%Title: Renormalization of 4-Quark Operators and QCD-Sum Rules
%Authors: L.-E. Adam and K.G. Chetyrkin
%Published: <EM> Phys. Lett. </EM> <B> B329 </B> (1994) 129-135. 
%hep-ph/9404331
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\begin{document}
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\begin{titlepage}
\noindent
\hfill TTP93--26 \\
\mbox{}
\hfill November 1993 \\
%
% Titel
%
\vspace{0.5cm}
\begin{center}
  \begin{Large}
    \begin{bf}
        Renormalization of 4--Quark Operators\\[0.4cm]  
        and QCD--Sum Rules  \\
    \end{bf}
  \end{Large}                                                                   
%                                                                               
% Author                                                                        
%                                                                               
 \vspace{1.2cm}                                                                
 \begin{large}                                                                 
L.-E.Adam   \\[3mm]                                        
    Institut f\"ur Theoretische Teilchenphysik\\                                
    Universit\"at Karlsruhe\\                                                   
    Kaiserstr. 12,    Postfach 6980\\ %[2mm]                                      
    76128 Karlsruhe, Germany\\ [5mm]
K.G.Chetyrkin  \\[3mm]                                                          
    Institute for Nuclear Research\\                                                
    Russian Academy of Sciences\\                                                   
    60th October Anniversary Prospect 7a\\                                          
    Moscow, 117312, Russia \\                  
 \end{large}                                                                   
%                                                                               
% Abstract                                                                      
%                                                                               
  \vspace{3cm}                                                                  
  {\bf Abstract}                                                                
\end{center}                                                                    
\begin{quotation}                                                               
\noindent                                                                       
We compute the renormalization mismatch  displayed in 1--loop 
approximation by classically equivalent 4-quark operators  and 
coming from different possible definitions of the $\gamma_5$ matrix 
in  dimensional regularization. The result is then employed to study 
the effect of the various treatments of  $\gamma_5$ upon the size of 
radiative corrections to 4-quark condensates in the QCD sum rules 
for $\rho$ and $A_1$ mesons. We find that a fully anticommuting 
$\gamma_5$ which automatically respects non-anomalous chiral 
Ward-Slavnov identities leads to considerably smaller corrections 
and reduces theoretical uncertainty in the QCD prediction for the 
$\tau$ hadronic decay rate.
\end{quotation}                         
\end{titlepage}                                                                 


\newpage

\section{Introduction} 

Dimensional regularization (DR) \cite{dim.reg} coupled with the 
minimal subtraction scheme (MS) \cite{ms} is widely acknowledged as 
theoretically sound and  extremely valuable in practice calculational 
framework.  In a theory like QCD with parity conserving amplitudes it 
respects all the symmetries of the theory such as gauge invariance, 
Bose symmetry and Ward-Slavnov identities. In addition, 
dimensionaly regularized divergent Feynman integrals are in many 
respects very similar to the convergent ones, which allowed to 
develop quite powerful methods of their  analytical calculation.

A notorious shortcoming of DR is the absence of a natural 
generalization of the Dirac matrix \GFt  \  to noninteger values 
$D\not=4$ of the dimension of space-time.  Indeed, the well-known  
relation  
\beq
  {\rm tr}(\GFm \g_{\mu_1}\g_{\mu_2}\g_{\mu_3}\g_{\mu_4})
  = - 4i\epsilon_{\mu_1\mu_2\mu_3\mu_4}
  \EQN{rel1}
\eeq
which in fact fixes \GFt  \  in 4 dimensions, is not compatible
with another property of \GFt  \  valid at $D=4$
\beq
  \{ \GFm,\gamma_\mu \} =0, %\ \ \  \mu= 1 \dots D
  \EQN{rel2}
\eeq
The problem is  that the totally antisymmetric $\epsilon$-tensor is 
a purely 4-dimensional object and thus can not be self-consistently
continued to $D$  dimensions. It may be shown that
if \r{rel2} is to be held literally for  all values of $\mu= 1 \dots D$ 
(and the standard properties of the trace operation like cyclycity
is respected) then  the rhs of \r{rel1} should vanish  identically 
\cite{BM77a}.    Such a defined  \GFt  \  does not, hence, go 
smoothly  at $D\to 4$ to the 4-dimensional 
\beq
 \GFm^{(4)} = - i\g^0\g^1\g^2\g^3
\EQN{gf4}
\eeq
in contradiction with general requirements to which  every 
self-consistent regularization must meet \cite{Bonneau90}.  From the 
same point of view the relation \r{rel2} may be freely  violated 
for the values of index $\mu$ lying outside of the physical 
4-dimensional subspace.  


