%Title: Complete QCD Corrections of Order $\alpha_s^2$ to the Z Decay Rate
%Authors: K.G.Chetyrkin, J.H.K\"uhn
%Published: Phys. Lett. B308 (1993) 127136.
%TTP93_8
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\begin{titlepage}
\noindent
%
% Datum
%
\hfill TTP938\\
\mbox{}
\hfill March 1993 \\
%\hfill \today \\
%
% Title
%
\vspace{0.5cm}
\begin{center}
\begin{Large}
\begin{bf}
Complete QCD Corrections of Order $\alpha_s^2$
\\
to
\\
the Z Decay Rate
\footnote{Work supported by DFG Grant Nr. Ku 502/31
and BMFT Contract 005 KA94P1}
\\
\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
The axial part of the Z decay rate into hadrons ($\G^A_Z$) gets
extra singlet contributions due to the huge splitting between top
and bottom quark masses. By treating the axial neutral current in
the large topmass limit we partially rederive in a systematic manner
the known results on the contribution of the $\as^2$ order anomalous
"double triangle" diagrams to $\G^A_Z$ and resolve an ambiguity
concerning the scale at which the strong coupling constant
$\alpha_s$ is to be taken in these diagrams.
We evaluate and sum all
leading log and next to leading log QCD corrections of orders
$\as^{1+n}(\ln(M_Z/4m^2_t))^n$ and $\as^{2+n}(\ln(M_Z/4m^2_t))^n$
respectively.
We examine the sensitivity of our results to a change of
the renormalization scales to estimate the size of yet unknown
corrections of order $\alpha_s^3$ to $\G^A_Z$. The analysis implies
that the corrections might change the magnitude of the singlet
term by up to 30\% while they hardly might be of noticeable
numerical significance in comparison with the total contribution from
nonsinglet terms.
\end{quotation}
\end{titlepage}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\em 1.Introduction}.
The current precision measurements of the Z decay rate into hadrons
$\G_Z^h$ have developed into an important experimental tool for a
reliable determination of $\as$. From the theoretical point of view
the QCD corrections to the former used to be treated in close
analogy to the corrections involved in studying the total
crosssection of $e^+e^$annihilation into hadrons through the
virtual photon. Within the standard model
the interaction of the Z boson to quarks is described
(in the lowest order approximation in the weak coupling constant)
by adding to the QCD Lagrangian an extra term of the form
$M_Z\left(\frac{G_F}{2\sqrt 2}\right)^{1/2}Z^\alpha J^0_\alpha $,
with
$
J^0_\al
=
\sum_i \ovl{\psi}_i\g_\al(g^V_i  g^A_i\g_5)\psi_i \
$
being the neutral axial quark current. The decay rate
$\G_Z^h$, including of all strong interaction corrections,
may be viewed as incoherent sum of vector
($\G^V_Z$) and axial ($\G^A_Z$) contributions. For
massless outgoing quarks the corresponding twopoint functions
are identical (apart from a simple modification
of some overall factors) to the electromagnetic current twopoint
function including terms of order $\alpha_s$.
This identity is not respected by mass terms of the order of
$m_b^2/M_Z^2$ due to the mass of the bquark. At present these
effects are known to third order in $\as$ for $\G_Z^V$ and to second
order for the axial part
\cite{ChetKuhn90,ChetKuhnKwiat92,ChetKwiat92}. An important feature
of these results is that the use of the $\msbar$  scheme
\cite{Hooft73,MSbar} to define and renormalize coupling constant and
quark masses has allowed to get rid of large logarithms of
$m_b^2/M^2_Z$ completely and, consequently, to obtain reliable
results.
However, as noted in \cite{Kniehl90a,Kniehl90b} there is a finer
subtlety which does differentiate between vector and axial cases even
for massless outgoing quarks. The difference appears starting
from the order of $O(\al^2_s)$ and is a consequence of the large mass
splitting between the masses of the top and bottom quarks. In
\cite{Kniehl90a,Kniehl90b} the singlet contribution to $\G_Z^A$
from the "double triangle" diagram (Fig.1a) was evaluated
analytically. The diagram is peculiar for the axial vector
current correlator since the contributions from its vector
counterpart vanish identically irrespective of
quark masses due to the Furry's theorem \cite{Furry37}. Their result
for the partial decay rate of the $Z$ boson to $b$ quarks and gluons
$\G^A(Z\to b \ovl b)$ is of the form (in the following we use the
effective couplant $a(\mu) \equiv \frac{\dsp\as(\mu)}{\dsp\pi}$)
\beq
\G^A(Z\to b \ovl b)=\G^{QPM}
\left\{
1 + \apis
+\left(\apis\right)^2(d^{NS}_2 + d^{S}_2 )
\right\}, \ \ \G^{QPM}=\frac{G_F M^3_Z}{8\pi\sqrt2}
\EQN{Kuhn1}
\eeq
Here $s=M_Z^2$ and $\G^{QPM}$ is the quarkparton model prediction
for the axial part of the $Z$ boson decay rate into of a pair of
free massless quarks. The nonsinglet contribution $d_2^{NS}=1.41$
was found in \cite{Sapirstein} while the singlet term
reads\footnote{In refs.~\cite{Kniehl90a,Kniehl90b} the
complete analytical result is derived for the function $I(r)$.
