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\begin{document}
\title{Massive Gauge Field Theory Without Higgs Mechanism\\
II. Ward-Takahashi Identities, Propagators and Vertices}
\author{Jun-Chen Su}
\address{Center for Theoretical Physics, Department of Physics,\\
Jilin University, Changchun 130023,\\
People's Republic of China}
\date{}
\maketitle

\begin{abstract}
Based on the BRST-symmetry of the quantum massive gauge field theory
described in the former paper, the Ward-Takahashi identities satisfied by
the generating functionals of full Green's functions, connected Green's
functions and proper vertex functions are successively derived. From such
identities, the Ward- Takahashi identities obeyed by the gauge boson
propagator and vertices are also derived and used to discuss the
renormalization of the propagator and vertices.

PACS: 11.15-q, 12.38-t
\end{abstract}

\setcounter{section}{1}

\section*{1.Introduction}

~~~~In the preceding paper(which will be referred to as paper I), it was
shown that the massive non-Abelian gauge field theory without Higgs
mechanism can perfectly be built up and the quantum theory has an important
property that the effective action appearing in the generating functional of
Green's functions is invariant with respect to a kind of BRST-transformations%
$^1$. Based on the BRST- symmetry, in this paper, we will derive various
Ward-Takahashi(W-T) identities$^{2-6}$ satisfied by the generating
functionals of Green's functions and proper vertices. From such identities,
we will derive a set of W-T identities obeyed by Green's functions and
vertices. The W-T identities are of special importance in proofs of
unitarity and renormalizability of the theory. The general proofs of these
problems will be presented in the next paper. In this paper, we confine
ourselves to discuss the renormalization of the gauge boson propagator and
vertices so as to demonstrate how the renormalization works for the massive
gauge field theory. By virtue of the W-T identity, it will be shown that the
renormalization constants for the massive gauge field theory respect the
same Slavnov-Taylor identity as given in the massless theory$^7$.

The arrangement of this paper is as follows. In Sect.2, we will derive the
W-T identities for the generating functionals. In doing this, we extend our
discussion by including fermions. In Sect.3, a W-T identity relating the
gauge boson propagator to the ghost particle propagator will be given and
the renormalization of the propagators will be discussed. Section 4 is used
to derive W-T identities respected by the gauge boson three and four-line
vertices and to discuss the renormalization of the vertices. we will end
this paper with some remarks in the last section.

\setcounter{section}{2}

\section*{2.Ward-Takahashi Identities}

\setcounter{equation}{0}

~~~This section serves to derive the W-T identities for the quantum massive
non-Abelian gauge field theory established in paper I on the basis of the
BRST-symmetry of the theory. Firstly, we summarize the main results derived
in Sect.3 of paper I. The effective Lagrangian for the quantum massive gauge
field is of the form 
\begin{equation}
{\cal L}_{eff}=-\frac 14F^{a\mu \nu }F_{\mu \nu }^a+\frac 12m^2A^{a\mu
}A_\mu ^a-\frac 1{2\alpha }(\partial ^\mu A_\mu ^a)^2+\bar C^a\partial ^\mu (%
{\cal D}_\mu ^{ab}c^b)
\end{equation}
where $A_\mu ^a$ are the vector potentials of a non-Abelian gauge field, $%
F_{\mu \nu }^a$ are the strength tensors of the field, $\bar C^a$ and $C^b$
denote the ghost field variables, $m$ is the gauge boson mass, $\alpha $ the
gauge parameter and 
\begin{equation}
{\cal D}_\mu ^{ab}=D_\mu ^{ab}+\delta ^{ab}\frac{\mu ^2}{\Box }\partial \mu
\end{equation}
here $\mu $ is defined by $\mu ^2=\alpha m^2$, and 
\begin{equation}
D_\mu ^{ab}=\delta ^{ab}\partial _\mu -gf^{abc}A_\mu ^c
\end{equation}
is the covariant derivative. The BRST-symmetry of the theory mentioned above
means that the action given by the Lagrangian in Eq.(2.1) is invariant under
the following BRST-transformations 
\begin{eqnarray}
\delta A^a(x)_\mu &=&\xi D_\mu ^{ab}C^b(x)  \nonumber \\
\delta \bar C^a(x) &=&\frac \xi \alpha \partial ^\mu A_\mu ^a(x)  \nonumber
\\
\delta C^a(x) &=&\int d^4y[\delta ^{ab}\delta ^4(x-y)-\mu ^2\triangle
^{ab}(x-y)]\delta C_0^b(y)  \nonumber \\
\delta C_0^b(y) &=&-\frac \xi 2gf^{bcd}C^c(y)C^d(y)
\end{eqnarray}
where $\xi $ is an infinitesimal anticommuting number, and $\Delta
^{ab}(x-y) $ represents the full ghost particle propagator. In the Landau
gauge ${(\mu =0)}$, the transformations shown above formally are identical
to those for the massless theory.

In the following, we would like to extend our discussions by including
fermion fields. Here, we take the Quantum Chromodynamics (QCD) as an
illustration. The effective Lagrangian of the QCD with massive gluons may be
written as$^{3-5}$ 
\begin{equation}
{\cal L}=\bar \psi \{i\gamma ^\mu (\partial _\mu -igT^aA_\mu ^a)-M\}\psi +%
{\cal L}_{eff}
\end{equation}
where $\psi (x)$ and $\bar \psi (x)$ are the quark field functions conjugate
to each other, $M$ is the quark mass and ${\cal L}_{eff}$ was shown in
Eq.(2.1) with the choice of SU(3) gauge group. the BRST-transformations of
the quark fields are 
\begin{eqnarray}
\delta \psi (x) &=&ig\xi T^aC^a(x)\psi (x)  \nonumber \\
\delta \bar \psi (x) &=&ig\xi \bar \psi (x)T^aC^a(x)
\end{eqnarray}
Since derivations of the W-T identities for the QCD with massive gluons are
much similar to those for the QCD with massless gluons, we only need here to
give a brief description of the derivations. Let us write down the
generating functional constructed by the Lagrangian in Eq.(2.5)$^{4-5}$ 
\begin{eqnarray}
Z[J_\mu ^a,\bar K^a,K^a,\bar \eta ,\eta ] &=&\frac 1N\int {\cal D}(A_\mu ^a,%
\bar C^a,C^a,\bar \psi ,\psi )exp\{iS[A_\mu ^a,\bar C^a,C^a,\bar \psi ,\psi ]
\nonumber \\
&&+i\int d^4x[J^{a\mu }A_\mu ^a+\bar K^aC^a+\bar C^aK^a  \nonumber \\
&&+\bar \psi \eta +\bar \eta \psi ]\}
\end{eqnarray}
where $S[A_\mu ^a,\bar C^a,C^a,\bar \psi ,\psi ]$ is the action given by the
Lagrangian in Eq.(2.5). When we make the BRST-transformations shown in
Eqs.(2.4) and (2.6) to the above generating functional and consider the
invariance of the generating functional , the action and the integration
measure under the transformations(the invariance of the integration measure
is easy to check), we obtain an identity such that 
\begin{eqnarray}
&&\ \ \frac 1N\int {\cal D}(A_\mu ^a,\bar C^a,C^a,\bar \psi ,\psi )\int
d^4x\{J^{a\mu }(x)\delta A_\mu ^a(x)+\delta \overline{C}^a(x)K^a(x)+\bar K%
^a(x)\delta C^a(x)  \nonumber \\
&&\ \ +\bar \eta (x)\delta \psi (x)+\delta \bar \psi (x)\eta (x)\}e^{iS+EST}
\\
\ &=&0  \nonumber
\end{eqnarray}
where EST is an abbreviation of the external source terms appearing in
Eq.(2.7). The Grassmann number $\xi $ contained in the BRST-transformations
in Eq.(2.8) may be eliminated by performing a partial differentiation of
Eq.(2.8) with respect to $\xi $. As a result, we get a W-T identity as
follows 
\begin{eqnarray}
&&\ \ \frac 1N\int {\cal D}(A_\mu ^a,\bar C^a,C^a,\bar \psi ,\psi )\int
d^4x\{J^{a\mu }(x)\Delta A_\mu ^a(x)+\triangle \overline{C}^a(x)K^a(x)-\bar K%
^a(x)\Delta C^a(x)  \nonumber \\
&&\ \ -\bar \eta (x)\Delta \psi (x)+\Delta \bar \psi (x)\eta (x)\}e^{iS+EST}
\\
\ &=&0  \nonumber
\end{eqnarray}
where 
\begin{eqnarray}
{\Delta }A_\mu ^a(x) &=&D_\mu ^{ab}(x)C^b(x)  \nonumber \\
{\Delta }\bar C^a(x) &=&\frac 1\alpha \partial ^\mu A_\mu ^a(x)  \nonumber \\
{\Delta }C^a(x) &=&\int d^4y[\delta ^{ab}\delta ^4(x-y)-\mu ^2\Delta
^{ab}(x-y)]{\triangle }C_0^b(y)  \nonumber \\
{\Delta }C_0^b(y) &=&-\frac 12gf^{bcd}C^c(y)C^d(y)  \nonumber \\
{\Delta }\psi (x) &=&igT^aC^a(x)\psi (x)  \nonumber \\
{\Delta }\bar \psi (x) &=&ig\bar \psi (x)T^aC^a(x)
\end{eqnarray}
These functions defined above are finite. Each of them differs from the
corresponding BRST-transformation written in Eq.(2.4) or Eq.(2.6) by an
infinitesimal Grassmann parameter $\xi .$