The existing solutions of the problem   may be divided into two 
classes.  The first  one  favours a nonvanishing anticommutator 
$\{\GFm,\gamma_\mu \}$ at $\mu \not=0,1,2,3$ 
and is discussed in general in \cite{gamma5mu} and \cite{Thompson85}. 
An explicit version of such a scheme was proposed by {}'t Hooft and 
Veltman and elaborated by Breitenlohner and Maison 
\cite{dim.reg,BM77a}. Another class keeps \r{rel2} intact at least 
for amplitudes without traces with odd numbers of \GFt  \  matrices 
while trying to reproduce  the identity \r{rel1} either by hand 
\cite{Chanowitz79} or, in a bit more sophisticated manner, by 
redefining the trace operation for the traces with odd number of \GFt  
\cite{Kreimer90,Korner92}.


Leaving aside a (rather academic) task  of finding an explicit formal 
proof of self-consistency of these solutions, we will study in this 
work another aspect of the \GFt  \ problem --- different 
possibilities of choosing between classically equivalent 4--quark 
operators which  begin to differ in higher orders due to different 
treatment of the \GFt  \  matrix. As we shall see below, different 
choices lead to different theoretical predictions of the QCD sum 
rule method \cite{SVZ79} because of a commonly accepted approximate 
procedure of the vacuum saturation in estimating the vacuum 
expectation values of 4-quark operators.  On considering  a number 
of particular physical problems, it will be demonstrated that the 
choice of fully anticommuting \GFt considerably reduces theoretical 
uncertainty coming from higher order corrections. 



\section{The $\gamma_5$ %
ambiguity in 4-quark operators 
%and its effect upon
%their vacuum expectation values
}


The problem under discussion appears even in cases when initial 
amplitudes comprise no \GFt  \  matrix at all. Indeed, let us consider 
the standard derivation of the QCD sum rules for the $\rho$ 
and $A_1$ mesons \cite{SVZ79}. It starts from constructing the  Wilson 
expansion \cite{Wilson69} for the correlator
%
\beq
  \Pi_{\mu\nu}(q) = i \, \int dx \, e^{iqx} 
               <Tj_{\mu}(x)j^\dagger_{\nu}(0)>\hspace{-0.5em}_{_0}
                   = (q_{\mu}q_{\nu} - g_{\mu\nu} q^2 ) \Pi(Q^2)
  \EQN{2.1}
\eeq
\beq
   \Pi(Q^2) = C_0 + \frac{C_4 <{\cal O}_4> 
                    \hspace{-0.5em}_{_0}}{Q^4} 
                  + \frac{C^{i,V/A}_6 <{\cal O}^i_6> 
                    \hspace{-0.5em}_{_0}}{Q^6} + \ldots
  \EQN{2.2}
\eeq
where $j_{\mu} =  \bar u \ \gtmu(\GFm)  d $
is the current for the $\rho (A_1)$ meson.  

