%Paper: 
%From: knetter@physf.uni-bielefeld.de
%Date: Thu, 13 Aug 1992 14:15:10 EDT

%
%
% This preprint is written in TeXsis.  If you cannot work with TeXsis
% we can send you a hardcopy of the paper
%
% The figures are not included. We can send them as PostScript files
% by E-Mail (2.5 MB) or as a hardcopy by normal Mail or Telefax
% if you order them by E-mail.
%
%
\def\lsim{\raise0.3ex\hbox{$<$\kern-0.75em\raise-1.1ex\hbox{$\sim$}}}
\def\gsim{\raise0.3ex\hbox{$>$\kern-0.75em\raise-1.1ex\hbox{$\sim$}}}
\font\ninerm=cmr9
\paper
\titlepage
\title
Two- and Three Vector Boson Production in ${\bf e^+e^-}$ Collisions
within the BESS Model
\endtitle
\author
G. Cveti\v c
Institut f\"ur Physik
Universit\"at Dortmund
W-4600 Dortmund 50
\endauthor
\author
C. Grosse-Knetter
\endauthor
\and
\author
R. K\"ogerler
Fakult\"at f\"ur Physik
Universit\"at Bielefeld
W-4800 Bielefeld 1
Germany
\endauthor
\abstract
The BESS model is the higgsless alternative to the standard model of
electroweak interaction with nonlinear realized spontaneous symmetry
breaking. Since it is non-renormalizable new couplings (not existing
in the SM) are induced at each loop order. On the basis of the one
loop induced gauge boson self-couplings we calculated the cross
sections for the two and three gauge boson production processes in
$e^+e^-$ collisions. Measurements of these cross sections in a planned
$e^+e^-$ linear collider at $\sqrt s = 500 GeV$ (NLC) will supply a
good empirical test of the gauge boson self interactions and thus
should enable to discriminate between SM and its alternatives (BESS).
\endabstract
\endtitlepage
\vfill\eject
\section{Introduction}
A thorough experimental
investigation of the gauge boson self-interactions is of
utmost importance for identification of the ``true'' (gauge-) theory
of electroweak interactions. The most powerful instrument for such
an analysis is certainly provided by production processes of (two
and three) gauge bosons in electron-positron annihilation at
sufficiently high energy. For the conclusiveness
of such experiments the available energy plays an important role,
since
most alternatives to the SM of electroweak interactions
are characterized by deviations from the Yang
Mills type self-couplings. Hence they lead to deviations from SM
predictions which in general increase with increasing energy.
Therefore, although LEP II is at the horizon and will certainly yield
interesting results, the efforts for establishing an
$e^+e^-$-collider at energies far beyond the $W^+W^-$ threshold (NLC)
[1] go into the right directions.
It is our conviction, and we will give indications within the present
paper, that new physics can be tested with sufficient reliability if
the energy of such a machine lies beyond $350 GeV$.\par
There are several possible ways for testing gauge boson self
interactions in boson production. One possibility starts from the
most general interaction Lagrangian (both for three and four boson
vertices) being expressed in terms of unspecified coupling constants.
By investigating sufficiently many physical quantities (cross sections
with different polarization configurations, asymmetries, density
matrices etc.) it should be possible in principle to determine those
coupling strengths numerically.
Due to the complexity of the general Lagrangians,
however, and because of possible conspiracies between different
terms, such a model-independent analysis will hardly be feasible
in practice [2]. Consequently, a realistic analysis has to be footed on
specific models, i.e. attainable experimental results are to
be interpreted within given models. In this way it should be possible
to discriminate between various theoretical possibilities for the
vector boson self-interactions and, in particular, decide how well
the SM of electroweak interaction is verified by experimental data.\par
Within the present and a forthcoming paper we are applying the latter
procedure for investigating vector boson self couplings within the
so-called BESS model [3,4]. This is considered to be one of the most
attractive alternatives to the SM, since it does not represent a
trivial extension of the latter one (by simply adding further gauge
groups together with the appropriate Higgs fields), but is footed
on a different mechanism for gauge boson mass generation
which completely avoids physical scalar (Higgs) particles. In a
more general sense it represents the most economical way of
parametrizing the effects of a strongly interacting sector of
(longitudinally polarized) gauge bosons.\par
Since any Higgs-less field theory of massive vector bosons is
non-renormaliz\-able and has to be considered as an effective
theory, it is of utmost importance to clarify in detail the
quantum effects (emerging from loop-generated interactions)
contributing to the different reactions.
