prime A}
 =\partial_\mu{\cal C}^{\prime A}
  +gf^{ABC}{\cal A}_\mu^{\prime B}{\cal C}^{\prime C},\\[2mm]
% 2nd line
\mbox{\boldmath$\delta$}_{\rm B}^\prime{\cal C}^{\prime A}
 =-\frac g2f^{ABC}{\cal C}^{\prime B}{\cal C}^{\prime C},\\[2mm]
% 3rd line
\mbox{\boldmath$\delta$}_{\rm B}^\prime\bar{\cal C}^{\prime A}
 =i{\cal B}^{\prime A},\\[2mm]
% 4th line
\mbox{\boldmath$\delta$}_{\rm B}^\prime{\cal B}^{\prime A}
 =0.
\end{array}
\end{equation}
%It is equivalent to the original BRST transformation~(\ref{eq:BRST}).
Then the Lagrangian~(\ref{eq:GF with SU2}) is rewritten as
\begin{equation}
{\cal L}_{\rm GF}
 =-i\gamma^\prime
   \mbox{\boldmath$\delta$}_{\rm B}^\prime
     \left[\bar{\cal C}^{\prime A}
           \left(\partial^\mu{\cal A}_\mu^{\prime A}
                 +\frac{\alpha^\prime}2{\cal B}^{\prime A}
                 -\zeta^\prime
                  gf^{ABC}\bar{\cal C}^{\prime B}{\cal C}^{\prime C}
                 \right)\right],
\end{equation}
where $\gamma^\prime:=\gamma/y$, $\alpha^\prime:=\alpha/y$ and
$\zeta^\prime:=\zeta/y$.
Therefore we can set $\gamma^\prime$ to 1 by requiring a condition
$y=\gamma$.

Here, we have introduced four rescaling parameters ($x$, $y$, $u$ and $v$)
and imposed three conditions ($x=1$, $y=uv$ and $\gamma=y$).
Then, we can require one more condition.
We consider the renormalization of fields and parameter as
\begin{equation}
{\cal C}=Z_C^{1/2}{\cal C}_{\rm R},\quad
\bar{\cal C}=Z_{\bar C}^{1/2}\bar{\cal C}_{\rm R},\quad
u=Z_uu_{\rm R},\quad
v=Z_vv_{\rm R},
\label{eq:renormalization1}
\end{equation}
while
\begin{equation}
{\cal C}^\prime=Z_{C^\prime}^{1/2}{\cal C}_{\rm R}^\prime,\quad
\bar{\cal C}^\prime=Z_{\bar C^\prime}^{1/2}\bar{\cal C}_{\rm R}^\prime.
\label{eq:renormalization2}
\end{equation}
In general, it is not necessary that the renormalization factors
$Z_C$ and $Z_{\bar C}$
(similarly $Z_{C^\prime}$ and $Z_{\bar C^\prime}$) are equivalent.
However, substituting~(\ref{eq:renormalization1})
and  (\ref{eq:renormalization2}) into (\ref{eq:rescaled fields}),
we have the relations:
\begin{equation}
Z_{C}^{1/2}Z_u{\cal C}_{\rm R}u_{\rm R}
 =Z_{C^\prime}^{1/2}{\cal C}_{\rm R}^\prime
\quad\mbox{and}\quad
Z_{\bar C}^{1/2}Z_v\bar{\cal C}_{\rm R}v_{\rm R}
 =Z_{\bar C^\prime}^{1/2}\bar{\cal C}_{\rm R}^\prime,
\end{equation}
and we can require the relation $Z_{C^\prime}=Z_{\bar C^\prime}$
by taking $u$ and $v$ appropriately.
Therefore we adopt it as the last condition.