For our aims it suffices to deal with the approximate expression
written below.}
\beq
\dsp
d^{S}_2=\frac{1}{3} I(r), \ \ I(r)=3 \ln(4r)  37/4 +(28/27)r
+ 0.632 \, r^2 + O(r^3), \ \ r\equiv s/(4m_t^2).
\EQN{Kuhn:I_2over3}
\eeq
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The function $(I(r))$ shows a remarkable behaviour with respect to
$r$ in that it increases logaritmically for vanishing $r$.
The reason for this behaviour was in fact clarified by
Collins, Wilczek and Zee \cite{Collins78} who demonstrated
that the triangle anomaly \cite{Adler69,BellJackiw69}
prevents the CarazonneAppelquist decoupling theorem
\cite{AppelquistCarazzone75} to operate in its naive form. The
latter demands that all effects of a heavy field
which are not vanishing in the infinite mass limit can be described
by a suitable redefinition of the parameters of the corresponding
effective low energy Lagrangian obtained by
omitting the terms depending on all heavy fields. Thus, even a very
heavy top quark will induce a noticeable effect on the Z decay rate.
It should be also noted that (i) starting from $m_t=M_Z$ the approximate
expression \r{Kuhn:I_2over3} agrees with the analytical result
to 1 part in $10^2$; (ii) the first two terms in
the $r$ expansion of $I(r)$ describe the function with the good
accuracy of about 5\% at $m_t=M_Z$; the accuracy is rapidly improving
for larger values of $m_t$.
%cite{Georgi84}
However, the appearance of these potentially large logarithms
calls for a more careful renormalization group analysis
of the result of \cite{Kniehl90a,Kniehl90b}. The standard trick of
absorbing large logs, which in general amounts to setting the
normalization point $\mu$ equal to a large mass scale inherent to
the process, is clearly not adequate since the problem at hand
involves two large (and generically different) mass scales, viz.,
$M_Z$ and $2m_t$. It should be also noted that from a theoretical
point of view the singlet and nonsinglet contributions may be
considered to be independent; in particular not only their sum but
each one individually must be renormalization group
invariant. This means that even if one refuses to distinguish
between the mass scales $M_Z$ and $2m_t$, the choice of the
normalization scale equal to $M_Z$ for the singlet term in \r{Kuhn1}
may not be inferred from the wellestablished prescription for the
nonsinglet contribution (where we know three successive
$\as$$dependent$ terms in the perturbation expansion!) and should be
considered merely an educated guess. At last, the technically
difficult completely analytical calculation of refs.
\cite{Kniehl90a,Kniehl90b} can hardly be extended in a
straightforward manner to include the order $\ac^3$ corrections
required to match the results already available for the vector
twopoint massless correlator \cite{{Gorishny1}}.
On the other hand, the very form of the rhs of \r{Kuhn:I_2over3}
along with the current experimental lower limit on $m_t$ ($m_t > M_Z$
\cite{Ellis92}) suggests that the standard methods of heavy mass
expansion with respect to the mass of the top quark \cite{Smi91} (or,
equivalently, those of the effective field theory \cite{Georgi84})
can not only lead to natural resummation of a series of powers of
$\alpha_s \ln(M_Z/4m^2_t)$ but could also provide a valuable
calculational scheme to proceed to higher orders.
In this work we shall use the effective field theory method
in order to explicitly separate contributions normalized at the scale
of order $m_t$ from the ones normalized at the scale of order
$M_Z$ and to analytically compute and sum all the nonsinglet
corrections of orders $\ac^{1+n}(\ln(M_Z/4m^2_t))^n$ and
$\ac^{2+n}(\ln(M_Z/4m^2_t))^n$ to $\G_Z^A$. We employ our results in
discussing the possible effects of higher order singlet corrections
to $\G^A_Z$.
%applications of our results to computing the anomalous contributions
%of order $\alpha_s^3$ to the axial part of the Z decay rate  the
%problem to be prohibitively difficult to solve by means of a direct
%calculation.
%Note that in a sense the present work can be considered
%as a further development of an early research \cite{Collins68}.
{2. \em Preliminaries}.
%{\bf 2.}
We start by introducing some notations.
In analogy to the vector case it is convenient to deal with
the axial vector correlator ( $A^i_\mu=\ovl\psi_i \g_5\g_\mu \psi_i$)
\beq
%\ba{c} \dsp
\Pi^{A,i,j}_{\mu\nu}\equiv
i\myint \ex^{iqx} d x
\langle0T({A}^{i}_\mu \>(x){A}^{j}_\nu)0\rangle
\equiv g_{\mu\nu}\Pi_1^{A,i,j}(Q^2)+
q_\mu q_\nu \Pi_2^{A,i,j}(Q^2),
%\ea
\EQN{2.1}
\eeq
whose spectral density
\beq
s R^A_{i,j}(s)=2 i \pi \lim_{\de\to 0}
(\Pi_1^{A,i,j}(Q^2=si\de)
\Pi_1^{A,i,j}(Q^2=s+i\de))
\EQN{2.2}
\eeq
is directly connected with the axial part of the Z decay rate:
\beq
\G^A_Z =\G^{QPM}\sum_{i,j}g^A_i g^A_{j}
R^A_{i\,j}(s=M_Z^2),
\EQN{2.3}
\eeq
where $g^A_i = \pm 1$ denotes the axial vector coupling constant
and the summation includes all six quark flavours.