The power corrections of order $O(1/Q^6)$  to  the rhs of the 
expansion \r{2.2} come from 4-quark condensates (that is the 
vacuum expectation values (VEV) of 4-quark operators) and lead 
to most important numerically non-perturbative corrections to  the 
sum rule for the $\rho$ meson. In the lowest order of 
perturbation theory the contributions of the  4-quark operators  
are well-known  and read \cite{SVZ79}
\beq
    \frac{C^{i,V}_6 Q^i_6}{g^2} 
  = -\frac{2}{9}
      (\bar u\ \gtmu t^a u +\bar d\ \gtmu t^a d)\cdot 
                      (\bar\Psi\  \ghmu t^a \Psi) 
   -  {2}
       \bar u\ \GFm \gamma_\mu t^a d 
    \, \bar d\ \GFm   \gamma_\mu t^a u      
     \EQN{2.3a}
\eeq
for the vector correlator
and 
\beq
    \frac{C^{i,A}_6 Q^i_6}{g^2} 
  = -\frac{2}{9}
      (\bar u\ \gtmu t^a u +\bar d\ \gtmu t^a d)\cdot 
                      (\bar\Psi\  \ghmu t^a \Psi) 
   -  {2}
      \bar u \gamma_\mu  t^a  d 
   \, \bar d  \gamma_\mu t^a  u      
     \EQN{2.3b}
\eeq
for the axial vector one.
Here  $t^a,\  a=1,\dots 8 $ are the colour Gell-Mann matrixes  
normalized as 
$\ Tr(t^a t^b) = \frac{1}{2} \delta^{ab}$; $ (\bar \Psi 
\Gamma \Psi) $ 
is a shorthand for $\sum_{f=u,d,s} (\bar \Psi_f \Gamma \Psi_f) $;
$g$ is the quark gluon coupling constant such that 
$\alpha_s = g^2/4\pi$.

Let us  consider in detail the structure of the second term in 
\r{2.3a} 
proportional to the 4-quark  operator
\beq
\calOm'_1 =
      \bar u\ \GFm \gamma_\mu t^a d \,
      \bar d\ \GFm \gamma_\mu t^a u.
\EQN{2.4a}
\eeq
In fact,  the very calculation  produces a bit  different 
operator
\beq
  \calOm_1 =
      \bar u\ \GDm t^a d \, \bar d\ \GDm t^a u, \ \  
      \GDm \equiv \frac{1}{2} (\gtmu \gtnu \gtl - \gtl \gtnu \gtmu ) 
  \EQN{2.5a}
\eeq
while the result \r{2.3a} is obtained after a use of the identities
\beq
  \GDm \bigotimes \GDm =  
  \,6 \GFm \gamma_\mu \bigotimes \GFm \gamma_\mu.
  \EQN{2.6}
\eeq
Similarly, the initial form of the second term in \r{2.3b}
\beq
\calOm'_2 =
      \bar u  \gamma_\mu t^a d \,\bar d \gamma_\mu t^a u.
\EQN{2.4b}
\eeq
is (with an appropriately changed coefficient function)
\beq
\calOm_1 =
      \bar u\ \GDm \GFm t^a d \, \bar d\ \GDm \GFm  t^a u, 
\EQN{2.5b}
\eeq

The relation \r{2.6} is valid in 4-dimensional space due to the 
very definition of the $\GFm^{(4)}$ matrix \r{gf4}.  However,  it 
loses any operational sence at generic $D\not=4$ within the framework 
of dimensional regularization.  This implies that if both 
operators $\calOm_1$ and $\calOm'_1$  are to be minimally renormalized, then  the 
difference 
\beq \delta \calOm_1 = \calOm_1 - 6\calOm'_1  = {\cal O}(\alpha_s)
\EQN{2.7} 
\eeq 
is an {\em evanescent} operator \cite{Collins84} which vanishes in 
the tree approximation but typically receives nonzero contributions 
in higher orders.  This also  means that higher order corrections to 
the coefficient functions (CF) $C_6^i$  also depend on specifying 
which operators are chosen as basic  ones.  It should be stressed 
that this $\GFm$ ambiguity  comes from the absence of a  unique 
canonical representation of 4-quark operators in dimensional 
regularization, which, in turn, does not allow to define the idea of 
minimal subtraction of UV poles in an unambiguous way.

The above discussion does not imply in any means that  the 
theoretical predictions for the contributions to the rhs of \r{2.2} 
from  4-quark operators are ill-defined in principle: the VEV's of 
the latter should, of course, also depend on the choice of the 
renormalization prescription in such a way that  the whole 
combination $\sum_i C^i_6 \calOm^i_6$  is  invariant   as it must be. 

