% CHIBA-EP-127 version 0.582
\documentstyle[letter,epsf]{ptptex}

\markboth%
%  {CHIBA-EP-127}%
  {Renormalizable Abelian-Projected Effective Gauge Theory II}
  {T.~Shinohara}

\title{%
  Renormalizable Abelian-Projected Effective Gauge Theory\\
  Derived from Quantum Chromodynamics II}

\author{%
  Toru S{\sc hinohara}%$^{1,}$%
    \footnote{E-mail: {\tt sinohara@cuphd.nd.chiba-u.ac.jp}}
  }

\inst{%
  %${}^1$
  Graduate School of Science and Technology, Chiba University,
         Chiba 263-8522 Japan
  }

\abst{%
In the previous paper,\cite{KS00b}
we have introduced the Abelian tensor field
together with two parameters $\rho$ and $\sigma$
to rewrite the non-Abelian gauge theory in maximal Abelian gauge
into an Abelian effective theory, and we determined the anomalous
dimensions of Abelian tensor field and a parameter $\rho$
by requiring the renormalizability
with respect to Abelian effective theory.
Then we obtained the renormalizable Abelian Projected Effective Gauge
Theory.
%
In this paper, the anomalous dimension of a parameter $\sigma$
is determined.
However, before proceeding to determine the anomalous dimension
of $\sigma$, we must modify the gauge fixing terms from the viewpoint of
renormalizability.
We obtain the anomalous dimensions under this modification.
}


\begin{document}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Introduction} %
%%%%%%%%%%%%%%%%%%%%%%%%

In the previous paper,\cite{KS00b} which is referred as (I) hereafter,
we have obtained Abelian Projected Effective Gauge Theory (APEGT)
by requiring the renormalizability of Green functions
with external Abelian legs.
Then we have obtained the anomalous dimensions of Abelian gluon field,
Abelian tensor field and a parameter $\rho$.
We have also obtained $\beta$-function that is exactly identical to that of
the ordinary Yang-Mills theory in Lorentz gauge by making use of an idea of
background field method.\cite{Abbott82}
To determine the anomalous dimension of parameter $\sigma$, we must
require the renormalizability of Green function with not only
Abelian external legs but also off-diagonal legs
since the parameter $\sigma$ always appears with off-diagonal gluons.

In the maximal (MA) gauge, we have an interaction term of
one off-diagonal ghost, one off-diagonal anti-ghost and
two off-diagonal gluons in the bare Lagrangian.
Due to existence of such an interaction term, undesirable divergent
contributions mentioned in \S 3 in (I) must be induced as quantum effect.
Thus a modified MA gauge was proposed by including the quartic off-diagonal
ghost and anti-ghost self interaction.\cite{KondoII,KS00a}
Moreover, in this paper, we modify the gauge fixing terms
for the residual Abelian gauge symmetry.
The most general gauge fixing terms with global $U(1)$ symmetry
in $SU(2)$ is discussed in Ref.~\citen{SIK01a}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Action and Feynman rules} %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%===================%
%\subsection{Action} %
%===================%
The action is given as
\begin{equation}
S=S_{\rm inv}
  +S_{\rm mMA}
  +S_{\rm Abel},
%  +S_{\rm ex}
\label{eq:S}
\end{equation}
where the first term $S_{\rm inv}$ is the gauge invariant part and
the other terms are gauge fixing parts.
By introducing an anti-symmetric Abelian tensor field $B_{\mu\nu}^i$
in the $SU(N)$ Yang-Mills action
\begin{equation}
S_{\rm YM}
  :=-\int d^4x\frac14F_{\mu\nu}^AF^{\mu\nu A},
\label{eq:S_YM}
\end{equation}
the gauge invariant part is given by
\begin{eqnarray}
S_{\rm inv}
 &=&\int d^4x\Bigl\{
    -\frac14\left[D_\mu A_\nu^a
                  -D_\nu A_\mu^a
                  +gf^{abc}A_\mu^bA_\nu^c\right]^2
    \nonumber\\
 & &-\frac{1-\rho^2}4\left(f_{\mu\nu}^i\right)^2
    -\frac{1-\rho\sigma}2gf_{\mu\nu}^if^{ibc}A^{\mu b}A^{\nu c}
    -\frac{1-\sigma^2}4g^2\left(f^{ibc}A_\mu^bA_\nu^c\right)^2
    \nonumber\\
 & &-\frac14\left(B_{\mu\nu}^i\right)^2
    -\frac14\epsilon^{\mu\nu\rho\sigma}B_{\mu\nu}^i
     \left[\rho f_{\mu\nu}^i+\sigma gf^{ibc}A_\mu^bA_\nu^c\right]
     \Bigr\},
\label{eq:S_inv}
\end{eqnarray}
where $\rho$ and $\sigma$ are arbitrary parameter.
By integrating Eq.~(\ref{eq:S_inv}) over $B_{\mu\nu}^i$, we can
reproduce the ordinary Yang-Mills action~(\ref{eq:S_YM}).