In order to represent the composite field functions $\Delta A_\mu ^a,\Delta
C^a,\Delta \bar \psi $ and $\Delta \psi $ in Eq.(2.9) in terms of
differentials of the functional Z with respect to external sources, we may,
as usual, construct a generalized generating functional by introducing new
external sources (called BRST sources later on) into the generating
functional written in Eq.(2.7), as shown in the following$^{3-5}$ 
\begin{eqnarray}
&&\ \ \ ~Z[J_\mu ^a,\bar K^a,K^a,\bar \eta ,\eta ;u^{a\mu },v^a,\bar \zeta
,\zeta ]  \nonumber \\
\ &&~=\frac 1N\int {\cal D}[A_\mu ^a,\bar C^a,C^a,\bar \psi ,\psi
]exp\{iS+i\int d^4x[u^{a\mu }\Delta A_\mu ^a+v^a\Delta C^a  \nonumber \\
&&\ \ \ ~+\Delta \bar \psi \zeta +\bar \zeta \Delta \psi +J^{a\mu }A_\mu ^a+%
\bar K^aC^a+\bar C^aK^a+\bar \eta \psi +\bar \psi \eta ]\}  \nonumber \\
&&\ \ \ ~
\end{eqnarray}
where $u^{a\mu },$ $v^a,$ $\stackrel{}{\overline{\varsigma }}$ and $%
\varsigma $ are the sources belong to the corresponding functions $\Delta
A_{\mu \text{ , }}^a\Delta C^a$, $\Delta \Psi $ and $\Delta \overline{\Psi 
\text{ }}$ respectively. Obviously, $u^{a\mu }$ and $\Delta A_\mu ^a$ are
anticommuting quantities, while $v^a,\bar \zeta ,\zeta ,\Delta C^a,\Delta 
\bar \psi $ and $\Delta \psi $ are commuting ones. We may start from the
above generating functional to rederive the W-T identity. In order that the
identity thus derived is identical to that as given in Eq.(2.9), it is
necessary to require the BRST-source terms $u_i\Delta \Phi _i$ where $%
u_{i=}u^{a\mu },v^a,\overline{\zeta }$ or $\zeta $ and $\Delta \Phi
_i=\Delta A_\mu ^a,\Delta C^a,\Delta \Psi $ or $\Delta \overline{\Psi \text{,%
}}$ to be invariant under the BRST-transformations. How to ensure the
BRST-invariance of the source terms? For illustration, let us introduce the
source terms in such a fashion 
\begin{eqnarray}
&&\ \ \int d^4x[\widetilde{u}^{a\mu }\delta A_\mu ^a+\widetilde{v}^a\delta
C^a+\overline{\widetilde{\zeta }}\delta \psi +\delta \overline{\psi }%
\widetilde{\zeta }]  \nonumber \\
\ &=&\int d^4x[u^{a\mu }\triangle A_\mu ^a+v^a\triangle C^a+\overline{\zeta }%
\triangle \psi +\triangle \overline{\psi }\zeta ]
\end{eqnarray}
where 
\begin{equation}
u^{a\mu }=\tilde u^{a\mu }\xi ,\;\;v^a=\tilde v^a\xi ,\;\;\bar \varsigma =%
\overline{\widetilde{\varsigma }}\xi ,\;\;\varsigma =-\tilde \varsigma \xi
\end{equation}
These external sources are defined by including the Grassmann number $\xi $
and hence products of them with $\xi $ vanish. This suggests that we may
generally define the sources by the following condition 
\[
u_i\xi =0 
\]
Considering that under the BRST-transformation, the variation of the
composite field functions given in the general gauges can be represented in
the form $\delta \Delta \Phi _i=\xi \widetilde{\Delta \Phi _i}$ where $%
\widetilde{\Delta \Phi _i}$ are functions without including the parameter $%
\xi $ , clearly, the definition in Eq.(2.14) for the sources would guarantee
the BRST- invariance of the BRST-source terms. When the BRST-transformations
in Eqs.(2.4) and (2.6) are made to the generating functional in Eq.(2.11),
we still obtain the identity as shown in Eq.(2.8) except that the external
source terms is now extended to include the BRST-external source terms. But,
due to the definition in Eq.(2.14) for the sources, the BRST-source terms
actually give a vanishing contribution to the identity in Eq.(2.8). This
fact indicates that we may directly insert the BRST-source terms into the
exponential in Eq.(2.8) without changing the identity itself. When
performing a partial differentiation of the identity with respect to $\xi ,$
we obtain a W-T identity which is the same as written in Eq.(2.9) except
that the BRST-source terms are now included in the identity and their
contribution to the identity does no longer vanish. Therefore, Eq.(2.9) may
be expressed as 
\begin{eqnarray}
&&\ \ \int d^4x[J^{a\mu }(x)\frac \delta {\delta u^{a\mu }(x)}-\bar K^a(x)%
\frac \delta {\delta v^a(x)}-\bar \eta (x)\frac \delta {\delta \bar \zeta (x)%
}  \nonumber \\
&&\ \ +\eta (x)\frac \delta {\delta \zeta (x)}+\frac 1\alpha K^a(x)\partial
_x^\mu \frac \delta {\delta J^{a\mu }(x)}]Z[J_\mu ^a,\cdots ,\zeta ] 
\nonumber \\
\ &=&0
\end{eqnarray}
This is the W-T identity satisfied by the generating functional of full
Green functions. On substituting in Eq.(2.15) the relation $^{3-5}$ 
\begin{equation}
Z=e^{iW}
\end{equation}
where W denotes the generating functional of connected Green's functions,
one may obtain a W-T identity expressed by the functional W 
\begin{eqnarray}
&&\ \ \int d^4x[J^{a\mu }(x)\frac \delta {\delta u^{a\mu }(x)}-\bar K^a(x)%
\frac \delta {\delta v^a(x)}-\bar \eta (x)\frac \delta {\delta \bar \zeta (x)%
}+\eta (x)\frac \delta {\delta \zeta (x)}  \nonumber \\
&&\ \ +\frac 1\alpha K^a(x)\partial _x^\mu \frac \delta {\delta J^{a\mu }(x)}%
]W[J_u^a,\cdots ,\zeta ]  \nonumber \\
\ &=&0
\end{eqnarray}
From this identity, one may get another W-T identity satisfied by the
generating functional of proper (one-particle-irreducible) vertex functions
. The functional $\Gamma $ is usually defined by the following Legendre
transformation$^{3-5}$ 
\begin{eqnarray}
\Gamma [A^{a\mu },\bar C^a,C^a,\bar \psi ,\psi ;u_\mu ^a,v^a,\bar \zeta
,\zeta ] &=&W[J_\mu ^a,\bar K^a,K^a,\bar \eta ,\eta ;u_\mu ^a,v^a,\bar \zeta
,\zeta ]  \nonumber \\
&&\ \ -\int d^4x[J_\mu ^aA^{a\mu }+\bar K^aC^a+\bar C^aK^a+\bar \eta \psi +%
\bar \psi \eta ]
\end{eqnarray}
where $A_\mu ^a,\bar C^a,C^a,\bar \psi $ and $\psi $ are field variables
defined by the following functional derivatives$^{3-5}$ 
\begin{eqnarray}
A_\mu ^a(x) &=&\frac{\delta W}{\delta J^{a\mu }(x)},\;\;\bar C^a(x)=-\frac{%
\delta W}{\delta K^a(x)},C^a(x)=\frac{\delta W}{\delta \bar K^a(x)}, 
\nonumber \\
\bar \psi (x) &=&-\frac{\delta W}{\delta \eta (x)},\;\;\psi (x)=\frac{\delta
W}{\delta \bar \eta (x)}
\end{eqnarray}
From Eq.(2.18), it is not difficult to get the inverse transformations$%
^{3-5} $ 
\begin{eqnarray}
J^{a\mu }(x) &=&-\frac{\delta \Gamma }{\delta A_\mu ^a(x)},\;\;\bar K^a(x)=%
\frac{\delta \Gamma }{\delta C^a(x)},K^a(x)=-\frac{\delta \Gamma }{\delta 
\bar C^a(x)},  \nonumber \\
\bar \eta (x) &=&\frac{\delta \Gamma }{\delta \psi (x)},\;\;\eta (x)=-\frac{%
\delta \Gamma }{\delta \bar \psi (x)}
\end{eqnarray}
It is obvious that 
\begin{eqnarray}
\frac{\delta W}{\delta u_\mu ^a}=\frac{\delta \Gamma }{\delta u_\mu ^a},\;\;%
\frac{\delta W}{\delta v^a}=\frac{\delta \Gamma }{\delta v^a},\;\;\frac{%
\delta W}{\delta \zeta }=\frac{\delta \Gamma }{\delta \zeta },\;\;\frac{%
\delta W}{\delta \bar \zeta }=\frac{\delta \Gamma }{\delta \bar \zeta }.
\end{eqnarray}
Upon employing Eqs.(2.20) and (2.21), the W-T identity in Eq.(2.17) will be
written as 
\begin{eqnarray}
&&\ \ ~\int d^4x\{\frac{\delta \Gamma }{\delta A_\mu ^a(x)}\frac{\delta
\Gamma }{\delta u^{a\mu }(x)}+\frac{\delta \Gamma }{\delta C^a(x)}\frac{%
\delta \Gamma }{\delta v^a(x)}+\frac{\delta \Gamma }{\delta \psi (x)}\frac{%
\delta \Gamma }{\delta \bar \zeta (x)}  \nonumber \\
&&\ \ ~+\frac{\delta \Gamma }{\delta \bar \psi (x)}\frac{\delta \Gamma }{%
\delta \zeta (x)}+\frac 1\alpha \partial _x^\mu A_\mu ^a(x)\frac{\delta
\Gamma }{\delta \bar C^a(x)}\}  \nonumber \\
\ &&~=0
\end{eqnarray}
This is the W-T identity satisfied by the generating functional of proper
vertex functions.

The above identity may be represented in another form with the aid of the
so-called ghost equation of motion. The ghost equation may easily be derived
by firstly making the translation transformation:$\bar C^a\rightarrow \bar C%
^a+\bar \lambda ^a$ in Eq.(2.7) where $\bar \lambda ^a$ is an arbitrary
Grassmann variable, then differentiating Eq.(2.7) with respect to the $\bar 
\lambda ^a$ and finally setting $\overline{\lambda }^a=0$. The result is $%
^{3-5}$ 
\begin{equation}
\frac 1N\int D(A_\mu ^a,\bar C^a,C^a,\bar \psi ,\psi )\{K^a(x)+\partial
_x^\mu ({\cal D}_\mu ^{ab}(x)C^b(x))\}e^{iS+EST}=0
\end{equation}
When we use the generating functional defined in Eq.(2.11) and notice the
relation in Eq.(2.2), the above equation may be represented as$^{3-5}$ 
\begin{equation}
\lbrack K^a(x)-i\partial _x^\mu \frac \delta {\delta u^{a\mu }(x)}-i{\mu }^2%
\frac \delta {\delta \bar K^a(x)}]Z[J_\mu ^a,\cdots ,\zeta ]=0
\end{equation}
By using the relation in Eq.(2.16), we may write a ghost equation for the
functional W, 
\begin{equation}
K^a(x)+\partial _x^\mu \frac{\delta W}{\delta u^{a\mu }(x)}+{\mu }^2\frac{%
\delta W}{\delta \bar K^a(x)}=0
\end{equation}
From this equation, the ghost equation obeyed by the functional $\Gamma $ is
easy to derived by virtue of Eqs.(2.19) - (2.21)$^{3-5}$ 
\begin{equation}
\frac{\delta \Gamma }{\delta \bar C^a(x)}-\partial _x^\mu \frac{\delta
\Gamma }{\delta u^{a\mu }(x)}-\mu ^2C^a(x)=0
\end{equation}
On applying the above equation to the last term in Eq.(2.20). the identity
in Eq.(2.20) will be rewritten as 
\begin{eqnarray}
&&\int d^4x\{\frac{\delta \Gamma }{\delta A_\mu ^a}\frac{\delta \Gamma }{%
\delta u^{a\mu }}+\frac{\delta \Gamma }{\delta C^a}\frac{\delta \Gamma }{%
\delta v^a}+\frac{\delta \Gamma }{\delta \psi }\frac{\delta \Gamma }{\delta 
\bar \zeta }+\frac{\delta \Gamma }{\delta \bar \psi }\frac{\delta \Gamma }{%
\delta \zeta }  \nonumber \\
&&+m^2\partial ^\nu A_\nu ^aC^a-\frac 1\alpha \partial ^\mu \partial ^\nu
A_\nu ^a\frac{\delta \Gamma }{\delta u^{a\mu }}\}  \nonumber \\
&=&0
\end{eqnarray}

Now, let us define a new functional $\hat \Gamma $ in such a manner 
\begin{equation}
\hat \Gamma =\Gamma +\frac 1{2\alpha }\int d^4x(\partial ^\mu A_\mu ^a)^2
\end{equation}
From this definition, it follows that 
\begin{equation}
\frac{\delta \Gamma }{\delta A_\mu ^a}=\frac{\delta \hat \Gamma }{\delta
A_\mu ^a}+\frac 1\alpha \partial ^\mu \partial ^\nu A_\nu ^a
\end{equation}
When inserting Eq.(2.28) into Eq.(2.27) and considering the relation in
Eq.(2.29), we arrive at 
\begin{eqnarray}
&&\int d^4x\{\frac{\delta \hat \Gamma }{\delta A_\mu ^a}\frac{\delta \hat 
\Gamma }{\delta u^{a\mu }}+\frac{\delta \hat \Gamma }{\delta C^a}\frac{%
\delta \hat \Gamma }{\delta v^a}+\frac{\delta \hat \Gamma }{\delta \psi }%
\frac{\delta \hat \Gamma }{\delta \bar \zeta }  \nonumber \\
&&+\frac{\delta \hat \Gamma }{\delta \bar \psi }\frac{\delta \hat \Gamma }{%
\delta \zeta }+m^2\partial ^\nu A_\nu ^aC^a\}  \nonumber \\
&=&0
\end{eqnarray}
The ghost equation represented through the functional ${\hat \Gamma }$ is of
the same form as Eq.(2.26) 
\begin{equation}
\frac{\delta \hat \Gamma }{\delta \bar C^a(x)}-\partial _x^\mu \frac{\delta 
\hat \Gamma }{\delta u^{a\mu }(x)}-{\mu }^2C^a(x)=0
\end{equation}
In the Landau gauge, since $\mu =0$ and ${\partial ^\nu A_\nu ^a=0}$,
Eqs.(2.30) and (2.31) respectively reduce to $^{3-5}$ 
\begin{equation}
\int d^4x\{\frac{\delta \hat \Gamma }{\delta A_\mu ^a}\frac{\delta \hat 
\Gamma }{\delta u^{a\mu }}+\frac{\delta \hat \Gamma }{\delta C^a}\frac{%
\delta \hat \Gamma }{\delta v^a}+\frac{\delta \hat \Gamma }{\delta \psi }%
\frac{\delta \hat \Gamma }{\delta \bar \zeta }+\frac{\delta \hat \Gamma }{%
\delta \bar \psi }\frac{\delta \hat \Gamma }{\delta \zeta }\}=0
\end{equation}
and 
\begin{equation}
\frac{\delta \hat \Gamma }{\delta \bar C^a}-\partial ^\mu \frac{\delta \hat 
\Gamma }{\delta u^{a\mu }}=0
\end{equation}
These equations formally are the same as those for the massless gauge field
theory.

