%Title:  Neutral Current in the Heavy Top Quark Limit and the Renormalization 
	of the Singlet Axial Current
%Authors:  K.G.Chetyrkin, J.H.K\"uhn
%Published:  <EM>Z. Phys.</EM><B> C60 </B><EM>(1993) 497.</EM>
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\begin{titlepage}
\noindent
%
% Datum
%
\hfill TTP93--7\\
\mbox{}
\hfill  March  1993   \\   %
%\hfill \today \\
%
% Title
%
\vspace{0.5cm}
\begin{center}
\begin{Large}
\begin{bf}
Neutral Current in the
 Heavy Top Quark Limit
\\
and
\\
the Renormalization of the Singlet Axial Current
\footnote{Work supported by DFG Grant Nr. Ku 502/3-1}
   \\
  \end{bf}
  \end{Large}
%
% Author
%
  \vspace{0.8cm}
  \begin{large}
K.G.Chetyrkin\footnote{On leave from Institute for Nuclear Research
of the Russian Academy of Sciences, Moscow, 117312, Russia},
J.H.K\"uhn    \\[3mm]
    Institut f\"ur Theoretische Teilchenphysik\\
    Universit\"at Karlsruhe\\
    Kaiserstr. 12,    Postfach 6980\\[2mm]
    7500 Karlsruhe 1, Germany\\
  \end{large}
%
% Abstract
%
  \vspace{1cm}
  {\bf Abstract}
\end{center}
\begin{quotation}
\noindent               %uniquely
We use a simple renormalization group argument to  demonstrate that
the large  top mass limit  of the axial neutral current in QCD is
unambiguously determined by the renormalization of the singlet axial
current and suggests a natural scheme independent normalization
condition for the latter.  By calculating the anomalous dimension of
the singlet axial current at three-loop level in the $\msbar$ scheme
we derive the result for the top quark decoupling from the axial
neutral current valid up to (and including) the terms of order
$\as^2$.

%We also prove the independence of the result on the specific
%presscription chosen to renormalize the singlet axial current.

\end{quotation}
\end{titlepage}


\section{Introduction}

Gauge theories including  axial vector currents are plagued with
potential anomalies which generically tend to spoil the
renormalizability. In the standard model  anomalies are avoided
through an accurate arrangement of quark and lepton multiplets.
For instance,  the very existence of the still missing t-quark is
strongly required  in a renormalizable theory.

Since the t-quark  must be rather heavy there appears the  quite
interesting physical problem of understanding how  the existence of
the very heavy top quark could manifest itself at lower energies.
The t-quark can not just neatly decouple from the low energy physics
because in that case the resulting effective theory would be
anomalous  and, thus, not renormalizable \cite{Collins78}.
% Indeed,
%there are examples of  pathological behaviour of some electroweak
%processes in the limit of very large mass of the top quark
%\cite{Veltman77,Sirlin80}.

On the other hand  QCD,   a vectorlike theory, allows for a
smooth decoupling of a heavy colored fermion and hence would tolerate
the absence of the t-quark.
At first sight  this property should persist if one  treats the
electroweak interactions in the lowest nonvanishing order.
This would imply, for instance,
that the weak neutral current in the effective QCD with five flavours
could  be obtained in the first approximation by simply
omitting the t-quark contribution. This reasoning is not correct though
nothing prevents us from considering an anomalous axial vector
current as an external current in QCD.
%Indeed, as we shall see shortly
The problem has its roots in two tightly interconnected  facts:
\begin{itemize}
\item

        Without the $t$-quark contribution the axial vector
        part of the neutral current ceases to be renormalization
        group invariant. The  effective
        neutral current in the effective QCD with 5 quark
        flavours would  then explicitly depend on
        the QCD renormalization
        scale, a result is physically unacceptable.
\item
        The topless neutral current leads to a new class of
        divergent diagrams (see Fig.1 a ) whose proper normalization
        may be performed only at the price of introducing
        an a priori unknown constant into  the physical prediction.
\end{itemize}


In the present paper  we settle  this  issue by  deriving  the effective
neutral current in the heavy top quark limit within the commonly
accepted framework of dimensional regularization and (modified) minimal
subtraction  \cite{Hooft72,dim.reg,Hooft73,MSbar}. We give an all-order
solution, which, in fact, proves to be completely independent from
particular prescriptions chosen to deal with $\g_5$ and to
renormalize various axial vector currents involved.

%In the present work we will undertake a regular all-order analysis of
%the decoupling of the heavy top quark from the neutral axial quark
%current  within a very convenient framework of the $\msbar$-scheme
%and the heavy mass expansion formalism in this scheme \cite{Smi91}.

The present paper  owes much to that of Collins,  Wilczek and Zee
\cite{Collins78} who studied a closely related problem. They
considered the kinematical situation when both the $t$- and
$b$-quarks were treated as having large masses $m_t$ and $m_b$
respectively with $m_t >> m_b$.  We shall comment on that work later.
%in conclusion.

This work is organized in the following way.  Section 2  deals with
the renormalization of singlet and non-singlet axial vector currents
in QCD. We discuss, in particularly, the additional renormalization
ambiguitiy which appears in the singlet case and suggest a natural
way to fix it. We also report on the  calculations of the
respective non-singlet and singlet anomalous dimensions in the
$\msbar$  scheme. The latter prove to be essential
ingredients in  our treating of both the renormalization ambiguities
and the top quark decoupling which is studied in Section 3.  The
subsequent short section is devoted to the construction of the
effective top- and bottomless neutral current  for the effective
theory obtained by the decoupling  of $t$ and $b$ quarks.  We
summarize and discuss our results in Section 5.