The gauge fixing parts $S_{\rm mMA}$ and $S_{\rm Abel}$ are given by
\begin{equation}
S_{\rm mMA}
 =\int d^4xi
    \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],
\label{eq:S_mMA}
\end{equation}
and
\begin{equation}
S_{\rm Abel}
 =-\int d^4x
     i\mbox{\boldmath$\delta$}_{\rm B}
     \left[\bar C^i
           \left(\kappa\partial^\mu A_\mu^i
                 +\frac\beta2B^i\right)\right],
\label{eq:S_Abel}
\end{equation}
where $\mbox{\boldmath$\delta$}_{\rm B}$ and
$\bar{\mbox{\boldmath$\delta$}}_{\rm B}$ are respectively BRST and
anti-BRST transformation.
The modified MA gauge fixing term $S_{\rm mMA}$ proposed
in Ref.~\citen{KondoII,KS00a} partially fixes
the gauge symmetry from $SU(N)$ to $U(1)^{N-1}$.
In addition, $S_{\rm Abel}$ fixes the residual $U(1)^{N-1}$ gauge symmetry.
We introduce a new parameter $\kappa$ in order to maintain the
renormalizability of Green functions with off-diagonal fields.
The necessity of such a parameter is discussed in Ref.~\citen{SIK01a}.
Of course, we can adopt more general gauge fixing terms than
Eqs.~(\ref{eq:S_mMA}) and (\ref{eq:S_Abel}).
The most general gauge fixing terms in $SU(2)$ obtained by requiring
the global $U(1)$ symmetry is also mentioned in Ref.~\citen{SIK01a}.

Integrating out $B^i$ and $B^a$, we begin with the total gauge fixing
terms
\begin{eqnarray}
S_{\rm GF}
 &=&\int d^4x\Bigl\{
    %-\frac14F^{\mu\nu A}F_{\mu\nu}^A
    -\frac1{2\alpha}(D^\mu A_\mu^a)^2
    -\frac{\kappa^2}{2\beta}(\partial^\mu A_\mu^i)^2
    -ig^2f^{adi}f^{cbi}\bar C^aC^bA^{\mu c}A_\mu^d
    \nonumber\\
 & &+i\bar C^aD^2C^a
    +ig\bar C^aD^\mu(f^{abc}A_\mu^bC^c)
    +\frac i2g(D^\mu A_\mu^a)(f^{abc}\bar C^bC^c)
    \nonumber\\
 & &+\frac\alpha8g^2f^{abe}f^{cde}\bar C^a\bar C^bC^cC^d
    +\frac\alpha4g^2f^{abi}f^{cdi}\bar C^a\bar C^bC^cC^d
    \nonumber\\
 & &+\frac\alpha8g^2(f^{abc}\bar C^bC^c)^2
    +i\kappa\bar C^i\partial^2C^i
    +i\kappa\bar C^i\partial^\mu(gf^{ibc}A_\mu^bC^c)
    \Bigr\}.
\label{eq:S_GF}
\end{eqnarray}
From this gauge fixing terms, we find that any interaction term with
diagonal ghost $C^i$ does not exit so that diagonal ghost does
not appear as internal legs in the perturbative calculation.