Now, we would like to give another form of the W-T identity. The ghost
equation (2.23) suggests that the first external source term in Eq.(2.12)
which appearing in the generating functional in Eq.(2.11) may be replaced by 
\begin{equation}
\int d^4x\tilde u^{a\mu }(x)\delta {\cal A}_\mu ^a(x)=\int d^4xu^{a\mu
}(x)\Delta {\cal A}_\mu ^a(x)
\end{equation}
where 
\begin{equation}
\delta {\cal A}_\mu ^a(x)=\xi {\cal D}_\mu ^{ab}(x)C^b(x)=\xi \Delta {\cal A}%
_\mu ^a(x)
\end{equation}
\begin{equation}
\Delta {\cal A}_\mu ^a(x)=\Delta A_\mu ^a+\frac{{\mu }^2}{\Box _x}\partial
_\mu ^xC^a(x)
\end{equation}
In this case, from the above relations, we see, the W-T identity in
Eq.(2.15) will be rewritten as 
\begin{equation}
\begin{array}{c}
\int d^4x\{J^{a\mu }(x)\frac \delta {\delta u^{a\mu }(x)}-\bar K^a(x)\frac 
\delta {\delta v^a(x)}-\bar \eta (x)\frac \delta {\delta \bar \zeta (x)}%
+\eta (x)\frac \delta {\delta \zeta (x)} \\ 
-J^{a\mu }(x)\frac{{\mu }^2}{\Box _x}\partial _\mu ^x\frac \delta {\delta 
\bar K^a(x)}+\frac 1\alpha K^a(x)\partial _x^\mu \frac \delta {\delta
J^{a\mu }(x)}\}Z[J_\mu ^a,\cdots ,\zeta ]=0
\end{array}
\end{equation}
and the ghost equation in Eq.(2.24) becomes 
\begin{equation}
\{K^a(x)-i\partial _x^\mu \frac \delta {\delta u^{a\mu }(x)}\}Z[J_\mu
^a,\cdots ,\zeta ]=0
\end{equation}
Repeating the derivations described In Eqs.(2.16)-(2.22) and (2.24)-(2.31),
one may obtain from Eqs.(2.37) and (2.38) the identities expressed by the
functional ${\hat \Gamma }$%
\begin{eqnarray}
&&\int d^4x\{\frac{\delta \hat \Gamma }{\delta A_\mu ^a(x)}\frac{\delta \hat 
\Gamma }{\delta u^{a\mu }(x)}+\frac{\delta \hat \Gamma }{\delta C^a(x)}\frac{%
\delta \hat \Gamma }{\delta v^a(x)}+\frac{\delta \hat \Gamma }{\psi (x)}%
\frac{\delta \hat \Gamma }{\delta \bar \zeta (x)}  \nonumber \\
&&+\frac{\delta \hat \Gamma }{\delta \bar \psi (x)}\frac{\delta \hat \Gamma 
}{\delta \zeta (x)}-\frac{\delta \hat \Gamma }{\delta A_\mu ^a}\frac{\mu ^2}{%
\Box _x}\partial _\mu ^xC^a(x)+m^2\partial _x^\mu A_\mu ^a(x)C^a(x)\} 
\nonumber \\
&=&0
\end{eqnarray}
and 
\begin{equation}
\frac{\delta \hat \Gamma }{\delta \bar C^a(x)}-\partial _x^\mu \frac{\delta 
\hat \Gamma }{\delta u^{a\mu }(x)}=0
\end{equation}
In comparison of Eqs.(2.39) and (2.40) with Eqs.(2.30) and (2.31), we see,
the advantage of using $\delta {\cal A}_\mu ^a$ to define the external
source is that the ghost equation (2.40) becomes homogeneous. However, the
price paying for this advantage is the increase of an inhomogeneous term
(the fifth term) in Eq.(2.39). In the Landau gauge and the zero-mass limit,
Eqs.(2.39) still reduces to the homogeneous equation (2.32).

From the W-T identities formulated above, we may derive various W-T
identities obeyed by Green's functions and vertices. as will be illustrated
in the next section. Particularly, these identities provide a firm basis for
the proof of renormalizability and unitarity problems of the quantum massive
gauge field theory as will be discussed in the next paper.