\section{Renormalization of  the singlet and non-singlet axial vector
currents at the three loop level}
%and $\g_5$ problem }

%We begin with introducing some notations.
The axial  part of the neutral currentis
will be conveniently written as sum over quark weak
isodublets $(\psi_i,\psi_{i'})$
\beq
A_\alpha = \sum_{i=u,c,t}
\D^{i}_\alpha, \ \  \D^{i}_\alpha= A^i_\alpha - A^{i'}_\alpha
\EQN{2.1}
\eeq
with
$A^i_\alpha=\ovl\psi_i \g_5\g_\alpha \psi_i$.
For the $\g_5$ matrix in dimensional regularization we
adopt  the definition proposed by t' Hooft and
Veltman \cite{Hooft72} with a  modification introduced in
\cite{{gamma5mu}}:
\beq
A^i_\alpha = \ovl{\psi}_i\g_\alpha\g_5 \psi_i \equiv
\epsilon _{\alpha\beta\nu\rho}
\ovl{\psi_i}\g_\beta\g_\nu\g_\rho \psi_i.
\EQN{axial.dim}
\eeq
The $\ep$-tensor is by definition a genuinely four-dimensional
object and, thus, the indices $\alpha,\beta,\nu,\rho$ are restricted
to  $0,1,2$ and $3$.   The Lorentz index $\alpha$ will
be dropped  in the rest of the paper.

Let us consider the minimally renormalized
current $A^i$ expressed in terms of bare operators:
\beq
[A^i]_{{}_{\sle ms}} =
\mbox{{\protect \large Z}}^{NS}_{}{\fbrs{\,0}{A^i}}
+\mbox{{\large Z}}^{\psi}_{}{\fbrs{\,0}{A^S}},
\EQN{2.2}
\eeq
with the singlet current $ A^S = \sum_i A^i$.
Both renormalization constants
$\mbox{{\protect \large Z}}^{NS}$ and
$\mbox{{\protect \large Z}}^{\psi}$
are chosen  to be
series in the (defined again via minimal subtraction procedure)
renormalized {\em couplant}
$a(\mu) = \alpha_s(\mu)/\pi \equiv g^2(\mu)/4\pi^2$ with
$\mu$ being the renormalization scale
and with coefficients containing pure poles in $\ep = (4 - D)/2$
viz.
\beq
\mbox{{\large Z}}^{NS}  = 1  + \sum^{i \geq 2}_{0<j\leq i}
\left( \mbox{{\large Z}}^{NS} \right)_{ij} a^i\ep^{-j}
\EQN{2.3}
\eeq
and similarly for $\mbox{{\protect \large Z}}^{\psi}$.
As is well known  one loop anomalous dimensions of all currents
are independ of the  renormalization prescription  and vanish
identically for vector and axial vector currents independent of
their flavour structure.
This explains why  the first non-trivial contributions to the rhs of
\r{2.3} start from  order  $a^2$.  The renormalization
constant $\mbox{{\large Z}}^{NS}_{}$ removes UV
divergencies appearing in non-singlet graphs  while
$\mbox{{\large Z}}^{\psi}$
performs the same task  for purely singlet
contributions.  It follows from \r{2.2} that the non-singlet current
$\D^i$ is  multiplicatively renormalized
\beq
[\D^i]_{{}_{\sle ms}} =
\mbox{{\large Z}}^{NS}_{}{\fbrs{\,0}{\D^i}}.
\EQN{2.4}
\eeq
$\mbox{{\large Z}}^{NS}$
is {\em independent}  of the renormalization prescription of
the singlet current, whereas the converse is not true
($n_f$ is the  number of quark flavours):
\beq
[A^S]_{{\sle ms}} =
\mbox{{\large Z}}^{S}_{}
{\fbrs{\,0}{A^S}}, \ \ \
\mbox{{\large Z}}^{S}_{}
=
\mbox{{\large Z}}^{NS}_{}
+
n_f \mbox{{\large Z}}^{\psi}_{}.
\EQN{2.5}
\eeq

At present $\mbox{{\large Z}}^{NS}_{}$ is computed up to and including
the terms of order $a^3$ \cite{Larin91}
\beq
\ba{c}
\dsp
        \mbox{{\large Z}}^{NS}_{}  =
\\
\dsp
          1 + \frac{\ac^2}{24\ep}( 11 C_F C_A - 2 C_F n_f )
        + \ac^3  \frac{C_F}{2592\epsilon^2} (6 (-121
        C_A^2 +44 C_A n_f -4 n_f^2 )
\\
\dsp
         +{\epsilon} ( -1386 C_F C_A + 1789 C_A^2
	+144 C_F n_f -416 C_A n_f +4 n_f^2 ) )
\ea
\EQN{Zns}
\eeq
where
$C_F$ and $C_A$ are the Casimir operators of the quark and the
adjoint representations of the color group.

\begin{figure}
\begin{picture}(50000,20000)(-11000,0)