The {\em real} 
problem lies  in our present inability to find the VEV's of 
4-quark operators in a sufficiently accurate way.
Indeed, the common practice is to use the vacuum     
saturation (VS)  procedure \cite{SVZ79}. It   states that  
the VEV of a 4-quark operator 
${\cal O} = (\bar \Psi \Gamma_1 \Psi) (\bar \Psi \Gamma_2 \Psi) $ 
may be estimated according to the following formula
\beq 
   <{\cal O}> \hspace{-0.5em}_{_0} = \frac{1}{144}\, \lbrack Tr(\Gamma_1)
       Tr(\Gamma_2) - Tr(\Gamma_1 \Gamma_2) \rbrack \,\cdot <\bar qq>^2
   \EQN{vd}
\eeq
where $\bar qq = \bar uu, \bar dd$ or $ \bar ss $.  

It is clear that  this procedure is too rough and does not feel in 
any way such fine details as the concrete mode of the renormalization 
of a 4-quark operator.  Hence, if one would like  to estimate the 
effect of higher order corrections to the CF's $C_6^i$ the numerical 
results will do depend on specifying 4-quark operators in $D$ 
dimensions. Unfortunately, these corrections prove to be of 
considerable size (see  \cite{CheSpi88}  and below) at least in 
the operator basis featuring objects like 
$\GDm  \bigotimes  
\GDm $.  
Let us try  to search for  a distinguished operator basis for which 
the VS would presumably work better.


Of course, from a general point of view there is an infinitely large 
variety  of possible choices of a (different at $D\not=4$ and 
equivalent at $D=4$)  4-quark operator. For instance, one 
could mix operators $\calOm_1$ and $\calOm'_1$ as follows 
\beq
  \calOm{''}_1   = \calOm_1\cdot sin^2 \alpha 
                   +  
                   6\,\calOm'_1\cdot cos^2 \alpha 
  \EQN{2.8}
\eeq
with an arbitrary angle $\alpha \in \{-\pi/2, \pi/2\}$.

On the other hand  a lot of calculational 
experience invariably demonstrates  
that if  amplitudes under investigation do not comprise traces with 
an odd number of \GFt  \  at all then the use of a fully 
anticommuting \GFt  \  is quite self-consistent and leads to great 
simplifications in doing practical calculations.  Much more 
important,  this choice is in a sense unique for the amplitudes 
from above described class (these will be referred to non-singlet ones 
in the following) as it  respects  the chiral                                                        
%
$SU_L(3) \bigotimes  SU_R(3)$ 
%
symmetry of the QCD lagrangian.  This is the case since on the purely 
diagrammatic level the chiral symmetry relies on the possibility of 
freely anticommutating of  \GFt with $\gamma_\mu$ at generic value of 
the index $\mu$.  It seems very reasonable to employ 
identities like \r{2.6} in order to maximally simplify 
kinematical structures appearing. Thus, we will
treat the operators
$\calOm'_1$ and $\calOm'_2$ 
with completely anticommutating \GFt as  ``natural'' ones and 
reexpress through them the operators\footnote{Note that we are 
concerned only with 4-quark operators appearing in the tree 
approximation; for operators which show up first  in 1-looop 
corrections the subtlety under discussion  becomes relevant starting 
at two loops.}  $\calOm_1$ and $\calOm_2$.


A simple calculation 
gives:
\beq
  \ba{l}    
  \dsp
     {\cal O}_1  
  %\rule[-4mm]{0mm}{10mm} 
  = \     ( 1 - 
        \frac{\alpha_s}{4\pi}\, \frac{22}{3})\cdot 6 \, {\cal O'}_1        
  %
      +  \frac{25\alpha_s}{4\pi}   
         \bar u \ \gta t^a  d \, \bar d\ \gha t^a u\ 
\\
\dsp
%
  +      \frac{40}{3}\, \frac{\alpha_s}{4\pi}                      
         \bar u \ \gta  d \,  \bar d\ \gta  u\ 
%
+        \frac{2}{9}\, \frac{\alpha_s}{4\pi} 
        (\bar u \ \gta t^a  u + \bar d\ \gha t^a d)\,
                                              (\bar \Psi \gha t^a \Psi)
,                        
\ea         
\EQN{d1}                                  
\eeq

