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\begin {document}
%HU-EP-00/57
\def\titleline{Fundamental Constants at High Energy\footnote{Partially
supported by VW--Stiftung Hannover (I-77495)}$^,$ \footnote{Invited talk given
at the Heisenberg symposium of the Alexander von Humboldt-Foundation
(Bamberg,\\September 2001)}
%\newtitleline
}
\def\authors{Harald Fritzsch}
%Author A\1ad and Author B \2ad }
\def\addresses{Theory Division, CERN, CH--1211 Geneva 23, Switzerland,\\
and\\
Ludwig--Maximilians--University Munich, Sektion Physik\\
Theresienstra{\ss}e 37, 80333 M\"unchen\\
%Address cont.\\
%{\tt e-mail Fritzsch@mppmu.mpg.de} \\
%\2ad Address B \\
%Address cont.\\
%{\tt e-mail B}
}
\def\abstracttext{The progress of Particle Physics is closely linked to the
progress in the understanding of the fundamental constants, like the
finestructure constant, the mass of the electron or nucleon, or the
electroweak mixing angle. The relation between the 18 fundamental constants
of the Standard Model and the elementary units used in other fields like
quantum optics or solid state physics is far from trivial and will be
discussed. Relations between the various constants might exist, providing
signals for the physics beyond the Standard Model. Recent observations in
astrophysics indicate a slight time variation of the finestructure constant.
If true, it has profound implications for many particle and nuclear
physics phenomena. In particular the nuclear mass scale should change in
time, a phenomenon which could be observed in the laboratory using
advanced methods of quantum optics.}
\large
\makefront

%\section{First chapter}
%Main text
The Standard Model of particle physics\cite{Pes} is a superposition
of QCD for the strong interactions and the electroweak gauge theory for the
electromagnetic and weak interactions. It gives a nearly complete
description
of all observed phenomena in atomic, nuclear and particle physics. It is
only nearly complete, since certain phenomena, among them the increasing
evidence for neutrino oscillations and the dominance of matter as compared
to antimatter in the
universe, cannot be described within the framework of the Standard Model.
The major drawback exhibited by the Standard Model is the fact that a
large number of constants, in particular many mass parameters, have to be
adjusted accor\-ding to the experimental measurements and cannot be
predicted within the theory. Many theoreticians in particle physics
believe for this reason that the Standard Model must be regarded only
as a first step towards a more complete understanding, and that in the
future it will be embedded in a larger and more complete theoretical
framework.

In this talk I cannot offer a solution of the problem of the
many fundamental constants, however I shall give a
critical overview and describe possible directions one might go. Let me
first point out that the Standard Model, which is based on a description
of the fundamental forces by the theoretical framework of quantum
theory, is a description of the local laws of nature. It does not say
anything about the possible boundary conditions which are imposed on
these local laws from the outside, in particular from cosmological
boundary conditions. It could well be that at least certain fundamental
constants are subject to such boundary conditions. If this is the case,
it would not help to find a description of these constants by
embedding the Standard Model in a more fundamental theory. Some of the
constants appearing in the
Standard Model could indeed be cosmic accidents, i. e. quantities which
were fluctuating wildly at the time of the creation of the universe,
but were frozen immediately afterwards. These constants could be called
``frozen accidents''\cite{Gell}. Other constants could eventually be determined by
dynamical laws, which go beyond the laws of the Standard Model. Such
constants could indeed be calculated in a future theory. It might also
be that some of the elementary constants are not constants at all, but are
slowly changing in time. Such ideas were pioneered by Dirac\cite{Dir}, who
once
proposed that the gravitational constant is a function of the cosmic time.
This idea, however, faded away during the course of the last century,
since no time variation of the gravitational constant has been observed.
Nevertheless one should keep in mind that not only the gravitational
constant, but also other constants appearing in the Standard Model might
turn out to be slowly varying functions of time.