A thorough analysis of these quantum induced interactions has been
completed recently [4,5] and the results will be converted into
predictions for various physical (boson production) processes within
the present paper. First results of this analysis have already
been published
[6]. Here, we present the full wealth of BESS-model
predictions (to one loop order) for those reactions which will be
feasible with $e^+e^-$-colliders working in an energy range of
above 350 GeV. A forthcoming paper [7] will be devoted to utilizing
the present results for a careful identification of the allowed
(BESS-model-) parameter range as restricted by the accuracy
which can be reached in such experiments.\par
In Chapter 2 we will motivate the BESS model and roughly sketch its
main features. Chapter 3 is devoted to a general discussion of its
phenomenological structure, mainly with respect to experiments at
high energy $e^+e^-$-colliders. Chapter 4 contains
the presentation and
discussion of our results. In Chapter 5 we draw some final conclusions.
\section{The BESS-Model}
We refrain from presenting the details of the model, since it has
been described already several times [3,4], and will only
sketch the theoretical motivations
and describe its main features concerning phenomenology.\par
The model can be motivated in several ways. Probably the most
fundamental and simple one starts from the sheer existence of massive
vector bosons $(W^\pm, Z)$ and continues with the following line of
reasoning: It is a basic fact of field theory that massive vector
bosons (with Yang-Mills-type self coupling) can -- by a suitable
field-enlarging point transformation -- be embedded in a theory with
local gauge symmetry [8]. The standard version of this
transformation is the St\"uckelberg formalism [9] based on
(unconstrained) vector and (unphysical) scalar fields. The local
gauge symmetry cannot, however, be completely Wigner-realized (in
the St\"uckelberg picture, e.g., the scalars transform \`a la
Goldstone-Nambu), i.e. this symmetry is necessarily spontaneously
broken. The canonical version of this
spontaneous symmetry breaking (SSB) is the Higgs mechanism
as applied in the Weinberg-Salam
model and in all its trivially extended versions.
If one prefers to avoid physical scalar (Higgs) particles -- as we
do here -- the only way is by assuming that the
symmetry (breaking) is realized nonlinearly (NL) (i.e. the scalar
components of the spin-1-field operator transforms nonlinearly under
local gauge transformations and, consequently, can be completely
gauged away). When this mechanism is formally applied to the
$SU(2) \times U(1)$ symmetry of SM
one obtains the gauged version of the
nonlinear $\sigma$-model [10]. Now it can be shown on the same field
theoretical footing as described before (field-enlarging point
transformation) that this theory is gauge-equivalent to theories
with additional (``hidden'') local symmetry groups [11]. Again,
these symmetries can become formally apparent when appropriate
numbers of unphysical scalar (would be Goldstone) fields are
introduced into the Lagrangian. In general, the gauge bosons
connected to the additional local gauge groups could, in principle,
be interpreted as purely auxiliary fields (combinations of
unphysical scalars) with no direct physical consequences -- in
accordance with the fact that they are ``produced'' by point
transformations. However, if the starting theory is a nonlinear
one there are strong indications [11] that these (a priori
hidden) gauge bosons will show up as physical particles. What has
been ``shown'' in fact [12], is the quantum-generation of kinetic terms
for these vector bosons which enables them to propagate as real
physical particles.\par
In the case of a starting (nonlinearly realized) $SU(2) \times U(1)$
gauge symmetry, the additional (``hidden'') gauge groups are bound
to be of SU(2)-type [13]. In the most simple case (which should
correctly describe physics at moderately low energies (up to 3 TeV,
as we will see later)) we thus have as a local gauge group
$SU(2)_L \times U(1)_Y \times SU(2)_V$ with the corresponding
gauge bosons $\vec {\tilde W}_\mu, \tilde Y_\mu, \vec {\tilde V}_\mu$
but with only 6 unphysical scalars (denoted by $\vec \pi$ and
$\vec \sigma$) and no physical Higgs bosons. The corresponding
(tree-level-) Lagrangian defines the BESS model [3,4].