In order to determine the anomalous dimension of $\sigma$, we
make use of the background field method\cite{Abbott82} as we did in (I).
We separate each fields ${\mit\Phi}$ into the background field
$\bar{\mit\Phi}$ and the quantum fluctuation field $\tilde{\mit\Phi}$
as ${\mit\Phi}=\bar{\mit\Phi}+\tilde{\mit\Phi}$, where ${\mit\Phi}$
stands for each field of $A_\mu^i$, $B_{\mu\nu}^i$, $A_\mu^a$,
$\bar C^i$, $\bar C^a$, $C^i$ and $C^a$.
After substituting ${\mit\Phi}=\bar{\mit\Phi}+\tilde{\mit\Phi}$ into
the action~(\ref{eq:S}) and expanding it in $\tilde{\mit\Phi}$,
we obtain
\begin{equation}
\textstyle
S[\bar{\mit\Phi}+\tilde{\mit\Phi}]
 =S[\bar{\mit\Phi}]
  +\frac12
   \left(\tilde{\mit\Phi}\frac\delta{\delta{\mit\Phi}}\right)^2
%   \frac{\delta^2S[{\mit\Phi}]}{\delta{\mit\Phi}^2}
   S[{\mit\Phi}]
   \bigr|_{{\mit\Phi}=\bar{\mit\Phi}}
   +\cdots,
\label{eq:expanded S}
\end{equation}
Here, we have used the equation of motion with respect to the background
field $\delta S[\bar{\mit\Phi}]/\delta\bar{\mit\Phi}=0$.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Action and Feynman rules} %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
After integrating out $\tilde B_{\mu\nu}^i$, we obtain the following
Feynman rules.
A part of Feynman rules have already been given in the \S 4.3.\ in (I).
%in the previous paper.\cite{KS00b}
Here, we list some of remaining parts that we need.
(See Fig.~\ref{fig:Feynman Rules})
\unitlength=0.001in
%%%%%%% BEGIN FIGURE (Feynman Rules) %%%%%%%%%%
\begin{figure}[tb]
\begin{center}
\begin{picture}(4300,2400)%(0,-3000)
%\put(0,2400){\tframe[500][100](4300,2400)}%
%===========%
% Graph (a) %
%===========%
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\put(550,2100){%
   \put(0,0){\epsfysize=5mm \epsfbox{apropa.eps}}%
   \put(-280,110){\mbox{$i,\mu$}}%
   \put(700,110){\mbox{$j,\nu$}}%
   \put(230,-50){\mbox{$p$}}%
   }%
%===========%
% Graph (b) %
%===========%
\put(0,1700){\mbox{\large(b)}}%
\put(200,1000){%
   \put(0,0){\epsfysize=20mm \epsfbox{aaa.eps}}%
   \put(170,200){\mbox{$p$}}%
   \put(630,750){\mbox{$q$}}%
   \put(900,150){\mbox{$r$}}%
   \put(100,500){\mbox{$a,\mu$}}%
   \put(900,600){\mbox{$b,\rho$}}%
   \put(500,30){\mbox{$c,\sigma$}}%
   }%
%===========%
% Graph (c) %
%===========%
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\put(1600,1000){%
   \put(0,0){\epsfysize=20mm \epsfbox{aaa.eps}}%
   \put(170,200){\mbox{$q$}}%
   \put(630,750){\mbox{$p$}}%
   \put(900,150){\mbox{$r$}}%
   \put(100,500){\mbox{$a,\rho$}}%
   \put(900,600){\mbox{$i,\mu$}}%
   \put(500,30){\mbox{$b,\sigma$}}%
   }%
%===========%
% Graph (d) %
%===========%
\put(2900,1700){\mbox{\large(d)}}%
\put(3100,1000){%
   \put(0,0){\epsfysize=20mm \epsfbox{acc.eps}}%
   \put(600,700){\mbox{$p$}}%
   \put(700,150){\mbox{$q$}}%
   \put(100,500){\mbox{$c,\mu$}}%
   \put(750,600){\mbox{$a$}}%
   \put(600,30){\mbox{$b$}}%
   }%
%===========%
% Graph (e) %
%===========%
\put(0,700){\mbox{\large(e)}}%
\put(400,0){%
   \put(0,0){\epsfysize=20mm \epsfbox{aaaa.eps}}%
   \put(-150,600){\mbox{$a,\mu$}}%
   \put(-150,100){\mbox{$b,\nu$}}%
   \put(700,600){\mbox{$i,\rho$}}%
   \put(700,100){\mbox{$j,\sigma$}}%
   }%
%===========%
% Graph (f) %
%===========%
\put(1450,700){\mbox{\large(f)}}%
\put(1800,0){%
   \put(0,0){\epsfysize=20mm \epsfbox{aaaa.eps}}%
   \put(-150,600){\mbox{$a,\mu$}}%
   \put(-150,100){\mbox{$b,\nu$}}%
   \put(700,600){\mbox{$c,\rho$}}%
   \put(700,100){\mbox{$d,\sigma$}}%
   }%
%===========%
% Graph (g) %
%===========%
\put(2900,700){\mbox{\large(g)}}%
\put(3150,0){%
   \put(0,0){\epsfysize=20mm \epsfbox{aacc.eps}}%
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   \put(-150,130){\mbox{$d,\nu$}}%
   \put(650,600){\mbox{$a$}}%
   \put(650,100){\mbox{$b$}}%
   }%
\end{picture}
\caption[]{%
    The (rapidly vibrating) wavy line denotes the fluctuation
    gluon and the broken line denotes the fluctuation ghost,
    while the (slowly vibrating) wavy line corresponds to the background
    gluon.
}
\label{fig:Feynman Rules}
\end{center}
\end{figure}
%%%%%%% END FIGURE (Feynman Rules) %%%%%%%%%%

