%Title: Standard Model Calculations for Precision Observables Beyond One Loop
%Author: Bernd Krause
%Title: Standard Model Calculations for Precision Observables Beyond One Loop
%Author: Bernd Krause


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\title{Standard Model Calculations for Precision Observables Beyond
       One Loop}

\author{B. Krause\address{Institut f\"ur Theoretische Teilchenphysik, \\
Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany. \hfill{Preprint TTP97-50}}
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\begin{document}

\begin{abstract}
This talk gives a review of recent two- and three-loop calculations to
electroweak precision observables.  Emphasis is put on the calculation
of low energy observables, in particular the muon anomalous magnetic
moment and the electric dipole moment of the neutron.
\end{abstract}

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\maketitle

\section{Introduction}

In order to match with the increasing precision of electroweak data
from experiments at LEP, hadron colliders or low energy experiments
theorists are forced to perform two- and three-loop calculations, thus
testing theory on the quantum level.
Important goals from the comparison between theory and experiment
include 
\begin{itemize}
\item precision tests of the Standard Model,
\item possible deviations as a signal for new physics,
\item improvement of indirect constraints on the Higgs mass.
\end{itemize}
Due to the high experimental precision, the following list of
electroweak input parameters is currently used: $\alpha(0)$, $G_\mu$,
$M_Z$, all observables being known at least to ${\cal O}(10^{-5})$. The
relation between the basic parameters  $\alpha(0)$, $G_\mu$, $M_Z$,
$M_W$ is
\be
G_\mu = {\pi\alpha\over \sqrt{2}}{1\over M_W^2 (1-M_W^2/M_Z^2)}
\left[1+\Delta r\right]
\label{eq:delta_r}
\ee
where $\Delta r$ parametrizes the radiative corrections. The
theoretical prediction for $\Delta r$ is
\be
\Delta r =\Delta \alpha -{c_W^2\over s_W^2}\Delta \rho +\Delta
r_{remainder}
\ee
Several improvements have been achieved on the SM radiative
corrections:

\begin{itemize}
\item
{\it Electroweak Precision Observables:}
Numerically the most important contribution to $\Delta r$ is from
$\Delta \alpha$ (see \cite{Hol97} for details). As for the
$\rho$-parameter, its contribution increases with the mass splitting
in each isospin doublet. For this reason the contribution from the
top-bottom-doublet is dominant. QCD-corrections to $\Delta \rho$ have
been computed 
including three loops \cite{AFMT95,CKS95}. These corrections are well
under control and have been implemented in the analysis of LEP data.

\item {\it Vector Boson Masses:}
In eq.(\ref{eq:delta_r}) the relation between $M_W$, $M_Z$ and the Fermi
constant $G_\mu$ is established. To $\Delta r$ one-loop and leading
parts ${\cal O}(G_\mu^2 m_t^4)$ of the two-loop contributions have been
available since some time already (see \cite{Hol97}). An estimate of
the theoretical uncertainty by investigation of the renormalization
scheme dependence 
in ref. \cite{yb95}  revealed,
however, that 
there was still a significant uncertainty ($\Delta M_W\approx 16\;{\rm
MeV}$, $\Delta s_W^2\approx 1.4\cdot 10^{-4}$ and a shift in the upper
bound of $M_H$ by up to 100 GeV) coming from the unknown non-leading
terms ${\cal O}(G_\mu^2 m_t^2 M_Z^2)$.
Two new calculations are now available which significantly reduce this
theoretical uncertainty: the complete ${\cal
  O}(G_\mu^2 m_t^2 M_Z^2)$ calculation of \cite{DGV96} shows that the
subleading terms in the $m_t$-expansion can be equally important as
the $m_t^4$ terms (see also \cite{DGPS97}). In the analysis of
\cite{BW97} the $M_H$ dependence 
of $\Delta r$ has been investigated. This is a subset of the diagrams
calculated in \cite{DGV96}, however in \cite{BW97} the full $m_t$,
$M_H$ dependence is retained. The comparison between \cite{DGV96} and
\cite{BW97} shows fair agreement, giving some confidence to the
assumption that it is enough to include the 
${\cal O}(G_\mu^2 m_t^2 M_Z^2)$ terms in order to control two-loop
electroweak corrections to $\Delta r$.

