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\begin{document}
\title{Massive Gauge Field Theory Without Higgs Mechanism\\
III. Proofs of Unitarity and Renormalizability}
\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}
~~It is proved that the quantum massive gauge field theory established in
the former papers is renormalizable. This conclusion is achieved with the
aid of the Ward-Takahashi identities satisfied by the generating functionals
which were derived in the preceding paper based on the BRST-symmetry of the
theory. By the use of the Ward-Takahashi identity, it is shown that the
S-matrix elements evaluated by the massive gauge field theory is
gauge-independent and hence unitary. This fact allows us to prove the
renormalizability of the theroy firstly in the physical Landau gauge and
then extend the proof to the other gauges. As a result of the proof, it is
found that the renormalization constants for the massive gauge field theory
comply with the same identity as that for the massless gauge field theory.

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

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\section*{1.INTRODUCTION}

~~~As we mentioned in the first paper (which will be referred to as paper I
hereafter), in the original attempts of setting up the massive gauge field
theory without the Higgs mechanism$^{1-6}$, the massive Yang-Mills
Lagrangian, i.e. the Yang-Mills Lagrangian with the mass term added was
regarded as to form a complete theoretical basis. In the theory, there are
two problems which were announced to be difficult to solve: one is the
gauge-non--invariance of the mass term in the action, another is the
nonrenormalizability of the theory. To evade the gauge-noninvariance of the
mass term, a formalism in which the mass term is given a gauge-invariant
form by taking the advantage of the Stueckelberg field was subsequently
proposed and attracted most attentions$^{7-10}$. However, it was argued that
the theory could not preserve the unitarity and renormalizability at the
same time$^{10}$. In paper I, the first problem has readily been
circumvented from the viewpoint that the massive gauge field only exists in
the physical space spanned by the transverse part of the vector potential.
In this space, the gauge boson mass term of the action is gauge-invariant.
If we want to represent the massive gauge field dynamics in the whole space
of the vector potential, the massive gauge field must be viewed as a
constrained system. The Lorentz gauge condition, acting as a constraint,
should initially be introduced and imposed on the Lagrangian expressed by
the full vector potential. From this point of view, it has been shown that
the massive gauge field theory can well be established on the basis of gauge
invariance.

In this paper, we are devoted to proving the renormalizability and unitarity
of the quantum massive gauge field theory described in paper I. From the
Feynman rules mentioned in Sect.4 of paper I, we have felt confident of that
the theory is renormalizable because the free massive gauge boson propagator
and the ghost particle one have the same behavior as the massless ones in
the ultraviolet limit and except for the propagators, the other Feynman
rules for the vertices are identical to those for the massless gauge field
theory. Theoretically, to give a rigorous proof of the renormalizability, we
need to utilize the Ward-Takahashi (W-T) identities satisfied by various
generating functionals. These identities have been derived in the second
paper (referred to as paper II later on) based on the BRST-invariance of the
theory. Before proving the renormalizability, we firstly examine whether the
S-matrix given by the massive gauge field theory is unitary? In Sect.2,
based on the W-T identity, it will be proved that the S-matrix is
independent of the gauge parameter. That is to say, the gauge-dependent
spurious pole appearing in the ghost particle propagator and the
longitudinal part of the gauge boson propagator would not contribute to the
S-matrix. This fact just ensures the unitarity of the S-matrix. The
gauge-independence of the S-matrix suggests that it is sufficient to prove
the renormalizability of the theory in one gauge. The Landau gauge is
favorable to be chosen in the proof of the renormalizability because in this
gauge, the W-T identities have the same forms as in the massless gauge field
theory. Therefore, we may simply cite the reasoning given in the latter
theory$^{6,11,12}$ to complete the proof as will be described in Sect.3 and
the results obtained can directly compare with the massless theory. Then, we
discuss how the proof can be extend to other gauges. From the proof, it will
be seen that the divergences occurring in perturbation calculations can
surely be eliminated by introducing a finite number of counterterms. As a
consequence of the proof. it will be found that the Slavnov-Taylor identity$%
^{13}$ for the renormalization constants which was derived in the massless
gauge field theory also holds for the massive gauge field theory. In the
last section we will comment on the problem of unrenormalizability raised in
the previous works and make some conclusions. In Appendix, to verify the
correctness of the W-T identity used in this paper to prove the unitarity of
theory, we will present a different derivation of the identity. %
\setcounter{section}{2}

\section*{2.Unitarity}

\setcounter{equation}{0}

~~~This section is used to prove the unitarity of the S-matrix calculated by
the quantum massive gauge field theory. For this purpose, it is necessary to
prove that the S-matrix is independent of the gauge parameter$^{6,11,12}$.
Let us start from the generating functional given in Eq.(2.7) in paper II.
It is well-known that the fermion field could not spoil the unitarity of the
theory because all components of the field are independent and physical.
Therefore, to prove the unitarity, we may, for simplicity of statement, omit
the fermion field functions in the generating functional and rewrite the
functional in the form 
\begin{eqnarray}
Z[J,\bar K,K] &=&\frac 1N\int {\cal D}[A.\bar C.C]exp\{iS  \nonumber \\
&&+i\int d^4x[-\frac 1{2\alpha }(\partial ^\mu A_\mu ^a)^2+J^{a\mu }A_\mu ^a+%
\bar K^aC^a+\bar C^aK^a]  \nonumber \\
&&+i\int d^4xd^4y\bar C^a(x)M^{ab}(x,y)C^b(y)\}
\end{eqnarray}
where 
\begin{equation}
S=\int d^4x[-\frac 14F^{a\mu \nu }F_{\mu \nu }^a+\frac 12m^2A^{a\mu }A_\mu^a]
\end{equation}
and 
\begin{equation}
M^{ab}(x,y)=\partial _x^\mu [{\cal D}_\mu ^{ab}(x)\delta ^4(x-y)]
\end{equation}
in which 
\begin{equation}
{\cal D}_\mu ^{ab}(x)=\delta ^{ab}(1+\frac{\mu ^2}{\Box })\partial _\mu
-gf^{abc}A_\mu ^c
\end{equation}
When we make the following translation transformations in Eq.(2.1) 
\begin{eqnarray}
C^a(x) &\to &C^a(x)-\int d^4y(M^{-1})^{ab}(x,y)K^b(y)  \nonumber \\
\bar C^a(x) &\to &\bar C^a(x)-\int d^4y\bar K^b(y)(M^{-1})^{ba}(y,x)
\end{eqnarray}
and complete the integration over the ghost field variables, Eq.(2.1) will
be expressed as $^{11,12}$ 
\begin{equation}
Z[J,\bar K,K]=e^{-i\int d^4xd^4y\bar K^a(x)(M^{-1})^{ab}(x,y,\delta /i\delta
J)K^b(y)}Z[J]
\end{equation}
where $Z[J]$ is the generating functional without the external sources of
the ghost field. 
\begin{equation}
Z[J]=\frac 1N\int {\cal D}(A)\Delta _F[A]exp\{iS+i\int d^4x[-\frac 1{2\alpha 
}(\partial ^\mu A_\mu ^a)^2+J^{a\mu }A_\mu ^a]\}
\end{equation}
in which 
\begin{equation}
\Delta _F[A]=detM[A]
\end{equation}
where the matrix M[A] was defined in Eq.(2.3). From Eq.(2.6), we may obtain
the ghost particle propagator in presence of the external source J 
\begin{eqnarray}
i\Delta ^{ab}[x,y,J] &=&\frac{\delta ^2Z[J,\bar K,K]}{\delta \bar K%
^a(x)\delta K^b(y)}|_{\bar K=K=0}  \nonumber \\
&=&i(M^{-1})_{ab}[x,y,\frac \delta {i\delta J}]Z[J]
\end{eqnarray}
The above result allows us to rewrite the W-T identity in Eq.(3.1) in paper
II in terms of the generating functional $Z[J]^{11,12}$ 
\begin{equation}
\frac 1\alpha \partial _x^\mu \frac{\delta Z[J]}{i\delta J^{a\mu }(x)}-\int
d^4yJ^{b\mu }(y)D_\mu ^{bd}[y,\frac \delta {i\delta J}](M^{-1})^{da}(y,x,%
\frac \delta {i\delta J})Z[J]=0
\end{equation}
where 
\begin{equation}
D_\mu ^{bd}(y)={\cal D}_\mu ^{bd}(y)-\frac{\mu ^2}{\Box }\partial _\mu
\delta ^{bd}
\end{equation}
is the usual covariant derivative. On completing the differentiations with
respect to the source J, Eq.(2.10) reads 
\begin{eqnarray}
&&\frac 1N\int {\cal D}[A]\Delta _F[A]exp\{iS+i\int d^4x[-\frac 1{2\alpha }%
(\partial ^\mu A_\mu ^a)^2+J^{a\mu }A_\mu ^a]\}  \nonumber \\
&&\times [\int d^4yJ^{b\mu }(y)D_\mu ^{bc}(y)(M^{-1})^{ca}(y,x)-\frac 1\alpha
\partial ^\nu A_\nu ^a(x)] \\
&=&0  \nonumber
\end{eqnarray}
By using Eqs.(2.3),(2.9) and (2.11), the ghost equation shown in Eq.(3.2) in
paper II may be written as $^{11,12}$ 
\begin{equation}
M^{ac}[x,\frac \delta {i\delta J}](M^{-1})^{cb}[x,y,\frac \delta {i\delta J}%
]Z[J]=\delta ^{ab}\delta ^4(x-y)Z[J]
\end{equation}
When the source J is turned off, we get 
\begin{equation}
M^{ac}(x)\Delta ^{cb}(x-y)=\delta ^{ab}\delta ^4(x-y)
\end{equation}
This equation only affirms the fact that the ghost particle propagator is
the inverse of the matrix M as was mentioned in Sect.3 of paper I.