The constant which plays the most significant role in atomic physics is the
finestructure constant $\alpha $ introduced by Arnold
Sommerfeld in 1917. Sommerfeld noted that the finestructure of the
atomic levels was determined by a dimensionless number whose value today is
given by:
\begin{equation}
\alpha^{-1} = 137.03599976(50) \, .
\end{equation}
Numerically it turned out that the inverse of $\alpha $ is quite close
to be an integer number. Sommerfeld himself did not indulge in philosophical
speculations about the nature of $\alpha $. Such speculations were started
in the thirties by Arthur Eddington, who speculated about an intrinsic
relation between the inverse of $\alpha $ and the total number of
different charged objects\cite{Gro}. He introduced a specific counting of
these objects,
including their spin and came up with the astonishing number 136,
which in Eddington's view was sufficiently close to the observed
number of 137.

Shortly after Eddington Werner Heisenberg came up with a proposal to
describe $\alpha $ by an algebraic formula, which works up to an
accuracy of $10^{-4}$:
\begin{equation}
\alpha = 2^{-4} 3^{-3} \pi \, ,
\end{equation}
Wyler found in 1971 an algebraic formula based on group--theory agruments
which works up to the level
of $10^{-6}$\cite{Gro}:
\begin{equation}
\alpha = \frac{9}{8 \pi^4} \left( \frac{\pi^5}{2^4 5!} \right)^{\frac{1}{4}}.
\end{equation}
Today we must interpret such attemps as useless. In particular in the
underlying theory of Quantum Electrodynamics the actual value of the
coupling constant changes if one changes the reference point, i. e.
changing the energy scale.

The theory of Quantum Electrodynamics is
still the most successful theory in science. It brings together
Electrodynamics, Quantum Mechanics and Special Relativity. QED is a
renormalizable theory and has been tested thus far up to a
level of one in 10 Million. Physical quantities like the anomalous
magnetic moment of the electron can be calculated in terms of powers of
$\alpha $, and $\alpha $ has been determined that way to a high degree
of accuracy.

The value of $\alpha $ describes the coupling strength of electrodynamics
at distances which are large compared to the Compton wavelength of the
electron. At smaller distances $\alpha $ changes slowly. In pure QED, i. e.
in the presence of only the photon field and the electron, the
effective value of $\alpha $ is given in momentum space by:
\begin{equation}
\alpha_{eff} \left( q^2 \right) = \frac{\alpha}{1 - \frac{\alpha}{3 \pi}
ln \left( \frac{- q^2}{Am_e^2} \right)}
\end{equation}
\begin{equation}
A = exp \left( 5/3 \right), -q^2 > 0 \, .
\end{equation}
The infinitesimal change of $\alpha $ is dictated by the renormalization
group equation:
\begin{equation}
\frac{d}{d ln (q / M)} e \left( q; e_r \right)
= \beta (e), \quad e \left( M; e_r \right) = e_r
\end{equation}
\begin{center}
($M$: renormalization point) \, .
\end{center}
At high energy not only virtual electron--positron pairs contribute, but
also myon pairs, $\tau $ pairs, quark--antiquark--pairs etc. At smaller
distances $\alpha $ is becoming larger. This effect can also be seen
directly in the experiments. At LEP the effective value of $\alpha $
given at an energy scale of 91 GeV (mass of $Z$--boson) is:
\begin{equation}
\alpha \left( M_z \right) \cong \left( 127.5 \right)^{-1} \, .
\end{equation}
The renormalization group requires that the strength of the electromagnetic
coupling increases at increasing energy and eventually reaches a point,
where perturbation theory breaks down. In pure QED, i. e. in
the theory of a photon field, interacting only with one charged fermion,
the electron, the critical energy (Landau singularity) is extremely high
and far above the energy scale given by gravity. Of course, this theory
is not realistic, since in the real
world there are 3 charged leptons and 6 charged quarks, and therefore the
increase
of the coupling constant happens at a much higher rate. Already at
energies, which were reached by the LEP--Accelerator, of the order of
200 GeV, the associated value of the finestructure constant is more than
10\% higher than at low energy. In any case this signifies that one should
not attach a specific fundamental meaning to the numerical value of the
finestructure constant.