\item {\it $Z$-Boson Observables:}
There is a new evaluation of non-factorizable QCD and electroweak 
corrections to the hadronic $Z$ boson decay rates \cite{CK96}. 
A negative correction to the $Z$ hadronic width was found reducing the 
SM prediction for $\Gamma_h$ by 0.59~MeV
after summing over the four light quark flavors. 
The corresponding effect for the partial width 
$\Gamma(Z\rightarrow `b\bar{b}\mbox{'})$ has not been evaluated. 
This shift leads to an enhancement of the 
$\alpha_s$ value extracted from the electroweak data by 0.001. 
%%%  
\end{itemize}

\section{Low-Energy Observables}
Although there is now clear evidence for electroweak loop effects in
high energy physics, it still remains a challenge to prove the
validity of the Standard Model on the quantum level also at very low
energies, ten orders of magnitude below the physics of the $Z$
resonance. Two observables are of particular interest for electroweak
physics: the anomalous magnetic moment of the muon and the electric
dipole moment (EDM) of the neutron, $d_n$. 

\subsection{Anomalous Magnetic Moment of the Muon}

One of the most precisely measured electroweak quantities is the 
anomalous magnetic moment of the muon, $a_\mu = (g_\mu -2)/2$. The
error of the current experimental average \cite{PDG}
\begin{eqnarray}
a_\mu^{\rm exp} = 116592300 (840) \times 10^{-11}
\end{eqnarray}
is going to be reduced by at least a factor of 20 in the new
Brookhaven National Laboratory experiment E821 to $\Delta
a_\mu^{exp}=\pm 40\cdot 10^{-11}$.
At that level of accuracy, the experiment will be sensitive to loop
effects of the $W$ and $Z$ bosons, thus testing the full 
Standard Model at very low energies.
The electroweak one-loop contribution to $a_\mu$ 
%\cite{ew72}
\be
a_\mu^{ew}(1-loop)=195\cdot 10^{-11}
\ee
has been supplemented with the corresponding two-loop corrections
\cite{ew2loop_ferm,ew2loop_bos}
\be
a_\mu^{ew}(2-loop)=-44\cdot 10^{-11}\;.
\ee
In fig.~\ref{fig:log_ew} the complete subset of bosonic diagrams which
contributes to the large logarithms $\ln {M_W^2\over m_\mu^2}$ is
depicted. Although only a small fraction of bosonic diagrams, they
account for the numerically most important contribution to
$a_\mu^{ew}(bos.)$. The diagrams which contain a closed fermion loop
have been discussed in detail in \cite{ew2loop_ferm,CK96}.
The electroweak two-loop contribution turned out to be of the same
size as the anticipated uncertainty of the BNL experiment, reducing
the total electroweak contribution to
\be
a_\mu^{ew}=151(4)\cdot 10^{-11}
\ee
with a very small theoretical uncertainty.

\noindent
It is mandatory to also include leading and higher order hadronic
contributions to $a_\mu$. The ${\cal O}(\alpha^3)$ hadronic
contributions fall into two classes: vacuum polarization insertions
into the photon propagator and the so-called
light-by-light-contribution. The former can be calculated 
by using experimental data for the ratio $R(s)=\sigma(e^+e^-\to
hadrons)/\sigma(e^+e^-\to \mu^+\mu^-)$ and has recently be
re-evaluated \cite{BK97}
\be
a_\mu^{had}({\rm higher\; order})=-101(6)\cdot 10^{-11}\;.
\ee
For the light-by-light-contribution,
however, there is no experimental data available. Instead, the
Extended Nambu-Jona-Lasinio model has been used \cite{light_enjl}.
The latest evaluation yields \cite{haki97}
\be
a_\mu^{had}({\rm light-by-light})=-79.2(15.4)\cdot 10^{-11}\;.
\ee
It should be kept in mind, however, that due to the model dependence
it is difficult to give an error estimate.