\THICKLINES
%
% diagram (a)
\drawline\fermion[\SW\REG](3000,14000)[4000]
\drawline\fermion[\SE\REG](\fermionfrontx,\fermionfronty)[4000]
\global\advance\pmidx by 500
\global\advance\pmidy by 500
\put(\pmidx,\pmidy){$h$}
\drawline\gluon[\S\CURLY](\pbackx,\pbacky)[4]
\drawline\fermion[\W\REG](\fermionbackx,\fermionbacky)[5657]
\drawline\gluon[\S\FLIPPEDCURLY](\pbackx,\pbacky)[4]
\drawline\fermion[\E\REG](\pbackx,\pbacky)[5657]
\global\advance\pmidy by -4000
\put(\pmidx,\pmidy){$(a)$}
\drawline\fermion[\S\REG](\pbackx,\pbacky)[2000]
\global\advance\fermionfrontx by -5657
\drawline\fermion[\S\REG](\fermionfrontx,\fermionfronty)[2000]
%
%
% diagram (b)
\drawline\fermion[\SW\REG](20000,15000)[4000]
\drawline\fermion[\SE\REG](\pfrontx,\pfronty)[4000]
\global\advance\pmidx by 500
\global\advance\pmidy by 500
\put(\pmidx,\pmidy){$l$}
\drawline\gluon[\S\CURLY](\pbackx,\pbacky)[6]
\drawline\fermion[\W\REG](\fermionbackx,\fermionbacky)[5657]
\drawline\gluon[\S\FLIPPEDCURLY](\pbackx,\pbacky)[2]
\global\advance\gluonbacky by -1000
\put(\gluonbackx,\gluonbacky){\circle{2000}}
\global\advance\gluonbackx by -2000
\put(\gluonbackx,\gluonbacky){$h$}
\global\advance\gluonbackx by 2000
\global\advance\gluonbacky by -1000
\drawline\gluon[\S\FLIPPEDCURLY](\gluonbackx,\gluonbacky)[2]
\drawline\fermion[\E\REG](\pbackx,\pbacky)[5657]
\global\advance\pmidy by -3000
\put(\pmidx,\pmidy){$(b)$}
\drawline\fermion[\S\REG](\pbackx,\pbacky)[2000]
\global\advance\fermionfrontx by -5657
\drawline\fermion[\S\REG](\fermionfrontx,\fermionfronty)[2000]
\end{picture}

\caption{
(a) The lowest order diagram peculiar to the axial vector current
and  contributing to  the anomalous dimension
$\g^\psi_{ns}$ and to the coefficient function ${\bf C}^h_{n}$ ;
(b) A lowest order diagram contributing to the coefficient function
${\bf C}^\psi_n$.
}
\label{fig1}
%
\end{figure}
The lowest order contribution to $\mbox{{\large Z}}^{\psi}_{}$
comes from  a two-loop diagram (see Fig. 1a). It essentially does not
depend on regularization and was first computed in Refs.
\cite{Adler69,Collins78}.
We have used the program MINCER  \cite{MINCER2}
written for the symbolic manipulation system FORM
\cite{FORM} to find the next  leading correction to
$\mbox{{\large Z}}^{\psi}_{}$ of order $\ac^3$
with the result
% This is a piece of TOPRED.RES
%(3*cf)/(16*ep)$
%
%DF(Zpsi,as,3)/6;
%
%
%(ep*cf*(109*ca - 54*cf + 2*nf) + 6*cf*( - 11*ca + 2*nf))/(576*ep**2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%(3*C_F)/(16*\ep)$
%
%DF(Zpsi,as,3)/6;
%
%
%(\ep*C_F*(109*C_A - 54*C_F + 2*nf) + 6*C_F*( - 11*C_A + 2*nf))/(576*\ep**2)$
\beq
\mbox{{\large Z}}^{\psi}_{} =
         \ac^2 \frac{ 3 C_F}{16\ep}
        + \ac^3  \frac{C_F}{(576\ep^2)}
                    (6 (-11 C_A +  2 n_f )
        +       {\epsilon}(
                      109C_A - 54 C_F + 2n_f
                          )
                     )
\EQN{Zpsi}
\eeq

From these definitions  it follows that the current
$[A^i]_{{\sle ms}}$
obeys the RG equation
\beq
\mu^2\frac{d}{\mu^2}[A^i]_{{\sle ms}}
=
\g^{NS}_{{}_{\sle ms}} [A^i]_{{\sle ms}}
+
\g^{\psi}_{{}_{\sle ms}} [A^S]_{{\sle ms}},
\EQN{2.6}
\eeq
with
\beq
\g^\psi_{{}_{\sle ms}} =
\frac{1}{n_f}(\g^S_{{}_{\sle ms}} - \g^{NS}_{{}_{\sle ms}}),
\ \ \
\g^{NS}_{{}_{\sle ms}} =
\mu^2\frac{d}{d\mu^2} \ln \mbox{{\large Z}}^{NS}_{},
\ \ \
\g^{S}_{{}_{\sle ms}} =
\mu^2\frac{d}{d\mu^2} \ln \mbox{{\large Z}}^{S}_{}.
\EQN{??}
\eeq
Here
$\mu$ is the  normalization point and
\beq
\mu^2\frac{d}{d\mu^2} = \mu^2\frac{\partial}{\partial\mu^2}
+ \beta_\ep(\ac) \frac{\partial}{\partial \ac}
+ 2{m}^2 \gm(\ac)\frac{\partial}{\partial {m}^2},
\EQN{2.7}
\eeq
where
$\dsp\beta(\ac)= - \ep a - \sum_{n=0}\beta_n
\left(\apis\right)^{n+2}$ and
$\dsp\g_m(\ac)=\sum_{n=0}-\g^m_n
\left(\apis\right)^{n+1}$
are the QCD $\beta$-function (in $D$ dimensions)
and the quark mass anomalous dimension
respectively.
The explicite expressions for
$\g^{NS}_{{}_{\sle ms}}$ and
$\g^{S}_{{}_{\sle ms}}$  read (
from now on we  specify  $C_F = 4/3$ and $C_A = 3$)
%These  are the results from TOPRED.RES
%GPSIms;
%as**3*nf*( - 1/72*nf - 85/48) - 1/2*as**2*nf$
%
%5GNSms;
%as**3*( - 1/162*nf**2 + 44/27*nf - 391/24) + as**2*(2/9*nf - 11/3)$
%
\beq
\ba{c}
\g^{NS}_{{}_{\sle ms}} = a^2 ( \frac{2}{9} n_f - \frac{11}{3})
               +
               a^3 ( -\frac{1}{162}n_f^2
                    + \frac{44}{27}n_f - \frac{391}{24}
                   ),
\\
\g^{\psi}_{{}_{\sle ms}} =
                       \ \  - a^2 \frac{1}{2}
                       - a^3 (\frac{1}{72} n_f
                              +
                             \frac{85}{48})
\ea
\EQN{2.8}
\eeq