The fact that in particle physics all phenomena can be described in
terms of a number of fundamental constants is, of course, directly
related to the Standard Model. The Standard Model is a superposition of
the quark--gluon gauge field theory (QCD) and of the electroweak gauge
theory, based on the gauge group $SU(2) \times SU(1)$. The Standard Model
is not merely a gauge field theory, like many others. It aims at a
complete description of all particle physics phenomena, and it is
extremely successful in doing so. Let me remind you that the experimental
program, using the LEP Accelerator at CERN, came to an end in the
year 2000. The outcome of the research done with the LEP Accelerator
constitutes a triumph in particular for the standard electroweak gauge
theory. The parameters of the theory, most notably the mass of the
$Z$--boson and the coupling parameters have been determined with an
impressive accuracy.

In the Standard Model the masses of the weak bosons
are generated by the coupling of the boson fields to the thus far hypothetical
Higgs--field. The model requires the existence of a Higgs particle, whose
mass, if it exists, is one of the basic parameters of the
Standard Model. The present limit of the mass of the Higgs particle, given
by the CERN experiments, is about 110 GeV.

In the Standard Model the number of basic parameters is 18, including
the three gauge coupling constants. Thirteen of these constants are
directly related to the fermion masses.

The 18 basic constants of the Standard Model can be listed as
follows:
\begin{equation}
\begin{array}{ccccccccc}
m_e & m_u & m_d; & m_{\mu} & m_c & m_s; & m_{\tau} &  m_t & m_b\\
& &  \, \, \, \,  \theta_u & \, \, \, \theta_d \, \, & \theta & \, \, \delta & & &\\
& & & M_w & M_h & & & &  \\
& & & \hspace*{-1cm} \alpha & \hspace*{-1cm} \alpha_s & \hspace*{-1cm}
\alpha_w  & & & .\\ 
\end{array}
\end{equation}
The nature of most of the fundamental constants seems to be intrinsically
related to the generation of masses. One of the peculiar features of the
Standard Model is the fact that two different types of mass generation
mechanisms seem to operate. On the one hand the masses of the weak bosons
and of the fermions are given by the coupling of these fields to the
scalar boson. On the other hand the masses of the nucleons and
moreover
the masses of all nuclei are predominantly due to a dynamical mass
generation. The generation of mass in QCD could be described as
``mass from no--mass''. In lowest order the behavior of the QCD coupling
constant $\alpha _s$ is given by:
\begin{equation}
\alpha _s \left( q^2 \right) = \frac{2 \, \pi}{b_0 ln \left( \frac{q}{\Lambda}
\right)}, \quad b_0 = 11 - \frac{2}{3} \, n_f
\end{equation}
\begin{center}
($n_f$: number of flavors, $q = \sqrt{q^2}$), $\Lambda$: scale parameter).
\end{center}
Formally the coupling constant becomes infinite, if the energy scale
involved approaches the critical value $\Lambda $. Through ``dimensional
transmutation'' the functional dependence of the
coupling constant on the energy leads to the appearance of a mass scale.
In the limit in which all the quark masses are set to zero, the masses
of the bound states (nucleons etc.) are proportional to
$\Lambda $. Using the experimental value $\alpha _s (M_z) = 0.1184 \pm
0.0031$, as given by the LEP--experiments, one obtains
$\Lambda = 213 +38/-35$ MeV\cite{Bet}.

In principle the nucleon mass, one of the
fundamental parameters of atomic physics, can be calculated in terms of
$\Lambda $, if the effects of the quark masses are neglected.
Thus far an exact determination of the nucleon mass in terms of $\Lambda $
has not been possible,
due to the complexity of the calculations, e. g. within the approach of
lattice QCD. However, simpler quantities, for example the pion decay
constant, have been calculated with success. The pion decay constant is
given by the matrix element of the axial vector current:
\begin{equation}
< 0 | A_{\mu} |  \pi > = i p_{\mu} F_{\pi} \, .
\end{equation}
It has the dimension of mass. The theoretical result is\cite{Lue}:
\begin{equation}
F_{\pi} / \Lambda = 0.56 \pm 0.05
\end{equation}
while the experiments give:
\begin{equation}
F_{\pi} / \Lambda = 0.62 \pm 0.10 \, .
\end{equation}
The good agreement between experiment and theory indicates
that QCD is able to describe not only perturbative features of the
strong interaction physics, but also indicates that in the future one
might be able to calculate more complicated quantities like the nucleon
mass with a good precision.