\beq
\ba{l}    
\dsp
   {\cal O}_2  
%\rule[-4mm]{0mm}{10mm} 
  = \     ( 1 - 
        \frac{\alpha_s}{4\pi}\, \frac{22}{3})\cdot  6\, {\cal O'}_2   
%
      +  \frac{25\alpha_s}{4\pi}   
         \bar u \ \gta \GFm t^a  d \, \bar d\ \gha \GFm t^a u\ 
\\
\dsp
%
  +      \frac{40}{3}\, \frac{\alpha_s}{4\pi}                      
         \bar u \ \gta \GFm  d \,  \bar d\ \gta \GFm  u\ 
%
+        \frac{2}{9}\, \frac{\alpha_s}{4\pi} 
        (\bar u \ \gta t^a  u + \bar d\ \gha t^a d)\,
                                                 (\bar \Psi \gha t^a \Psi).
\ea         
\EQN{d2}                                  
\eeq

To finish this section, a few technical words about the calculation. 
It is clear that the  very difference $\calOm_i - \calOm'_i$  vanishes   in 
the classical limit and consequently  may come  only from   UV 
divergences which manifest themselves as poles in $\epsilon$ in DR 
\cite{Collins84}. Thus, the calculation is conveniently  performed 
hand in hand with the standard procedure of evaluating one-loop 
anomalous dimensions of 4-quark operators 
\protect\cite{SVZ79} (see also \cite{Buras89}, where similar 
calculations were performed in the  context of evaluating 
QCD corrections to   the effective weak Hamiltonian.



%\section{The $\gamma_5$ %
%ambiguity and the QCD sum rule for the $\rho$ meson}
\section{ QCD sum rules for the $\rho$ and $A_1$ mesons
        }


An  explicit calculation of one-loop 
corrections to 4-quark condensates  in \r{2.3a}  
gives  \cite{CheSpi88}
(in the commonly accepted $\overline{MS}$ scheme \cite{MSbar};
$L = \ln\frac{\mu^2}{Q^2}$ )
\bea
    \frac{C^{i,V}_6 Q^i_6}{g^2} 
  &=& -\frac{2}{9}\,\left(1+\frac{\alpha_s}{\pi} \left\lbrack \frac{95}{72} L 
                             + \frac{107}{48} \right\rbrack \right) \
      (\bar u\ \gtmu t^a u +\bar d\ \gtmu t^a d)\cdot (\bar\Psi\ \ghmu t^a\Psi) 
     \label{S1}    \label{r18} \\
  & & -\frac{1}{3}\,\left(1+\frac{\alpha_s}{\pi}\left \lbrack \frac{9}{8} L 
                             + \frac{431}{96} \right\rbrack \right) \
     \bar u\ \GDm t^a d \, \bar d\ \GDm t^a u      
     \label{S2}    \label{r19}  \\
  & & -\frac{\alpha_s}{24 \pi} \, 
      \left\{ \begin{array}{lll} 
\rule[-4mm]{0mm}{10mm}        & ( 16 L - 12 )  
       & \bar u \Gamma_{\mu} d \, \bar d \ \Gamma_{\mu} u               \\
\rule[-4mm]{0mm}{10mm}       +& (30 L - \frac{45}{2}) 
       & \bar u \Gamma_{\mu} t^a d\,  \bar d \ \Gamma_{\mu} t^a u      \\
\rule[-4mm]{0mm}{10mm}       +& ( \frac{16}{9} \, L - \frac{8}{27} )
       & (\bar \Psi \ \gtmu t^a \Psi )^2                                   \\
\rule[-4mm]{0mm}{10mm}       +& (\frac{16}{9} \, L + \frac{56}{27} )
       & (\bar u\ \gtmu \GFm  u + \bar d\ \gtmu \GFm  d)
        (\bar\Psi\ \ghmu \GFm  \Psi)                                       \\
\rule[-4mm]{0mm}{10mm}       +& (\frac{10}{3} \, L + \frac{35}{9} )
       & (\bar u\ \gtmu \GFm t^a u + \bar d\ \gtmu \GFm t^a d)
         (\bar\Psi\ \ghmu \GFm t^a \Psi)                                    
      \end{array} \right\} 
\label{3.1}
.
\eea
For the axial vector correlator the corrections look exactly the 
same, only the matrices \GDt \ and 
$\Gamma_{\mu}\,\,\,{}$ should be multiplied 
by \GFt. 
After the use of the VS approximation we get for the $\rho$ meson
\beq 
   C^{i,V}_6<{\cal O}^i_6> \hspace{-0.5em}_{_0} = -\frac{224 \pi}{81} 
   \left(
         1 + (\frac{705}{112}+ \frac{13}{252}L)\frac{\alpha_s}{\pi}
   \right) 
   (\alpha_s <\bar qq>^2 \hspace{-0.5em}_{_0})
\EQN{3.2a}
\eeq
%
and for the $A_1$ meson
\beq 
   C^{i,A}_6<{\cal O}^i_6> \hspace{-0.5em}_{_0} = \frac{352 \pi}{81} 
\left(
1 + (\frac{777}{176} + \frac{149}{396}L)\frac{\alpha_s}{\pi}
\right) 
   (\alpha_s <\bar qq>^2 \hspace{-0.5em}_{_0})
\EQN{3.3a}
\eeq
At last, a  straightforward application 
of \r{d1},\r{d2} leads us to a 
new result for  (\ref{r18}-\ref{3.1}) and consequently for that of 
\r{3.2a}: 
\beq 
   C^{i,V}_6<{\cal O}^{i}_6> \hspace{-0.5em}_{_0} = -\frac{224 \pi}{81} 
\left(
1 + (\frac{685}{336}+ \frac{13}{252}L)\frac{\alpha_s}{\pi}
\right)
   (\alpha_s <\bar qq>^2) \hspace{-0.5em}_{_0}
\EQN{3.2b}
\eeq
%
and for the $A_1$ meson
\beq 
   C^{i,A}_6<{\cal O}^{i}_6> \hspace{-0.5em}_{_0} = \frac{352 \pi}{81} 
   \left(
         1 + (\frac{917}{528} + \frac{149}{396}L)\frac{\alpha_s}{\pi}
   \right) 
   (\alpha_s <\bar qq>^2 \hspace{-0.5em}_{_0})
\EQN{3.3b}
\eeq
Thus, we observe that uncomfortably large radiative corrections in 
\r{3.2a}and \r{3.3a} are transformed 
into  quite moderate ones after the above described 
transition to the new basis of 4-quark operators. Moreover, now 
the $\alpha_s$ corrections get of approximately the same (relative) 
size for the vector and axial correlators  --- a fact that will be of 
special importance in the next section.