Of course, the use of the $\msbar$ scheme to renormalize an  external
operator is by no means  mandatory even if one sticks (as we and
almost everybody do nowadays) to this prescription  for the
renormalization of the QCD Lagrangian.
In full generality the renormalized
current $A^i$ may be written as
\beq
[A^i] =
\xi^{NS}_{} \mbox{{\large Z}}^{NS}{\fbrs{\,0}{A^i}}
+
\left\{
\frac{1}{n_f}
(\xi^S_{}  - \xi^{NS}_{})
\mbox{{\large Z}}^{NS}_{}
 +
\xi^{S}_{} \mbox{{\large Z}}^{\psi}_{}
\right\}
{\fbrs{\,0}{A^{S}}}.
\EQN{5}
\eeq
The two {\em finite} normalization
constants $\xi_{}^{NS}$ and  $\xi_{}^{S}$
 -- both series in
$a$ starting from $1$ and not depending on $\ep$ --
describe  the finite arbitrariness in the renormalization
of  the non-singlet current
$$
[A^i - \frac{1}{n_f}A^S]
=
\xi_{}^{NS}
[A^i - \frac{1}{n_f}A^S]_{{}_{\sle ms}}
$$
and the singlet current
$$
[A^S]
=
\xi_{}^{S}
[A^S]_{{}_{\sle ms}}
$$

In fact, there is a very good reason to make use of this freedom
at least  for non-singlet diagrams. Within the
standard model the normalization of the anomaly free non-singlet
current $\D^i$ is rigidly connected to that of the quark
propagator through the corresponding axial  Ward identity. On the other
hand it is well-known (see, e.g. \cite{Collins84}) that this identity
is violated because within  our definitions the anticommutator
$\{\g_5,\g_\beta\}$ ceases in general to  be zero.  In order to
effectively restore the  Ward-Takakashi identity it is necessary to
invoke an extra finite renormalization or, equivalently,  to
choose the non-singlet normalization constant in a such a way that
the resulting non-sinhlet anomalous dimension will vanish identically
\cite{Trueman79,Larin91}. This requirement fixes $\xi^{NS}_{}$ to
\beq
z^{NS}_{{}_{\sle ms}}\equiv
\exp
-\myint^{\ac(\mu)}_{0}
\ \frac{\g^{NS}_{ms}(a')}{\beta(a')} d a'
\EQN{zns.ms}
\eeq
and leads to  a RG independent non-singlet current
$[A^i - \frac{1}{n_f} A^S]_{{{}_{\sle I}}}$.

This reasoning does not, generally speaking, impose any constraint on
the singlet current renormalization since neither $\xi^S_{}$ nor
$\mbox{{\large Z}}^S_{}$ are involved in the relation between
$[\D^i]_0$ and $[\D^i]_I$.
Yet it proves
extremely useful to fix $\xi^{S}_{}$ in the same vein: that is to set
it to
\beq z^{S}_{{}_{\sle ms}}
\equiv
\exp -\myint^{\ac(\mu)}_{0}
\frac{\g^{S}_{ms}(a')}{\beta(a')} d a'
\EQN{zs.ms}
\eeq
in order to nullify the  singlet anomalous dimension $\g^{S}$ of the
resulting  singlet current $[A^S]_{{}_{\sle I}}$  and, as a
consequence, that of $[A^i]_{{}_{\sle I}}$.
{
\renewcommand{\arraystretch}{2.3}
Up to order $a^2$ one finds
\beq
\ba{c}
\dsp
z^{NS}_{ms} =
1 + -\frac{4}{3} a
+
a^2(- \frac{19}{36} + \frac{1}{54} n_f),
\ \ \
z^S_{ms} =
1
+
\ac\frac{( - \frac{5}{18} n_f - \frac{11}{3})}{\beta_0}
\\
\dsp
+
\ac^2\left(
 \frac{1}{\beta_0^2}
 (
   - \frac{185}{2592} n_f^2
   + \frac{391}{864} n_f
   + \frac{2651}{144}
 )
-\frac{1}{\beta_0}(\frac{13}{1296} n_f^2
 + \frac{61}{864} n_f
 + \frac{391}{48})
\right),
\ea
\EQN{???}
\eeq
}
%BT0:=(11-2/3*F)/4$
where $\beta_0 = (11 - \frac{2}{3} n_f)/4$.
Thus, starting from the minimally renormalized current
\r{2.2} one can unambiguously construct the
RG invariant axial vector
currents $[A^i]_{{{}_{\sle I}}}$  and $[A^S ]_{{{}_{\sle I}}}$. In fact, this construction
is quite general and can be formulated without  reference
to a concrete renormalization  prescription.
Let us define
\beq
[A^i - \frac{1}{n_f} A^S]_{{{}_{\sle I}}} \equiv
z^{NS} [A^i - \frac{1}{n_f} A^S],
\ \ \
[A^S ]_{{{}_{\sle I}}} \equiv
z^{S} [A^S]
\EQN{2.10}
\eeq
and, finally,
\beq
[A^i]_{{{}_{\sle I}}} = z^{NS}[A^i] +
                        z^\psi  [ A^S], \ \
 z^\psi = \frac{(z^S - z^{NS})}{n_f}.
\EQN{2.11}
\eeq
Here the functions $z^{NS}$ and $z^{S}$ are defined by by the rhs  of
\r{zns.ms} and \r{zs.ms} respectively with the minimal anomalous
dimensions $\g^S_{{}_{\sle ms}}$ and $\g^{NS}_{{}_{\sle ms}}$ being  substituted by the
corresponding generic ones $\g^S$ and $\g^{NS}$.
It is a straightforward exercise to check that the definitions
\r{2.10} and \r{2.11} do not depend on the specific choice of
$\xi^{NS}$ and $\xi^S$ and, thus, coincide with
the ones introduced above within the minimal subtraction
prescription.