An equivalent representation for the axial decay rate $\G^A_Z$
reads
\beq
\G^A_Z =\G^{QPM}\sum_{i,j}
R^{\D}_{i,j}(s=M_Z^2),
\EQN{2.4}
\eeq
where $i$ and $j$ refer to the upper members of quark weak
isodoublets and, by definition, for every pair of weak isodoublets
$(\psi_i,\psi_{i'})$ and $(\psi_j,\psi_{j'})$
$$R^{\D}_{i,j}= R^A_{i,j}  R^A_{i,j'} R^A_{i',j}+R^A_{i',j'}.$$
Eq. \r{2.4} is easily obtained by noting that
the axial part of the neutral quark weak current
$A^0_\mu$ may be written as the following sum over weak doublets:
$$
A^0_\mu = \sum_i \D^{i}_\mu, \ \ \D^{i}_\mu= A^i_\mu  A^{i'}_\mu
$$
and that the spectral density $R^\D_{i,j}$ is nothing but the
spectral density for the transverse part of the following correlator:
\beq
\Pi^{\D,i,j}_{\mu\nu} =
i\myint \ex^{iqx} d x
\langle0T[{\D}^{i}_\mu \>(x){\D}^{j}_\nu(0)]0\rangle.
\EQN{2.5}
\eeq
As a wellknown consequence of the triangle anomaly
the axial quark current $A_\mu^i$, unlike the vector
one, is neither conserved
nor invariant with respect to the renormalization
group transformations (even in the case of $m_i\equiv0$).
The latter implies that the corresponding anomalous dimension
\beq
\mu^2\frac{d}{\mu^2} A_\nu^i = \g^{\psi} (\ac) A^\psi_\nu
=
\left
(\sum_{n=0}\g^{\psi}_n \left(\apis\right)^{n+2}
\right)A^\psi_\nu, \ \ A^\psi_\nu \equiv \sum_i A^i_\nu
\EQN{2.6}
\eeq
is different from zero.
Here
$\mu$ is the normalization point and
\beq
\mu^2\frac{d}{d\mu^2} = \mu^2\frac{\partial}{\partial\mu^2}
+ \beta(\ac) \frac{\partial}{\partial \ac}
+ 2{m}^2 \gm(\ac)\frac{\partial}{\partial {m}^2},
\EQN{2.7}
\eeq
where
$\dsp\beta(\ac)=\sum_{n=0}\beta_i
\left(\apis\right)^{n+2}$ and
$\dsp\g_m(\ac)=\sum_{n=0}\g^m_i
\left(\apis\right)^{n+1}$
are the QCD $\beta$function and
the quark mass anomalous dimension respectively.
It should be stressed that in order to be able to treat the axial
vector quark current selfconsistently within dimensional
regularization we must properly define the $\g_5$ matrix in $D$dimensions.
We employ here essentially the definition put forward by t' Hooft and
Veltman \cite{Hooft72}, and formalized by
Breitenlohner and Maison \cite{BM77a}
with a modification introduced in
\cite{{gamma5mu}}:
\beq
A^i_\alpha = \ovl{\psi}_i\g_\alpha\g_5 \psi_i \equiv
\frac{\xi^A_5(\ac)}{6}\epsilon _{\alpha\beta\nu\rho}
\ovl{\psi_i}\g_\beta\g_\nu\g_\rho \psi_i.
\EQN{axial.c.def}
\eeq
%
The indices $\alpha,\beta,\nu,\rho$ are restricted to be $0,1,2$ and
$3$. The finite renormalization factor $\xi_5^A = 1 4{\Ac}/{3}
+O(\Ac^2)$ is usually introduced
\cite{Trueman79,Collins84,Larin91} in order
to effectively restore the anticommutativity property of $\g_5$ for
the case of an (anomaly free) flavour nonsinglet current
$A^{ij}_\alpha = \ovl{\psi}_i\g_\alpha\g_5 \psi_j$, $i\not=j$ defined
with the help of a straightforward generalization of
\r{axial.c.def}. We have found it very useful to keep the factor
also in our case of diagonal flavour conserving currents. One of the
reasons is that within such a convention all the nonsinglet graphs
contributing to, say, the correlator $\Pi^{A,i,i}_{\mu\nu}$ (that is
ones which do not comprise a single $\g_5$ sitting in a closed quark
loop) coincide in the massless limit with the corresponding graphs
contributing to the vector twopoint correlator
$\Pi^{V,i,i}_{\mu\nu}$.