Now we are in a position to describe the proof of the unitarity mentioned in
the beginning of this section. To do this, it is suitable to use the
generating functional written in Eq.(2.7) and the W-T identity shown in
Eq.(2.12) because the S-matrix only has gluon external lines, without ghost
particle external lines. For simplifying statement of the proof, in the
following, we use the matrix notation$^{12}$ to represent the integrals. In
this notation, Eqs.(2.7) and (2.12) are respectively written as 
\begin{equation}
Z[J]_F=\frac{1}{N}\int {\cal D}(A)\Delta_F[A]e^{i\{S[A]-\frac{1}{2}%
F^2+J\cdot A\}}
\end{equation}
and 
\begin{equation}
\frac{1}{N}\int {\cal D}(A)\Delta_F[A]e^{i\{S[A]-\frac{1}{2}F^2+J\cdot
A\}}[J_bD_{bc}(M^{-1}_F)_{ca} -\frac{1}{\sqrt{\alpha}}F_a]=0
\end{equation}
where we have defined 
\begin{equation}
F_a\equiv \frac{1}{\sqrt{\alpha}}\partial^{\mu}A^a_{\mu}(x)
\end{equation}
with F corresponding to the gauge $\alpha$, the subscript a,b or c stands
for the color and/or Lorentz indices and the space-time variable, and the
repeated indices imply summation and/or integration.

Let us consider the generating functional in the gauge $\alpha +\Delta
\alpha $ where $\Delta \alpha $ is taken to be infinitesimal 
\begin{equation}
Z[J]_{F+\Delta F}=\frac 1N\int {\cal D}(A)\Delta _{F+\Delta F}[A]e^{i\{S[A]-%
\frac 12(F+\Delta F)^2+JA\}}
\end{equation}
In the above, 
\begin{eqnarray}
e^{-\frac i2(F+\Delta F)^2} &=&e^{-\frac i2F^2}[1+\frac{i\Delta \alpha }{%
2\alpha }F^2] \\
\Delta _{F+\Delta F}[A] &=&detM_{F+\Delta F}
\end{eqnarray}
According to the definition given in Eqs.(2.3) and (2.4), it is seen that 
\begin{equation}
M_{F+\Delta F}^{ab}=M_F^{ab}+\delta ^{ab}\Delta \alpha m^2
\end{equation}
Therefore 
\begin{eqnarray}
\Delta _{F+\Delta F}[A] &=&det[M_F(1+\Delta \alpha m^2M_F^{-1})]  \nonumber
\\
&=&detM_Fe^{Trln(1+\Delta \alpha m^2M_F^{-1})}  \nonumber \\
&=&\Delta _F[A][1+\Delta \alpha m^2TrM_F^{-1}]
\end{eqnarray}
Upon substituting Eqs.(2.19) and (2.22) into Eq.(2.18). we obtain 
\begin{eqnarray}
Z_{F+\Delta F}[J] &=&\frac 1N\int {\cal D}(A)\Delta _F[A]e^{i\{S[A]-\frac i2%
F^2+JA\}}  \nonumber \\
&&\times \{1+\frac{i\Delta \alpha }{2\alpha }F^2+\Delta \alpha
m^2TrM_F^{-1}\}
\end{eqnarray}
For further derivation, it is necessary to employ the W-T identity described
in Eq.(2.16). Acting on Eq.(2.16) with the operator $\frac 12\Delta \alpha
\alpha ^{-\frac 12}F_a[\frac \delta {i\delta J}]$ and noticing 
\begin{eqnarray}
iF_\alpha [\frac \delta {i\delta J}]J_be^{iJcAc} &=&iF_a[\frac \delta {%
i\delta J}]\frac \delta {i\delta A_b}e^{iJ_cA_c}  \nonumber \\
&=&e^{iJ\cdot A}\{iJ_bF_a[A]+\frac{\delta F_a[A]}{\delta A_b}\}
\end{eqnarray}
we have 
\begin{eqnarray}
&&~~\frac 1N\int {\cal D}(A)\Delta _F[A]e^{i\{S[A]-\frac 12F^2+JA\}}\frac{%
\Delta \alpha }{2\sqrt{\alpha }}\{iJ_bD_{bc}[A](M_F^{-1})_{ca}F_a[A] 
\nonumber \\
&~&+\frac{\delta F_a[A]}{\delta A_b}D_{bc}[A](M_F^{-1})_{ca}-\frac i{\sqrt{%
\alpha }}F^2\}=0
\end{eqnarray}
Adding Eq.(2.25) to Eq.(2.23) and considering 
\begin{equation}
\frac{\delta F_a[A]}{\delta A_b}D_{bc}(M_F^{-1})_{ca}=\frac 1{\sqrt{\alpha }}%
Tr[1-\mu ^2M_F^{-1}]
\end{equation}
One may reach the result 
\begin{equation}
Z_{F+\Delta F}[J]=\frac 1N\int {\cal D}(A)\Delta _F[A]e^{i\{S[A]+\Delta S-%
\frac 12F^2+J\cdot A^{\prime }\}}
\end{equation}
where 
\begin{eqnarray}
A_a^{\prime } &=&A_a+\frac{\Delta \alpha }{2\sqrt{\alpha }}%
D_{ab}(M_F^{-1})_{bc}F_c \\
\Delta S &=&-\frac{i\Delta \alpha }{2\alpha }Tr[1+\mu ^2M_F^{-1}]
\end{eqnarray}
in which 
\begin{equation}
TrM_F^{-1}=\int d^4x\Delta ^{aa}(0)=const.
\end{equation}
Since the $\Delta S$ is a constant (even though it is infinite), it may be
taken out from the integral sign and put in the normalization constant N.
Thus, Eq.(2.27) will finally be represented as $^{11,12}$ 
\begin{equation}
Z_{F+\Delta F}[J]=\frac 1N\int {\cal D}(A)\Delta _F[A]e^{i\{S[A]-\frac 12%
F^2+J\cdot A^{\prime }\}}
\end{equation}
In comparison of Eq.(2.31) with Eq.(2.15), it is clear to see that the
difference between the both generating functionals merely comes from the
vector potentials in the external source terms, while, the remaining terms
belong to the dynamical part in the both generating functionals are
completely the same. This indicates that any change of the gauge can only
affect the source term in the generating functional.