To conclude this section  we note that  minimal renormalization
combined with the definition of $\gamma_5$ in  eq.(2)
is inconvenient as  reference  for the axial vector current. It
is in disagreement with  the ``naive''  prescription  which amounts
to using  an anticommuting   $\g_5$ for all non-anomalous cases
(That  is, roughly speaking, for diagrams
without  a single $\g_5$ sitting
inside a closed fermion loop\footnote{See ref.  \cite{Korner92} where
such a scheme is elaborated in  detail.}.) -- or better, with the
requirement  of chiral invariance. To some extent this
rule can be simulated if we define the $n$-normalization by fixing
$\xi^{NS} = \xi^{S}= z^{NS}_{ms}$, and hence
\beq
[A^i]_{{}_{\sle n}} \equiv z^{NS}_{{}_{\sle ms}}.
[A^i]_{{}_{\sle ms}}
\EQN{2.12}
\eeq
It is also worthwhile to note that for a non-singlet current
the $n$-normalization coincides  with the $I$-one
\beq
[\D^i]_{{}_{\sle I}} \equiv
[\D^i]_{{}_{\sle n}}
\EQN{2.13}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Heavy top limit of the neutral current from the
                 renormalization group
        }


Now we address the following question: what happens with an axial
vector current after integrating out a heavy  quark $h$?  Without
essential loss of generality we may deal with the RG invariant
combinations $[A^i]_{{}_{\sle I}}$ exclusively   as  the results for
a generic current $[A^i]$ can be afterward read from
\r{2.10} and \r{2.11}. But the most important advantage of the choice is
that the explicit RG invariance simplifies a lot all derivations  and
makes them almost trivial!

Indeed, in the limit of the mass of the heavy  quark
$m_h$ going to infinity there must  exist
an operator  relation expesssing an operator of the full theory
in terms of some effective operators constructed from remaining
$n'_f= n_f - 1$ light quark fields of the resulting effective
field theory. On purely dimensional  grounds we can write
\beq
[A^i]^{(n)}_{{}_{\sle I}}
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{m_h\to\infty}}^{}
(1 + {\rm \bf C}^\ell_{{}_I}(\ac(\mu),\mu/m_t) )
[A^i]^{(n'_f)}_{{}_{\sle I}}
+
{\rm \bf C}^\psi_{{}_I}(\ac(\mu),\mu/m_t)  [A^S]^{(n'_f)}_{{}_{\sle I}},
\ \ \
i \not = h
\EQN{3.1}
\eeq
\beq
[A^h]^{(n)}_{{}_{\sle I}}
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{m_h\to\infty}}^{}
{\rm \bf C}^h_{{}_I}(\ac(\mu),\mu/m_t)     [A^S]^{(n_f')}_{{}_{\sle I}}
\EQN{3.2}
\eeq
with the coefficient functions
%${\bf C_{{}_I}}^i(\ac(\mu),\mu/m_t)$
%and ${\bf C_{{}_I}}^\psi (\ac(\mu),\mu/m_t)$
being series in the couplant $\ac(\mu)$ of the complete  theory.
The  upper and  subscripts $(6)$ and $(5)$ here and in what  follows
indicate that the running coupling constant, an operator  or a
function is to be taken  in the full QCD  with $n_f$ or in the
effective theory with $n'_f=n_f-1$  ``effective''
flavours respectively.  Recall that by definition the asymptotic
expansion \r{3.1}  means that any Green function of light fields with
an insertion of, say, the  operator $A^h$ can be approximated (up
to power coorections ) by an "effective" Green function to be
obtained through replacing the heavy current by the rhs of \r{3.2}.
The corresponding equations for the coefficient functions  are
usually called  matching equations.  It is also understood that the
Green functions are calculated  by means of the effective QCD
Lagrangian   with the properly changed coupling constant and quark
masses \cite{Bernreuther,Marciano}.