The lowest order diagram contributing to the rhs of \r{2.6}
is shown in Fig.~1b. Its contribution to the anomalous dimension
$\g^{\psi}(\ac)$ is known since long \cite{Adler69}. The next term
(of order $\ac^3$!) is crucial for the
summation over nexttoleading logs in $\G^A_Z$ and
recently has been
evaluated \cite{Larin92,CK3} in
the $\msbar$  scheme with the result ($n_f$ is
the number of quark flavours):
\beq
\g^{\psi}_0 = 1/2, \ \ \ \ \g^{\psi}_1 = 85/48 + n_f /72.
\EQN{anomax}
\eeq
Note that, within the $\msbar$ scheme, $\g^{\psi}$ does not depend on
the dimensionfull parameters $\mu$ and $m$
\cite{Collins75}. Hence we are led to the conclusion that the
combination $\D^i_\mu$ and, correspondingly, the full neutral quark current do
not possess nontrivial anomalous dimensions and, thus,
are independent of $\mu$ as expected. On the other hand,
the existence of a new (in comparison with the nondiagonal current)
class of divergent diagrams for the diagonal current means that
there appears an extra freedom in renormalizing the current, and
consequently in the anomalous dimension $\g^{\psi}$.
This ambiguity starts from $\g_1^\psi$.
A close examination shows \cite{CK3} that the difference
$\D^i_\mu$ does not suffer from this uncertainty provided we
choose a flavour symmetric renormalization prescription.
{\em 3. Axial vector neutral current in large topmass
limit.}
%{\bf 3.}
To be precise, let us consider the difference $\D^t_\al$ and
the corresponding diagonal correlator $\Pi^{\D,{t,t}}_{\al\be}$. Note that
up to and including terms order $\ac^2$ and {\em before} summing
higher order logs of $s/m^2_t$ the following identity (see e.g. a
discussion of what is to be meant by $\G^A(Z\to b \ovl b)$ in
\cite{KuhnZervas92})
is fulfilled
\beq
\G^A(Z\to b \ovl b)=\G^{QPM} \ R^{\D}_{t,t}
\EQN{3.1}
\eeq
Beyond this order the separation from different quark
species is no longer meaningful; the distinction between
singlet and nonsinglet terms, however, remains valid.
We shall see soon that equation \r{3.1} has to be modified
if nexttoleading logs are taken into account.
The generalization of our results for the case
of the correlator $\Pi^{\D,{t,j}}_{\mu\nu}$ and the
spectral density $R^{\D}_{t,j}$ is straightforward and will be
presented below.
The expansion we are interested in is
written in the form:
\beq
A^t_\al
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{m_t\to\infty}}^{}
{\bf C}_h(\ac(\mu),\mu/m_t) \cdot A^L_\al +O(1/m_t),
\EQN{3.2}
\eeq
where the $A^L_\al = \sum_{i=u,d,s,c,b}A^i_\al $ is the singlet axial
current in the effective QCD with five ($L\equiv$ light) quark
flavours. The exact meaning of the expansion \r{3.2} is as follows:
any Green function of light fields with an insertion of the operator
$A^t_\al$ can be approximated (with the indicated accuracy) by an
"effective" Green function to be obtained through replacing the
heavy current by the rhs of \r{3.2}. The corresponding equation for
${\bf C}_h $ is usually called the matching equation. It is
understood that this Green function is calculated by means of the
effective QCD Lagrangian with $n_f=5$ and with the properly changed
coupling constant and quark masses \cite{Bernreuther,Marciano}.
Note that the
%here I ended my spell  cheking
coefficient function
${\bf C}_h $ receives, by definition, contributions from
singlet graphs only. (Nonsinglet
diagrams have at least two external
heavy top quarks lines and are thus excluded from the matching
equation.) The lowest order singlet diagram contributing to ${\bf
C}_h $ is the same graph of Fig.1b which gives the leading
contribution to \r{anomax}. A simple calculation
gives
\beq
{\bf C}_h = \left(\frac{1}{2} \ln (\frac{\mu^2}{m_t^2})
+\frac{1}{8}\right)
(\Ac(\mu))^2 +O(\Ac^3).
\EQN{3.4}
\eeq
The cofficient of the logarithm is scheme independent, can
be identified with $\g^{\psi}_0$, and was calculated in
\cite{Collins78}. The corresponding
expansion for the current $A^b_\alpha$ reads
\beq
A^b_\al
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{m_t\to\infty}}^{}
A^b_\al +
{\bf C}_\psi(\ac(\mu),\mu/m_t)\cdot A^L_\al +O(1/m_t).
\EQN{3.5}
\eeq
The operators at the lhs and rhs refer to the 5 and 6 flavour theory
respectively.
The coefficient ${\bf C}_\psi $ originates from singlet graphs containing
$t$ quarks loops. The lowest order contribution to $\bf C_\psi$ s is
of order $\ac^3$ (Fig.~1c) and the first term is due to
nonsinglet contributions. It is wortwhile to note that the resulting
unit coefficient function (in all orders in $\ac$!) follows from our
normalization condition for nonsinglet graphs and the
accepted definition of $\g_5$ \cite{CK3}.