Note also that the terms proportional to $L$ 
in \r{3.2b} and \r{3.3b} are quite small.
To our opinion,   this fact demonstrates that
the effects due to  two-loop  anomalous dimensions of operators
in \r{3.1} are presumably small and may be  neglected.
On the other hand,  the combination of operators
$\alpha_s(\mu)  <\bar qq>(\mu)   <\bar qq>(\mu)$
has only very weak dependence on $\mu$ as its anomalous dimensions
is fortunately rather small. This means that if the VS
approximation is a good one at some scale $\mu$, then it should
be considered as a reasonable approximation at 
a diffrent scale $\mu'$.


\section{The tau hadronic width
        }


As was pointed out on \cite{SchTra84,Bra88,NarPic88}
some time ago,  the methods of perturbative QCD
can be applied to estimate the decay rate ratio
\beq   
R_{\tau} = \frac{\Gamma(\tau\rightarrow\nu_{\tau} {\rm hadrons})}
           {\Gamma(\tau\rightarrow\nu_{\tau}e \bar{\nu}_e)}.
\EQN{4.1}
\eeq
%Further work has been dealing  with  non-perturbative effects
%\cite{SchTra84,NarPic88}
%as well as with  electroweak corrections \cite{MarSir88,BraLi90} to this 
%quantity. 
An updated theoretical discussion of  the ratio 
\r{4.1} within the QCD frameworks  was recently given in  
\cite{BraNarPic91}. 