Next, we consider the case of the gauge fixing term with global U(1)
gauge symmetry alone in $SU(2)$ Yang-Mills theory.
Such a term is given by
\begin{equation}
{\cal L}_{\rm GF}
 :=i\gamma\partial^\mu\bar C^a\partial_\mu C^a
   +i\kappa\partial^\mu\bar C^3\partial_\mu C^3
   +\cdots,
\end{equation}
where ``$\cdots$'' denotes the interaction terms given
in section~\ref{sec:The most general gauge fixing terms}.
Decomposing the BRST transformation~(\ref{eq:BRST}) into
diagonal and off-diagonal components explicitly, we obtain
\begin{equation}
\begin{array}{ll}
% a
\mbox{\boldmath$\delta$}_{\rm B}a_\mu
 =\partial_\mu C^3
    +g\epsilon^{ab}A_\mu^aC^b,
% A
&\mbox{\boldmath$\delta$}_{\rm B}A_\mu^a
 =\partial_\mu C^a
    +g\epsilon^{ab}A_\mu^bC^3
    -g\epsilon^{ab}a_\mu C^b,\\[2mm]
% C^3
%\displaystyle
\mbox{\boldmath$\delta$}_{\rm B}C^3
 =-\frac12g\epsilon^{ab}C^aC^b,
% C^a
&\mbox{\boldmath$\delta$}_{\rm B}C^a
 =-g\epsilon^{ab}C^bC^3,\\[2mm]
% $\bar C^3$
\mbox{\boldmath$\delta$}_{\rm B}\bar C^3
 =iB^3,
% $\bar C^a$
&\mbox{\boldmath$\delta$}_{\rm B}\bar C^a
 =iB^a,\\[1mm]
% B^3
\mbox{\boldmath$\delta$}_{\rm B}B^3
 =0,
% B^a
&\mbox{\boldmath$\delta$}_{\rm B}B^a
 =0.
\end{array}
\label{eq:BRST U1}
\end{equation}
After rescaling the fields as
\begin{equation}
\begin{array}{llll}
a_\mu^\prime=ka_\mu,
&B^{\prime3}=lB^3,
&C^{\prime3}=mC^3,
&\bar C^{\prime3}=n\bar C^3,\\[2mm]
A_\mu^{\prime a}=xA_\mu^a,
&B^{\prime a}=yB^a,
&C^{\prime a}=uC^a,
&\bar C^{\prime a}=v\bar C^a,
\end{array}
\end{equation}
the BRST transformation is rewritten as
\begin{equation}
\begin{array}{l}
\ \mbox{\boldmath$\delta$}_{\rm B}a_\mu^\prime
 =\left(\frac km\right)
  \partial_\mu C^{\prime3}
  +\left(\frac k{xu}\right)
   g\epsilon^{ab}A_\mu^{\prime a}C^{\prime b},\\[2mm]
% A
\ \mbox{\boldmath$\delta$}_{\rm B}A_\mu^{\prime a}
 =\left(\frac xu\right)
  \partial_\mu C^{\prime a}
  +\left(\frac1m\right)
   g\epsilon^{ab}A_\mu^{\prime b}C^{\prime3}
  -\left(\frac x{ku}\right)
   g\epsilon^{ab}a_\mu^\prime C^{\prime b},\\[2mm]
\begin{array}{ll}
% C^3
\mbox{\boldmath$\delta$}_{\rm B}C^{\prime3}
 =-\left(\frac m{u^2}\right)
   \frac12g\epsilon^{ab}C^{\prime a}C^{\prime b},
% C^a
&\mbox{\boldmath$\delta$}_{\rm B}C^{\prime a}
 =-\left(\frac1m\right)
   g\epsilon^{ab}C^{\prime b}C^{\prime3},\\[2mm]
% $\bar C^3$
\mbox{\boldmath$\delta$}_{\rm B}\bar C^{\prime3}
 =\left(\frac nl\right)
   iB^{\prime3},
% $\bar C^a$
&\mbox{\boldmath$\delta$}_{\rm B}\bar C^{\prime a}
 =\left(\frac vy\right)
   iB^{\prime a},\\[1mm]
% B^3
\mbox{\boldmath$\delta$}_{\rm B}B^{\prime3}
 =0,
% B^a
&\mbox{\boldmath$\delta$}_{\rm B}B^{\prime a}
 =0.
\end{array}
\end{array}
\end{equation}
Similarly to the previous case, imposing the conditions:
\begin{equation}
\frac km
 =\frac k{xu}
 =\frac xu
 =\frac1m
 =\frac x{ku}
 =\frac m{u^2}
 =\frac nl
 =\frac vy,
\end{equation}
or
\begin{equation}
k=1,\quad
x^2=1,\quad
m^2=u^2,\quad
l=mn,\quad
y=uv,
\label{eq:5conditions}
\end{equation}
we can obtain the some BRST transformation as (\ref{eq:BRST U1})
for the rescaled fields:
\begin{equation}
\begin{array}{ll}
% a
\mbox{\boldmath$\delta$}_{\rm B}^\prime a_\mu^\prime
 =\partial_\mu C^{\prime3}
    +g\epsilon^{ab}A_\mu^{\prime a}C^{\prime b},
% A
&\mbox{\boldmath$\delta$}_{\rm B}^\prime A_\mu^{\prime a}
 =\partial_\mu C^{\prime a}
    +g\epsilon^{ab}A_\mu^{\prime b}C^{\prime3}
    -g\epsilon^{ab}a_\mu^\prime C^{\prime b},\\[2mm]
% C^3
%\displaystyle
\mbox{\boldmath$\delta$}_{\rm B}^\prime C^{\prime3}
 =-\frac12g\epsilon^{ab}C^{\prime a}C^{\prime b},
% C^a
&\mbox{\boldmath$\delta$}_{\rm B}^\prime C^{\prime a}
 =-g\epsilon^{ab}C^{\prime b}C^{\prime3},\\[2mm]
% $\bar C^3$
\mbox{\boldmath$\delta$}_{\rm B}^\prime\bar C^{\prime3}
 =iB^{\prime3},
% $\bar C^a$
&\mbox{\boldmath$\delta$}_{\rm B}^\prime\bar C^{\prime a}
 =iB^{\prime a},\\[1mm]
% B^3
\mbox{\boldmath$\delta$}_{\rm B}^\prime B^{\prime3}
 =0,
% B^a
&\mbox{\boldmath$\delta$}_{\rm B}^\prime B^{\prime a}
 =0,
\end{array}
\end{equation}
where we have defined the new BRST transformation
$\mbox{\boldmath$\delta$}_{\rm B}^\prime$ as
$\mbox{\boldmath$\delta$}_{\rm B}^\prime
  :=m\mbox{\boldmath$\delta$}_{\rm B}$
by using the rescaling factor $m$ of the diagonal ghost $C^3$.