One must keep in mind that the quark masses are non--zero and will
influence the numerical value of the nucleon mass. Unfortunately the
uncertainties imposed by our ignorance about the contribution of the
quark mass terms to the nucleon mass is high. The matrix element of
the non--strange quark mass term, the $\sigma $--term, is only poorly
known:
\begin{equation}
< p | m_u \bar uu + m_d \bar d d | p > \approx 45 MeV \pm 25\% \, .
\end{equation}
Also the mass term of the strange quarks plays an important role for the
nucleon mass. Typical estimates give:
\begin{equation}
< p | m_s \bar ss | p > \sim 40 MeV \, ,
\end{equation}
with an error which is not less than about 50\%. Note that the $u-d$
contribution and the $s$--contribution to the nucleon mass are of
similar order.

In general we can say that the nucleon mass is a dual entity. The
dominant part of it (about 90\%) is due to the dynamical mechanism
offered uniquely by QCD, i. e. due to the field energy of the confined
quarks and gluons. About 10\% of the nucleon mass arises due to the
nonvanishing masses of the $u, d$ and $s$--quarks. The strange part of
this contribution is about as large as the non--strange part. Moreover
there is a small electromagnetic term of about 2\% (of order
$\alpha \cdot \Lambda$).

In particle physics we are confronted at the beginning of the new
millenium with the unsolved problem of the spectrum of the lepton and
quark masses. The mass eigenvalues show a remarkable mass hierarchy.
As an example I mention the masses of the charge 2/3 quarks:
$u:c:t \approx 5:1150:174000$ (the masses are given in MeV).

Most of the quark masses and all of the lepton masses are much smaller
than the mass scale of the weak interactions given in the Standard Model
by the vacuum expectation value $v$ of the scalar field $v \approx $
246 GeV. Only the mass of the $t$--quark is of the same order of
magnitude as the weak interaction mass scale. It is remarkable that
the mass of the $t$--quark is within the allowed errors equal to the
vacuum expectation value divided by $\sqrt{2}$:
\begin{equation}
v / \sqrt{2} \approx 174 \, GeV = m_t \, .
\end{equation}
Such a mass relation might be a hint towards an interpretation as a
Clebsch--Gordan--relation, related to an internal symmetry. However,
no such symmetry has been identified thus far, and the question remains
whether
the relation above is an accident or not. Another interesting feature
of the quark mass spectrum is the fact that for each charge channel the
mass ratios seem to be universal:
\begin{eqnarray}
m_d : m_s & = & m_s : m_b \nonumber \\
m_u : m_c & = & m_c : m_t \, .
\end{eqnarray}
Again a deeper understanding of this scaling feature is missing.

In the Standard Model the transitions between the various families of
quarks (and possibly also of the leptons) arise because the states
entering the weak interactions are not identical to the mass eigenstates.
The transition strengths are in general given by complex amplitudes,
however, it is well--known that the multitude of the flavor transitions
is given by 3 mixing angles, which I like to denote by $\theta $,
$\theta_u$ and $\theta _d$, and a complex phase parameter $\delta $. All
transition strengths can be expressed in terms of these four
parameters. For example, the Cabbibo transition between the up quark and
the strange quark, often denoted as $V_{12}$, is given in the complex
plane by $\theta _u$, $\theta _d$ and the phase $\delta $, which is the
relative phase between the two angles
\begin{equation}
V_{12} \cong \Theta_u - \Theta_d e^{-i\delta} \, .
\end{equation}
In the complex plane $V_{12}$,
$\theta _u$ and $\theta _d$ form a triangle, which is congruent to the
so called ``unitarity triangle''. Since the absolute value of $V_{12}$
is given with very high precision, a good determination of the angles
$\theta _u$ and $\theta _d$ would allow us to determine the shape of the
triangle. Thus far only one of the angles of the
triangle, denoted usually by $\beta $, has been determined by the
experiments, since it is related to the observed strength of the
$CP$ violation in the decay of $B$--mesons. However, the allowed ranges
are still large: $sin 2 \beta \approx 0.45 \ldots 1$.