It is well-known that the Green's functions computed by the generating
functionals $Z_F[J]$ and $Z_{F+\Delta F}[J]$ have different external lines,
but the same internal structure. The difference of the external lines can
only affects the renormalization of the Green's functions and wave
functions. For the gauge field functions or operators, we can write$^{11}$ 
\begin{eqnarray}
A^{a\mu }(x) &=&Z_F^{\frac 12}A_R^{a\mu }(x) \\
A^{\prime }{}^{a\mu }(x) &=&Z_{F+\Delta F}^{\frac 12}A_R^{a\mu }(x)
\end{eqnarray}
where $A_R^{a\mu }(x)$ denotes the renormalized vector potential and $Z_F$
and Z$_{F+\Delta F}$ are the renormalization constants given in the gauges $%
F $ and $F+$ $\triangle F$ respectively. According to the equivalence theorem%
$^{6,11,12}$, however, the difference mentioned above does not influence on
the S-matrix, that is to say, the S-matrix elements do not depend on the
gauge parameter because the gauge-dependent renormalization constants for
the Green's functions and wave functions would be cancelled out in the
S-matrix elements. Speaking specifically, in accordance with the
renormalization shown in Eqs.(2.31) and(2.32), in the gauges $F$ and $F+$ $%
\triangle F$, the renormalization of gauge boson wave functions, which
represent the states of transverse polarization, can be in momentum space
written as 
\begin{equation}
A_F(k_{i)}=Z_F^{\frac 12}A_R(k_i),
\end{equation}
\begin{equation}
A_{F+\triangle F}=Z_{F+\triangle F}^{\frac 12}A_R(k_{i)}
\end{equation}
where $A_R(k_i)$ stands for the renormalized wave function and $k_i$
designates the momentum and other quantum numbers, including implicitly the
Lorentz and color indices, for i-th particle. The renormalization of n-point
Green's functions of gauge bosons in the different gauges can be represented
as follows 
\begin{equation}
G_F(k_{1,}k_2,...,k_n)=Z_F^{\frac n2}G_R(k_{1,}k_2,...,k_n)
\end{equation}
\begin{equation}
G_{F+\triangle F}(k_{1,}k_2,...,k_n)=Z_{F+\triangle F}^{\frac n2%
}G_R(k_{1,}k_2,...,k_n)
\end{equation}
where $G_R(k_{1,}k_2,...,k_n)$ denotes the renormalized Green's function.
Correspondingly, for the gauge boson propagator, we may write 
\begin{equation}
D_F(k_i)=Z_FD_R(k_i)=D_F^T(k_i)+R_F(k_i)
\end{equation}
\begin{equation}
D_{F+\triangle F}(k_i)=Z_{F+\triangle F}D_R(k_i)=D_{F+\triangle
F}^T(k_i)+R_{F+\triangle F}(k_i)
\end{equation}
where $_{}$%
\begin{equation}
D_F^T(k_i)=\frac{Z_F}{k_i^2-m_R^2}
\end{equation}
\begin{equation}
D_{F+\triangle F}^T(k_i)=\frac{Z_{F+\triangle F}}{k_i^2-m_R^2}
\end{equation}
In the above, the $D_F^T(k_i)$ and $D_{F+\triangle F}^T(k_i)$ come from the
transverse parts of the propagators $D_F(k_i)$ and $D_{F+\triangle F}(k_i)$
and have a physical pole at $k_i^2=m_R^2.$ The $R_F(k_i)$ and $%
R_{F+\triangle F}(k_i)$ represent the remaining parts of the propagators $%
D_F(k_i)$ and $D_{F+\triangle F}(k_i)$ which are regular at the pole.
According to the reduction formula for the S-matrix$^{6,11}$, a S-matrix
element can be given from the corresponding Green's function by the
procedure that on the place of the external lines (propagators) of the
Green's function, which are cut off from the Green's function, we put the
external wave functions with setting all the external momenta on the
mass-shell. By this procedure, the gauge boson scattering matrix elements
evaluated in the gauges $F$ and $F+\triangle F$ may be respectively
expressed as 
\begin{equation}
S_F(k_1,...,k_n)_{}=\prod_{i=1}^n\lim_{k_i^2\rightarrow
m_R^2}A_F(k_i)D_F^T(k_i)^{-1}G_F(k_1,...,k_n)
\end{equation}
\begin{equation}
S_{F+\triangle F}(k_1,...,k_n)_{}=\prod_{i=1}^n\lim_{k_i^2\rightarrow
m_R^2}A_{F+\triangle F}(k_i)D_{F+\triangle F}^T(k_i)^{-1}G_{F+\triangle
F}(k_1,...,k_n)
\end{equation}
When substituting Eqs.(2.33), (2.35) and (2.37) into Eq.(2.41) and
Eqs.(2.34), (2.36) and (2.38) into Eq.(2.42), it is easy to see that the
gauge-dependent renormalization constants are all cancelled in Eqs.(2.41)and
(2.42). As a result, we have $^{6.11.12}$ 
\begin{equation}
S_F(k_1,\cdots ,k_n)_{}=S_{F+\triangle F}(k_1,\cdots
,k_n)_{}=S_R(k_1,...,k_n)
\end{equation}
where 
\begin{equation}
S_R(k_1,...,k_n)_{}=\prod_{i=1}^n\lim_{k_i^2\rightarrow
m_R^2}A_R(k_i)D_R^T(k_i)^{-1}G_R(k_1,...,k_n)
\end{equation}
is the renormalized S-matrix element which is independent of the gauge
parameter. The gauge-independence of the S-matrix implies nothing but the
unitarity of the S-matrix because the gauge-dependent spurious poles which
appear in the longitudinal part of the gluon propagator and the ghost
particle propagator and represent the unphysical excitation of the massive
gauge field in the intermediate states are eventually cancelled out in the
S-matrix. From the construction of the theory, the cancellation seems to be
natural. In fact, in the original Lagrangian written through the transverse
vector potential as shown in Eq.(2.27) in paper I, except for the residual
gauge degrees of freedom, there are not the unphysical longitudinal degrees
of freedom. The occurrence of the longitudinal degrees of freedom in the
theory arises from the formulation of the Lagrangian in terms of the full
vector potential. However, all the unphysical degrees of freedom are
restricted by the constraint conditions imposed on the gauge field and the
gauge group. When these constraint conditions are incorporated in the
Lagrangian, the theoretical principle we based on would automatically
guarantee the cancellation of the unphysical excitations. The situation as
shown in this section is just as we expect. The conclusion drawn from the
above general proof can be easily checked in practical perturbative
calculations, as will be illustrated in the next paper.