Once  relations \r{3.1} and \r{3.2}  are established, one can
freely  put down similar expansion for a different normalalization
convention, say, for the $n$-normalization, viz.
\beq
[A^i]^{(n)}_{{}_{n}}
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{m_h\to\infty}}^{}
(1 + {\rm \bf C}^\ell_{{}n}(\ac(\mu),\mu/m_t) )
      [A^i]^{(n'_f)}_{{}_{n}}
+
{\rm \bf C}^\psi_{{}n}(\ac(\mu),\mu/m_t)  [A^S]^{(n'_f)}_{{}_{n}},
\ \ \
i \not = h
\EQN{3.1n}
\eeq
\beq
[A^h]^{(n)}_{{}_{n}}
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{m_h\to\infty}}^{}
{\rm \bf C}^h_{{}n}(\ac(\mu),\mu/m_t)     [A^S]^{(n_f')}_{{}_{n}},
\EQN{3.2n}
\eeq
These differently
normalized coefficient functions
are\footnote{In deriving \r{3.2nn} we anticipate that
\protect
$C^\ell_n\equiv C^\ell_{{}_{\sle I}}\equiv 0$
up to power corrections,
see below.}
\beq
\ba{c}
\dsp
C^h_{{}_I} = C^h_n
\frac{ 1 + z_n^{\psi(n_f)}}{ 1 + n'_f z_n^{\psi(n'_f)}}
+
z_n^{\psi(n'_f)}
\frac{ 1 + n'_f C^{\psi}_{n}}{ 1 + n'_f z_n^{\psi(n'_f)}}
\\
\dsp
C^\psi_{{}_I} = C^\psi_n
\frac{ 1 + n_f z_n^{\psi(n_f)}}{ 1 + n'_f z_n^{\psi(n'_f)}}
+
\frac{(C^h_n +1) z_n^{\psi(n_f)} - z_n^{\psi(n'_f)}}%
{ 1 + n'_f z_n^{\psi(n'_f)}}
\ea
\EQN{3.2nn}
\eeq

Next, as the operators $[A^i]^{(n'_f)}_{{}_{\sle I}}$ and
$[A^\psi]^{(n'_f)}_{{}_{\sle I}}$ are
linearly independent and RG invariant
we immediately find that the coefficent functions in \r{3.1} are RG
invariant and, thus, must fulfill the relation
\beq
\mu^2\frac{d}{d\mu^2} { \bf C_{{}_I}^\ell}      =
\mu^2\frac{d}{d\mu^2} { \bf  C_{{}_I}^h}       =
\mu^2\frac{d}{d\mu^2} { \bf  C_{{}_I}^\psi}     = 0.
\EQN{3.3}
\eeq
At last, having in mind the asymptotic freedom, we conclude that at
the limit of $m_h\to\infty$ or that of $\ac(m_h)=0$
all the coefficient functions
%${\rm \bf C_{{}_I}}^i(\ac(m_t),1)$ and
${\rm \bf C}^\psi_{{}_I} (\ac(\mu),\mu/m_t)$
are just constants which can be found without any loop calculations,  just
from the tree level matching equations!
The result is self-evident  and reads
\beq
{\bf C}^\ell_{{}_I} = {\bf C_{{}_I}}^h
= {\bf C_{{}_I}}^\psi  = 0,
\EQN{3.4}
\eeq
whence
\beq
A^{i(n_f)}_I
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{m_h\to\infty}}^{}
A^{i(n'_f)}_I
\ \ \ \
\mbox{{\rm and}}
\ \ \ \
A^{h(n_f)}_I
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{m_h\to\infty}}^{}
0.
\EQN{3.4a}
\eeq
To avoid  confusion it is worthwhile to stress that the rhs
of \r{3.4} and \r{3.4a}  vanish   only provided one neglects
power suppressed   corrections and  corrections proportional
to $\ac(m_h)$. The first type of corrections is discarded  through
all the work while the lowest order contributions of the
second type will be computed at the end of this section the next section.

Let  us apply \r{3.4} to derive the heavy top limit of the
neutral current.
Foe the moment we shall set $a(m_h)=0$.
As we have discussed above the correctly normalized
axial part of the quark neutral current reads\footnote{Recall that if
$\psi,\psi'$ stands for a quark  weak isodublet then
$\D^{i}= A^i - A^{i'}$.}
\beq
A =  \sum_{i=u,c,t} [\D^i]^{(6)}_{{}_{\sle I}}
\EQN{3.5}
\eeq
First of all we observe that if $i\not=t$ then
\beq
[\D^i]^{(6)}_{{}_{\sle I}}
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{m_t\to\infty}}^{}
[\D^i]^{(5)}_I
\equiv
\left\{
z^{NS}_{{}_{\sle ms}} [\D^i]_{{}_{\sle ms}}
\right\}^{(5)}
\EQN{3.6}
\eeq
and
\beq
[\D^t]^{(6)}_{{}_{\sle I}}
\bbuildrel{=\!=\!=\!=\!\Longrightarrow}_{\scriptstyle{m_t\to\infty}}^{}
-[A^b]^{(5)}_I
\equiv
\left\{
- z^{NS}_{{}_{\sle ms}} [A^b]_{{}_{\sle ms}}
- z^{\psi}_{{}_{\sle ms}} [A^L]_{{}_{\sle ms}}
\right\}^{(5)},
\EQN{3.6a}
\eeq
with
$
A^L = \sum_{i=u,d,s,c,b} A^i .
$
Explicitly we have
%(after taking again
%$C_F = 4/3$ and $C_A = 3$)
\beq
[\D^t]^{(6)}_{{}_{\sle I}}
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{m_t\to\infty}}^{}
\left\{
-(1 - \frac{4}{3} a - \frac{47}{108} a^2)[A^b]_{{}_{\sle ms}}
+ a \frac{6}{23}
(1 - \frac{2569}{3312} a)
[A^L]_{{}_{\sle ms}}
\right\}^{(5)}.
\EQN{3.7}
\eeq