One can show that the angle $\Theta_u$, which describes essentially the
mixing between the $u$-- and the $c$--quarks is essentially 0 in the
limit $m_u \rightarrow 0$. Likewise $\Theta_d$ is essentially 0 for
$m_d \rightarrow 0$. In simple models for the mass generation based on
symmetries beyond the Standard Model one finds simple relations
between the mass eigenvalues and the mixing angles\cite{Fri}.
\begin{equation}
tan \, \Theta_u \approx \sqrt{\frac{m_u}{m_c}} \qquad tan \, \Theta_d
\approx \sqrt{\frac{m_d}{m_s}} \, .
\end{equation}
If these relations hold the unitarity triangle is determined with rather
high precision. In particular the angle $\alpha$ which is equal to the
phase parameter $\delta $ is essentially $\frac{\pi}{2}$, which would
imply the $CP$ violation in nature to be maximal\cite{Fri}.
Relations between the mass eigenvalues and the mixing angles are of high
interest since such relations would reduce the number of fundamental
parameters of the Standard Model. It is conceivable that all the 3 mixing
angles as well as the phase parameters are fixed by such relations,
although the exact structure of the relations is still unclear.
Further relations, in particular mass relations between the leptons and
quarks and relations among the coupling constants can be obtained if the
Standard Model is viewed as a low energy limit of a grand unified theory,
based on large symmetry groups, e. g. $SO(10)$.

The coupling constants of
the gauge groups $SU(3)$ and $SU(2)$ both decrease at high energies,
while the coupling constant of the $U(1)$--sector decreases. At very high
energies they become of comparable magnitude. If one uses
the observed magnitudes of the coupling constants, one finds that they do
converge at high energies, however do not meet exactly at one point.
The energy scale were they approach each other is about $10^{15}$ GeV.
If the gauge groups of the Standard Model are indeed subgroups of a
bigger symmetry group and if the symmetry breaking of the grand unified
theory happens at one specified energy, one would expect that the three
coupling constants meet at one particular point on the energy scale.
One but not the only possibility to reach a convergence of the coupling
constants is to introduce supersymmetry. In supersymmetry for each fermion
of the Standard Model a corresponding boson is introduced, and each
boson of the Standard Model is accompanied by a corresponding fermion.
Since the partners of the fermions and bosons have not been observed,
their masses must be sufficiently high, typically above about 200 GeV.

The supersymmetric partners of the fermions and bosons do contribute to
the renormalization of the coupling constants. If one choses a
symmetry breaking for the supersymmetry at an energy scale of the order
of about one TeV, one finds that the two coupling constants
converge
at an energy scale of $1.5 \cdot 10^{16}$ GeV. In such a theory the three
different
coupling constants for the strong, electromagnetic and weak interactions
are fixed just by one coupling constant, the unified gauge coupling
constant at high energies.

In grand unified theories one typically finds also a parellelism between
the quarks of charge $-\frac{1}{3}$ and the charged leptons, implying
that at the grand unified energy scale the mass of the charged lepton
and of the corresponding quark should be equal, e. g.: $m_b = m_{\tau}$.
Indeed, such a relation works quite well for the b$-{\tau}$--system.
The observed fact that the $b$--quark mass is about a factor of 3 larger
than the $\tau $--lepton mass comes from the renormalization effect,
mostly due to the QCD interaction. Similar relations between the
$\mu$--mass and the $s$--quark mass or between the electron mass and
the $d$--mass do not seem to hold. The relations between
these masses must be more complicated than the one given above.
Nevertheless it is conceivable that such mass relations exist.