All the operators entering into expansions \r{3.2} and \r{3.5} are
assumed to be minimally renormalized at the scale $\mu$. The
operator difference $\D^t_\alpha$ is renormalization group
invariant. Hence the standard convenient way of deriving its expansion
which, first, includes only operators composed from light quark
fields and renormalized at the scale $\mu=\mu_Z \approx M_Z$ and,
second, does not contain ``large" logs of $M_Z^2/4m_t^2$ is as
follows: We begin from expansions \r{3.2} and \r{3.5} at a higher
normaization point $\mu=\mu_t \approx m_t$ and then scale the light
quark operators down to $\mu=\mu_Z\approx M_Z$ with the help of the
RG equation \r{2.6}. One obtains
\beq
\ba{c}
\dsp
\left[\D^t_\al\right]^{(6)} \trpsbs{\mu}{t} =
\\
\dsp
\left(
{\bf C}_h(\ac_{(6)}(\mu_t),\frac{\mu_t}{m_t})
{\bf C}_\psi(\ac_{(6)}(\mu_t),\frac{\mu_t}{m_t})
\frac{1}{5}
\right)
\left\{\
\exp
\myint^{\Ac^{(5)}(\mu_t)}_{\Ac^{(5)}(\mu_Z)}
\ \frac{5 \g^{\psi}_{(5)}(x)}{\beta_{(5)}(x)} dx
\right\}
\left[
A^L_\al
\right]^{(5)}\trpsbs{\mu}{Z}
\\
\dsp
+ \frac{1}{5} %\cdot
\left[
A^L_\al
\right]^{(5)}\trpsbs{\mu}{Z}
\left[A^b_\al\right]^{(5)}\trpsbs{\mu}{Z} +O(1/m_t).
\ea
\EQN{3.6}
\eeq
The superscripts and subscripts $(6)$ and $(5)$ indicate that the
running
coupling constant, an operator or a RG function is to be taken with
$n_f=6$ or $n_f=5$ ``effective'' flavours respectively. The
notation $[ O ]_\mu$ indicates that the operator $[O]$ is renormalized
at scale $\mu$.
In order to derive \r{3.6} it suffices to observe that
according to equation \r{2.6}
\[
\left\{\
\exp
\myint^{\Ac^{(5)}(\mu_t)}_{\Ac^{(5)}(\mu_Z)}
\ \frac{5 \g^{\psi}_{(5)}(x)}{\beta_{(5)}(x)} dx
\right\}
\left[
A^L_\al
\right]^{(5)}\trpsbs{\mu}{Z}
\equiv
\left[
A^L_\al
\right]^{(5)}\trpsbs{\mu}{t}
\]
and that
the combination
\[
+ \frac{1}{5}
\left[
A^L_\al
\right]^{(5)}\trpsbs{\mu}{Z}
\left[A^b_\al\right]^{(5)}\trpsbs{\mu}{Z}.
\]
does not depend on the normalization scale $\mu_Z$.
Thus we conclude that the rhs of \r{3.6}
does not depend on $\mu_Z$ at all and may be evaluated
at $\mu_Z=\mu_t$, with the result being just the
the difference
$\left[\D^t_\al\right]^{(6)} \trpsbs{\mu}{t}$
expanded according to \r{3.2} and \r{3.5}.
The rhs of eq. \r{3.6} expanded up to and including order $\ac^2$
assumes the form (if not stated otherwise, all quantities below are
to taken in the effective QCD with $n_f=5$)
\beq
\ba{c}
\dsp
[\D^t_\al]^{(6)} =
\left\{{\bf C}_h(\ac_{(6)}(\mu_t),\frac{\mu_t}{m_t})

\frac{\g^{A}_0}{\be_0}\left(\Ac(\mu_t)  \Ac(\mu_Z)\right)
\right\}
\left[A^L_\al\right] \trpsbs{\mu}{Z}
 \left[A^b_\al\right]\trpsbs{\mu}{Z}
\\
\dsp
\left\{
\frac{5}{2}\left(\frac{\g^{A}_0}{\be_0}\right)^2
\left(\Ac(\mu_t)  \Ac(\mu_Z)\right)^2
\left.
+\frac{1}{2}
\left(
\frac{\g^{A}_1}{\be_0}

\frac{\g^{A}_0\be_1}{\be^2_0}
\right)
\left(\Ac^2(\mu_t)  \Ac^2(\mu_Z)\right)
\right\}
\left[ A^L_\alpha \right]\trpsbs{\mu}{Z}.
\right.