\setcounter{section}{3}

\section*{3.Renormalizability}

\setcounter{equation}{0}

~~~It was mentioned in the Introduction that the renormalizability of the
massive non-Abelian gauge field theory may be seen from the intuitive
observation that the Feynman rules derived from the effective Lagrangian
presented in Eq.(2.1), except for the gluon and ghost particle propagators,
are the same as those given in the massless gauge field theory, and the
massive propagators have the same behavior as the massless ones in the large
momentum limit. In particular, the primitively divergent diagrams are
completely the same in the both theories. These facts suggest that the power
counting argument of analyzing the renormalizability for the massive gauge
field theory is as useful as for the massless gauge field theory.
Theoretically, to accomplish a rigorous proof of the renormalizability of
the massive non-Abelian gauge field theory, it is necessary to implement a
subtraction procedure to see whether the divergences occurring in the
Green's function or the S-matrix can be removed by introduction of a finite
number of counterterms in the action in a perturbation calculation$^{6,11,12}
$. This procedure, as one knows, amounts to the well-known R-operation
invented by Bogoliubov, Parasiuk, Hepp and Zimmermann$^{12,14}$. In this
section, for clearness, we firstly restrict ourselves to describe the
subtraction procedure in the physical transverse gauge, i.e. the Landau
gauge. Actually, it is enough to do so because the gauge-independence of the
renormalized S-matrix mentioned in the preceding section tells us that if we
are able to prove the renormalizability of the theory in one gauge, it would
not be problematical in other gauges. Since all of vertices and even Green's
functions may be derived from the proper vertex generating functional
defined in Eq.(2.18) in paper II, we only need to deal with the
renormalization of the vertex generating functional. The principal idea of
proving the renormalizability of the theory under consideration is the usage
of the W-T identities formulated in Sect.2 of paper II. In the Landau gauge,
as mentioned before, these identities formally are identical to those for
the massless gauge field theory. Therefore, the proof of the
renormalizability almost is a repeat of the reasoning given in the massless
theory. It is, of course, adequate to give here a brief description for the
proof.

For simplifying statement, we write, as usual, the W-T identity in the
Landau gauge, which was given in Eq.(2.32) in paper II, in the form 
\begin{equation}
\hat \Gamma *\hat \Gamma =0
\end{equation}
with defining 
\begin{equation}
\hat \Gamma *\hat \Gamma =\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 }\}
\end{equation}
Let us make use of the loop diagram expansion,which is a power series in $%
\hbar $, the Planck constant, for the proper vertex generating functional$%
^{6.12}$ 
\begin{equation}
\hat \Gamma =\sum_{n=0}^\infty \hat \Gamma _n
\end{equation}
In the tree diagram approximation, as one knows, the proper vertex
generating functional $\Gamma _0$ just is the generalized action as was used
in Eq.(2.11) in paper II and, according to the definition given in Eq.(2.28)
in paper II, it will be rewritten as 
\begin{equation}
\Gamma _0=S_0=\hat S_0-\frac 1{2\alpha }\int d^4x(\partial ^\mu A_\mu ^a)^2
\end{equation}
where 
\begin{eqnarray}
\hat \Gamma _0 &=&\hat S_0=\int d^4x\{\bar \psi [i\gamma ^\mu (\partial _\mu
-igA_\mu ^aT^a)]\psi -M\bar \psi \psi -\frac 14F^{a\mu \nu }F_{\mu \nu }^a 
\nonumber \\
&&+\frac 12m^2A_T^{a\mu }A_{T\mu }^a+\bar C^a\partial ^\mu ({\cal D}_\mu
^{ab}C^b)+\bar \zeta \Delta \psi +\Delta \bar \psi \zeta +u^{a\mu }\Delta
A_\mu ^a  \nonumber \\
&&+v^a\Delta C^a\}
\end{eqnarray}
In the above, except for the gluon mass term, we still use $A_\mu ^a$ to
represent the gauge field and keep the operator ${\cal D}_\mu ^{ab}$ in the
ghost field term for later convenience although the gauge field should be
transverse and the ${\cal D}_\mu ^{ab}$ should be replaced by the ordinary
covariant derivative $D_\mu ^{ab}$ in the Landau gauge. Certainly, for this
operator and the gauge field as well as the last term in Eq.(3.4), the limit 
$\alpha \to 0$ should be understood in the following statements.
Particularly, we emphasize that according to the additional renormalization
scheme, all the quantities in Eqs.(3.4) and (3.5) are considered to be
renormalized ones. In perturbative calculations. each loop term in Eq.(3.3)
has to be regularized by an appropriate regularization scheme which preseves
the BRST-symmetry in the whole process of regularization. Suppose the n-th
term $\hat \Gamma _n$ in Eq.(3.3) has been separated into a finite part $%
\hat \Gamma _n^f$ and a divergent part $\hat \Gamma _n^d$ through the
regularization procedure$^{6,12}$ 
\begin{equation}
\hat \Gamma _n=\hat \Gamma _n^f+\hat \Gamma _n^d
\end{equation}
In perturbative calculations, the divergences included in Eq.(3.3) may be
eliminated order by order through a recursive construction of counterterms
in the action. For instance, to eliminate the one-loop divergence $\hat 
\Gamma _1^d$ which is generated by using the action shown in Eq.(3.5) in the
perturbative calculation, we may choose a counterterm $\Delta S_0$ such that 
\begin{equation}
\Delta \hat S_0=-\hat \Gamma _1^d
\end{equation}
whose concrete form will be given later. It is apparent that when we use the
action 
\begin{equation}
\hat S_1=\hat S_0+\Delta \hat S_0
\end{equation}
to recalculate the functional $\hat \Gamma _1$ of order $\hbar $, the
divergence in it disappears. In general, to remove the divergent part $\hat 
\Gamma _n^d$ in the n-loop term $\hat \Gamma _n$ of order $\hbar ^n$, we
need to introduce a counterterm like this 
\begin{equation}
\Delta \hat S_{n-1}=-\hat \Gamma _n^d
\end{equation}
Adding it to the action, we have 
\begin{equation}
\hat S_n=\hat S_{n-1}+\Delta \hat S_{n-1}
\end{equation}
where the $\hat S_{n-1}$ has included the counterterms up to the order $%
\hbar ^{n-1}$. The action $\hat S_n$ used to calculate the $\hat \Gamma _n$
will lead to a finite result in the n-th order perturbation. The
counterterms mentioned above may be determined with the help of the W-T
identity. In fact, on substituting Eq.(3.3) into Eq.(3.1), we get 
\begin{equation}
\sum_{p+q=n}\hat \Gamma _p*\hat \Gamma _q=0
\end{equation}
When Eq.(3.6) is inserted into Eq.(3.11), one may obtain a series of
identities satisfied by the finite and divergent parts of the $\hat \Gamma _n
$. Each of the identities contains terms which are of the same order of
divergence ( reference to Eq.(3.42) given in the latter part of this
section). Furthermore, the action constructed in Eq.(3.10) is also required
to fulfill the W-T identity 
\begin{equation}
\hat S_n*\hat S_n=0
\end{equation}
From the above requirements, one may derive the following equations$^{6.12}$ 
\begin{equation}
\rho (\hat S_0)\hat \Gamma _n^d=0
\end{equation}
and from Eqs.(3.12), (3.10) and (3.9), it is easy to find 
\begin{equation}
\rho (\hat S_{n-1})\hat \Gamma _n^d=0
\end{equation}
In the above, the operator $\rho (\hat S_0)$ is defined as$^{6.12}$ 
\begin{equation}
\rho (\hat S_0)=\frac{\delta \hat S_0}{\delta \varphi _i}\cdot \frac \delta {%
\delta u_i}+\frac{\delta \hat S_0}{\delta u_i}\cdot \frac \delta {\delta
\varphi _i}
\end{equation}
here $\varphi _i$ and $u_i(i=A,\bar \psi ,\psi ,C)$ stand for the field
variables $A_\mu ^a,\bar \psi ,\psi ,C^a$ and source variables $u_\mu
^a,\zeta ,\bar \zeta ,v^a$, respectively, and the symbol ''.'' in each term
on the RHS of Eq.(3.15) is an abbreviation notation of the integration as
shown in Eq.(3.2). The definition of $\rho (\hat S_{n-1})$ is similar to
Eq.(3.15) with replacing $\hat S_0$ by $\hat S_{n-1}$. On the other hand,
substitution of Eqs.(3.3) and (3.6) in Eq.(2.33) in paper II gives the ghost
equation$^{6.12}$ 
\begin{equation}
\frac{\delta \hat \Gamma _n^d}{\delta \bar C^a}-\partial _\mu (\frac{\delta 
\hat \Gamma _n^d}{\delta u_\mu ^a})=0
\end{equation}
The divergence $\hat \Gamma _n^d$ and thus the counterterm $\Delta \hat S%
_{n-1}$ may be determined by solving Eq.(3.13) and (3.16) together, or,
instead, by solving Eq.(3.14) and (3.16) provided that the action $\hat S%
_{n-1}$ has been given in the former n-1 steps of recursion. The general
solution to the above equations was already found in the literature$^{6,12}$%
. It consists of two parts as shown below 
\begin{equation}
\Delta \hat S_{n-1}=-\hat \Gamma _n^d=\sum_aa_a^nH_a+\rho (\hat S_{n-1})F_n
\end{equation}
where the $H_a$ in the first term are functionals of the field variables $%
A_\mu ^a,\bar \psi $ and $\psi $ which are invariant with respect to the
gauge transformation. Therefore, they obviously satisfy Eq.(3.13) or (3.14).
These functionals, actually, can only be of the forms as the first four
terms in Eq.(3.5). The second term in Eq.(3.17) follows from the nilpotency
property of the operator $\rho ,\rho ^2=0$ (see Ref.(6)), while, the
functional $F_n$ is arbitrary. However, the term $\Delta \hat S_{n-1}$ as a
part of the action demands that the $F_n$ must be a functional with minus
mass dimension and minus ghost number. Moreover, the $F_n$, as easily seen,
must satisfy the ghost equation 
\begin{equation}
\frac{\delta F_n}{\delta \bar C^a}-\partial _\mu \frac{\delta F_n}{\delta
u_\mu ^a}=0
\end{equation}
With the requirements stated above, the form of the functional $F_n$ will be
uniquely determined, as given in the following$^{6,12}$ 
\begin{equation}
F_n=\int d^4x\{b_A^nA^{a\mu }(u_\mu ^a-\partial _\mu \bar C^a)+\sum_{i\ne
A}b_i^n\varphi _iu_i\}
\end{equation}
It is noted that the coefficients $a_a^n$ in Eq.(3.17) and $b_i^n$ in
Eq.(3.19) all depend on the regularization parameter, say, the $\varepsilon
=2-n/2$ (which tends to zero, when $n\to 4$) in the dimensional
regularization$^{15}$. The operator $\rho (\hat S_{n-1})$ in Eq.(3.17)
implies that we have chosen the counterterm $\Delta \hat S_{n-1}$ to be the
solution of Eq.(3.14) for convenience of later recursion.

Up to the present, the counterterm $\Delta \hat S^{n-1}$ appearing in
Eq.(3.10) has explicitly been constructed as given in Eqs.(3.17) and (3.19).
The action $\hat S_{n-1}$ constructed in the foregoing steps of recursion
has the same functional structure as that for the $\hat S_0$ given in
Eq.(3.5). This can be seen from the fact that Eq.(3.14) has the same form as
Eq.(3.13). An interesting thing is that the function of the counterterm in
Eq.(3.10) is only to make a change to the variables in the first term $\hat S%
_{n-1}$. If the coefficients in Eq.(3.19) are assumed to be infinitesimal,
we have the following variations: 
\begin{eqnarray}
&~&\delta \varphi _i=\frac{\delta F_n}{\delta u_i},i=A,\bar \psi ,\psi ,C 
\nonumber \\
&~&\delta u_i=-\frac{\delta F_n}{\delta \varphi _i},i=\bar \psi ,\psi ,C 
\nonumber \\
&~&\delta u_\mu ^a-\partial _\mu \delta \bar C^a=-\frac{\delta F_n}{\delta
A_\mu ^a},i=A
\end{eqnarray}
(Note: we assume here that the field variables are commuting and the source
variables are anticommuting for convenience of statement). According to the
definition given in Eq.(3.15) and noticing the ghost equation obeyed by the $%
\hat S_{n-1}$ , Eq.(3.20) allows us to write 
\begin{eqnarray}
\hat S_{n-1}[\varphi _i,u_i] &&+\rho (\hat S_{n-1})F_n[\varphi _i,u_i] 
\nonumber \\
=\hat S_{n-1}[\varphi _i,u_i] &&+\sum_{i=A,\bar \psi ,\psi ,C}(\delta
\varphi _i\cdot \frac{\delta \hat S_{n-1}}{\delta \varphi _i}+\delta
u_i\cdot \frac{\delta \hat S_{n-1}}{\delta u_i})+\delta \bar C^a\cdot \frac{%
\delta \hat S_{n-1}}{\delta \bar C^a}  \nonumber \\
=\hat S_{n-1}[\varphi _i^{\prime },u_i^{\prime }] &&
\end{eqnarray}
in which 
\begin{eqnarray}
\varphi _i^{\prime } &=&\varphi _i+\delta \varphi _i=Y_i^n\varphi
_i,\;\;\;i=A,\bar \psi ,\psi ,C  \nonumber \\
u_i^{\prime } &=&u_i+\delta u_i=Y_i^{n^{-1}}u_i,\;\;\;i=A,\bar \psi ,\psi ,C
\nonumber \\
\bar C^{\prime }{}^a &=&\bar C^a+\delta \bar C^a=Y_A^{n^{-1}}\bar C^a
\end{eqnarray}
where the coefficients $Y_i^n$ are defined as 
\begin{equation}
Y_i^n=1+b_i^n,i=A,\bar \psi ,\psi ,C
\end{equation}
In the above, the expression written in Eq.(3.19) has been used to evaluate
the differentials. Considering that the functionals in the first term of
Eq.(3.17) are gauge-invariant and of the same functional structure as those
terms in the $\hat S_{n-1}$ which are the functionals of the fields $A_\mu
^a,\bar \psi $ and $\psi $, we are allowed to change the field variables
from $\varphi _i$ to $\varphi _i^{\prime }$ in the functionals in the first
term of Eq.(3.17) and combine these functionals with the corresponding terms
in the $\hat S_{n-1}[\varphi _i^{\prime },u_i^{\prime }]$ shown in Eq.(3.21)
together. The results is just to redefine the variables and physical
constants in the action $\hat S_{n-1}$. Thus, Eq.(3.10) can be written as 
\begin{equation}
\hat S_n[\varphi _i,u_i]=\hat S_{n-1}[\sqrt{Z_i^n}\varphi _i,\sqrt{\tilde Z%
_i^n}u_i]
\end{equation}
where $Z_i^n$ and $\tilde Z_i^n$ are the n-th order multiplicative
renormalization constants for the field and source variables respectively.
Eq.(3.24) establishes a recursive relation of the renormalization. When the
order n tends to infinity, we obtain from Eq.(3.24) by recursion the
following result$^{6,12}$ 