There are five fundamental parameters: $g, g^\prime,
g^{\prime\prime}$ (the gauge coupling constants connected with the
three fundamental gauge groups $SU(2)_L, U(1)_Y, SU(2)_V$,
respectively), $f^2$ (the overall scale parameter measuring the
size of SSB) and $\lambda^2$ (the relative strength of the additional
``hidden'' symmetry).
An important part of the BESS model Lagrangian are the couplings of
(unphysical) scalars to the vector bosons. They provide both mass-
and mixing-terms of bosons. Consequently, the physical
(mass-eigenstate) vector-boson-fields (denoted by $W^\pm, Z, A, V^\pm,
V^0$) are mixtures of the original (unmixed) fields $\vec {\tilde W},
\tilde Y, \vec {\tilde V}$, the connection being given by two
mixing matrices (for charged and neutral particles separately):
$$
{g \tilde W^\pm \choose {{g^{\prime\prime}} \over 2}\tilde V^\pm} =
{\cal C} {W^\pm
\choose  V^\pm}~~~~~~~{\rm and}~~~~~~~
\left( \matrix{ g^\prime \tilde Y \cr
g \tilde W_3 \cr
{{g^{\prime\prime}} \over 2} \tilde V_3\cr}\right) =
{\cal N}
\left( \matrix{ A \cr
Z \cr
V^0\cr}\right)\eqno(2.1 a,b)
$$
with
$$
{\cal C} = \left( \matrix{ g \cos \varphi &g \sin \varphi \cr
- {{g^{\prime\prime}} \over 2}
\sin \varphi &{{g^{\prime\prime}}\over 2} \cos
\varphi\cr}\right)\eqno(2.2)
$$
and
$$
{\cal N} = {{gg^\prime} \over G} \left( \matrix{
\cos \psi &\sin \xi \sin \psi - {{g^\prime} \over g} \cos \xi
&- \cos \xi \sin \psi - {{g^\prime} \over g} \sin \xi \cr
\cos \psi &\sin \xi \sin \psi + {g \over {g^\prime}} \cos \xi
&- \cos \xi \sin \psi + {g \over {g^\prime}} \sin \xi \cr
\cos \psi &- \sin \xi \sin \psi \cot^2 \psi &\cos \xi \sin \psi
\cot^2 \psi \cr}\right)~~.\eqno(2.3)
$$
The masses and mixing angles can be expressed in terms of the
model's parameters in the following compact way:\par
First define three quantities $p_i, c_i, d_i$ (for both the charged
$(i=1)$ and neutral $(i=2)$ sector) by
$$
\eqalignno{
p_1 &= 2 \lambda gg^{\prime\prime}~~~~~~~~~~~~~p_2 = 2 \Lambda
GG^{\prime\prime} &(2.4a,b)\cr
c_1 &= g^2 (1 + \lambda^2)~~~~~~~~
c_2 = G^2 (1 + \Lambda^2) &(2.5a,b)\cr
d_1 &= g^{\prime\prime 2} \lambda^2~~~~~~~~~~~~~
d_2 = G^{\prime\prime 2}
\Lambda^2 &(2.6a,b)\cr}
$$
(with $\Lambda = \lambda {{g^2 - g^{\prime 2}} \over
{g^2 + g^{\prime 2}}},~~~G^2 = g^2 + g^{\prime 2},~~~G^{\prime\prime
2} = {{\lambda^2} \over {\Lambda^2}} (g^{\prime\prime 2} + 4
{{g^2 g^{\prime 2}} \over {G^2}})$) and
$$
e_i = c_i + d_i~~,~~~~~x_i = \sqrt{1- ({{p_i} \over {e_i}})^2}~~~~~
(i = 1,2).\eqno(2.7)
$$
Then the masses are given by
$$
\eqalignno{
M^2_{W^\pm} = {{f^2} \over 8} \cdot e_1 (1-x_1)~~~~~~~
&M^2_Z = {{f^2} \over 8} e_2 (1-x_2) &(2.8a,b)\cr
M^2_{V^\pm} = {{f^2} \over 8} \cdot e_1 (1+x_1)~~~~~~~
&M^2_{V^0} = {{f^2} \over 8} e_2 (1 + x_2) &(2.9a,b)\cr}
$$
and the mixing angles by
$$
\eqalignno{
tg 2 \varphi = \lambda {{p_1} \over {c_1 - d_1}}~~~~~~~
&tg 2 \xi = \Lambda {{p_2} \over {c_2 - d_2}} &(2.10a,b)\cr
{}~ &tg \psi = {{2gg^\prime} \over G} \cdot {1 \over {g^{\prime\prime}}}~.