\ea
\EQN{3.7}
\eeq
{\em 4. Evaluation of the function $I$ in the nexttoleading
logarithmic approximation.}
%{\bf 4.}
Equation \r{3.7} can be conveniently used to compute the correlator
$\Pi^{\D,{t,t}}_{\mu\nu}$ and consequently to find the spectral
density $R^\D_{t,t}(s)$ in the region of $s=M_Z^2 << 4 m^2_t$ we are
interested in. Indeed, on substituting \r{3.6} into \r{2.5} one
immediately arrives at the conclusion that $R^\D_{t,t}(s)$ is
entirely expressed in terms of various functions appearing in the rhs
of \r{3.6} (that is the coefficient functions ${\bf C}_h$, ${\bf C}_\psi$,
the QCD $\beta$function $\beta(\ac)$ and the anomalous
dimension $\g^{\psi}(\ac)$) and the spectral densities $R^A_{i,j}(s)$,
$i,j=u,d,s,c$ and $b$ which are to be computed in the effective massless
QCD with $n_f=5$. Thus, we get
\beq
\ba{c}
R^\D_{t,t}(s) =
\dsp
R^A_{b,b}
 2{\bf C}_h(\ac_{(6)}(\mu_t),\frac{\mu}{m_t})
R^A_{L,b}
\\
\dsp
+ 2 \frac{\g^{A}_0}{\be_0}
(\Ac^{}(\mu_t)  \Ac(\mu_Z))R^A_{L,b}
+ \left(
\frac{\g^{A}_1\be_0  \g^{\psi}_0\be_1}{\be^2_0}
\right)
\left(\Ac^2(\mu_t)  \Ac^2(\mu_Z)\right)R^A_{L,b}
\\
\dsp
+ \left(\frac{\g^{A}_0}{\be_0}\right)^2
\left(\Ac(\mu_t)  \Ac(\mu_Z)\right)^2
\left(5 R^A_{L,b} + R^A_{L,L}\right),
\ea
\EQN{3.8}
\eeq
where we put $\dsp R^A_{L,b}= \sum_{i=u,d,s,c,b} R^A_{i,b}$
and $\dsp R^A_{L,L} = \sum_{i,j=u,d,s,c,b} R^A_{i,j}$, and
all the spectral densities in the rhs of eq. \r{3.8} are to
be evaluated at the normalization point $\mu =\mu_Z$.
The singlet (Fig.~1a) and nonsinglet graphs have to
be evaluated with the prescriptions detailed above.
Finally, it remains to observe that due to the $SU^f(5)$ symmetry
the spectral density $R^A_{i,j}$ can be written
(in the limit of the massless $u,d,s,c,$
and $b$ quarks)
%and discarding the terms of order
%$\ac^3$ and higher)
in the following form
(with
$
\ell = \ln\frac{\mu^2}{s},
$)
\beq
R^A_{i,j}(s) =
\delta_{ij} R^{A,NS}(s)
+
R^{A,S}(s)
\EQN{3.9}
\eeq
where
\beq
R^{A,NS}(s) = \sum_{0 \leq j < i}^{i \geq 0}
d^{NS}_{i,j} a^i \ell^j
\ \
\mbox{{\rm and}}
\ \
R^{A,S}(s) = \sum_{0 \leq j < i}^{i \geq 2}
d^{S}_{i,j} a^i \ell^j.
\EQN{3.9a}
\eeq
The nonsinglet contribution $R^{A,NS}(s)$ is renormalization
group invariant and is known up to and including terms of order
$\ac^3$ \cite{Gorishny1}.
The coefficients up to second order are given \cite{Sapirstein}
by
$$d^{NS}_{0,0} = 1, \ d^{NS}_{1,0} = 1, \
d^{NS}_{2,0} = 1.986  0.115 n_f, \ d^{NS}_{2,1} =\beta_0;
$$
those of order $\ac^3$ can be found in \cite{Gorishny1}.
Next, as a consequence of \r{2.6}
one has
\beq
\mu^2\frac{d}{d\mu^2} R^{A,S}
= 2 \g^{\psi} ( R^{A,NS} +n_f R^{A,S}),
\EQN{3.9b}
\eeq
whence it immediately follows that
\beq
d^{S}_{2,1} = 2g^A_0,
\ \
d^{S}_{3,2} = \beta_0 d^{S}_{2,1}/2,
\ \
d^{S}_{3,1} =  2 (g^A_0 + g^A_1) + \beta_0 d^{S}_{2,0}.
\EQN{3.9c}
\eeq
The coefficents $d^S_{i,0}$ are not constrained by \r{3.9b}
and should be computed independently.
We have used the program MINCER \cite{MINCER2}
written for the symbolic manipulation system FORM
\cite{FORM} to find $d^S_{2,0}$ with the result
\beq
d^S_{2,0}
= 17/6.
\EQN{3.9d}
\eeq
Combining \r{anomax} and \r{3.9c} we also have (setting $n_f=5$)
\beq
d^{S}_{2,1} = 1,
\ \
d^{S}_{3,2} = 23/24,
\ \
d^{S}_{3,1} = 91/9.
\EQN{3.9e}
\eeq
The calculation of $d^S_{3,0}$ is in progress.
It is of importance to note that the singlet contribution to
\r{3.8} coming from $R^{A}_{i,i}$ is to be naturally assigned to
the decay rate $\G(Z \to \ovl q_i q_i)$ while those from
$R^{A}_{i,j}$ with $i \not = j $ contributes equally to
$\G(Z \to \ovl q_i q_i)$ and $\G(Z \to \ovl q_j q_j)$.