%%%%%%%%%%%%%%%%%
% Feynman rules %
%%%%%%%%%%%%%%%%%
\begin{enumerate}
%---------
\item[(a)] diagonal gluon propagator:
\begin{equation}
iD_{\mu\nu}^{ij}
  =-\frac i{p^2}
    \left[g_{\mu\nu}-\left(1-\hat\beta\right)
                     \frac{p_\mu p_\nu}{p^2}\right]
    \delta^{ij},
\end{equation}
where we have defined $\hat\beta:=\beta/(\kappa^2)$ for convenience.

%---------
\item[(b)] Vertex of three off-diagonal gluons:
\begin{equation}
i\left<\bar A_\mu^a(p)\tilde A_\rho^b(q)
                  \tilde A_\sigma^c(r)\right>_{\rm bare}
  =gf^{abc}[(q-r)_\mu g_{\rho\sigma}
            +(r-p)_\rho g_{\sigma\mu}
            +(p-q)_\sigma g_{\mu\rho}].
\end{equation}

%---------
\item[(c)] Vertex of one diagonal and two off-diagonal gluons:
\begin{eqnarray}
&&i\left<\tilde A_\mu^i(p)\bar A_\rho^a(q)
                 \tilde A_\sigma^b(r)\right>_{\rm bare}
  \nonumber\\
&&\textstyle
 =gf^{iab}\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].%\qquad
\end{eqnarray}

%---------
\item[(d)] Vertex of one off-diagonal gluon,
           one off-diagonal ghost and one anti-ghost:
\begin{equation}
i\left<\tilde{\bar C}{}^a(p)\tilde C^b(q)\bar A_\mu^c\right>_{\rm bare}
 =\textstyle
  \frac i2g(p+q)_\mu f^{acb}.
\end{equation}

%---------
\item[(e)] Two diagonal gluons and two off-diagonal gluons:
\begin{eqnarray}
&&\textstyle
  i\left<\bar A_\mu^a\bar A_\nu^b
       \tilde A_\rho^i\tilde A_\sigma^j\right>_{\rm bare}
  \nonumber\\
&&\textstyle
  =ig^2f^{aic}f^{cjb}
    \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{eqnarray}

%---------
\item[(f)] Vertex of four off-diagonal gluons:
\begin{eqnarray}
&&\textstyle
  i\left<\bar A_\mu^a\bar A_\nu^b
       \tilde A_\rho^c\tilde A_\sigma^d\right>_{\rm bare}
     \nonumber\\
&&\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{eqnarray}
where
$I_{\mu\nu,\rho\sigma}
  :=(g_{\mu\rho}g_{\nu\sigma}
     -g_{\mu\sigma}g_{\nu\rho})/2
$.