In terms of $[A^i]_{{}_{\sle n}}$  the asymptotic expansions
\r{3.6}, \r{3.6a} and  \r{3.7}  are rewritten as follows
\beq
[\D^i]^{(6)}_{{}_{\sle I}}
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{m_t\to\infty}}^{}
[\D^i]^{(5)}_{{}_{\sle ms}},
\EQN{3.9}
\eeq
\beq
[\D^t]^{(6)}_{{}_{\sle I}}
\bbuildrel{=\!=\!=\!=\!\Longrightarrow}_{\scriptstyle{m_t\to\infty}}^{}
\left\{
-  [A^b]_{{}_{\sle ms}}
- z^{\psi}_{{}_{\sle n}} [A^L]_{{}_{\sle ms}}
\right\}^{(5)},
\EQN{3.9a}
\eeq
and
\beq
[\D^t]^{(6)}_{{}_{\sle I}}
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{m_t\to\infty}}^{}
\left\{
-[A^b]_{{}_{\sle n}}
+ a \frac{6}{23}
(1 + \frac{1847}{3312} a)
[A^L]_{{}_{\sle n}}
\right\}^{(5)}.
\EQN{3.10}
\eeq

However the most important and useful feature of the  the $n$-normalization
is that  eq. \r{3.9} remains valid (up to power terms) even if one does
take into account  all the corrections  of order $O(\ac(m_t))$ and higher!
This is the case  because  due to its very definition  the current
$[\D^i]_n$ with $i\not= h$ must meet the standard non-anomalous
Ward identity\footnote{As we neglect all power corrections in any case
we may freely assume that all quarks except for $h$ are massless in
our discussion.} (below $j\not=h$)
\beq
\ba{c}
%\dsp
i\frac{\prd}{\prd y}
\langle T
\left(
\psi_j (x) [A^i(y) - A^{i'}(y)]^{(n_f)} \ovl{\psi}_j (z)
\right)
\rangle
=
\\
%\dsp
\left\{
\left(
\delta_{ij}\delta(x-y)
-
\delta_{ij}\delta(y-z)
\right)
\langle T
\left(
\psi_j (x) \ovl{\psi}_j (z)
\right)
\rangle
\right\}
-
\left\{
i \to i'
\right\}
\ea
\EQN{3.11}
\eeq
The identity must be true  in the complete QCD and in the effective
theory  with the heavy quark $h$ decoupled.  On the  other hand, one
should be able to begin from \r{3.11} in the full theory and then to
find its counterpart in the effective theory by using \r{3.9}. The
result is of course the absence of higher order corrections in
$\ac(m_t)$
to the rhs of \r{3.9},  which in turn implies that
the coefficient function $C^\ell_n \equiv 0$.

Moreover, the first order corrections  in $\ac(m_h)$ to $C^h_I$ and
$C^\psi_I$ can as well be found without explicit calculations of
massive diagrams.  We begin by  observing that according to the very
idea of decoupling we need to know the operators and the coefficient
functions in the rhs of \r{3.9a} at a low normalization scale
$\mu =\mu_{(n'_f)} \ll m_h$.  However,
a direct evaluation of $C^h_n$ and $C^\psi_n$
at such a low scale is absolutely meaningless  because of the
inevitable appearence of large logs of the ratio of $m_h^2/\mu^2$ in the
result. A correct approach is to compute  the {\em RG invariant}
functions $C^h_{{}_I}$ and   $C^\psi_{{}_I}$ choosing
the scale $\mu= m_h$
%and {\em then}  to  construct
%$C^h_n$ and   $C^\psi_n$ {at $\mu = \mu_{(n'_f}$} from
%the system of linear equations \r{3.2nn}.
In fact, we may drop both the
coefficient functions $C^h_n$ and   $C^\psi_n$ from the rhs of \r{3.2nn}
as they start  from the $\ac(m_h)^2$ terms (see Fig.1 (a),(b)) while  we are
interested  in the lowest order corrections in  $\ac(m_h)$.
The results read
\beq
\ba{l}
C^h_{{}_I} =
 (z_n^{\psi(n_f)})_1 a^{(n_f)}(m_h) + O((\ac(m_h))^2),
\\
C^\psi_{{}_I} =
 (z_n^{\psi(n_f)})_1 a^{(n_f)}(m_h)
-
 (z_n^{\psi(n'_f)})_1 a^{(n'_f)}(m_h) + O((\ac(m_h))^2),
\ea
\EQN{3.12}
\eeq
with
\beq
z^{{\psi}(n_f)}_{n}  =  \sum_{i \geq 1}
(z^{{\psi}(n_f)}_{n})_i  (a^{(n_f)}))^i
= -\frac{1}{2\beta_0} + \dots
= -\frac{6}{33-2n_f} a^{(n_f)} + \dots
\EQN{3.13}
\eeq
Inclusion of of the term of order  $\ac(m_t)$
therefore leads to the following modifications of
\r{3.9a}  and \r{3.10}
\beq
\ba{c}
\dsp
[\D^t]^{(6)}_{{}_{\sle I}}
\bbuildrel{=\!=\!=\!=\!\Longrightarrow}_{\scriptstyle{m_t\to\infty}}^{}
\\
\dsp
-  [A^b]^{(5)}_{{}_{\sle ms}}
-
\left\{
z^{\psi}_{{}_{\sle n}} [A^L]_{{}_{\sle ms}}
\right\}^{(5)}
+\ac^{(5)}(m_t) \frac{6}{23}
\left\{
(1+ 5 z^{\psi}_{{}_{\sle n}}) [A^L]_{{}_{\sle ms}},
\right\}^{(5)},
\ea
\EQN{3.14}
\eeq
\beq
\ba{c}
\dsp
[\D^t]^{(6)}_{{}_{\sle I}}
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{m_t\to\infty}}^{}
\\
\dsp
-[A^b]^{(5)}_{{}_{\sle n}}
+
\left\{
 a \frac{6}{23}
(1 + \frac{1847}{3312} a)
[A^L]_{{}_{\sle n}}
\right\}^{(5)}
%\\
%\dsp
- \ac^{(5)}(m_t) \frac{6}{23}
\left\{
(1 + \frac{30}{23} a)
[A^L]_{{}_{\sle n}}
\right\}^{(5)}.
\ea
\EQN{3.15}
\eeq