The resulting expression
for $R^{\D,S}_{t,t}$  the singlet contribution to the rhs
of eq. \r{3.8}  which is our improved (via
summing up higher order logs) result for the product $\Ac^2 I_2(r)/3$
reads:
\beq
\ba{c}
\dsp
R^{\D,S}_{t,t} =
\\
\dsp
(
 \frac{17}{6}  \ln(\frac{\mu_Z^2}{M_Z^2})
+ \frac{7}{81}\frac{M_Z^2}{m_t^2}
+ 0.0132\frac{M_Z^4}{m_t^4}
)\Ac^2(\mu_Z) +
\frac{12}{23}(\AcI{\mu_t} \AcI{\mu_Z})(1 + \AcI{\mu_Z})
\\
\dsp
+ \frac{216}{529}\left(\AcI{\mu_t}  \AcI{\mu_Z}\right)^2
%\\
%\dsp
+ \frac{4007}{6348}(\Ac^2(\mu_t)  \Ac^2(\mu_Z))
%\\
%\dsp
 (\frac{1}{4}  \ln(\frac{\mu_t^2}{m_t^2}))
(\Ac_{(6)}(\mu_t))^2
\\
\dsp
+ \frac{144}{529}\left(\AcI{\mu_t}  \AcI{\mu_Z}\right)^2
\ea
\EQN{3.10}
\eeq
Here the expression on the last line comes from the terms proportional
to $R^{A}_{i,i}$ with $i=u,d,s,c$ in \r{3.8}
while the rest should be interpreted as
the improved singlet contribution to the decay rate
$\G^A(Z \to \ovl{b} b)$, that is
\beq
\ba{c}
\G^{A,S}(Z \to \ovl{b} b) = \G^{QPM}\times
\\
\dsp
\left\{
(
 \frac{17}{6}  \ln(\frac{\mu_Z^2}{M_Z^2})
+ \frac{7}{81}\frac{M_Z^2}{m_t^2}
+ 0.0132\frac{M_Z^4}{m_t^4}
)\Ac^2(\mu_Z) +
\frac{12}{23}(\AcI{\mu_t} \AcI{\mu_Z})(1 + \AcI{\mu_Z})
\right.
\\
\dsp
\left.
+ \frac{216}{529}\left(\AcI{\mu_t}  \AcI{\mu_Z}\right)^2
+ \frac{4007}{6348}(\Ac^2(\mu_t)  \Ac^2(\mu_Z))
 (\frac{1}{4}  \ln(\frac{\mu_t^2}{m_t^2}))
a^2_{(6)}(\mu_t)
\right\}
\ea
\EQN{3.11}
\eeq
The logarithmic terms of of order $\ac^3$ as given in \r{3.9e}
can be incorporated into \r{3.11} by replacing the the brackets
on the rhs by
\beq
\left\{ \right\}
\longrightarrow
\left\{ \right\}
+ \left(
23/24 (\ln(\frac{\mu_Z^2}{M_Z^2}))^2
91/9 \ln(\frac{\mu_Z^2}{M_Z^2})
\right)
\ac^2(\mu_Z)
\EQN{3.12}
\eeq
In deriving eqs. \r{3.10}, \r{3.11} we have taken into account the power
supressed terms of order $M_Z^2/m_t^2$ and $M_Z^4/m_t^4$ as
these appear in \r{Kuhn:I_2over3}. In principle the power suppressed
contributions could be obtained and improved directly by extending
the expansion \r{3.5} to include the operators of dimension $5$ and $7$
respectively. Having in mind that the terms of order $r$ and $r^2$
in \r{Kuhn:I_2over3} amount to less than 10\% of the function $I(r)$
unless $m_t < M_Z $, we believe that higher logarithmic corrections
to these terms can be safely ignored.
{\em 6. Discussion.}
%{\bf 6.}
In order to compare our result \r{3.11} to the previous one
\r{Kuhn:I_2over3} we first set the normalization scales
$\mu_Z$ and $\mu_t$ to their ``natural'' values of $M_Z$ and $m_t$
respectively.
Then we use the wellknown matching equation
$\ac^{(6)}(m_t) = \ac(m_t)$
and express the couplant $\ac(m_t)$ in terms of
$\dsp L=\ln(\frac{m_t^2}{M_Z^2})$ and
$\ac(M_Z)$ (the expansion may be easily inferred from the
renormalization group equation for the couplant):
\beq
%\ba{c}
\dsp
\ac(m_t)
=
\ac(M_Z)  \ac(M_Z)^2\frac{23}{12} L
 ( \frac{29}{12}L

\frac{529}{144}L^2)
\ac(M_Z)^3 +O(\ac(M_Z)^4).
%\ea
\EQN{6.1}
\eeq
to get
\beq
\ba{c}
\G^{A,S}(Z \to \ovl{b} b) = \G^{QPM}\times
\\
\dsp
\left\{
(
 \frac{37}{12}  \ln(\frac{M_t^2}{M_Z^2})
+ \frac{7}{81}\frac{M_Z^2}{m_t^2}
+ 0.0132\frac{M_Z^4}{m_t^4}
)\Ac^2(M_Z) +
\right.
\\
\dsp
\left.
\left(
\frac{23}{12}\ln(\frac{M_t^2}{M_Z^2})^2
 \frac{67}{18}\ln(\frac{M_t^2}{M_Z^2})
+ d^S_{3}
\right)\Ac^3(M_Z)
\right\}
\ea
\EQN{6.2}
\eeq
Thus, we observe agreement between our result and
\r{Kuhn:I_2over3} concerning the (numerically most important)
terms of order $\ac^2$ that are not suppressed by powers of $m_t$.