%---------
\item[(g)] Vertex of two off-diagonal gluons, one off-diagonal ghost
           and one off-diagonal anti-ghost :
\begin{equation}
i\left<\tilde{\bar C}{}^a\tilde C^b
       \bar A_\mu^c\bar A_\nu^d\right>_{\rm bare}
  =g^2g_{\mu\nu}(f^{iad}f^{icb}+f^{iac}f^{idb}).
\end{equation}

\end{enumerate}

In order to show the (multiplicative renormalizability
and determine the anomalous dimensions we define the renormalized fields
and parameters as
\begin{equation}
\begin{array}{c}
\begin{array}{ccc}
% 1st line
\bar A_\mu^i=Z_a^{1/2}\bar A_{{\rm R}\mu}^i, &
C_{\rm cl}^i=Z_c^{1/2}(C_{\rm R}^{\rm cl})^i, &
\bar C_{\rm cl}^i=Z_c^{1/2}(\bar C_{\rm R}^{\rm cl})^i, \\[2mm]
% 2nd line
\bar A_\mu^a=Z_A^{1/2}\bar A_{{\rm R}\mu}^a, &
C_{\rm cl}^a=Z_C^{1/2}(C_{\rm R}^{\rm cl})^a, &
\bar C_{\rm cl}^a=Z_C^{1/2}(\bar C_{\rm R}^{\rm cl})^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},
\\[2mm]
% 4th line
\bar B_{\mu\nu}^i=Z_BB_{{\rm R}\mu\nu}^i
\quad
\rho=Z_\rho\rho_{\rm R},
\quad
\sigma=Z_\sigma\sigma_{\rm R}.
\end{array}
\label{eq:renormalization}
\end{equation}
Here, there is no need to take into account renormalization of each
fluctuation field explicitly.
The anomalous dimensions of $A_\mu^i$, $B_{\mu\nu}^i$, $\hat\beta$,%
  \footnote{%
  The anomalous dimension of $\beta$ in (I) is identical to
  the anomalous dimension of $\hat\beta$ in this paper.
  }
$\rho$ and $g$ ($\beta$ function) have already been termined in (I).
The anomalous dimensions of the remaining two parameters $\alpha$
and $\sigma$ are obtained by requiring the renormalizability for the
propagator of off-diagonal gluons $A_\mu^a$ and the three
vertex of one background tensor field $\bar B_{\mu\nu}^i$ and
two off-diagonal gluons $\bar A_\mu^a$.
Substituting the renormalization relation~(\ref{eq:renormalization}) into
(\ref{eq:expanded S}), we obtain the counterterms:
\begin{enumerate}
\item[(h)] Counterterm of two off-diagonal gluons:
\begin{equation}
i\left<\bar A_\mu^a\bar A_\nu^b\right>_{\rm counter}
  =-i\delta_T(p^2g_{\mu\nu}-p_\mu p_\nu)\delta^{ab}
   -\frac i{\alpha_{\rm R}}\delta_Lp_\mu p_\nu\delta^{ab},
\label{eq:AA_counter}
\end{equation}
where $\delta_{\rm T}=Z_A-1$ and $\delta_{\rm L}=Z_A/Z_\alpha-1$.

\item[(i)] Counterterm of vertex of one diagonal tensor field and
           two off-diagonal gluons:
\begin{equation}
i\left<{}^\ast\!\bar B_{\mu\nu}^i\bar A_\xi^a
       \bar A_\eta^b\right>_{\rm counter}
  =-2\sigma_{\rm R}g_{\rm R}\delta_\sigma
    I_{\mu\nu,\rho\sigma}f^{iab},
\label{eq:BAA_counter}
\end{equation}
where $\delta_\sigma=Z_B^{1/2}Z_AZ_\sigma Z_g-1$.
\end{enumerate}
We can obtain the anomalous dimensions of $\alpha$ and $\sigma$
by calculating three factors $\delta_{\rm T}$,
$\delta_{\rm L}$ and $\delta_\sigma$.