Schematically,   $R_\tau$   may  represented in the following form  
(in the chiral limit of the massless $u$, $d$ and $s$ quarks and 
neglecting the logarithmic dependence of the Wilson coefficients)
%
\beq
R_\tau =    R^0_\tau 
\left\{1+\frac{\alpha_s}{\pi} 
+5.202(\frac{\alpha_s}{\pi})^2 +26.37
(\frac{\alpha_s}{\pi})^3
+ \delta^{D =6}
+ \dots
\right\}.
\EQN{4.2}
\eeq
Here  the perturbative coefficients are appropriate for the number of 
flavours $f=3$ and $\alpha_s =  \alpha_s(M_\tau)$
and  the power correction of order $1/M_\tau^6$
is directly related to the contribution of 4-quark operators to
\r{2.2}, viz.
\beq
\delta^{D =6} = (\delta^{D =6}_V + \delta^{D = 6}_A)/2, \ \    
\delta^{D =6}_{V/A}
=
24\pi^2 \frac{\dsp -\sum_i 
( C^{i,V/A}_{6}) < \calOm >_i )}%
{M_\tau^6}
\EQN{4.3}
\eeq
As was argued  in \cite{BraNarPic91} the only significant 
sources of  uncertainty in the QCD prediction for $R_\tau$ 
are due to the uncalculated perturbative QCD correction of order
$\alpha_s^4(M_\tau)$ and  due to the  power correction of order
$1/M_\tau^6$. In this section we will concentrate on the latter.

The estimation of the power correction  made in  
\cite{BraNarPic91} amounts essentially to keeping the lowest order 
contribution to the CF $\delta^{D =6}$ and using  the VS 
approximation. The result is  
\beq
\delta^{D =6}_{V/A} =       
\left(                                          
      \ba{c}                                
        7                                    
        \\                                     
       -11                                    
       \ea                              
\right)
\frac{256\pi^3}{27} 
\frac{\rho \alpha_s <\bar{\psi}\psi>^2}{M_\tau^6}.
\EQN{4.4}
\eeq
Here 
$
\rho \alpha_s <\bar{\psi}\psi>^2 
\approx (3.8 \pm 2.0)\times
10^{-4} GeV^6
$ is an effective scale
invariant matrix element which is determined phenomenologically 
\cite{rho1,rho2,rho3}. It is introduced in order to partially take 
into account deviations from the VS approximation. An important 
observation is that when the vector and  axial vector terms are 
averaged to give $\delta^{D=6}$, there is a large mutual cancellation 
between these contributions. It is the  assumption ---
that this cancellation also holds for the 
uncertainties of separate 
contributions\footnote{This would become true if
the prescription  \r{vd} could be made exact by 
multiplying its rhs with  an (operator independent!)   correction
factor $\rho$.} 
 ---,  which is   crucial in producing a rather  accurate   prediction
for $\alpha_s$ given in    \cite{BraNarPic91}, viz. 
\beq
\alpha_s(M_\tau) = 0.34 \pm 0.004 .
\EQN{4.5}
\eeq
This assumption is certainly quite optimistic  and has 
already been criticized in  \cite{Altarelli92,Chetyrkin93}.  
From   formal point of view it is self-consistent {\em 
provided}  one  may neglect higher order corrections to the 
coefficient functions involved.  However, the above discussion 
of the leading 1-loop corrections to the CF 
$C_6^{i,V/A}$ clearly demonstrates 
that  this neglection is very questionable   unless  one  deals with 
the primed basis of 4-quark operators and avoids objects like 
$\Gamma^{[3]}\bigotimes \Gamma^{[3]}$ in favour of those
featuring the completely anticommuting   $\GFm$.



\section{Conclusions}

In this work we have studied the problem of the dependence of 
radiative corrections to the coefficient functions of 4-quark 
condensates on the exact way of treating the $\gamma_5$ matrix and 
related objects specifying a 4-quark operator.  It was argued  that 
a choice of a fully anticommutating $\gamma_5$ matrix which respects
the chiral symmetry of the QCD lagrangian is theoretically preferred
and leads to better convergent perturbation series. 
It also gives an extra argument in favour of the stability
of \r{4.4} and \r{4.5} with respect to radiative corrections.

We  would like to thank J. H. K\"uhn and A. Kwiatkowski for their 
support and helpful discussions.

%\small
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\end{document}
 