\begin{equation}
\hat S[\varphi _i,u_i]=\hat S_0[\varphi _i^0,u_i^0]
\end{equation}
where 
\begin{equation}
\varphi _i^0=\sqrt{Z_i}\varphi _i,u_i^0=\sqrt{\tilde Z_i}u_i,  \nonumber \\
g^0=Z_gg,M^0=Z_MM,m^0=Z_mm
\end{equation}
are the bare quantities appearing in the unrenormalized action $\hat S_0$,
and the renormalization constants are given by$^{6,12}$ 
\begin{eqnarray}
Z_i &=&\prod_{n=1}^\infty Z_i^n,\tilde Z_i=\prod_{n=1}^\infty \tilde Z%
_i^n,Z_g=\prod_{n=1}^\infty Z_g^n,  \nonumber \\
Z_M &=&\prod_{n=1}^\infty Z_M^n,Z_m=\prod_{n=1}^\infty Z_m^n
\end{eqnarray}
Eq.(3.25) shows us that the renormalized action has the same functional
structure as the unrenormalized one.

To be more specific, let us firstly describe the one-loop renormalization of
the functional $\Gamma _1$ starting from the action written in Eqs.(3.4) and
(3.5). As pointed out before, in this case, we have to introduce a
counterterm as mentioned in Eqs.(3.7) and (3.8) whose general form was given
in Eq.(3.17) with the order label n=1,. In the first term of Eq.(3.17), the
BRST-invariant functionals of the variables $A_\mu ^a,\bar\psi $ and $\psi $
can only be 
\begin{eqnarray}
H_G&=&-\frac{1}{4}\int
d^4x(\partial_{\mu}A^a_{\nu}-\partial_{\nu}A^a_{\mu}+gf^{abc}A^b_{\mu}A^c_{%
\nu})^2 \\
H_F&=&\int d^4x\bar\psi i\gamma^{\mu}(\partial_{\mu}-igA^a_{\mu}T^a)\psi \\
H_M&=&\int d^4xM\bar\psi\psi \\
H_m&=&\frac{1}{2}m^2\int d^4x(A^{a\mu}_T)^2
\end{eqnarray}
The corresponding coefficients in Eq.(3.17) will be written as $a_G,a_F,a_M$
and ${a_m}$. In the following, the order label will be suppressed and the
source terms in Eq.(3.5) and (3.17) will be omitted for simplicity because
these terms act only in the intermediate stages of the proof. As
demonstrated in Eq.(3.21), the variables of the action $\hat S_0$ in
Eq.(3.8) which was explicitly written in Eq.(3.5) will be changed to the
ones shown in Eq.(3.22) owing to the effect of the counterterm given by the
second term in Eq.(3.17) where the functional F being represented in
Eq.(3.19) and the variables in the counterterms given by the first term in
Eq.(3.17) which are explicitly written in Eqs.(3.28)-(3.31) may also be made
such changes due to the gauge-invariance of the functionals. Thus, the
action in Eq.(3.8) without the source terms may be written as

\begin{eqnarray}
&&~~\hat S[A_\mu ^a,\bar \psi ,\psi ,\bar C^a,C^a]  \nonumber \\
&&~=\int d^4x[Y_FY_{\bar \psi }Y_\psi \bar \psi i\gamma ^\mu (\partial _\mu
-igY_AA_\mu ^aT^a)\psi -MY_MY_{\bar \psi }Y_\psi \bar \psi \psi  \nonumber \\
&&~-\frac 14Y_A^2Y_G(\partial _\mu A_\nu ^a-\partial _\nu A_\mu
^a+gY_Af^{abc}A_\mu ^bA_\nu ^c)^2+\frac 12Y_mY_A^2m^2A_T^{a\mu }A_{T\mu }^a 
\nonumber \\
&&~-Y_A^{-1}Y_C\partial ^\mu \bar C^a[\delta ^{ab}\partial _\mu
-gY_Af^{abc}A_\mu ^c]C^b]
\end{eqnarray}
where 
\begin{equation}
Y_F=1+a_F,Y_M=1+a_M,Y_G=1+a_G,Y_m=1+a_m
\end{equation}
When we define renormalization constants as below 
\begin{eqnarray}
Z_2 &=&Y_FY_{\bar \psi }Y_\psi ,Z_3=Y_GY_A^2,\tilde Z_3=Y_CY_A^{-1} 
\nonumber \\
Z_1^F &=&Y_FY_{\bar \psi }Y_\psi Y_A,Z_1=Y_GY_A^3,Z_4=Y_GY_A^4  \nonumber \\
\tilde Z_1 &=&Y_C,Z_M=Y_MY_F^{-1},Z_m^2=Y_mY_G^{-1}=Z_3^{-1}
\end{eqnarray}
and noticing the relation given in Eq.(3.4), We may write the full action as
follows 
\begin{eqnarray}
S &=&\int d^4x\{Z_2\bar \psi (i\gamma ^\mu \partial _\mu -MZ_M)\psi +gZ_1^F%
\bar \psi \gamma ^\mu A_\mu ^aT^a\psi  \nonumber \\
&&-\frac 14Z_3(\partial _\mu A_\nu ^a-\partial _\nu A_\mu ^a)^2-\frac 12%
gZ_1f^{abc}(\partial _\mu A_\nu ^a-\partial _\nu A_\mu ^a)A^{b\mu }A^{c\nu }
\nonumber \\
&&-\frac 14g^2Z_4f^{abc}f^{ade}A^{b\mu }A^{c\nu }A_\mu ^dA_\nu ^e+\tilde Z_3%
\bar C^a\Box C^a  \nonumber \\
&&+g\tilde Z_1f^{abc}\partial ^\mu \bar C^aC^bA_\mu ^c+\frac 12m^2A_\mu
^aA^{a\mu }-\frac 1{2\alpha }(\partial ^\mu A_\mu ^a)^2\}
\end{eqnarray}
where the subscript T has been suppressed in the gluon mass term, but, it
should be understood that the vector potential in Eq.(3.35) is restricted in
the Landau gauge. It is noted here that the last equality in Eq.(3.34) is
necessarily required so as to be consistent with the renormalization of
gluon propagator, as demonstrated in Sect.3 of paper II, there the gluon
mass renormalization constant is the inverse of the gluon wave function
renormalization constant. This equality implies ${Y_mY_A^2}$=1 so that the
mass term of the transverse field $A_T^{a\mu }$ in Eq.(3.35) keeps unchanged
in the process of subtraction. The action in Eq.(3.35) with the
renormalization constants being defined in Eq.(3.34) just gives the
recursive relation shown in Eq.(3.24) with the label n=1. The action in
Eq.(3.35) would eliminate the divergence appearing in the generating
functional $\Gamma $ evaluated in the one-loop approximation.

From Eq.(3.34), it is clear to see that 
\begin{equation}
\frac{Z_1}{Z_3}=\frac{Z_1^F}{Z_2}=\frac{\tilde Z_1}{\tilde Z_3}=\frac{Z_4}{%
Z_1}
\end{equation}
This precisely is the Slavnov-Taylor identity$^{13}$ given in the massless
QCD and obtained partly in paper II..

If we define the bare quantities as 
\begin{eqnarray}
A^{a\mu}_0&=&\sqrt{Z_3}A^{a\mu},\;\psi_0=\sqrt{Z_2}\psi,\;\bar\psi_0=\sqrt{%
Z_2}\bar\psi  \nonumber \\
C^a_0&=&\sqrt{\tilde Z_3}C^a,\;\bar C^a_0=\sqrt{\tilde Z_3}\bar C%
^a,\;g_0=Z_1Z_3^{-3/2}g  \nonumber \\
M_0&=&Z_MM,\;m_0=Z_mm=m/ \sqrt{Z_3},\;\alpha_0=Z_3\alpha
\end{eqnarray}
and use the identity in Eq.(3.36), we arrive at 
\begin{equation}
S[A_\mu ^a,\bar \psi ,\psi ,\bar C^a,C^a]=S_0[A_0^{a\mu },\bar \psi _0,\psi
_0,\bar C_0^a,C_0^a]
\end{equation}
where 
\begin{eqnarray}
&~&S_0=\int d^4x\{\bar\psi_0[i\gamma_{\mu}(\partial^{\mu}-ig_0A^{a%
\mu}_0T^a)-M_0]\psi_0  \nonumber \\
&~&-\frac{1}{4}(\partial^{\mu}A^{a\nu}_0-\partial^{\nu}A^{a\mu}_0
+g_0f^{abc}A^{b\mu}_0A^{c\nu}_0)^2  \nonumber \\
&~&+\bar C^a_0\Box C^a_0+g_0f^{abc}\partial_{\mu}\bar C^a_0C^b_0A^{c\mu}_0 
\nonumber \\
&~&+\frac{1}{2}m^2_0A^{a\mu}_0A^a_{0\mu}-\frac{1}{2\alpha_0}%
(\partial_{\mu}A^{a\mu}_0)^2\}
\end{eqnarray}
is the unrenormalized action. Eqs.(3.4), (3.5), (3.38) and (3.39) indicate
that the actions, renormalized and unrenormalized, have the same structure
and thus the same symmetry, just as we met in the massless gauge field
theory.

We note here that although the above results are obtained in the one-loop
renormalization, they can, actually, be considered to be the exact ones. In
fact, for removing the two-loop divergence in the $\hat \Gamma $, obviously,
the second cycle of recursion of the renormalization can be carried out in
the same way as stated in Eqs.(3.28)-(3.35) starting from the action shown
in Eq.(3.32) and higher order recursions can be continued further along the
same line. The results given in Eqs.(3.32)-(3.35) formally remain unchanged
for each cycle of the recursion. Therefore, by the recursive procedure, all
the results denoted in Eqs.(3.34)-(3.39) can be regarded as the ones as
shown in Eqs.(3.25)-(3.27).

Now, we are in a position to describe the renormalizability of the theory in
the general gauges. We firstly note that the results in the Landau gauge as
given before can directly be extended to the other gauges. To see this
point, let us analyze the W-T identity for the general gauge which was given
in Eq. (2.28) in paper II and is rewritten as 
\begin{equation}
\hat \Gamma *\hat \Gamma +\omega =0
\end{equation}
where we have set 
\begin{equation}
\omega =\int d^4xm^2\partial ^\mu A_\mu ^a(x)C^a(x)
\end{equation}
and the symbol ''*'' was defined in Eq.(3.2). In the loop expansion, when
Eqs.(3.3) and (3.6) are substituted in Eq.(3.40), one may obtain a series of
equations among which the equations of order ${\hbar }^n$ are 
\begin{eqnarray}
\sum_{p+q=n}\hat \Gamma _p^f*\hat \Gamma _q^f+\omega \delta _{n0} &=&0 
\nonumber \\
\hat \Gamma _{n-1}^f*\hat \Gamma _1^d+\hat \Gamma _1^d*\hat \Gamma _{n-1}^f
&=&0  \nonumber \\
&&\cdots \cdots  \nonumber \\
\hat \Gamma _1^f*\hat \Gamma _{n-1}^d+\hat \Gamma _{n-1}^d*\hat \Gamma _1^f
&=&0  \nonumber \\
\rho (\hat S_0)\hat \Gamma _n^d+\sum_{p+q=n}\hat \Gamma _p^d*\hat \Gamma
_q^d &=&0
\end{eqnarray}
where the symbol $\rho (\hat S_0)$ was defined in Eq.(3.15) and the fact
that the $\omega $ is of zeroth order of ${\hbar }$ has been noticed. This
fact is obvious because we start from renormalized field functions and
parameters in the additional renormalization. The action defined in
Eq.(3.10) with the $\Delta \hat S_{n-1}$ being given in Eq.(3.9) is also
required to satisfy the W-T identity 
\begin{equation}
\hat S_n*\hat S_n+\omega =0
\end{equation}
When Eq.(3.6) is inserted into the above equations, it is easy to find an
equation which shows that the last term in Eq.(3.42) equals to zero.
Therefore, we still have the equation as shown in Eq.(3.13) and the one
written in Eq.(3.14). In addition, when Eq.(3.6) is substituted in the ghost
equation 
\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}
which was given in Eq.(2.29) in paper II, obviously, we still have the
equation written in Eq.(3.16) for the divergent part $\hat \Gamma _n^d$.
Therefore, the counterterm, as the solution to Eqs.(3.14) and (3.16), is
still expressed by Eqs.(3.17), (3.19) and (3.28)-(3.31) with a note that
except for the gluon mass counterterm written in Eq.(3.31) which is still
given for the transverse field, the vector potential in the other
counterterms now becomes full one. It is clear that the statements in
Eqs.(3.20)-(3.27) completely hold for the general gauges. The results
described in Eqs.(3.32)-(3.39), as easily seen, except for a few supplements
for the gluon and ghost particle mass terms, are also preserved in the
present case. The gluon mass term in Eq.(3.5) is now written for the full
vector potential, or say, a longitudinal field mass term is supplemented to
Eq.(3.5). When the countertems in Eq.(3.17) with their explicit expressions
written in Eqs.(3.19) and (3.28)-(3.31) are added to the action, the mass
term becomes 
\begin{equation}
\int d^4x\frac 12Y_mY_A^2m^2[A_T^{a\mu }A_{T\mu }^a+Y_m^{-1}A_L^{a\mu
}A_{L\mu }^a]
\end{equation}
This term should replace the corresponding term in Eq.(3.32) to appear in
the action. In the above, the factor $Y_A^2$ arises from the variable change
generated by the counterterm given in the second term in Eq.(3.17), as shown
in Eqs.(3.21) and (3.22) and the factor $Y_m$ comes from the counterterm for
the transverse field mass term as denoted in Eq.(3.31). When we notice the
last equality in Eq.(3.34) and define 
\begin{equation}
Z_3^{\prime }=Y_m^{-1}
\end{equation}
\begin{equation}
A^{a\mu }=A_T^{a\mu }+\sqrt{Z_3^{\prime }}A_L^{a\mu }
\end{equation}
Eq.(3.45) can be written as 
\begin{equation}
\int d^4xm^2[A_T^{a\mu }A_{T\mu }^a+Z_3^{\prime }A_L^{a\mu }A_{L\mu
}^a]=m^2\int d^4xA^{a\mu }A_\mu ^a
\end{equation}
where the orthonormality between the transverse and longitudinal variables
has been considered. With the above expression, the gluon mass term in
Eq.(3.35) can be understood for the full vector potential in the general
gauge. Eq.(3.48) shows that the renormalization of the longitudinal part of
the vector potential is different from that for the transverse part by an
extra renormalization constant $\sqrt{Z_3^{\prime }}$. This result is
consistent with the renormalization of gluon propagator shown in Eq.(3.27)
in paper II where the longitudinal part of the propagator has an extra
renormalization constant $Z_3^{\prime }$. According to the definition given
in Eq.(3.37), the gluon mass term may come to the form expressed by the bare
quantities as shown in Eq.(3.39).

Let us turn to the renormalization of the ghost particle mass term. From the
last equality in Eq.(3.37), we see, the renormalization of the gluon mass
and the gauge parameter renders the ghost particle mass to be a
renormalization- invariant quantity, similar to the kinetic operator ${\Box }
$ in the Landau gauge. Therefore, in the general gauge, the ghost particle
kinetic term in Eq.(3.32) may directly be extended to the form 
\begin{equation}
Y_A^{-1}Y_C\bar C^a(\Box +{\mu }^2)C^a
\end{equation}
where the factors ${Y_A^{-1}}$ and ${Y_C}$ arise respectively from the
change of the variables ${\bar C^a}$ and ${C^a}$ which are caused by the
counterterm contained in the second term in Eq.(3.17), as described in
Eqs.(3.21) and (3.22). With the definitions given in Eqs.(3.34) and (3.37),
Eq.(3.49) can be represented as 
\begin{equation}
\tilde Z_3\bar C^a(\Box +{\mu }^2)C^a=\bar C_0^a(\Box +{\mu }_0^a)C_0^a
\end{equation}
In the general gauge, the ghost particle kinetic terms in Eqs.(3.35) and
(3.39) should be replaced by the terms on the LHS and RHS of Eq.(3.50),
respectively.

The derivations and results stated in this section clearly show that the
divergences appearing in the perturbative calculations for the massive gauge
field theory can indeed be eliminated by introducing a finite number of
counterterms as shown in Eqs.(3.17),(3.19) and (3.28)-(3.31). Saying
equivalently, these divergences may be absorbed into a finite number of
renormalization constants and thus be removed by redefining the wave
functions and the physical parameters. In view of this. we may say, the
renormalizability of the massive non-Abelian gauge field theory without
Higgs mechanism is absolutely no problem.

\setcounter{section}{4}

\section*{4. COMMENTS AND CONCLUSIONS}

\setcounter{equation}{0}