$\delta_{\rm T}$ and $\delta_{\rm L}$ are determined
by requiring the renormalizability for the propagators
of off-diagonal gluons.
The divergent graphs for the vacuum polarization
of off-diagnoal gluons at the one-loop level appear
in Fig.~\ref{fig:vacuum polarization}.
Each contribution of three graphs (a4), (a5) and (a6)
in Fig.~\ref{fig:vacuum polarization} is not divergent
%\unitlength=0.001in
%%%%%%% BEGIN FIGURE (Vacuum Polarization) %%%%%%%%%%
\begin{figure}[tb]
\begin{center}
\begin{picture}(4700,2000)%(0,-3000)
%\put(0,2000){\tframe[500][100](4700,2000)}%
%===========%
% Graph (a1) %
%===========%
\put(0,1900){\mbox{(a1)}}%
\put(100,1200){%
   \put(0,0){\epsfysize=17mm\epsfbox{aa1.eps}}%
%   \put(-80,430){\mbox{$a,\mu$}}%
%   \put(1050,400){\mbox{$b,\nu$}}%
%   \put(70,130){\mbox{$q$}}%
%   \put(550,-80){\mbox{$p$}}%
%   \put(420,490){\mbox{$p+q$}}%
   }%
%===========%
% Graph (a2) %
%===========%
\put(1600,1900){\mbox{(a2)}}%
\put(1700,1150){%
   \put(0,0){\epsfysize=18mm\epsfbox{aa2.eps}}%
%   \put(-80,470){\mbox{$a,\mu$}}%
%   \put(1050,420){\mbox{$b,\nu$}}%
%   \put(70,220){\mbox{$q$}}%
%   \put(620,-50){\mbox{$p$}}%
%   \put(420,450){\mbox{$p+q$}}%
   }%
%===========%
% Graph (a3) %
%===========%
\put(3200,1900){\mbox{(a3)}}%
\put(3300,1150){%
   \put(0,0){\epsfysize=18mm\epsfbox{aa2.eps}}%
%   \put(-80,470){\mbox{$a,\mu$}}%
%   \put(1050,420){\mbox{$b,\nu$}}%
%   \put(70,220){\mbox{$q$}}%
%   \put(620,-50){\mbox{$p$}}%
%   \put(420,450){\mbox{$p+q$}}%
   \put(620,720){\mbox{$i$}}%
   }%
%===========%
% Graph (a4) %
%===========%
\put(0,900){\mbox{(a4)}}%
\put(220,130){%
   \put(0,0){\epsfysize=19mm\epsfbox{aa3.eps}}%
%   \put(-100,170){\mbox{$a,\mu$}}%
%   \put(900,160){\mbox{$b,\nu$}}%
%   \put(450,600){\mbox{$p$}}%
%   \put(70,-50){\mbox{$q$}}%
   }%
%===========%
% Graph (a5) %
%===========%
\put(1600,900){\mbox{(a5)}}%
\put(1820,130){%
   \put(0,0){\epsfysize=20mm\epsfbox{aa4.eps}}%
%   \put(-100,150){\mbox{$a,\mu$}}%
%   \put(900,160){\mbox{$b,\nu$}}%
%   \put(420,550){\mbox{$p$}}%
%   \put(50,-70){\mbox{$q$}}%
   }%
%===========%
% Graph (a6) %
%===========%
\put(3200,900){\mbox{(a6)}}%
\put(3420,130){%
   \put(0,0){\epsfysize=20mm\epsfbox{aa4.eps}}%