Here, we have introduced five conditions~(\ref{eq:5conditions})
for eight parameters ($k$, $l$, $m$, $n$, $x$, $y$, $u$, $v$).
Therefore, we can impose three more conditions.
Note that there are four options:
\begin{enumerate}
%-----------
\item[(i)]
An absorption of a parameter $\gamma$,

%-----------
\item[(ii)]
An absorption of a parameter $\kappa$,

%-----------
\item[(iii)]
An equivalence of renormalization factor of $C^a$ and $\bar C^a$,

%-----------
\item[(iv)]
An equivalence of renormalization factor of $C^3$ and $\bar C^3$.

\end{enumerate}
Since we can impose only three conditions,
one of the four options is never satisfied.
It is possible to discard a condition (iii) or (iv)
as done in Refs.~\cite{MLP85} and \cite{HN93}.
However, in order to deal with the ghost and anti-ghost on equal footing,
for example, FP conjugation or BRST--anti-BRST field formalism,
it is useful to retain the parameter $\gamma$ or $\kappa$
as we have adopted in this paper.

Thus we can restrict the gauge fixing terms with global $U(1)$
without losing generality to
\begin{equation}
{\cal L}_{\rm GF}
 :=i\partial^\mu\bar C^a\partial_\mu C^a
   +i\kappa\partial^\mu\bar C^3\partial_\mu C^3
   +\cdots,
\end{equation}
where the renormalization factors of $\bar C^a$ and $C^a$ are
identical to each other and this is also the case for $\bar C^3$ and $C^3$.
It is remarkable that the parameter $\kappa$ (or $\gamma$)
cannot be absorbed.













































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\end{document}
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