~~~Contrary to the prevailing concept that it is impossible to build a
renormalizable and/or unitary massive gauge field theory without recourse to
the Higgs mechanism $^{1-10}$, we have succeeded in establishing such a
theory that it is renormalizable and unitary. The basic idea to achieve this
success is the consideration that the massive gauge field is a constrained
system in the whole space of the full vector potential. It is a criterion
that a quantum theory for a constrained field system must be renormalizable
and unitary if the theory is established on the faithful theoretical
principle and the unphysical variables contained in the theory have been
completely eliminated by appropriate constraint conditions. For the massive
gauge fields discussed in this paper, the quantum theory was set up from
beginning to end on the basis of gauge-invariance principle and the
unphysical degrees of freedom appearing in the massive Yang-Mills Lagrangian
have been quenched by the constraint on the gauge field (the Lorentz
condition) and the constraint on the gauge group (the ghost equation). Such
a theory could not be nonrenormalizable and nonunitary. This essential point
was not realized clearly and handled correctly in some previous studies. In
the earlier works of investigating the massive gauge field theory$^{1-6}$,
authors all started with the massive Yang-Mills Lagrangian and considered
that this Lagrangian forms a complete description of the massive gauge field
dynamics. When using this Lagrangian to construct the quantum theory, they
found that except for the neutral vector meson field in interaction with a
conserved current, the theory is nonrenormalizable because of the presence
of the mass term. The typical arguments are the following. In Ref.(3), the
authors showed an equivalence theorem by which they gave a Hamiltonian
derived from the Lagrangian by introducing an auxiliary Stueckelberg field.
When making an unitary transformation to the Schrodinger equation, the mass
term in the Hamiltonian becomes dependent on the auxiliary field and
contains an infinite number of terms in its expansion of power series which
leads to bad unrenormalizability. In Ref.(4), the equivalence theorem was
given in the form of S-matrix. The author also introduced the Stueckelberg
field and used it to make a finite gauge transformation to the fields
involved in the theory. As a result. the mass term in the S-matrix contains
an exponential function of the auxiliary field which gives rise to an
infinite variety of distinct primitively divergent graphs that can not be
eliminated by the introduced conditions imposed on the gauge transformation.
Later, the authors in Refs.(5) and (6) made the usual finite gauge
transformations to the generating functional of Green's functions which is
built by the Lagrangian mentioned above and obtained the same result that
the gauge boson mass term depends on the parametric function of the gauge
group and contains various unrenormalizable infinities. Similarly, the
theory constructed by introducing the group-valued Stueckelberg field to the
mass term was also shown to be nonrenormalizable due to the nonpolynomiality
of the Stueckelberg function$^{7-10}$.