%   \put(-100,150){\mbox{$a,\mu$}}%
%   \put(900,160){\mbox{$b,\nu$}}%
%   \put(420,550){\mbox{$p$}}%
%   \put(50,-70){\mbox{$q$}}%
   \put(480,800){\mbox{$i$}}%
   }%
\end{picture}
\end{center}
\caption[]{Vacuum polarization graphs that are necessary to obtain
           propagator of off-diagonal gluon at the one-loop level.
           Wavy lines labeled by index $i$ represent diagonal gluons.
           }
\label{fig:vacuum polarization}
\end{figure}
%%%%%%% END FIGURE (Vacuum Polarization) %%%%%%%%%%
since we are making use of dimensional regularization.
Calculating the contributions from the graphs (a1), (a2) and (a3),
we obtain two factors $\delta_{\rm T}$ and $\delta_{\rm L}$ as
\begin{eqnarray}
\delta_{\rm T}
 &=&%\textstyle
    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}
        +\left(\frac53+\frac{1-\alpha_{\rm R}}2\right)(C_2(G)-2)
    \right],
\label{eq:delta_T}
    \\
\delta_{\rm L}
 &=&%\textstyle
    \frac{(g_{\rm R}\mu^{-\epsilon})^2}{(4\pi)^2\epsilon}
    \left[\frac3{\alpha_{\rm R}}
          +\frac32
          +\frac{\alpha_{\rm R}}2-\hat\beta_{\rm R}
          -\frac34{\alpha_{\rm R}}(C_2(G)-2)
    \right],
\label{eq:delta_L}
    \\
Z_\alpha^{(1)}
 &=&%\textstyle
    \frac{(g_{\rm R}\mu^{-\epsilon})^2}{(4\pi)^2\epsilon}
    \left[\frac43-\alpha_{\rm R}-\frac3{\alpha_{\rm R}}
          +\left(\frac53+\frac{1-\alpha_{\rm R}}2
                 -\frac34\alpha_{\rm R}\right)(C_2(G)-2)
    \right],
\label{eq:Z_alpha}
\end{eqnarray}
where $\epsilon=(d-4)/2$ and $Z^{(n)}$ is $n$-loop contribution to
$Z$-factor:
\begin{equation}
Z=1+Z^{(1)}+Z^{(2)}+\cdots,
\end{equation}
and $C_2(G)$ is a quadratic Casimir operator defined by
$C_2(G)\delta^{AB}=f^{ACD}f^{BCD}$ so that the terms proportional to
$(C_2(G)-2)$ vanish in the case of $SU(2)$.
Both the Feynman rules used here
and the counterterm~(\ref{eq:AA_counter}) do not include neither
$\rho$ nor $\sigma$.
Therefore the quantum corrections~(\ref{eq:delta_T}), (\ref{eq:delta_L})
and (\ref{eq:Z_alpha}) do not depend on these parameter.
In other words, the anomalous dimensions of off-diagonal gluons and
parameter $\alpha$ are not affected by introduction
of Abelian tensor field $B_{\mu\nu}$.