\setcounter{section}{3}

\section*{3.Propagators}

\setcounter{equation}{0}

~~~In this section, we take a particular interest in deriving the W-T
identities satisfied by the massive gluon and ghost particle propagators and
then discussing their renormalization. To derive the mentioned W-T
identities, it is appropriate to start from the W-T identity in Eq.(2.37)
and the ghost equation in Eq.(2.38). While, we would rather here to use the
corresponding equations shown in Eqs.(2.15) and (2.24). Let us perform
differentiations of the identities represented in Eqs.(2.15) and (2.24) with
respect to the external sources $K^a(x)$ and $K^b(y)$ respectively and then
set all the sources except for the source $J_\mu ^a(x)$ to be zero. In this
way, we obtain the following identities$^{3,6}$ 
\begin{eqnarray}
\frac 1\alpha \partial _x^\mu \frac{\delta Z[J]}{\delta J^{a\mu }(x)}+\int
d^4yJ^{b\nu }(y)\frac{\delta ^2Z[J,K,u]}{\delta K^a(x)\delta u^{b\nu }(y)}%
|_{K=u=0}=0
\end{eqnarray}
and 
\begin{equation}
\begin{array}{c}
i\partial _\mu ^x\frac{\delta ^2Z[J.K.u]}{\delta u_\mu ^a(x)\delta K^b(y)}%
|_{K=u=0}+i{\mu }^2\frac{\delta ^2Z[J,\bar K,K]}{\delta \bar K^a(x)\delta
K^b(y)}|_{\bar K=K=0} \\ 
+\delta ^{ab}\delta ^4(x-y)Z[J]=0
\end{array}
\end{equation}
Furthermore, on differentiating Eq.(3.1) with respect to $J_\nu ^b(y)$ and
then letting the source J vanish, we may get an identity which is, in
operator representation, of the form $^{4-6}$ 
\begin{eqnarray}
\frac 1\alpha \partial _x^\mu &<&0^{+}|T[\hat A_\mu ^a(x)\hat A_\nu
^b(y)]|0^{-}>  \nonumber \\
&=&<0^{+}|T^{*}[\hat {\bar C^a}(x)\hat D_\nu ^{bd}(y)\hat C^d(y)]|0^{-}>
\end{eqnarray}
where $\hat A_\nu ^a(x)$,$\hat C^a(x)$ and $\bar C^a(x)$ stand for the gauge
field and ghost field operators and $T^{*}$ symbolizes the covariant
time-ordering product. When the source J is set to vanish, Eq.(3.2) will
give such an equation$^{4-6}$ 
\begin{equation}
\begin{array}{c}
i\partial _y^\nu <0^{+}|T^{*}\{\hat {\bar C^a}(x)\hat D_\nu ^{bd}(y)\hat C%
^d(y)\}|0^{-}> \\ 
+i{\mu }^2<0^{+}|T[\bar C^a(x)\hat C^b(y)]|0^{-}>=\delta ^{ab}\delta ^4(x-y)
\end{array}
\end{equation}
Upon inserting Eq.(3.3) into Eq.(3.4), we have 
\begin{equation}
\partial _x^\mu \partial _y^\nu D_{\mu \nu }^{ab}(x-y)-\alpha \mu ^2\Delta
^{ab}(x-y)=-\alpha \delta ^{ab}\delta ^4(x-y)
\end{equation}
where 
\begin{equation}
iD_{\mu \nu }^{ab}(x-y)=<0^{+}|T\{\hat A_\mu ^a(x)\hat A_\nu ^b(y)\}|0^{-}>
\end{equation}
which is the familiar full gluon propagator and 
\begin{equation}
i\Delta ^{ab}(x-y)=<0^{+}|T\{\hat C^a(x)\hat {\bar C^b}(y)\}|0^{-}>
\end{equation}
which is the full ghost particle propagator. Eq.(3.5) just is the W-T
identity respected by the gluon propagator which establishes a relation
between the longitudinal part of gluon propagator and the ghost particle
propagator. Particularly, in the Landau gauge, Eq.(3.5) reduces to the form
which exhibits the transversality of the gluon propagator. By the Fourier
transformation, Eq.(3.5) will be converted to the form given in the momentum
space as follows 
\begin{equation}
k^\mu k^\nu D_{\mu \nu }^{ab}(k)-\alpha \mu ^2\Delta ^{ab}(k)=-\alpha \delta
^{ab}
\end{equation}
The ghost particle propagator may be determined by the ghost equation shown
in Eq.(3.4). However, we would rather here to derive its expression by the
perturbation method. According to this method, as one knows, we may write
out a Dyson-Schwinger equation$^8$ like this 
\begin{equation}
\Delta ^{ab}(k)=\Delta _0^{ab}(k)+\Delta _0^{aa^{\prime }}(k){\Sigma }%
^{a^{\prime }b^{\prime }}(k)\Delta ^{b^{\prime }b}(k)
\end{equation}
where 
\begin{equation}
i\Delta _0^{ab}(k)=i\delta ^{ab}\Delta _0(k)=\frac{-i\delta ^{ab}}{k^2-\mu
^2+i\varepsilon }
\end{equation}
is the free ghost particle propagator obtained in paper I and $-i\Sigma
^{ab}(k)=-i\delta ^{ab}\Sigma (k)$ denotes the proper self-energy operator
of ghost particle. From Eq.(3.9), it is easy to solve that 
\begin{equation}
i\Delta ^{ab}(k)=\frac{-i\delta ^{ab}}{(k^2-\mu ^2+i\varepsilon )[1+\Omega
(k^2)]}
\end{equation}
where the self-energy has preferably been expressed by 
\begin{equation}
\Sigma (k)=(k^2-\mu ^2)\Omega (k^2)
\end{equation}
Similarly, we may write a Dyson-Schwinger equation for the gluon propagator
by the perturbation procedure$^8$ 
\begin{equation}
D_{\mu \nu }(k)=D_{\mu \nu }^0(k)+D_{\mu \lambda }^0(k)\Pi ^{\lambda \rho
}(k)D_{\rho \nu }(k)
\end{equation}
where 
\begin{equation}
iD_{~~~\mu \nu }^{(0)ab}(k)=i\delta ^{ab}D_{\mu \nu }^{(0)}(k)=-i\delta
^{ab}[\frac{g_{\mu \nu }-k_\mu k_\nu /k^2}{k^2-m^2+i\varepsilon }+\frac{%
\alpha k_\mu k_\nu /k^2}{k^2-\mu ^2+i\varepsilon }]
\end{equation}
is the free gluon propagator as derived in paper I and $-i\Pi _{\mu \nu
}^{ab}(k)=-i\delta ^{ab}\Pi _{\mu \nu }(k)$ stands for the gluon proper
self-energy operator. Let us decompose the propagator and the self-energy
operator into transverse and longitudinal parts: 
\begin{equation}
D^{\mu \nu }(k)=D_T^{\mu \nu }(k)+D_L^{\mu \nu }(k),\Pi ^{\mu \nu }(k)=\Pi
_T^{\mu \nu }(k)+\Pi _L^{\mu \nu }(k)
\end{equation}
where 
\begin{eqnarray}
D_T^{\mu \nu }(k) &=&(g^{\mu \nu }-\frac{k^\mu k^\nu }{k^2}%
)D_T(k^2),D_L^{\mu \nu }(k)=\frac{k^\mu k^\nu }{k^2}D_L(k^2),  \nonumber \\
\Pi _T^{\mu \nu }(k) &=&(g^{\mu \nu }-\frac{k^\mu k^\nu }{k^2})\Pi
_T(k^2),\Pi _L^{\mu \nu }(k)=\frac{k^\mu k^\nu }{k^2}\Pi _L(k^2)
\end{eqnarray}
Considering these decompositions and the orthogonality between the
transverse and longitudinal parts, Eq.(3.13) will be split into two
equations 
\begin{equation}
D_{T\mu \nu }(k)=D_{T\mu \nu }^0(k)+D_{T\mu \lambda }^0(k)\Pi _T^{\lambda
\rho }(k)D_{T\rho \nu }(k)
\end{equation}
and 
\begin{equation}
D_{L\mu \nu }(k)=D_{L\mu \nu }^0(k)+D_{L\mu \lambda }^0(k)\Pi _L^{\lambda
\rho }(k)D_{L\rho \nu }(k)
\end{equation}
Solving the equations (3.17) and (3.18) , one can get 
\begin{eqnarray}
iD_{\mu \nu }^{ab}(k) &=&-i\{\frac{g_{\mu \nu }-k_\mu k_\nu /k^2}{k^2[1+\Pi
_1(k^2)]-m^2+i\varepsilon }  \nonumber \\
&&+\frac{\alpha k_\mu k_\nu /k^2}{(k^2-\mu ^2+i\varepsilon )[1+\Pi _2(k^2)]}%
\}
\end{eqnarray}
where 
\begin{equation}
\Pi _T(k^2)=k^2\Pi _1(k^2),\;\;\alpha \Pi _L(k^2)=(k^2-\mu ^2)\Pi _2(k^2)
\end{equation}
Substitution of Eqs.(3.11) and (3.19) into Eq.(3.8) yields 
\begin{equation}
\Pi _2(k^2)=\frac{\mu ^2\Omega (k^2)}{k^2+(k^2-\mu ^2)\Omega (k^2)}
\end{equation}
From this relation, we see, either in the Landau gauge or in the zero-mass
limit, the $\Pi _2(k^2)$ vanishes.

Now let us discuss the renormalization. The operator $\Omega (k^2)$ in
Eq.(3.11) and operators $\Pi _1(k^2)$ and $\Pi _2(k^2)$ in Eq.(3.19) are
generally divergent in higher order perturbative calculations. According to
the conventional procedure of renormalization, the divergences included in
the functions $\Omega (k^2),\Pi _1(k^2)$ and $\Pi _2(k^2)$ may be subtracted
at a renormalization point, say, $k^2=\nu ^2$. Thus, we may write$^{4-6}$ 
\begin{eqnarray}
\Omega (k^2) &=&\Omega (\nu ^2)+\Omega ^c(k^2),\;\;\Pi _1(k^2)=\Pi _1(\nu
^2)+\Pi _1^c(k^2),  \nonumber \\
\Pi _2(k^2) &=&\Pi _2(\nu ^2)+\Pi _2^c(k^2)
\end{eqnarray}
where $\Omega (\nu ^2),\Pi _1(\nu ^2),\Pi _2(\nu ^2)$ and $\Omega
^c(k^2),\Pi _1^c(k^2),\Pi _2^c(k^2)$ are respectively the divergent parts
and the finite parts of the functions $\Omega (k^2),\Pi _1(k^2)$ and $\Pi
_2(k^2)$. The divergent parts can be absorbed in the renormalization
constants $\tilde Z_3,Z_3$ and $Z_3^{\prime }$ which are defined as$^{4-6}$ 
\begin{eqnarray}
\tilde Z_3^{-1} &=&1+\Omega (\nu ^2),\;\;Z_3^{-1}=1+\Pi _1(\nu ^2), 
\nonumber \\
Z_3^{^{\prime }-1} &=&1+\Pi _2(\nu ^2)
\end{eqnarray}
With these definitions, on inserting Eq.(3.22) into Eqs.(3.11) and (3.19) ,
the ghost particle propagator and gluon propagator are able to be
renormalized, respectively, in such a way 
\begin{equation}
i\Delta ^{ab}(k)=\tilde Z_3i\Delta _R^{ab}(k)
\end{equation}
and 
\begin{equation}
iD_{\mu \nu }^{ab}(k)=Z_3iD_{R\mu \nu }^{~~ab}(k)
\end{equation}
where 
\begin{equation}
i\Delta _R^{ab}(k)=\frac{-i\delta ^{ab}}{(k^2-\mu _R^2+i\varepsilon
)[1+\Omega _R(k^2)]}
\end{equation}
and 
\begin{eqnarray}
iD_{R\mu \nu }^{ab}(k) &=&-i\delta ^{ab}\{\frac{g_{\mu \nu }-k_\mu k_\nu /k^2%
}{k^2-m_R^2+\Pi _R^1(k^2)+i\varepsilon }  \nonumber \\
&&+\frac{Z_3^{\prime }\alpha _Rk_\mu k_\nu /k^2}{(k^2-\mu _R^2+i\varepsilon
)[1+\Pi _R^2(k^2)]}\}
\end{eqnarray}
are the renormalized propagators in which $m_R$ is the renormalized mass, $%
\alpha _R$ the renormalized gauge parameter, $\Omega _R(k^2),\Pi _R^1(k^2)$
and $\Pi _R^2(k^2)$ denote the finite corrections coming from the loop
diagrams. They are defined as 
\begin{eqnarray}
m_R &=&\sqrt{Z_3}m,\;\alpha _R=Z_3^{-1}\alpha ,\;\mu _R=\sqrt{\alpha _R}%
m_R,\;\Omega _R(k^2)=\tilde Z_3\Omega ^c(k^2)  \nonumber \\
\Pi _R^1(k^2) &=&Z_3k^2\Pi _1^c(k^2),\;\Pi _R^2(k^2)=Z_3^{\prime }\Pi
_2^c(k^2)
\end{eqnarray}
The finite corrections above are zero at the renormalization point $\nu $.
As we see from Eq.(3.27), the longitudinal part of gluon propagator, except
for in the Landau gauge, needs to be renormalized and has an extra
renormalization constant ${Z}_3^{\prime }$. This fact coincides with the
general property of the massive vector boson propagator (see Ref.(4)
Chap.V). From Eqs.(3.21)-(3.23) , it is easy to find 
\begin{equation}
Z_3^{\prime }=1-\frac{\mu _R^2}{\nu ^2}(1-\tilde Z_3)
\end{equation}
If we choose $\nu =\mu _R$ , then we have $Z_3^{\prime }=\tilde Z_3$.

\setcounter{section}{4}

\section*{4.Vertices}

\setcounter{equation}{0} ~~~~In this section, we take the gluon three-line
and four-line proper vertices as examples to show how to derive W-T
identities satisfied by the vertices involved in the theory and discuss
their renormalization. Firstly, let us begin with derivation of an identity
satisfied by the gluon three-point Green's function. The identity may be
obtained by successive differentiations of Eq.(3.1) with respect to the
sources $J_\nu ^b(y)$ and $J_\lambda ^c(z)$ and then setting the sources to
vanish. The result is $^5$ 
\begin{eqnarray}
\frac 1\alpha \partial _x^\mu G_{\mu \nu \lambda }^{abc}(x,y,z)
&=&<0^{+}|T^{*}[\hat {\bar C^a}(x)\hat D_\nu ^{bd}(y)\hat C^d(y)\hat A%
_\lambda ^c(z)]|0^{-}>  \nonumber \\
+ &<&0^{+}|T^{*}[\hat {\bar C^a}(x)\hat A_\nu ^b(y)\hat D_\lambda ^{cd}(z)%
\hat C^d(z)]|0^{-}>
\end{eqnarray}
where 
\begin{equation}
G_{\mu \nu \lambda }^{abc}(x,y,z)=<0^{+}|T[\hat A_\mu ^a(x)\hat A_\nu ^b(y)%
\hat A_\lambda ^c(z)]|0^{-}>
\end{equation}
is the three-point Green's function mentioned above. The identity in
Eq.(4.1) will be simplified by a ghost equation which may be derived by
differentiating Eq.(3.2) with respect to the source $J_\lambda ^c(z)$ 
\begin{equation}
\begin{array}{c}
\partial _x^\mu <0^{+}|T^{*}{\hat D}_\mu ^{ad}(x)\hat C^d(x)\hat {\bar C^b}%
(y)\hat A_\lambda ^c(z)\}|0^{-}> \\ 
+{\mu }^2<0^{+}|T[\hat C^a(x)\bar C^b(y)\hat A_\lambda ^c(z)]|0^{-}]>=0
\end{array}
\end{equation}
Taking derivatives of Eq.(4.1) with respect to y and z and employing
Eq.(4.3), we get 
\begin{equation}
\partial _x^\mu \partial _y^\nu \partial _z^\lambda G_{\mu \nu \lambda
}^{abc}(x,y,z)=\alpha \mu ^2\{\partial _y^\nu G_{~~\nu
}^{cab}(z,x,y)+\partial _z^\lambda G_{~~\lambda }^{bac}(y,x,z)\}
\end{equation}
where 
\begin{equation}
G_{~~\mu }^{abc}(x,y,z)=<0^{+}|T\{\hat C^a(x)\hat {\bar C^b}(y)\hat A_\mu
^c(z)\}|0^{-}>
\end{equation}
In the Landau gauge or the zero-mass limit, Eq.(4.4) reduces to 
\begin{equation}
\partial _x^\mu \partial _y^\nu \partial _z^\lambda G_{\mu \nu \lambda
}^{abc}(x,y,z)=0
\end{equation}
From Eq.(4.4), we may derive a W-T identity for the gluon three-line vertex.
For this purpose, it is necessary to use the following irreducible
decompositions of the Green's functions$^5$ 
\begin{eqnarray}
G_{\mu \nu \lambda }^{abc}(x,y,z) &=&-i\int d^4x^{\prime }d^4y^{\prime
}d^4z^{\prime }D_{\mu \mu ^{\prime }}^{aa^{\prime }}(x-x^{\prime }) 
\nonumber \\
&&\ \times D_{\nu \nu ^{\prime }}^{bb^{\prime }}(y-y^{\prime })D_{\lambda
\lambda ^{\prime }}^{cc^{\prime }}(z-z^{\prime })\Gamma _{a^{\prime
}b^{\prime }c^{\prime }}^{\mu ^{\prime }\nu ^{\prime }\lambda ^{\prime