Let us make comments on the above argument. Firstly, we note that the
Lagrangian they started with, as was pointed out in paper I, can not serve a
complete description for the massive gauge field dynamics because it
contains redundant unphysical degrees of freedom arising from the
longitudinal part of the vector potential, the residual gauge degrees of
freedom and/or the Stueckelberg field. If the unphysical degrees of freedom
are not restricted by appropriate constraint conditions, the Lagrangian can
not be used to construct a correct theory. For example, from the theory
given by the massive Yang-Mills Lagrangian, as shown in paper I, one can
only get a wrong gauge boson propagator 
\begin{equation}
iD_{\mu \nu }^{ab}(k)=\frac{-i}{k^2-m^2+i{\varepsilon }}(g_{\mu \nu }-k_\mu
k_\nu /m^2)
\end{equation}
in which the term proportional to ${k_\mu k_\nu /m^2}$ leads to horrible
nonrenormalizability. In our theory, the good renormalizability originates
from the fact that in the effective Lagrangian, the gauge-fixing term and
the ghost term coming from the constraints just play the role of quenching
the effect of the unphysical degrees of freedom contained in the Lagrangian.
Another point worthy of note is that in the previous works mentioned above,
the finite gauge transformations were made and used to demonstrate the
nonrenormalizability of the theory. However, as was pointed out in Sect.2 of
paper I, in the physical space restricted by the Lorentz condition, the
infinitesimal gauge transformation is only needed to be considered. The
present quantum gauge field theories such as the standard model, actually,
are set up on this basis. Otherwise, the theories would be different and
troublesome. In view of this, let us comment on some aspects of the previous
works further. In Ref.(5), the author found a relation by which any
non-Abelian vector field may be represented as a gauge transformation of the
transverse field and tried to separate the gauge degrees of freedom from the
transverse ones. He eventually failed to do it because the coupling between
the both degrees of freedom does not vanish upon integration. Nevertheless,
we note, under the infinitesimal transformation, the coupling disappears.
Especially, the renormalizability conditions introduced in Ref.(4) will be
fulfilled for the non-Abelian gauge field if the infinitesimal gauge
transformation is concerned only. As for the problem of nonrenormalizability
argued in Ref.(6), it was pointed out in the last section of paper I that if
the operation of quantization is performed in a correct way, the
nonrenormalizable and nonunitary terms can not appear in the effective
Lagrangian. For the unitarity problem, we will give more detailed
discussions and comments in the next paper.

The massive gauge field theory presented in paper I and this paper fulfills
the original belief$^{16}$ that such a theory should naturally go over into
the massless theory in the zero-mass limit. At present, the massless QCD has
widely been recognized to be the candidate of the strong interaction theory
and has been proved to be compatible with the present experiments. However,
we think, the massive QCD would be more favorable to explain the strong
interaction phenomenon, particularly, at the low energy domain because the
massive gluon would make the force range more shorter than that caused by
the massless gluon. Especially, in some phenomenological investigations,
such as the calculation of glueball spectra$^{17}$ and the studies of
hadron-antihadron low energy annihilation phenomena$^{18}$, the gluon mass
was necessarily introduced by hand so as to get fairly reasonable results.
As for the high energy and large momentum transfer phenomena, as seen from
the massive gluon propagator, the gluon mass gives little influence on the
theoretical result so that the massive QCD could not conflict with the
well-established results gained from the massless QCD. The formalism
provided in the former papers and this paper suggests that the Higgs
mechanism may be unnecessary in the weak interaction theory. But, this does
not mean that the spontaneous symmetry-breaking mechanism is useless. The
first example is the chiral $\sigma $-model originally proposed by Gell-Mann
and Levy$^{19}$. Because it is quite successful in explaining the hadron
interaction phenomena, this model still is widely applied in the today's
nuclear physics. As one knows, even though the $\sigma $ field
phenomenologically is useful. there is no any real $\sigma $-meson
discovered in experiments. The $\sigma $-field can only be viewed as a
phenomenological field$^{6}$ which incorporates some nonperturbative effects
through the vacuum symmetry-breaking mechanism. The situation presumably is
similar for the present weak interaction model where the Higgs field
appearing in the model has the same property as the $\sigma $-field. With
the belief that the basic dynamics for a massive gauge field must be
formally simple and of an exact gauge symmetry, we think, it is a meaningful
attempt to establish a gauge-symmetric weak interaction theory without
involving the Higgs particle in it. Such a theory, is likely to be
constructed starting from a SU(2) gauge-invariant action written for the
chiral fermions and the massive gauge bosons. What is the connection between
the theory with an exact gauge-symmetry and the model with a broken
gauge-symmetry? This is an interesting subject worthy of pursuing in the
future.