Next, we determine the anomalous dimension of $\sigma$.
We can obtain the factor $\delta_\sigma$ by calculating
the graphs that appear in Fig.~{\ref{fig:three vertex}}.
%\unitlength=0.001in
%%%%%%% BEGIN FIGURE (Vacuum Polarization) %%%%%%%%%%
\begin{figure}[tb]
\begin{center}
\begin{picture}(4500,900)%(0,-3000)
%\put(0,900){\tframe[500][100](4500,900)}%
%===========%
% Graph (a1) %
%===========%
\put(0,800){\mbox{(a1)}}%
\put(100,100){%
   \put(0,0){\epsfysize=17mm\epsfbox{baa1.eps}}%
   }%
\put(1500,800){\mbox{(a2)}}%
\put(1600,100){%
   \put(0,0){\epsfysize=17mm\epsfbox{baa2.eps}}%
%   \put(1050,400){\mbox{$$}}%
   }%
\put(3000,800){\mbox{(a3)}}%
\put(3100,100){%
   \put(0,0){\epsfysize=17mm\epsfbox{baa2.eps}}%
   \put(980,280){\mbox{$i$}}%
   }%
\end{picture}
\end{center}
\caption[]{The graphs that contribute the vertex of one diagonal
           tensor field and two off-diagonal gluons
           at the one-loop level.
           The zig-zag line corresponds to the
           background anti-symmetric tensor field.}
\label{fig:three vertex}
\end{figure}
%%%%%%% END FIGURE (Vacuum Polarization) %%%%%%%%%%
Requiring the renormalizability for three-point vertex of
one diagonal anti-ghost, one off-diagonal ghost and
one off-diagonal gluon, we obtain
\begin{eqnarray}
\delta_\sigma
 &=&%\textstyle
    \frac{(g_{\rm R}\mu^{-\epsilon})^2}{(4\pi)^2\epsilon}
    \frac12
    \left[1+\frac32\alpha_{\rm R}
          -\frac2{\alpha_{\rm R}}
          -\frac32\hat\beta_{\rm R}
          -\frac{\hat\beta_{\rm R}}{\alpha_{\rm R}}
          +\frac32(C_2(G)-2)
    \right],
    \\
Z_\sigma^{(1)}
 &=&%\textstyle
    \frac{(g_{\rm R}\mu^{-\epsilon})^2}{(4\pi)^2\epsilon}
    \Bigl[\frac43
           +\frac{1+\alpha_{\rm R}}2\sigma_{\rm R}^2
           +\frac54\alpha_{\rm R}
           -\frac1{\alpha_{\rm R}}
           +\frac{\hat\beta_{\rm R}}4
           -\frac{\hat\beta_{\rm R}}{2\alpha_{\rm R}}
    \nonumber\\
 & &%\textstyle
           +\left(\frac5{12}
                  +\frac{\alpha_{\rm R}}2
                  +\frac{1+\alpha_{\rm R}}4\sigma_{\rm R}^2
                  \right)(C_2(G)-2)
  \Bigr],
\end{eqnarray}
Thus, at one-loop level, the anomalous dimensions of off-diagonal gluons
and two parameters $\alpha$ and $\sigma$ are given by
\begin{eqnarray}
\gamma_A
 &=&%\textstyle
    -\frac{g_{\rm R}^2}{(4\pi)^2}
    \left[\frac{17}6-\frac{\alpha_{\rm R}}2-\hat\beta_{\rm R}
          +\left(\frac53+\frac{1-\alpha_{\rm R}}2\right)(C_2(G)-2)
    \right],
    \\
\gamma_\alpha
 &=&%\textstyle
    \frac{2g_{\rm R}^2}{(4\pi)^2}
    \left[\frac43\alpha_{\rm R}
          -\alpha_{\rm R}^2
          -3
          +\alpha_{\rm R}
           \left(\frac53+\frac{1-\alpha_{\rm R}}2
                 -\frac34\alpha_{\rm R}\right)(C_2(G)-2)
    \right],
    \\
\gamma_\sigma
 &=&%\textstyle
    \frac{2g_{\rm R}^2\sigma_{\rm R}}{(4\pi)^2}
    \Bigl[\frac43
           +\frac{1+\alpha_{\rm R}}2\sigma_{\rm R}^2
           +\frac54\alpha_{\rm R}
           -\frac1{\alpha_{\rm R}}
           +\frac{\hat\beta_{\rm R}}4
           -\frac{\hat\beta_{\rm R}}{2\alpha_{\rm R}}
    \nonumber\\
 & &%\textstyle
           +\left(\frac5{12}
                  +\frac{\alpha_{\rm R}}2
                  +\frac{1+\alpha_{\rm R}}4\sigma_{\rm R}^2
                  \right)(C_2(G)-2)
  \Bigr],
\end{eqnarray}
As a result, we conclude that anomalous dimensions of $A_\mu^i$,
$A_\mu^a$, $\alpha$, $\hat\beta$ and $g$ do not depend on
both new parameters $\rho_{\rm R}$ and $\sigma_{\rm R}$.
In the special case $\rho_{\rm R}=\sigma_{\rm R}=0$, our starting
action is identical to the ordinary Yang-Mills theory.
Therefore the anomalous dimensions of the fields and parameter
included in original Yang-Mills theory are not affected by
introducing the Abelian anti-symmetric tensor field $B_{\mu\nu}^i$
and two parameters $\rho$ and $\sigma$.

In this paper, we can obtained the all new anomalous dimension induced
by introduction of $B_{\mu\nu}$ after modification of gauge fixing terms,
then we have obtained the renormalizable Abelian-projected effective
gauge theory.\cite{KondoI}
The most general form of MA gauge fixing terms
and the necessity of $\kappa$
will be discussed in a subsequent paper.~\cite{SIK01a}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements} %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The author would like to thank Dr.~Kei-Ichi Kondo for useful discussions
and helpful advices.



\begin{thebibliography}{99}%%%%%%%%%%%%%%%%%%%%


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\end{thebibliography}



\end{document}