}}(x^{\prime },y^{\prime },z^{\prime },)
\end{eqnarray}
and 
\begin{eqnarray}
G_{~~\nu }^{abc}(x,y,z) &=&i\int d^4x^{\prime }d^4y^{\prime }d^4z^{\prime
}\Delta ^{aa^{\prime }}(x-x^{\prime })\Gamma ^{a^{\prime }b^{\prime
}c^{\prime },\nu ^{\prime }}(x^{\prime },y^{\prime },z^{\prime })  \nonumber
\\
&&\ \times \Delta ^{b^{\prime }b}(y^{\prime }-y)D_{\nu ^{\prime }\nu
}^{c^{\prime }c}(z^{\prime }-z)
\end{eqnarray}
where the proper gluon three-line vertex $\Gamma _{abc}^{\mu \nu \lambda
}(x,y,z)$ and the proper ghost vertex $\Gamma _{~~\lambda }^{abc}(x,y,z)$
are defined as$^{4,5}$ 
\begin{eqnarray}
\Gamma _{abc}^{\mu \nu \lambda }(x,y,z) &=&i\frac{\delta ^3\Gamma }{\delta
A_\mu ^a(x)\delta A_\nu ^b(y)\delta A_\lambda ^c(z)}|_{J=0} \\
\Gamma _{~~\lambda }^{abc}(x,y,z) &=&i\frac{\delta ^3\Gamma }{\delta \bar C%
^a(x)\delta C^b(y)\delta A^{c\lambda }(z)}|_{J=0}
\end{eqnarray}
here J stands for all the external sources. Substituting Eqs.(4.7) and (4.8)
into Eq.(4.4) and transforming Eq.(4.4) into the momentum space. one can
derive an identity which establishes the relation between the longitudinal
part of the gluon three-line vertex and the ghost three-line vertex, as
follows 
\begin{eqnarray}
p^\mu q^\nu k^\lambda \Lambda _{\mu \nu \lambda }^{abc}(p,q,k) &=&\frac{\mu
^2}\alpha \chi (p^2)[\chi (k^2)q^\nu \Lambda _{~~\nu }^{cab}(k,p,q) 
\nonumber \\
&&\ +\chi (q^2)k^\lambda \Lambda _{~~\lambda }^{bac}(q,p,k)]
\end{eqnarray}
where we have defined 
\begin{eqnarray}
\Gamma _{\mu \nu \lambda }^{abc}(p,q,k) &=&(2\pi )^4\delta ^4(p+q+k)\Lambda
_{\mu \nu \lambda }^{abc}(p,q,k) \\
\Gamma _{~~\lambda }^{abc}(p,q,k) &=&(2\pi )^4\delta ^4(p+q+k)\Lambda
_{~~\lambda }^{abc}(p,q,k)
\end{eqnarray}
and 
\begin{equation}
\chi (p^2)=[1+\Pi _2(p^2)][1+\Omega (p^2)]^{-1}
\end{equation}
In the lowest order approximation, owing to 
\begin{equation}
\chi (p^2)=1
\end{equation}
and 
\begin{equation}
\Lambda _{~~~~~\mu }^{(0)abc}(p,q,k)=-gf^{abc}p_\mu
\end{equation}
(Note: the minus sign in Eq.(4.16) is consistent with the special definition
in Eq.(4.10) for the ghost vertex.) the right hand side (RHS) of Eq.(4.11)
vanishes, therefore, we have 
\begin{equation}
p^\mu q^\nu k^\lambda \Lambda _{~~~\mu \nu \lambda }^{(0)abc}(p,q,k)=0
\end{equation}
This result is consistent with that given by the Feynman rule for the gluon
three-line vertex.

By the similar procedure, we may derive W-T identities for more than
three-line gluon vertices. For example, the W-T identity obeyed by the gluon
four-point Green's function may be derived by differentiating Eq.(3.1) with
respect to the sources $J_\mu ^b(y),J_\lambda ^c(z)$ and $J_\tau ^d(u)$, as
given in the following$^{5}$ 
\begin{eqnarray}
&~&~\frac{1}{\alpha}\partial^{\mu}_xG^{abcd}_{\mu\nu\lambda\tau}(x,y,z,u) 
\nonumber \\
&~&=<0^+|T^*[\hat{\bar C^a}(x)\hat D^{be}_{\nu}(y)\hat C^e(y)\hat A%
^c_{\lambda}(z)\hat A^d_{\tau}(u)]|0^->  \nonumber \\
&~&+<0^+|T^*[\hat{\bar C^a}(x)\hat A^b_{\nu}(y)\hat D^{ce}_{\lambda}(z)\hat C%
^e(z)\hat A^d_{\tau}(u)]|0^->  \nonumber \\
&~&+<0^+|T^*[\hat{\bar C^a}(x)\hat A^b_{\nu}(y)\hat A^c_{\lambda}(z)\hat D%
^{de}_{\tau}(u)\hat C^e(u)]|0^->  \nonumber \\
\end{eqnarray}
where 
\begin{equation}
G_{\mu \nu \lambda \tau }^{abcd}(x,y,z,u)=<0^{+}|T[\hat A_\mu ^a(x)\hat A%
_\nu ^b(y)\hat A_\lambda ^c(z)\hat A_\tau ^d(u)]|0^{-}>
\end{equation}
is the gluon four-point Green's functions. The accompanying ghost equation
may be obtained by differentiating Eq.(3.2) with respect to the sources $%
J_\lambda ^c(z)$ and $J_\tau ^d(u)$. The result is 
\begin{eqnarray}
\partial^{\mu}_x<0^+|T^*[\hat D^{ae}_{\mu}(x)\hat C^e(x)\hat {\bar C^b}(y)%
\hat A^c_{\lambda}(z)\hat A^d_z(u)]|0^->  \nonumber \\
+{\mu}^2G^{abcd}_{~~\lambda\tau}(x,y,z,u)=-\delta^{ab}\delta^4(x-y)D^{cd}_{%
\lambda\tau}(z-u)
\end{eqnarray}
where 
\begin{equation}
G_{~~\lambda \tau }^{abcd}(x,y,z,u)=<0^{+}|T[\hat C^a(x)\hat {\bar C^b}(y)%
\hat A_\lambda ^c(z)\hat A_\tau ^d(u)]|0^{-}>
\end{equation}
is the four-point gluon-ghost particle Green's function. Differentiation of
Eq.(4.18) with respect to the coordinates y,z,u and use of Eq.(4.20) lead to 
\begin{eqnarray}
&~&~\frac{1}{\alpha}\partial^{\mu}_x\partial^{\nu}_y\partial^{\lambda}_z%
\partial^{\tau}_uG^{abcd}_{\mu\nu\lambda\tau}(x,y,z,u)
=\delta^{ab}\delta^4(x-y)\partial^{\lambda}_z\partial^{\tau}_uD^{cd}_{%
\lambda\tau}(z-u)  \nonumber \\
&~&+\delta^{ac}\delta^4(x-z)\partial^{\nu}_y\partial^{\tau}_uD^{bd}_{\nu%
\tau}(y-u)
+\delta^{ad}\delta^4(x-u)\partial^{\nu}_y\partial^{\lambda}_zD^{bc}_{\nu%
\lambda}(y-z)  \nonumber \\
&~&+\mu^2\{\partial^{\lambda}_z\partial^{\tau}_uG^{bacd}_{~~\lambda%
\tau}(y,x,z,u)
+\partial^{\nu}_y\partial^{\tau}_uG^{cabd}_{~~\nu\tau}(z,x,y,u)  \nonumber \\
&~&+\partial^{\nu}_y\partial^{\lambda}_zG^{dabc}_{~~\nu\lambda}(u,x,y,z)\}
\end{eqnarray}

It is noted that the four-point Green's functions appearing in the above
equations are unconnected. Their decompositions to connected Green's
functions are not difficult to be found by the conventional procedure $^5$.
The result is as follows 
\begin{eqnarray}
G_{\mu \nu \lambda \tau }^{abcd}(x,y,z,u) &=&G_{\mu \nu \lambda \tau
}^{abcd}(x,y,z,u)_c-D_{\mu \nu }^{ab}(x-y)D_{\lambda \tau }^{cd}(z-u) 
\nonumber \\
&&-D_{\mu \lambda }^{ac}(x-z)D_{\nu \tau }^{bd}(y-u)-D_{\mu \tau
}^{ad}(x-u)D_{\nu \lambda }^{bc}(y-z) \\
G_{~~\lambda \tau }^{abcd}(x,y,z,u) &=&G_{~~\lambda \tau
}^{abcd}(x,y,z,u)_c+\Delta ^{ab}(x-y)D_{\lambda \tau }^{cd}(z-u)
\end{eqnarray}
The first terms marked by the subscript ''c'' in Eqs.(4.23) and (4.24) are
connected Green's functions. When inserting Eqs.(4.23) and (4.24) into
Eq.(4.22) and using Eq.(3.5), one may find 
\begin{eqnarray}
\partial _x^\mu \partial _y^\nu \partial _z^\lambda \partial _u^\tau G_{\mu
\nu \lambda \tau }^{abcd}(x,y,z,u)_c &=&\alpha \mu ^2\{\partial _y^\nu
\partial _z^\lambda G_{~~\nu \lambda }^{dabc}(u,x,y,z)_c  \nonumber \\
+\partial _y^\nu \partial _u^\tau G_{~~\nu \tau }^{cabd}(z,x,y,u)_c
&+&\partial _z^\lambda \partial _u^\tau G_{~~\lambda \tau
}^{bacd}(y,x,z,u)_c\}
\end{eqnarray}
This is the W-T identity satisfied by the connected four-point Green's
functions. In the Landau gauge, we have 
\begin{equation}
\partial _x^\mu \partial _y^\nu \partial _z^\lambda \partial _u^\tau G_{\mu
\nu \lambda \tau }^{abcd}(x,y,z,u)_c=0
\end{equation}
which shows transversality of the Green's function.

The W-T identity for the four-line gluon proper vertex may be derived from
Eq.(4.25) with the help of the following irreducible decompositions of the
connected Green's functions$^{5}$. 
\begin{eqnarray}
&~&G^{abcd}_{\mu\nu\lambda\tau}(x_1,x_2,x_3,x_4)_c  \nonumber \\
&~&=\int\Pi^4_{i=1}d^4y_iD^{aa^{\prime}}_{\mu\mu
^{\prime}}(x_1-y_1)D^{bb^{\prime}}_{\nu\nu ^{\prime}}(x_2-y_2)\Gamma^{\mu
^{\prime}\nu ^{\prime}\lambda ^{\prime}\tau
^{\prime}}_{a^{\prime}b^{\prime}c^{\prime}d^{\prime}}(y_1,y_2,y_3,y_4) 
\nonumber \\
&~&\times D^{c^{\prime}c}_{\lambda
^{\prime}\lambda}(y_3-x_3)D^{d^{\prime}d}_{\tau ^{\prime}\tau}(y_4-x_4) 
\nonumber \\
&~&+i\int\Pi^3_{i=1}d^4y_id^4z_i\{D^{aa^{\prime}}_{\mu\mu
^{\prime}}(x_1-y_1)D^{bb^{\prime}}_{\nu\nu ^{\prime}}(x_2-y_2)\Gamma^{\mu
^{\prime}\nu ^{\prime}\rho}_{a^{\prime}b^{\prime}e}(y_1,y_2,y_3)  \nonumber
\\
&~&\times D^{ee^{\prime}}_{\rho\rho ^{\prime}}(y_3-z_1)\Gamma^{\rho
^{\prime}\lambda ^{\prime}\tau
^{\prime}}_{e^{\prime}c^{\prime}d^{\prime}}(z_1,z_2,z_3)D^{c^{\prime}c}_{%
\lambda ^{\prime}\lambda}(z_2-x_3)D^{d^{\prime}d}_{\tau
^{\prime}\tau}(z_3-x_4)  \nonumber \\
&~&+D^{aa^{\prime}}_{\mu\mu
^{\prime}}(x_1-y_1)D^{cc^{\prime}}_{\lambda\lambda
^{\prime}}(x_3-y_2)\Gamma^{\mu ^{\prime}\lambda
^{\prime}\rho}_{a^{\prime}c^{\prime}e}(y_1,y_2,y_3)D^{ee^{\prime}}_{\rho\rho
^{\prime}}(y_3-z_1)  \nonumber \\
&~&\times \Gamma^{\rho ^{\prime}\nu ^{\prime}\tau
^{\prime}}_{e^{\prime}b^{\prime}d^{\prime}}(z_1,z_2,z_3)D^{b^{\prime}b}_{\nu
^{\prime}\nu}(z_2-x_2)D^{d^{\prime}d}_{\tau ^{\prime}\tau}(z_3-x_4) 
\nonumber \\
&~&+D^{bb^{\prime}}_{\nu\nu
^{\prime}}(x_2-y_1)D^{cc^{\prime}}_{\lambda\lambda
^{\prime}}(x_3-y_2)\Gamma^{\nu ^{\prime}\lambda
^{\prime}\rho}_{b^{\prime}c^{\prime}e}(y_1,y_2,y_3)D^{ee^{\prime}}_{\rho\rho
^{\prime}}(y_3-z_1)  \nonumber \\
&~&\times \Gamma^{\rho ^{\prime}\mu ^{\prime}\tau
^{\prime}}_{e^{\prime}a^{\prime}d^{\prime}}(z_1,z_2,z_3)D^{a^{\prime}a}_{\mu
^{\prime}\mu}(z_2-x_1)D^{d^{\prime}d}_{\tau ^{\prime}\tau}(z_3-x_4)\}
\end{eqnarray}
and 
\begin{eqnarray}
&~&G^{abcd}_{~~\lambda\tau}(x_1,x_2,x_3,x_4)_c  \nonumber \\
&~&=-i\int\Pi^4_{i=1}d^4y_i\Delta^{aa^{\prime}}(x_1-y_1)\Gamma_{a^{%
\prime}b^{\prime}c^{\prime}d^{\prime}}^{~~~~\lambda ^{\prime}\tau
^{\prime}}(y_1,y_2,y_3,y_4)\Delta^{b^{\prime}b}(y_2-x_2)  \nonumber \\
&~&\times D^{c^{\prime}c}_{\lambda
^{\prime}\lambda}(y_3-x_3)D^{d^{\prime}d}_{\tau ^{\prime}\tau}(y_4-x_4) 
\nonumber \\
&~&+i\int\Pi^3_{i=1}d^4y_id^4z_i\{\Delta^{aa^{\prime}}(x_1-y_1)\Gamma_{a^{%
\prime}ed^{\prime}}^{~~~\tau
^{\prime}}(y_1,y_2,y_3)\Delta^{ee^{\prime}}(y_2-z_1)  \nonumber \\
&~&\times D_{\tau
^{\prime}\tau}^{d^{\prime}d}(y_3-x_4)\Gamma_{e^{\prime}b^{\prime}c^{%
\prime}}^{~~~~\lambda
^{\prime}}(z_1,z_2,z_3)\Delta^{b^{\prime}b}(z_2-x_2)D^{c^{\prime}c}_{\lambda
^{\prime}\lambda}(z_3-x_3)  \nonumber \\
&~&+\Delta^{aa^{\prime}}(x_1-y_1)\Gamma_{a^{\prime}ec^{\prime}}^{~~~\lambda
^{\prime}}(y_1,y_2,y_3)\Delta^{ee^{\prime}}(y_2-z_1)D^{c^{\prime}c}_{\lambda
^{\prime}\lambda}(y_3-x_3)  \nonumber \\
&~&\times \Gamma_{e^{\prime}b^{\prime}d^{\prime}}^{~~~\tau
^{\prime}}(z_1,z_2,z_3)\Delta^{b^{\prime}b}(z_2-x_2)D^{d^{\prime}d}_{\tau
^{\prime}\tau}(z_3-x_4)  \nonumber \\
&~&-\Delta^{aa^{\prime}}(x_1-y_1)\Gamma_{a^{\prime}b^{\prime}e}^{~~~%
\rho}(y_1,y_2,y_3)\Delta^{b^{\prime}b}(y_2-x_2)D^{ee^{\prime}}_{\rho\rho
^{\prime}}(y_3-z_1)  \nonumber \\
&~&\times \Gamma^{\rho ^{\prime}\lambda ^{\prime}\tau
^{\prime}}_{e^{\prime}c^{\prime}d^{\prime}}(z_1,z_2,z_3)D^{c^{\prime}c}_{%
\lambda ^{\prime}\lambda}(z_2-x_3)D^{d^{\prime}d}_{\tau
^{\prime}\tau}(z_3-x_4)\}
\end{eqnarray}
where the four-line proper vertices are defined as $^{4,5}$ 
\begin{eqnarray}
\Gamma^{abcd}_{\mu\nu\lambda\tau}(x_1,x_2,x_3,x_4)=i\frac{\delta^4\Gamma} {%
\delta A^{a\mu}(x_1)\delta A^{b\nu}(x_2)\delta A^{c\lambda}(x_3)\delta
A^{d\tau}(x_4)}|_{J=0} \\
\Gamma^{abcd}_{~~\lambda\tau}(x_1,x_2,x_3,x_4)=\frac{\delta^4\Gamma} {\delta 
\bar C^{a}(x_1)\delta C^{b}(x_2)\delta A^{c\lambda}(x_3)\delta A^{d\tau}(x_4)%
}|_{J=0}
\end{eqnarray}
When substituting Eqs.(4.27) and (4.28) into Eq.(4.25) and transforming
Eq.(4.25) into the momentum space, one can find the following identity
satisfied by the gluon four-line proper vertex 
\begin{eqnarray}
&~&~k^{\mu}_1k^{\nu}_2k^{\lambda}_3k^{\tau}_4\Lambda^{abcd}_{\mu\nu\lambda%
\tau}(k_1,k_2,k_3,k_4) =\Psi\left(\matrix{a&b&c&d\cr k_1&k_2&k_3&k_4\cr}%
\right)  \nonumber \\
&~&+\Psi\left(\matrix{a&c&d&b\cr k_1&k_3&k_4&k_2\cr}\right) +\Psi\left(%
\matrix{a&d&b&c\cr k_1&k_4&k_2&k_3\cr}\right)
\end{eqnarray}
where 
\begin{eqnarray}
&~&~\Psi\left(\matrix{a&b&c&d\cr k_1&k_2&k_3&k_4\cr}\right)  \nonumber \\
&~&=-ik^{\mu}_1k^{\nu}_2\Lambda^{abe}_{\mu\nu\sigma}(k_1,k_2,-(k_1+k_2))D^{%
\sigma\rho}_{ef}(k_1+k_2)  \nonumber \\
&~&\times
k^{\lambda}_3k^{\tau}_4\Lambda^{fcd}_{\rho\lambda\tau}(-(k_3+k_4),k_3,k_4) 
\nonumber \\
&~&+\frac{i\mu^2}{\alpha}\chi (k^2_1)\chi
(k^2_2)[k^{\lambda}_3k^{\tau}_4\Lambda^{bacd}_{~~\lambda%
\tau}(k_2,k_1,k_3,k_4)  \nonumber \\
&~&+\Lambda^{bae}_{~~\sigma}(k_2,k_1,-(k_1+k_2))D^{\sigma%
\rho}_{ef}(k_1+k_2)k^{\lambda}_3k^{\tau}_4\Lambda^{fcd}_{\rho\lambda%
\tau}(-(k_3+k_4),k_3,k_4)  \nonumber \\
&~&-k^{\tau}_4\Lambda^{bed}_{~~\tau}(k_2,-(k_2+k_4),k_4)%
\Delta^{ef}(k_2+k_4)k^{\lambda}_3\Lambda^{fac}_{~~%
\lambda}(-(k_1+k_3),k_1,k_3)  \nonumber \\
&~&-k^{\lambda}_3\Lambda^{bec}_{~~\lambda}(k_2,-(k_2+k_3),k_3)%
\Delta^{ef}(k_2+k_3)  \nonumber \\
&~&\times k^{\tau}_4\Lambda^{fad}_{~~~\tau}(-(k_1+k_4),k_1,k_4)]
\end{eqnarray}
The second and third terms in Eq.(4.31) can be written out from Eq.(4.32)
through cyclic permutations. In the above, we have defined 
\begin{eqnarray}
\Gamma^{abcd}_{\mu\nu\lambda\tau}(k_1,k_2,k_3,k_4)=(2\pi)^4\delta^4(%
