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\begin{document}

\begin{flushright}
        \small
%        \\
        MPI-PhT-96-85\\
        August 1996
\end{flushright}
%\vspace{.2cm} 
\begin{center}
{\large\bf 
Two-loop heavy top effects   \\
           on precision observables\footnote{Invited talk at
 the Third International Symposium on 
Radiative Corrections,
Cracow, Poland, 1-5 August 1996. 
%To appear in the Proceedings in Acta Phys. Pol.
}}
 \\
\vspace{.62cm}
{\sc\small  Paolo Gambino}\\
\vspace{.2cm}
{\em\small Max Planck Institut f\"ur Physik, Werner Heisenberg Institut,\\
 F\"ohringer Ring 6, D80805 M\"unchen, Germany}
\end{center}
\vspace{.2cm} 
\begin{center}
{\bf\small Abstract}
\end{center}
\vspace{-.7cm}
\begin{quotation}
\small
\noindent
The corrections induced by a heavy top on the main precision observables 
are now available up to $O(\alpha^2 \mt^2/\mw^2)$.
The new results  
 imply a significant reduction of the 
theoretical uncertainty and can have a sizable
impact on the determination of \sineff.
%  due to  unknown higher order effects.
\end{quotation}
\vspace{.5cm}
\noindent
The very precise measurements carried out at LEP and SLC in the recent past
have made the study of higher order radiative corrections 
necessary in order to test the Standard Model, 
and possibly to  uncover hints of new physics.
 The one-loop corrections to all the relevant
\ew\ observables are by now 
very well established \cite{YB},
and  two and three-loop effects have been investigated in several  cases. 
The dominance of a heavy top quark in the one-loop \ew\ corrections, 
which depend quadratically on the top mass, has allowed to predict
with good approximation
the  mass of the heaviest quark before its actual discovery. 

Among the 
higher order effects connected with these large non-decoupling
contributions, the QCD corrections are now known 
through $O(\alpha_s^2)$ \cite{QCD}.
As for the  purely \ew\
effects originated at higher orders by the large Yukawa coupling 
of the top,
reducible contributions  have been first studied in 
\cite{CHJ}, while a  thorough investigation of leading irreducible 
two-loop
contributions has been initiated in \cite{van},
 in the limit of a
massless Higgs, and later continued by Barbieri {\em et al.} and others
for arbitrary $\mh$ \cite{barb}.
In this talk I will illustrate 
some implications of the new calculation of the \amtd\
corrections to the main precision observables \cite{physlett,zako}.

The result of the calculation of the leading \gmtq\ effects 
on the $\rho$
parameter \cite{barb}  is shown in Fig.1 (upper curve). The 
correction  is relatively sizable
and in the  heavy Higgs case 
 reaches the permille level in the prediction of $\sineff$, 
comparable to the present experimental accuracy \cite{war}.
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\vspace{-.6cm}
\caption{\small\sf Two-loop heavy top corrections to $\drhoc$ in 
units $10^{-4}$.  The upper  line 
represents the leading $O(G_\mu^2 m_t^4)$ correction 
of Ref. [5].}  %\cite{barb}.}
\vspace{-4mm}
\end{figure}
We observe that the correction is extremely small for a small Higgs mass, 
due to large cancellations. We  can naively expect that setting 
the masses of the vector boson different from zero (and so going beyond 
the pure Yukawa theory considered in \cite{barb}) might spoil the 
cancellations and lead to relevant deviations from the upper curve of 
Fig.1 in the light Higgs region.

In addition,   the theoretical \, uncertainty\, coming from unknown
higher order effects is dominated by terms \amtd\ \cite{YB}. 
  Indeed, the  renormalization 
scheme ambiguities and the different resummation options examined in
\cite{YB} led to an estimate of the uncertainty of the theoretical 
predictions which was in a few cases  disturbingly sizable, i.e.
comparable to the present experimental error.
In particular, for $\sineff$, the estimated uncertainty was 
$\delta\,\sineff (th)\lequiv\, 1.4\times 10^{-4}$, 
while the present experimental
average is $\sineff= 0.23165\pm 0.00024$ \cite{war}.
For $\mw$ the uncertainty, $\delta{\mw} (th)\lequiv\,
16$ MeV,  was much  smaller than the  error 
on the present world average, 125 MeV.
 A different analysis based on the explicit two-loop calculation 
of the $\rho$ parameter in low-energy processes has also reached 
very similar conclusions \cite{us}.

Motivated  by the previous observations, the complete 
analytic calculation of the 
two-loop quadratic top effects has been performed  
for the relation between the 
vector boson masses and the muon decay constant $G_\mu$ \cite{physlett}
and for the effective leptonic mixing angle, $\sineff$ \cite{zako,prep}.
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\vspace{-11mm}
\caption{\sf\small Scale dependence of the prediction for $\mw$ 
in the $\msbar$
scheme for $\mt=180$GeV, $\mh=300$GeV, including only the leading
\gmtq\ correction (dotted curve)
or all the available two-loop contributions, through
\amtd\ (solid curve).}
\label{mudep}
\vspace{-3mm}
\end{figure}
 Corrections to the 
 $Z^0$ decay width are also under study.
All these calculations have been done in 
the $\msbar$ scheme introduced in \cite{msbar}, i.e. using $\msbar$ 
couplings and on-shell masses, a  particularly convenient framework. 
Besides the leading \gmtq\ result, I have displayed in Fig.1 the
two-loop heavy top correction to $\hat{\rho}$ up to \amtd. 
In the $\msbar$ scheme, $\hat{\rho}$ is defined as 
$\mw^2/\mz^2 \ 1/\cos^2\hat{\theta}_{\msbar}(\mz)$, 
and represents the most obvious 
process-independent analogue of the low-energy $\rho$ parameter. 
The deviation from the leading \gmtq\ result is striking.