\subsection{Electric Dipole Moment of the Neutron}

The EDM of the neutron is a quantity of great interest since, if
non-zero, would be the first manifestation of CP-violation outside of
the $K^0-\bar{K}^0$ system. Current experimental bounds are already
very stringent: $|d_n|< 1.1\times 10^{-25} e\cdot cm$
\cite{edm:exp}. In the Standard Model, EDM of
elementary particles are highly suppressed, the first non-vanishing
contribution for quarks arising only at the three-loop level.
The corresponding  final result for the Standard Model 
calculation has been 
published in \cite{CK97}. A particularly important issue which has
been clarified in the explicit calculation of \cite{CK97} was the
r\^ole of the top quark. Here we report on details of the top contribution.
Previous estimates \cite{S80} and the calculation in the Fermi theory
\cite{Kr86} date back to the early 80s and were based on the
assumption $M_W> m_t$. Since we know that the top quark is very heavy
and that some electroweak observables are enhanced with
$m_t^2/M_W^2$ a reinvestigation of the EDM of the neutron seems to be
in order. The relevant diagrams with contributions to $d_n$ from
the top-quark are depicted in fig.\ref{3loop}. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\mbox{
\psfig{figure=capri_fig.ps,width=7.3cm,bbllx=90pt,bblly=620pt,%
                                     bburx=430pt,bbury=780pt}
     }
\parbox{7.2cm}{
\caption{\label{3loop} \small \it 
Contribution of the top-quark to the EDM of up- and down-quarks,
respectively. The coupling of the external photon to all charged lines
has been omitted for brevity. Wiggled lines denote $W$ bosons, spiral
lines the Gluon. A sum over all light quarks in the loop is understood.
        }    }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The sum of the diagrams
containing the top quark is separately finite and yields for the EDM of
the down-and up-quark respectively:
\be
{d_d(top)\over e} &=&
- {m_d m_c^2 \alpha_s G_F^2   \tilde\delta\over 48\pi^5} \times \\
&& \hspace{-.6cm}\times\left\{ - {13\over 3} + {37\over 27} 
    \cdot {\pi^2\over 6}
       + {59\over 27} L_ {bc } +{37\over 54} L ^2 _{bc }
\right\}  \nonumber 
\label{eq:d_d_top}
\ee
The corresponding result for the up-quark is 
\be
{d_u(top)\over e} &=&
- {m_u m_s^2 \alpha_s G_F^2   \tilde\delta\over 48\pi^5} \times \\
&\times&
\left\{
 {95\over 27}\left[ {\pi^2\over 6} - {1\over 2} 
  - {1\over 2} L _{bs } \right] - {37\over 108} L^2_{bs }
\right\} \nonumber
\label{eq:d_u_top}
\ee
where $\tilde\delta$ denotes the $CP$-violating invariant
\be
\tilde\delta = s_1^2s_2s_3c_1c_2c_3\delta\;.
\ee
and $L_{ab}\equiv \ln(m_a^2/m_b^2)$.
In eq.(\ref{eq:d_d_top}) and (\ref{eq:d_u_top}) terms suppressed with
additional powers of light quark masses have been neglected.
The main observation is that terms enhanced with $m_t^2$ are actually
absent. This is true also in next order in the expansion in $1/M_W$,
i.e. there are no terms of the type $m_t^2m_c^2m_d/M_W^6$ (for the
down-quark) or $m_t^2m_s^2m_u/M_W^6$ (for the up-quark).
In leading order, the top quark contribution does not depend on the
top mass at all. Its numerical impact on $d_n$ is not very important,
and as a consequence the Standard Model contribution to the electric
dipole moment of the neutron will be definitively unobservable for the
experiments in the near future.


\section{Conclusions}
Higher order Standard Model loop calculations of the past few years
have been reviewed. Theoretical progress generally matches the present
and anticipated experimental accuracy of precision observables like
e.g. $M_W$ or $a_\mu$. It is interesting to note that in most of the
recent higher order loop calculations the method of asymptotic
expansions (for a recent review see e.g. \cite{Smi94}) has been
applied and proven to be a convenient tool for practical
calculations. The theoretical uncertainties in electroweak physics are
now dominated by the experimental error in $\sigma(e^+ e^-\to
Hadrons)$. Reduction of this error is crucial for improved sensitivity
to new physics and more stringent bounds on $M_H$.

\section{Acknowledgements}

The author would like to thank A. Czarnecki and W. Marciano,
with whom part of the results presented here have been worked out.  
I am also grateful to W. Hollik and the organizers of the WIN'97
Conference for the invitation, the perfect organization and kind
hospitality during the Conference. 
This work has been supported by ``Graduiertenkolleg
Elementarteilchenphysik'' at the University of Karlsruhe.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\begin{center}
\mbox{
\psfig{figure=abb_KKSS.ps,width=7.4cm,height=11cm,bbllx=50pt,bblly=140pt,%
                                     bburx=440pt,bbury=790pt}
     }
\parbox{7.7cm}{\vspace{-0.8cm}
\caption{\label{fig:log_ew} \small \it 
Electroweak two-loop corrections to the anomalous magnetic moment of
the muon:
the complete set of bosonic diagrams with contributions to the leading
logarithms  $\ln {M_W^2\over m_\mu^2}$.
Wiggled lines denote photons, spiral lines the massive vector bosons
$W,Z$ and dashed lines the Goldstone bosons.
        }    }
\end{center}
\end{figure}
\clearpage
%\newpage
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\end{document}