\sum^4_{i=1}k_i)\Lambda^{abcd}_{\mu\nu\lambda\tau}(k_1,k_2,k_3,k_4) \\
\Gamma^{abcd}_{~~\lambda\tau}(k_1,k_2,k_3,k_4)=(2\pi)^4\delta^4(%
\sum^4_{i=1}k_i)\Lambda^{abcd}_{~~\lambda\tau}(k_1,k_2,k_3,k_4)
\end{eqnarray}
In the lowest order approximation, we have checked that except for the first
term in Eq.(4.32) which was encountered in the massless theory, the
remaining mass-dependent terms are cancelled out with the corresponding
terms contained in the second and third terms in Eq.(4.31). Therefore, the
identity in Eq.(4.31) leads to a result in the lowest order approximation
which is consistent with the Feynman rule.

Lastly, let us discuss the renormalization of the vertices mentioned before.
At first, we note that the renormalization of the gluon and ghost particle
propagators as described in Eqs.(3.24)-(3.27) implies that the
renormalization relations for the field variables should be as follows 
\begin{eqnarray}
A^{a\mu }(x) &=&\sqrt{Z_3}A_R^{a\mu }(x) \\
C^a(x) &=&\sqrt{\tilde Z_3}C_R^a(x),\bar C^a(x)=\sqrt{\tilde Z_3}\bar C%
_R^a(x)
\end{eqnarray}
The relation in Eq.(4.35) is dominantly written for the transverse part of
the vector potential. In the next paper, it will be proved that in general
gauges, we have $A_R^{a\mu }(x)=A_{TR}^{a\mu }(x)+\sqrt{Z_3^{\prime }}%
A_{LR}^{a\mu }(x)$ where the longitudinal part of the vector potential is
renormalized in a different fashion characterized by an extra
renormalization constant $\sqrt{Z_3^{\prime }}$ (refer to Ref.(4), Chap.V);
while, in Landau gauge, we have $A_R^{a\mu }(x)=A_{TR}^{a\mu }(x)$.
According to the definitions given in Eqs.(4.9) and (4.10) and the relation
shown in Eqs.(4.35) and (4.36), the renormalization transformations for the
three-line proper vertices can immediately be written as 
\begin{eqnarray}
\Lambda _{\mu \nu \lambda }^{abc}(p,q,k) &=&Z_3^{-3/2}\Lambda _{R\mu \nu
\lambda }^{~~abc}(p,q,k) \\
\Lambda _{~~\lambda }^{abc}(p,q,k) &=&\tilde Z_3^{-1}Z_3^{-1/2}\Lambda
_{R~~\lambda }^{~~abc}(p,q,k)
\end{eqnarray}
Applying these relations. the renormalized version of the identity written
in Eq.(4.11) will be 
\begin{eqnarray}
p^\mu q^\nu k^\lambda \Lambda _{R\mu \nu \lambda }^{~~abc}(p,q,k) &=&\frac{%
\mu _R^2}{\tilde \alpha _R}{\chi }_R(p^2)[{\chi _R}(k^2)q^\nu \Lambda
_{R~~\nu }^{~~cab}(k,p,q)  \nonumber \\
&&\ \ \ +\chi _R(q^2)k^\lambda \Lambda _{R~~\lambda }^{~~bac}(q,p,k)]
\end{eqnarray}
where $\tilde \alpha _R=Z_3^{\prime }\alpha _R$ and 
\begin{equation}
\chi _R(k^2)=\frac{\tilde Z_3}{\tilde Z_3+(1-\mu _R^2/k^2)[1-\tilde Z%
_3+\omega _R(k^2)]}
\end{equation}
is the renormalized expression of the function $\chi (k^2)$ which is
obtained by substituting Eq.(3.21) into Eq.(4.14) and then using Eqs.(3.22)
and (3.23). At the renormalization point chosen to be $p^2=q^2=k^2=\mu _R^2$%
, we see, $\chi _R(\mu _R^2)=1$. In this case, when the renormalized ghost
vertex takes the form of the bare vertex, the RHS of Eq.(4.39) vanishes.
therefore, Eq.(4.39) becomes 
\begin{equation}
p^\mu q^\nu k^\lambda \Lambda _{R\mu \nu \lambda
}^{~~abc}(p,q,k)|_{p^2=q^2=k^2=\mu _R^2}=0
\end{equation}
Ordinarily, we are interested in discussing the renormalization of such
three-line vertices that which are defined by extracting a coupling constant
g from the vertices defined in Eqs.(4.9) and (4.10). These vertices are
denoted by $\tilde \Lambda _{\mu \nu \lambda }^{abc}(p,q,k)$ and $\tilde 
\Lambda _{~~\lambda }^{abc}(p,q,k)$. Commonly, they are renormalized in such
a fashion$^{4,5}$. 
\begin{equation}
\tilde \Lambda _{\mu \nu \lambda }^{abc}(p,q,k)=Z_1^{-1}\widetilde{\Lambda }%
_{R\mu \nu \lambda }^{abc}(p,q,k)
\end{equation}
\begin{equation}
\tilde \Lambda _{~~\lambda }^{abc}(p,q,k)=\widetilde{Z}_1^{-1}\widetilde{%
\Lambda }_{R~~\lambda }^{abc}(p,q,k)
\end{equation}
where $Z_1$ and $\tilde Z_1$ are referred to as the renormalization
constants for the gluon three-line vertex and the ghost vertex,
respectively. It is clear that the W-T identity shown in Eq.(4.11) also
holds for the vertices $\tilde \Lambda _{\mu \nu \lambda }^{abc}(p,q,k)$ and 
$\tilde \Lambda _{~~\lambda }^{abc}(p,q,k)$. So, when the vertices $\Lambda
_{\mu \nu \lambda }^{abc}(p,q,k)$ and $\Lambda _{~~\lambda }^{abc}(p,q,k)$
in Eqs(4.11) are replaced by $\tilde \Lambda _{\mu \nu \lambda }^{abc}(p,q,k)
$ and $\tilde \Lambda _{~~\lambda }^{abc}(p,q,k)$ respectively and then
Eqs.(4,42) and (4.43) are inserted to such an identity, we obtain a
renormalized version of the identity as follows 
\begin{eqnarray}
p^\mu q^\nu k^\lambda \tilde \Lambda _{R\mu \nu \lambda }^{~~abc}(p,q,k) &=&%
\frac{Z_1\tilde Z_3}{Z_3\tilde Z_1}\frac{\mu _R^2}{\tilde \alpha _R}\chi
_R(p^2)[\chi _R(k^2)  \nonumber \\
&&\ \ \ \times q^\nu \tilde \Lambda _{R~~\nu }^{~~cab}(k,p,q)+\chi
_R(q^2)k^\lambda \tilde \Lambda _{R~~\lambda }^{~~bac}(q,p,k)]
\end{eqnarray}
When multiplying the both sides of Eq.(4.44) with a renormalized coupling
constant $g_R$ and absorbing it in the vertices, noticing 
\[
\Lambda _{R\mu \nu \lambda }^{abc}(p,q,k)=g_R\widetilde{\Lambda }_{R\mu \nu
\lambda }^{abc}(p,q,k)
\]
\begin{equation}
\Lambda _{R~~\lambda }^{abc}(p,q,k)=g_R\widetilde{\Lambda }_{R~~\lambda
}^{abc}(p,q,k)
\end{equation}
we have 
\begin{eqnarray}
p^\mu q^\nu k^\lambda \Lambda _{R\mu \nu \lambda }^{~~abc}(p,q,k) &=&\frac{%
Z_1\tilde Z_3}{Z_3\tilde Z_1}\frac{\mu _R^2}{\tilde \alpha _R}\chi
_R(p^2)[\chi _R(k^2)  \nonumber \\
&&\ \ \ \ \times q^\nu \Lambda _{R~~\nu }^{~~cab}(k,p,q)+\chi
_R(q^2)k^\lambda \Lambda _{R~~\lambda }^{~~bac}(q,p,k)]
\end{eqnarray}
In comparison of Eq.(4.46) with Eq.(4.39), we see, except for the factor $Z_1%
\tilde Z_3Z_3^{-1}\tilde Z_1^{-1}$, the both identities are identical to
each other. From this observation, we deduce 
\begin{equation}
\frac{Z_1}{Z_3}=\frac{\tilde Z_1}{\tilde Z_3}
\end{equation}
This is the Slavnov-Taylor (S-T) identity we are familiar with in the
massless gauge field theory$^7$.

For the renormalization of the four-line vertices, discussions are similar
to those for the three-line vertices. From the definitions given in
Eqs.(4.29), (4.30), (4.35) and (4.36), it is clearly seen that the four-line
vertices should be renormalized in such a manner 
\begin{eqnarray}
\Lambda _{\mu \nu \lambda \tau }^{abcd}(k_1,k_2,k_3,k_4) &=&Z_3^{-2}\Lambda
_{R\mu \nu \lambda \tau }^{~~abcd}(k_1,k_2,k_3,k_4) \\
\Lambda _{~~\lambda \tau }^{abcd}(k_1,k_2,k_3,k_4) &=&\tilde Z%
_3^{-1}Z_3^{-1}\Lambda _{R~~\lambda \tau }^{~~abcd}(k_1,k_2,k_3,k_4)
\end{eqnarray}
On inserting these relations into Eqs.(4.31) and (4.32), one may obtain a
renormalized identity similar to Eq.(4.39), that is 
\begin{eqnarray}
\ &~&~k_1^\mu k_2^\nu k_3^\lambda k_4^\tau \Lambda _{R\mu \nu \lambda \tau
}^{abcd}(k_1,k_2,k_3,k_4)=\Psi _R\left( \matrix{a&b&c&d\cr
k_1&k_2&k_3&k_4\cr}\right)  \nonumber \\
&&\ ~+\Psi _R\left( \matrix{a&c&d&b\cr k_1&k_3&k_4&k_2\cr}\right) +\Psi
_R\left( \matrix{a&d&b&c\cr k_1&k_4&k_2&k_3\cr}\right)
\end{eqnarray}
\begin{eqnarray}
\ &&~  \nonumber \\
&&\ 
\end{eqnarray}
where 
\begin{eqnarray}
&&\ \ ~~\Psi _R\left( \matrix{a&b&c&d\cr k_1&k_2&k_3&k_4\cr}\right) 
\nonumber \\
\ &&~=-ik_1^\mu k_2^\nu \Lambda _{R\mu \nu \sigma
}^{abe}(k_1,k_2,-(k_1+k_2))D_{Ref}^{\sigma \rho }(k_1+k_2)  \nonumber \\
&&\ \ ~\times k_3^\lambda k_4^\tau \Lambda _{R\rho \lambda \tau
}^{fcd}(-(k_3+k_4),k_3,k_4)  \nonumber \\
&&\ \ ~+\frac{i\mu _R^2}{\widetilde{\alpha }_R}\chi _R(k_1^2)\chi
_R(k_2^2)[k_3^\lambda k_4^\tau \Lambda _{R~~\lambda \tau
}^{bacd}(k_2,k_1,k_3,k_4)  \nonumber \\
&&\ \ ~+\Lambda _{R~~\sigma }^{bae}(k_2,k_1,-(k_1+k_2))D_{Ref}^{\sigma \rho
}(k_1+k_2)k_3^\lambda k_4^\tau \Lambda _{R\rho \lambda \tau
}^{fcd}(-(k_3+k_4),k_3,k_4)  \nonumber \\
&&\ \ ~-k_4^\tau \Lambda _{R~~\tau }^{bed}(k_2,-(k_2+k_4),k_4)\Delta
_R^{ef}(k_2+k_4)k_3^\lambda \Lambda _{R~~\lambda }^{fac}(-(k_1+k_3),k_1,k_3)
\nonumber \\
&&\ \ ~-k_3^\lambda \Lambda _{R~~\lambda }^{bec}(k_2,-(k_2+k_3),k_3)\Delta
_R^{ef}(k_2+k_3)  \nonumber \\
&&\ \ ~\times k_4^\tau \Lambda _{R~~~\tau }^{fad}(-(k_1+k_4),k_1,k_4)]
\end{eqnarray}
\[
\]
We can also define the vertices $\tilde \Lambda _{\mu \nu \lambda \tau
}^{abcd}(k_1,k_2,k_3,k_4)$ and $\tilde \Lambda _{~~\lambda \tau
}^{abcd}(k_1,k_2,k_3,k_4)$ by taking out the coupling constant squared from
the vertices $\Lambda _{\mu \nu \lambda \tau }^{abcd}(k_1,k_2,k_3,k_4)$ and $%
\Lambda _{~~\lambda \tau }^{abcd}(k_1,k_2,k_3,k_4)$, respectively. The
renormalization of these vertices are usually defined by$^{4,5}$ 
\begin{eqnarray}
\tilde \Lambda _{\mu \nu \lambda \tau }^{abcd}(k_1,k_2,k_3,k_4) &=&Z_4^{-1}%
\tilde \Lambda _{R\mu \nu \lambda \tau }^{~~abcd}(k_1,k_2,k_3,k_4) \\
\tilde \Lambda _{~~\lambda \tau }^{abcd}(k_1,k_2,k_3,k_4) &=&\tilde Z_4^{-1}%
\tilde \Lambda _{R~~\lambda \tau }^{~~abcd}(k_1,k_2,k_3,k_4)
\end{eqnarray}
Obviously, the identity in Eqs.(4.31) and (4.32) remains formally unchanged
if we replace all the vertices $\Lambda _i$ in the identity with the ones $%
\widetilde{\Lambda _i}$ .Substituting Eqs.(3.24), (3.25), (4.42), (4.43),
(4.50) and (4.51) into such an identity, one may write a renormalized
identity similar to Eq.(4.44), that is 
\begin{eqnarray}
\ &~&~k_1^\mu k_2^\nu k_3^\lambda k_4^\tau \widetilde{\Lambda }_{R\mu \nu
\lambda \tau }^{abcd}(^{}k_1,k_2,k_3,k_4)=\widetilde{\Psi }_R\left( %
\matrix{a&b&c&d\cr k_1&k_2&k_3&k_4\cr}\right)  \nonumber \\
&&\ \ ~+\widetilde{\Psi }_R\left( \matrix{a&c&d&b\cr k_1&k_3&k_4&k_2\cr}%
\right) +\widetilde{\Psi }_R\left( \matrix{a&d&b&c\cr k_1&k_4&k_2&k_3\cr}%
\right)
\end{eqnarray}

where 
\begin{eqnarray}
&&\ \ \ ~~\widetilde{\Psi }_R\left( \matrix{a&b&c&d\cr k_1&k_2&k_3&k_4\cr}%
\right)   \nonumber \\
\  &&~=\frac{Z_4Z_3}{Z_{1^{}}^2}\{-ik_1^\mu k_2^\nu \widetilde{\Lambda }%
_{R\mu \nu \sigma }^{abe}(k_1,k_2,-(k_1+k_2))D_{Ref}^{\sigma \rho }(k_1+k_2)
\nonumber \\
&&\ \ \ ~\times k_3^\lambda k_4^\tau \widetilde{\Lambda }_{R\rho \lambda
\tau }^{fcd}(-(k_3+k_4),k_3,k_4)\}  \nonumber \\
&&\ \ \ ~+\frac{i\mu _R^2}{\widetilde{\alpha }_R}\chi _R(k_1^2)\chi
_R(k_2^2)\{\frac{\widetilde{Z}_3Z_4}{Z_3\widetilde{Z_4}}k_3^\lambda k_4^\tau 
\widetilde{\Lambda }_{R~~\lambda \tau }^{bacd}(k_2,k_1,k_3,k_4)  \nonumber \\
&&\ \ \ ~+\frac{Z_4\widetilde{Z}_3}{Z_1\widetilde{Z_1}}\widetilde{\Lambda }%
_{R~~\sigma }^{bae}(k_2,k_1,-(k_1+k_2))D_{Ref}^{\sigma \rho
}(k_1+k_2)k_3^\lambda k_4^\tau \widetilde{\Lambda }_{R\rho \lambda \tau
}^{fcd}(-(k_3+k_4),k_3,k_4)  \nonumber \\
&&\ \ \ ~-\frac{Z_4\widetilde{Z}_3^2}{Z_3\widetilde{Z}_1^2}[k_4^\tau 
\widetilde{\Lambda }_{R~~\tau }^{bed}(k_2,-(k_2+k_4),k_4)\Delta
_R^{ef}(k_2+k_4)k_3^\lambda \widetilde{\Lambda }_{R~~\lambda
}^{fac}(-(k_1+k_3),k_1,k_3)  \nonumber \\
&&\ \ \ ~+k_3^\lambda \widetilde{\Lambda }_{R~~\lambda
}^{bec}(k_2,-(k_2+k_3),k_3)\Delta _R^{ef}(k_2+k_3)  \nonumber \\
&&\ \ \ ~\times k_4^\tau \widetilde{\Lambda }_{R~~~\tau
}^{fad}(-(k_1+k_4),k_1,k_4)]\}
\end{eqnarray}
\[
\]
Multiplying the both sides of Eq.(4.52) by $g_R^2$, according to the
relations given in Eqs.(4.45) and in the following 
\begin{eqnarray}
\Lambda _{R\mu \nu \lambda \tau }^{abcd}(k_{1,}k_2,k_3,k_4) &=&g_R^2\tilde 
\Lambda _{R\mu \nu \lambda \tau }^{~~abcd}(k_1,k_2,k_3,k_4)  \nonumber \\
\Lambda _{R~~\lambda \tau }^{abcd}(k_1,k_2,k_3,k_4) &=&g_{R^{}}^2\widetilde{%
\Lambda }_{R~~\lambda \tau }^{~~abcd}(k_1,k_2,k_3,k_4)
\end{eqnarray}
\[
\]
we have an identity which is of the same form as the identity in Eqs.(4.52)
and (4.53) except that the vertices $\widetilde{\Lambda _R^i}$ in Eqs.(4.52)
and (4.53) are all replaced by the vertices $\Lambda _R^i$. Comparing this
identity with that written in Eqs (4.48) and (4.49), one may find 
\begin{equation}
\frac{Z_3Z_4}{Z_{1^{}}^2}=1,\frac{\widetilde{Z}_3Z_4}{Z_3\widetilde{Z}_4}=1,%
\frac{Z_4\widetilde{Z}_3}{Z_1\widetilde{Z}_1}=1,\frac{Z_4\widetilde{Z}%
_{3^{}}^2}{Z_3\widetilde{Z}_1^2}=1
\end{equation}
which lead to 
\begin{equation}
\frac{Z_1}{Z_3}=\frac{\widetilde{Z}_1}{\widetilde{Z}_3}=\frac{Z_4}{Z_1},%
\frac{Z_1}{\widetilde{Z}_1}=\frac{Z_3}{\widetilde{Z}_3}=\frac{Z_4}{%
\widetilde{Z}_4}
\end{equation}
This just is the S-T identity as found previously in the massless theory$^7$.

\setcounter{section}{5}

\section*{5.Remarks}