In order to gauge the impact of the new calculation, however, 
it is best to 
consider   physical observables, and to study the scheme and scale 
dependence of their predictions from $\alpha$, $G_\mu$, and $\mz$.
 I will do that for the two most relevant 
precision quantities: the mass of the $W$ boson and $\sineff$.
Concerning the scale dependence of the $\msbar$ predictions, 
the situation is 
exemplified in Fig.\ref{mudep}. Over a wide range of $\mu$ values, the
scale dependence of  $\mw$ 
is  significantly reduced by the inclusion
of the \amtd\ contribution.

\renewcommand{\arraystretch}{1.2}
\begin{table}[t] 
\[
\begin{array}{|c||c|c||c|c|}\hline
\mh & \Delta \,{s^2}_{eff}^{lead}  
&\Delta \,{s^2}_{eff}  & \Delta\mw^{lead} & \Delta\mw  \\  \hline\hline
65  &-0.90 & -0.14 & 8.4 &1.5  \\ \hline
100  & -0.90& -0.12 & 8.2& 1.3\\ \hline
300   & -0.87& -0.08 &7.6 &0.5 \\ \hline
600  & -0.83&-0.05   &7.0 &0.1 \\ \hline
1000 & -0.79&-0.03 & 6.5 &-0.3 \\ \hline
\end{array}            
\]
\caption{\sf\small Scheme 
dependence of the prediction of $\sineff$ before and 
after the inclusion of the new \amtd\ correction for $\mt=175$GeV.
$\Delta s^2_{eff}$ is in units $10^{-4}$,  $\Delta\mw$ in MeV and 
$\mh$ in GeV.}
\vspace{-2mm}
\end{table}
After translating  \cite{prep} the results of the two-loop 
calculation into the on-shell (OS)
scheme \cite{si80},  I have compared the 
predictions for $\mw$ and $\sineff$ 
in the $\msbar$  and  OS scheme,
before and after the inclusion of the \amtd\ contribution.
 The results  for one particular OS implementation
are shown in Table 1,
where $\Delta s_{eff}^2\equiv \sineff(\msbar)-
\sineff(\rm OS)$ and $\Delta\mw\equiv \mw(\msbar) - \mw(\rm OS)$.
 As expected, the scheme dependence
of the predicted values of $\sineff$ and $\mw$ is drastically reduced.
After considering alternative OS options,
we can safely conclude that the inclusion of the \amtd\ correction reduces
the scheme dependence in the two cases considered 
by  at least a factor corresponding to the expansion 
parameter $\mw^2/\mt^2\approx 0.2$.

Finally, let me consider the impact of  the new calculations on 
the precise predictions of $\sineff$ and $\mw$. 
It is clear that  the shifts
induced depend very strongly on the actual implementations
of OS and $\msbar$ scheme. The results shown in Table \ref{impact}
 refer to a typical $\msbar$ implementation \cite{zako,msbar}
and to two
different OS implementations, OSI and OSII.
Using $\delta\mw\equiv
\mw - \mw^{lead}$ and $\delta \,s^2_{eff}\equiv
\delta s^2_{eff} - \delta {s^2}_{eff}^{lead}$ 
for the shifts induced by the \amtd\ corrections on $\mw$ and $\sineff$,
we see that $\delta\mw$ and $\delta\, s^2_{eff}$ 
tend to be  larger in the light Higgs region, 
and that  $\delta\, s^2_{eff}$ 
can be quite sizable, more than 1$\times 10^{-4}$.
\begin{table}[t] 
\[
\begin{array}{|c||c|c|c||c|c|c|}\hline
\mh  & \delta \,{s^2}_{eff}^{\rm OSI}  
& \delta \,{s^2}_{eff}^{\rm OSII}  
&\delta \,{s^2}_{eff}^{\msbar}& \delta\mw^{\rm OSI} & \delta\mw^{\rm OSII} 
&\delta\mw^{\msbar} 
\\  \hline\hline
65  &0.04& 1.56 &0.80 &  -6.5& -14.5 &-13.4 \\ \hline
100 & -0.02& 1.27 &0.76 & -5.9 &-12.7 &-12.9 \\ \hline
300   & -0.14& 0.35 &0.65 & -4.1 &-6.7 &-11.1 \\ \hline
600   & -0.20& -0.33&0.58  & -2.6 &-1.9 &-9.5  \\ \hline
1000 & -0.30 &-0.93& 0.46  & -0.5 &2.9 &-7.3 \\ \hline
\end{array}            
\]
\caption{\sf\small Shifts induced  by the \amtd\ corrections
 in the OS and $\msbar$ scheme for $\mt=175$GeV.
$\delta s^2_{eff}$ is in units $10^{-4}$,  $\delta\mw$ in MeV.}
\label{impact}
\vspace{-3mm}
\end{table}
 Because of the sign of the shifts, in general the \amtd\ correction 
further enhances the screening of the top quark
 contribution by higher order effects.

In summary, two-loop \ew\ $\mt^2$ effects are now available
in analytic form
for the main precision observables in {\em both} $\msbar$ and OS 
frameworks. The new contributions consistently reduce the scheme
and scale dependence of the predictions by {\em at least}
a factor $\mw^2/\mt^2\approx 0.2$, suggesting 
a relevant improvement in the 
theoretical accuracy. The impact on 
the value of the effective sine can be sizable, up to 1.5$\times 10^{-4}$,
but it is highly sensitive to  the scheme adopted.  



\vspace{.4cm}
I wish to thank the organizers for the excellent organization and
the pleasant atmosphere during the Symposium.
Useful conversations with G. Degrassi, W. Hollik, and A. Sirlin are 
gratefully acknowledged.

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\end{document}