The term $d^S_3\ac^3$ is not yet known and shall be neglected in the
subsequent discussion. It is worthwhile to note that
the terms proportional to $L^2$ and $L$ respectively cancel
partially, leaving a numerically quite small
correction to the result \r{Kuhn:I_2over3}.
It is also of interest to discuss
the asymptotic behaviour of \r{3.11} in the limit of very
heavy top quark. A straightforward calculation gives
\beq
\G^{A,S}(Z \to \ovl{b} b)
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{m_t\to\infty}}^{}
\G^{QPM}\times
\dsp
\left\{
(
 \frac{12}{23}\Ac(\mu_Z)

\left(
\ln(\frac{\mu_Z^2}{M_Z^2})
+ \frac{7571}{2116}
\right)
\Ac(\mu_Z)^2
\right\}.
\EQN{6.3}
\eeq
We observe that in the limit under consideration expression
initially second order in $\ac$
converts into aterm of firstorder. In fact, given
$\Lambda_{QCD}$ below $1 GeV$, this asymptotic
formula can be applied with 30\% accuracy
at astronomically high values of $m_t$ as large as
$10^3  10^4 TeV $ only!
{\em 7. Numerical comparisons and conclusions.}
%{\bf 7.}
Let us compare the improved result \r{3.11} for
$\G^{A,S}(Z \to \ovl{b} b)$ and the lowest order
one as it is given by \r{Kuhn1}, \r{Kuhn:I_2over3}.
First of all we note that our result explicitly depend on
both the scales $\mu_Z$ and $\mu_t$ while \r{Kuhn:I_2over3}
depends only on the scale at which
the effective couplant is taken. The scale is naturally to be identified
with $\mu_Z$.
According to general principles
of the renormalization theory a physical result must not
depend on any renormalization scales. Still, due to unknown higher
corrections an approximation taking into account only a limited number
of terms of its perturbation series terms may exhibit such a dependence.
A numerical analysis reveals that the rhs of eq. \r{3.11}
depends only weakly on $\mu_t$ but has a rather
strong dependence on $\mu_Z$.
%(see pictures ... )
In other words, the ratio of
$\G^{A,S}(Z \to \ovl b b)/
\left(\G^{QPM}\frac{1}{3}I(r)\ac^2(M_Z)\right)$
depends strongly on $\mu_Z$ but is quite insensitive
towards the values of
$\Lambda$, $m_t$, $\mu_Z$ and $\mu_t$.
Varying $\mu_Z$ between $M_Z/2$ and $2 M_Z$
and $\mu_t$ between $m_t$ and $2 m_t$ respectively, we find that
this ratio varies between
$0.75$ and $1.25$ (Fig. 2)
%when compared to
%the same function taken at the "central" point
%$\mu_Z=M_Z,\ \mu_t = m_t$ and at the same value of
%$\Lambda_{QCD}^{(5)}$
%(the latter was varied from $150 $ to
%$660$ MeV).
A close examination of \r{3.11}
shows that the notable $\mu_Z$ dependence may be severely suppressed
if one takes into account higher logarithmic contributions
of order $\ac^3$ to $R^{A,S}$ which were presented above in
\r{3.12}.
One observes that the dependence of
the corresponding ratio
on the choice of $\mu_t$ and $\mu_Z$
is weakened as indicated by the dashed curve in Fig. 2
and the function can be can be considered as stable
to within 10\% \ for the variations $\mu_t$
and $\mu_Z$ within the limits stated above.
These observations indicate that corrections
of order $\alpha_s^3$ may well lead to a change of the
theoretical prediction for $\G^{A,S}(Z \to \ovl{b} b)$,
of about $\pm$10\% . Of course this estimate
should be considered as an optimistic one; on the
other hand it seems that in the worst case the change
might hardly larger than $\pm$30\% .
To summarize, we have developed a new calculational approach to
evaluating of higher order singlet corrections to the hadronic
decay rate of the $Z$ boson. The approach is based on effective
field theory methods and is applicable starting from
values of the top quark masses as low as that of the $Z$ boson. We
have employed the method to rederive in a systematic manner and
with much less effort the known lowest order (up to power suppressed
terms) singlet contribution to $\G^h_Z$ [1,2]. We have also found
leading log and next to leading log QCD corrections of orders
$\as^{1+n}(\ln(M_Z/4m^2_t))^n$ and $\as^{2+n}(\ln(M_Z/4m^2_t))^n$
respectively to the result.
Most importantly, the approach may well be extended to analytically
compute all the corrections of order $\alpha_s^3$ to $\G^h_Z$,
including the constant term as well power suppressed ones.
{\em Acknowledgement.}
We are indebted to A.K. Kwiatkowski for his help in making
pictures by means of a \LaTeX \ routine FEYNMAN \cite{FEYNMAN}.
\newpage
%\begin{verbatim}
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%\end{verbatim}
\input{ckpics2.tex}
\end{document}