~~~~From the previous sections, it is clearly seen that the W-T identities
held for the massive gauge field theory are different from the corresponding
ones for the massless theory in some mass-related terms. However, in the
zero-mass limit, all the W-T identities naturally go over to the massless
results. This fact shows the consistency of the theory built up in the
former paper. It is interesting that in the Landau gauge, all the W-T
identities satisfied by the generating functionals and Green's functions
formally are identical to those given in the massless theory. For the
massive gauge fields, the Landau gauge is the truly physical gauge which we
need to work in only in practical calculations. In this gauge, the gauge
boson longitudinal excitation disappears ,but the residual gauge degrees of
freedom still exist and need to be counterbalanced by the ghost terms
included in the theory. Even though the Landau gauge is physical, the other
gauges, are still of advantage for theoretical development. For instance, in
the derivations of Eq.(4.11) from Eq.(4.4) and Eq.(4.31) from Eq.(4.25), we
have used the inverse of the gluon propagator. The propagator has its
inverse only in the general gauge ($\alpha \ne 0$). As shown in Sect.3 of
paper I, the general gauges arise from the extension of the Lorentz
condition:$\partial ^\mu A_\mu ^a+\alpha \lambda ^a=0$. Such an extension
allows us in a natural way to put the Lorentz condition into the effective
action, giving the gauge-fixing term in the action. The general gauge can be
viewed as an approximation to the Lorentz gauge in the sense of the limit:$%
\alpha \to 0$. Therefore, in order to get results in the Landau gauge, we
may firstly derive the corresponding results in the general gauge, then come
to the Landau gauge by taking the above limit.

~~~~Another point we would like to emphasize is that for the massive gauge
field theory, the ghost particle acquires a spurious mass $\mu ^2$ in
general gauges. This mass term is necessary to be introduced so as to
compensate the gauge-non-invariance of the gauge boson mass term and
preserve the action to be gauge-invariant. As we see, the occurrence of this
mass term in the theory is essential to guarantee the theory to be
self-consistent. Otherwise, for example, if lack of this mass term in the
ghost particle propagator, the W-T identities shown in Eqs.(4.11) and
(4.31)-(4.32), which are derived from Eqs.(4.4) and (4.25) respectively,
will have different expressions. In the lowest order approximation, these
expressions can not be converted to the results which coincide with the
Feynman rules. At this point, we can say, some previous works$^{4,9-13}$ are
unreasonable and even not correct because these theories did not give the
Lagrangian a ghost particle mass term in the general gauges.