Taking into account the relations between the fundamental parameters of
the Standard Model discussed above, one may ask how many independent
parameters might finally remain. The most optimistic answer is 7: one
coupling constant for the unified interaction, the 3 masses of the
charged leptons and the 3 masses of the charge $\frac{2}{3}$ quarks.
Note that the $t$--mass is supposed to describe also the energy scale
of the weak interaction, fixing at the same time the $W$-- $Z$--masses
and the mass of the scalar boson. The strength of the unified coupling
constant can be related to the scale parameter $\Lambda $ of QCD.

Gravity does not have a place in the Standard Model. The gravitational
interaction is characterized by a critical energy scale, the Planck--mass:
$\Lambda _p = 1.221047 \times 10^{19}$ GeV. The interplay between the
gravitational interaction and the Standard Model gauge interactions can
be described by dimensionless ratios like:
\begin{equation}
\Lambda / \Lambda_p  = 0.17 \times 10^{-19} \, .
\end{equation}
Such ratios are not fixed by the considerations made above. They are
candidates for a time variation on a cosmological time scale.
Furthermore the unified coupling constant might also depend on the
time, implying that the finestructure constant $\alpha $ becomes a
function of time.

Recently one has found indications that the finestructure constant
was perhaps smaller in the past. Studying the finestructure of various
lines in distant gas clouds, one found\cite{Webb}:
$\delta \alpha / \alpha = (-0.72 \pm 0.18) \cdot 10^{-5}$.
It remains to be seen whether this effect holds up in future
observations. If the finestructure constant $\alpha $ undergoes a
cosmological shift, one should expect similar shifts also to affect
the strong interaction coupling constant, in other words the
$\Lambda $--scale, which in turn would affect the magnitude of the
proton mass. Furthermore the neutron--proton--mass difference, which
has an electromagnetic contribution, would change. This would be
important for the nucleosynthesis of the light elements. Thus far a
systematic study of all effects of a time variation
of $\alpha $ has not been carried out.

If one takes the idea of a Grand Unification of the gauge forces seriously,
a time shift of $\alpha $ would make sense only if the unified coupling
constant undergoes a time shift as well. But this would imply, as recently
pointed out\cite{Cal} that the QCD scale $\Lambda $ would also change in
time. As a result the nucleon mass and all nuclear mass scales would
be time--dependent. Grand unification implies that the relative change
of the nucleon mass is about two orders of magnitude larger than the
relative change of 
$\alpha $. Using advanced methods of quantum optics, a time variation
of $\alpha $ and of the nuclear mass scale could be observed by
monitoring the atomic finestructure and molecular rotational or vibrational
frequencies\cite{Cal}.

In general we can expect that the problem of the fundamental constants
of the Standard Model will remain in the focus of research in particle
physics at least for the ten years. The success of any new direction in
theoretical research should be measured in terms of its power to make
predictions about the fundamental constants or about relations
among them. 

%\cite{HW} 
%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{77}
\bibitem{Pes}See e. g.:
C. Quigg, Gauge Theories of the Strong, Weak and Electromagnetic
Interactions (Advanced Book Classics) (Reprint Series);
M. Peskin and U. Schroeder, Elementary Particle
Physics: Concepts and Phenomena (Texts and Monographs in Physics);
O. Nachtmann, A. Lahee and W. Wetzel, Elementary Particle Physics, Berlin
Heidelberg (1990)
\bibitem{Gell} M. Gell--Mann, private communication
\bibitem{Dir} P. Dirac, Nature {\bf 192} (1987) 235
\bibitem{Gro} For a discussion see: D. Gross, Phys. Today, Vol. 42, Nr. 12
(1989)
\bibitem{Bet} S. Bethke, J. Phys. {\bf G26} R27 (2000)
\bibitem{Lue} M. Luescher, Phys. Bl. {\bf 56} (2000) 65
\bibitem{Fri} H. Fritzsch and Z. Xing, Prog. Part. and
Nucl. Phys. {\bf 45} (2000) 1--81 
\bibitem{Webb} J.K. Webb et al., Phys. Rev. Lett. {\bf 87}, 091301 (2000)
\bibitem{Cal} X. Calmet und H. Fritzsch, CERN--TH / 2001--33

\end{thebibliography}
\end{document}