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

The author wishes to thank professor Shi-Shu Wu for valuable discussions.
Thanks are also owed to professor Ze-Sen Yang for stimulating discussions.
This project was supported in part by National Natural Science Foundation of
China.

\begin{center}
{\bf APPENDIX A:Another derivation of the W-T identity used in section 2 }
\end{center}

As shown in Sect.2, the W-T identity written in Eq.(2.12) provides a basis
to prove the unitarity of the massive gauge field theory. This identity,
which is equivalent to the one denoted in Eq.(2.1), was derived originally
from the W-T identity given in Eq.(2.15) of paper II. As pointed out in
paper II, the correctness of the latter identity implies that the argument
about the BRST-invariance of the source terms introduced into the generating
functional is reasonable. To confirm this point, we start here from the
generating functional written in Eq.(2.7) to rederive the identity in
Eq.(2.12). The generating functional in Eq.(2.7) may be more directly
derived by the Faddeev-Popov method of quantization. In fact, when the
source term $J^{a\mu }A_\mu ^a$ is introduced to the transition amplitude
shown in Eq.(3.43) in paper I, we immediately obtain that generating
functional.

Let us make the ordinary gauge transformation $\delta A_\mu ^{a_{}}=D_\mu
^{ab}\theta ^b$ to the generating functional in Eq.(2.7). Considering the
gauge invariance of the functional integral, the integration measure and the
functional $\triangle _F[A]=\det M[A],$ we get 
\begin{eqnarray}
\delta Z[J] &=&\frac 1N\int D(A)\triangle _F[A]\int d^4y[J^{b\mu
}(y)+m^2A^{b\mu }(y)  \nonumber \\
&&\ -\frac 1\alpha \partial ^\nu A_\nu ^b\partial _y^\mu ]D_\mu
^{bc}(y)\theta ^c(y)\exp \{iS \\
&&\ +i\int d^4x[-\frac 1{2\alpha }(\partial ^\mu A_\mu ^a)^2+J^{a\mu }A_\mu
^a]\}  \nonumber \\
\  &=&0  \nonumber
\end{eqnarray}
According to the well-known procedure$^{11,12}$, the group parameter $\theta
^a(x)$ in Eq.(A.2) may be determined by the following equation 
\begin{equation}
M^{ab}(x)\theta ^b(x)\equiv \partial _x^\mu ({\cal D}_\mu ^{ab}(x)\theta
^b(x))=\lambda ^a(x)
\end{equation}
where $\lambda ^a(x)$ is an arbitrary function. When setting $\lambda
^a(x)=0,$ Eq.(A.3) will be reduced to the constraint condition on the gauge
group as derived in paper I. which is used to determine the $\theta ^a(x)$
as a functional of the vector potential $A_\mu ^a(x)$. However, when the
constraint condition is incorporated into the action by the Lagrange
undetermined multiplier method and then generates the gauge-fixing term in
the generating functional, the $\theta ^a(x)$ should be treated as arbitrary
according to the spirit of Lagrange multiplier method. That is why we may
use Eq.(A.3) to determine the functions $\theta ^a(x)$ in terms of the
function $\lambda ^a(x)$ . From Eq.(A.3), we solve 
\begin{equation}
\theta ^a(x)=\int d^4x(M^{-1})^{ab}(x-y)\lambda ^b(y)
\end{equation}
Upon substituting the above expression into Eq.(A.2) and then taking
derivative of Eq.(A.2) with respect to $\lambda ^a(x),$ we obtain 
\begin{eqnarray}
&&\frac 1N\int D(A)\triangle _F[A]\int d^4y[J^{b\mu }(y)+m^2A^{b\mu }(y) 
\nonumber \\
&&\ \ -\frac 1\alpha \partial _y^\nu A_\nu ^b(y)\partial _y^\mu ]D_\mu
^{bc}(y)(M^{-1})_{}^{ca}(y-x)\exp \{iS \\
&&\ \ +i\int d^4x[-\frac 1{2\alpha }(\partial ^\mu A_\mu ^a)^2+J^{a\mu
}A_\mu ^a]\}  \nonumber \\
\  &=&0  \nonumber
\end{eqnarray}
From the relation given in Eq.(2.11) with the definition denoted in Eq.(2.4)
and the identity $f^{bcd}A^{c\mu }A_\mu ^{d_{}}=0$, it is easy to see 
\begin{equation}
A^{b\mu }(y)D_\mu ^{bc}(y)(M^{-1})_{}^{ca}(y-x)=A^{b\mu }(y)\partial _\mu
^y(M^{-1})_{}^{ba}(y-x)
\end{equation}
By making use of the relation in Eq.(2.11), the definition in Eq.(2.3) and
the equation in Eq.(2.18 ), we deduce 
\begin{eqnarray}
\frac 1\alpha \partial _y^\nu A_\nu ^b(y)\partial _y^\mu D_\mu
^{bc}(y)(M^{-1})_{}^{ca}(y-x) &=&\frac 1\alpha \partial ^\nu A_\nu
^b(y)\delta ^4(x-y)  \nonumber \\
&&-m^2\partial _y^\nu A_\nu ^b(y)(M^{-1})_{}^{ba}(y-x)
\end{eqnarray}
On inserting Eqs.(A.6) and (A.7) into Eq.(A.5), we obtain an identity which
is exactly identical to that given in Eq.(2.12) although the derivation
described above is different from that given in paper II. In the above
derivation, we started from the generating functional without containing the
ghost field functions and the BRST-sources, therefore, the derivation does
not concern the nilpotency of the composite field functions appearing in the
BRST-source terms. This derivation and the result strongly indicate that the
W-T identity derived in Sect.2 of paper II is correct and hence the
procedure of introducing the BRST-invariant source terms into the generating
functional is completely reasonable.

\begin{references}
\bibitem{1}  J.Sakurai, Ann. Phys. 11, (1960) 1;\\ M.Gell-Mann, Phys. Rev.
125, (1962) 1067.

\bibitem{2}  S.L.Glashow and M.Gell-Mann, Ann. Phys. 15, (1961) 437;\\ %
S.L.Glashow, Nucl. Phys. 22, (1961) 579.

\bibitem{3}  H.Umezawa and S.Kamefuchi, Nucl. Phys. 23, (1961) 399;\\ %
P.A.Ionides, Nucl. Phys. 23, (1961) 662.

\bibitem{4}  A.Salam, Nucl. Phys. 18, (1960) 681; Phys. Rev. 127, (1962) 331.

\bibitem{5}  D.G.Boulware, Ann. Phys. 56, (1970) 140.

\bibitem{6}  C.Itzykson and F-B. Zuber, Quantum Field Theory, McGraw-Hill,
New York (1980).

\bibitem{7}  T.Kunomasa and T.Goto, Prog. Theor. Phys. 37 (1967) 452;\\ %
K.Shizuya, Nucl. Phys. B94 (1975) 26; ibid. B121 (1977) 125.

\bibitem{8}  A.Salam and J.Strathdee, Phys. Rev. D2 (1970) 2869.

\bibitem{9}  R.Delbourgo and G Thompson, Phys. Rev. Lett. 57 (1986) 2610.

\bibitem{10}  A.Burnel, Phys. Rev. D33 (1986) 2985.

\bibitem{11}  L.D.Faddeev and A.A.Slavnov, Gauge Fields:Introduction to
Quantum Theory, The Benjamin Cummings Publishing Company Inc.(1980).

\bibitem{12}  E.S.Abers and B.W.Lee, Phys. Rep. C9 (1973) 1;\\B.W.Lee, In
Methods in Field Theory (1976), ed.R.Balian and Zinn-Justin.

\bibitem{13}  A.Slavnov, Theor. and Math. Phys. 10, (1972) 99, (English
translation);\\J.C.Taylor, Nucl. Phys. B33, (1971) 436.

\bibitem{14}  N.N.Bogoliubov and O. Parasiuk, Acta. Math. 97,(1957) 227;\\%
K.Hepp, Commun. Math. Phys. 2, (1966) 301;\\W.Zimmermann, in Lectures on
Elementary Particles and Quantum Field Theory, edited by S.Deser et al. (MIT
Press, Combride, Mass, 1970).

\bibitem{15}  G.t'Hooft and M.Veltman, Nucl. Phys. B44 (1972) 189.

\bibitem{16}  R.P.Feynman, Acta. Phys. Polonica 24 (1961) 276.

\bibitem{17}  M.H.Thoma, M.luse and H.J.Mang,J.Phys.G: Nucl. Part. Phys. 18,
(1992) 1125 ,references therein.

\bibitem{18}  A.Faessler, G.Lubeck and K.Shimizu, Phys. Rev. D26 (1982) 3280;%
\\M.Kohno and W.Weise ,Nucl. Phys. A479 (1988) 433.

\bibitem{19}  M.Gell-Mann and M.Levy, Nuov. Cim, 16, (1960) 705;\\B.W.Lee,
Chiral Dynamics, Gordon and Breach (1972).

\bibitem{20}  C.G.Callan, Phys. Rev. D2 (1970) 1541;\\K.Symanzik, Commun.
Math. Phys. 18 (1970) 227.
\end{references}

\end{document}