~~~~At last, we make some remarks on the BRST external source terms
introduced in Eq.(2.11). Ordinarily, to guarantee the BRST-invariance of the
source terms, the composite field functions are required to have the
nilpotency property $\delta \Delta \Phi _i=0$ under the
BRST-transformations. For the massless gauge field theory, as one knows, the
composite field functions are indeed nilpotent. This nilpotency property is
still preserved for the massive gauge field theory established in the
physical Landau gauge because in this gauge the BRST- transformations are
the same as for the massless theory. However, for the massive gauge field
theory set up in the general gauges. we find $\delta \Delta \Phi _i$ $\neq 0$%
, the nilpotency loses, since in these gauges the ghost field acquires a
spurious mass $\mu .$ In this case, as pointed out in Sect.2, to ensure the
BRST-invariance of the source terms, we may simply require the sources $u_i$
to satisfy the condition denoted in Eq.(2.14).. The definition in Eq.(2.14)
for the sources is reasonable. Why say so? Firstly, we note that the
original W-T identity formulated in Eq.(2.9) does not involve the BRST-
sources. This identity is suitable to use in practical applications.
Introduction of the BRST source terms in the generating functional is only
for the purpose of representing the identity in Eq.(2.9) in a convenient
form, namely, to represent the composite field functions in the identity in
terms of the differentials of the generating functional with respect to the
corresponding sources. For this purpose, we may start from the generating
functional defined in Eq.(2.11) to rederive the identity in Eq.(2.9). In
doing this, it is necessary to require the source terms $u_i\triangle \Phi
_i $ to be BRST-invariant so as to make the derived identity coincide with
the identity given in Eq.(2.9). How to ensure the source terms to be
BRST-invariant? If the composite field functions $\triangle \Phi _i$ are
nilpotent under the BRST-transformation, $\delta \Delta \Phi _i=0$, the
BRST-invariance of the source terms is certainly guaranteed. Nevertheless,
the nilpotency of the functions $\triangle \Phi _i$ is not an uniquely
necessary condition to ensure the BRST- invariance of the source terms,
particularly , in the case that the functions $\triangle \Phi _i$ are not
nilpotent. In the latter case, considering that under the BRST-
transformations, the functions $\triangle \Phi _i$ can be , in general,
expressed as $\delta \Delta \Phi _i=\xi \widetilde{\Phi }_i$ where the $%
\widetilde{\Phi }_i$ are some nonvanishing functions, we may alternatively
require the sources $u_i$ to satisfy the condition shown in Eq.(2.14) so as
to guarantee the source terms to be BRST- invariant. Actually, this is a
general trick to make the source terms to be BRST-invariant in spite of
whether the functions $\triangle \Phi _i$ are nilpotent or not. As mentioned
before, the sources themselves have no physical meaning. They are, as a
mathematical tool, introduced into the generating functional just for
performing the differentiations. For this purpose, only a certain
algebraically and analytical properties of the sources are necessarily
required. Particularly, In the differentiations, only the infinitesimal
property of the sources are concerned. Therefore, the sources defined above
are mathematically suitable for the purpose of introducing them. The
reasonability of the arguments stated above for the source terms introduced
into the generating functional is substantiated by the correctness of the
W-T identities derived in Sect.2. Even though these identities are
represented in terms of the BRST-sources, they give rise to correct
relations among the Green functions or the vertices as shown in sections (3)
and (4). For example, the correctness of the identity written in Eq.(3.8)
may easily be verified by the free propagators shown in Eqs.(3.10) and (3
14). These propagators were derived in paper I by employing the perturbation
method, without concerning the BRST source terms and the nilpotency of the
BRST- transformations. Similarly, the W-T identities derived in Sect.(4) for
the vertices all automatically reduce to the lowest order results derived in
paper I perturbatively, exhibiting the theoretical consistency of the
identities derived. A powerful argument of proving the correctness of the
way of introducing the BRST-sources is that after completing the
differentiations in Eq.(2.15) and setting the BRST-sources to vanish, we
immediately obtain the W-T identity in Eq.(2.9) which is irrelevant to the
BRST-sources. Therefore, all identities or relations derived from the W-T
identity in Eq.(2.15) are completely the same as those derived from the
identity in Eq.(2.9). An important example of showing this point will be
presented in the next paper where an identity used to prove the unitarity of
our theory will be derived respectively from the W-T identities shown in
Eqs.(2.9) and (2.15).

\begin{center}
{\bf ACKNOWLEDGMENTS }
\end{center}

The author is deeply indebted to professor Shi-Shu Wu and professor Ze-Sen
Yang (Beijing University) for helpful discussions. This work is supported in
part by National Natural Science Foundation of China.

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\end{document}