\section{Heavy top and bottom  limit of the neutral current
        }


The outlined technique can be easily applied for sequential decoupling
of more then one heavy quarks from  the axial vector neutral current.
Indeed, considering   scales less then that given by the mass
of the bottom quark $m_b$  we can derive the effective top and bottomless
neutral current  by elaborating \r{3.5}
with the  help   of  \r{3.1}, \r{3.2} and \r{3.12}.
The result is written as follows
\beq
A^0
\bbuildrel{=\!=\!=\!=\!\Longrightarrow}_%
{\scriptstyle{m_t\gg m_b\gg \mu^{(4)}}}^{}
[\D^u]^{(4)}_I
+
[\D^c]^{(4)}_I
+
\frac{6}{23}
(\ac^{(5)}(m_b) - \ac^{(5)}(m_t)
[A^L]^{(4)}_{I}.
\EQN{4.1}
\eeq

It is instructive to compare this expresssion
with an  expansion obtained in \cite{Collins78}
\beq
A^0
\bbuildrel{=\!=\!=\!=\!\Longrightarrow}_%
{\scriptstyle{m_t\gg m_b\gg \mu^{(4)}}}^{}
[\D^u]^{(4)}_I
+
[\D^c]^{(4)}_I
+
\frac{1}{2}
\ovl{a}(m_t) \ovl{a}(m_b)
\ln\frac{m_t^2}{m_b^2}
[A^L]^{(4)}.
\EQN{4.2}
\eeq
Here $\mu^{(4)}$ is the  normalization scale
of the 4-quark effective theory and
the running couplant $\ovl{\ac}(\mu)$
is defined as follows:
\beq
\ovl{a}(\mu) =
\left\{
\ba{l}
\dsp
a^{(6)}(\mu) \ \ \mbox{{\rm if}} \ \ \mu \geq m_t
\\
\dsp
a^{(5)}(\mu) \ \ \mbox{{\rm if}} \ \ m_t > \mu \geq m_b
\\
\dsp
a^{(4)}(\mu) \ \ \mbox{{\rm if}} \ \ \mu \leq m_b
\ea
\right.
\EQN{4.3}
\eeq
As \r{4.2} describes the lowest order singlet correction to the
effective  axial vector  neutral current
one  may use a   simple formula
\beq
a^{(n_f)}(\mu) = \frac{6}{(33 - 2 n_f)
\ln\frac{\scriptstyle \mu}{\scriptstyle \Lambda^{(n_f)}}}
\EQN{?}
\eeq
to find that
after splitting
$\ln\frac{m_t^2}{m_b^2} =
\ln\frac{m_t^2}{\mu^2} - \ln\frac{m_b^2}{\mu^2}$ with
$m_t > \mu > m_b  $
\r{4.2}  transforms to
\beq
A^0
\bbuildrel{=\!=\!=\!=\!\Longrightarrow}_%
{\scriptstyle{m_t\gg m_b\gg }}^{}
[\D^u]^{(4)}_I
+
[\D^c]^{(4)}_I
+
\frac{6}{23}
(\ac^{(5)}(m_b) - \ac^{(5)}(m_t)
[A^L]^{(4)}.
\EQN{4.5}
\eeq
in complete agreement with \r{4.1}.

Note that
in \cite{Collins78} it was not specified  which concrete
renormalization scheme should be employed to renormalize
the current  $[A^L]^{(4)}$  in \r{4.2}.
This means that \r{4.2} is valid only provided one neglects
all the corrections of order  \\
$(\ac^{(4)}(\mu_0))^n\ovl{a}(m_t) \ovl{a}(m_b)$ , $n \ge 1$
as a change  of the  scheme
would generically lead to  corrections of just this order.
On the other hand, our formula \r{4.1} -- explicitly scheme
independent one --  takes effectively in  account all these
corrections.

\section{Conclusion}


To summarize, we have proved that the large top mass limit of the
axial part of  the quark neutral current can be evaluated (for a
given renormalization   scheme) completely in terms the anomalous
dimension of the singlet axial vector current.  We have calculated
the anomalous dimension up to and including three-loop corrections
and have used the result to explicitly construct the effective
top-less neutral current including the terms of order $\alpha_s^2$.
It has also been demonstrated that  the heavy top limit of the neutral
current does not, in fact, depend on any renormalization ambiguities
involved in defining of the singlet axial current and can be used to
introduce a physically motivated scheme-independent renormalization
condition for this operator.  As a spin-off we also have found the
effective top and bottom-less neutral current for the effective four
flavour QCD and have checked  that our result is in agreement with
that of Ref.  \cite{Collins78}



Our results can be used in evaluating of hadronic contributions to
the $Z$-boson propagator for momentum transfers far below the mass of
the top quark.  Present bounds from hadron colliders and radiative
corrections seem to favour $m_t$ larger than the $Z$-boson mass
the approximations developed in this paper
may well be employed in studying the $Z$-boson rate into hadrons
within  effective topless QCD.  We hope to  return to
these issues in future publications.  \vspace{1cm}







\noindent
{\large \bf Acknowledgement}

After the completion of the evaluation of  the renormalization constant
$Z^S$ we received  the preprint  \cite{Larin92} where
the  anomalous dimension of the singlet axial current was  computed
in the $\msbar$ scheme.
K.G.Ch. would like to  thank S.A. Larin for a useful   discussion
and sending  him a copy of the preprint  \cite{Larin92} .

\newpage

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\end{document}