&(2.10c)\cr}
$$
Note that the masses of the heavy bosons can be expressed in terms
of the corresponding light boson masses by means of the master
formula
$$
M_{V_i} = M_{W_i} {{1 + x_i} \over {1 - x_i}}~~~~~i = 1 ({\rm charged})~
{\rm or}~2 ({\rm neutral})~.\eqno(2.11)
$$
It allows an easy understanding of how the theory behaves as
$M_{V_i} \to \infty$. This limit is reached when $x_i \to 1$ or,
equivalently ${{p_i} \over {e_i}} \to 0$. The latter condition can
be realized in three alternative ways:
\item{(i)} $p_i \to 0$, i.e. $\lambda \to 0$ ($g, g^\prime$ finite
$\not= 0$, $g^{\prime\prime}$ finite or $0$)\footnote{*}
{\ninerm \baselineskip 11 pt Note that
the electromagnetic coupling constant
${\scriptstyle e = \sqrt{4 \pi \alpha} =
{{gg^\prime} \over G} \cos \psi}$, therefore neither ${\scriptstyle g}$
nor
${\scriptstyle g^\prime}$ nor
${\scriptstyle g^{\prime\prime}}$ can be equal to zero.}\hfill\break
In that case
$\varphi \to 0, \xi \to 0$; $\psi$ remains finite, but $V^{\pm 0}$
decouple from fermionic interactions.
\item{(ii)} $e_i \to \infty$ implied by
$g^{\prime\prime} \to \infty ~(\lambda^2 \not= 0)$\hfill\break
Here all mixing
angles vanish and we obtain the SM ($V's$ decouple completely).
\item{(iii)} $e_i \to \infty$ implied by $\lambda^2 \to \infty$
($g^{\prime\prime}$ finite)\hfill\break
In this case all mixing angles are nonzero, i.e. the existence of
the (infinitely heavy) $V's$ manifest
themselves in the fermionic couplings. Furthermore,
as will be seen, the induced interactions increase with increasing
values of $M_V$. So, in this limit, the $V$-bosons do not decouple from
the low energy physics.\par
The effect of non-decoupling can be easily understood also by
remembering that for sufficiently heavy vector mass particles
$V^{\pm, 0}$ their masses are approximately given by
$$
M_{V^0} \simeq M_{V^\pm} \simeq M_W ({{g^{\prime\prime}} \over g})
\lambda, \eqno(2.12)
$$
i.e. the V-masses are driven by the (dimensionless) coupling constant
$\lambda$. This non-decoupling nature of V will be of importance in
particular for quantum-induced interactions as will be shown in the
following.\par
By phenomenological reasons (low energy
experiments are well reproduced by $SU(2)_L\break \times U(1)_Y$ alone!)
the additional vector boson $V^\pm, V^0$ have to be heavier than
350 GeV or so, which implies effectively that $g^{\prime\prime}/g$
has to be larger than 10.\par
The couplings of fermions to vector bosons are specified by their
transformation property under the fundamental symmetry group, in
particular by the fact that they are singlets under the ``hidden''
group $SU(2)_V$. Thus fermions are coupled -- in the tree level
Lagrangian -- only to $\vec {\tilde W}_\mu$ and $\tilde Y_\mu$,
i.e. they interact with (physical) $V^{\pm, 0}$ only via mixing.
Similarly, the tree level structure of vector boson self interactions
is determined by the fact that the unmixed bosons $\vec {\tilde V}$
couple only among themselves (in Yang-Mills-manner) and thus
interactions between light and heavy (physical) bosons are mediated
by mixing only.\par
A particularly momentuous feature of the BESS model, which is
directly connected to its NL nature and to the consequent absence
of physical Higgs particles, is its lack of formal renormalizability,
which implies that the theory has to be understood as an effective
one, with finite validity range defined by a cut-off $\Lambda$.
As a consequence, cut-off dependent terms arise at 1- or
higher-loop level. They can be divided into two families: One group
consists of terms which can be fully absorbed into the starting
(tree-level) Lagrangian by renormalizing its field and/or coupling
constants. The remaining ones (which cannot be absorbed by
renormalization) constitute new observable interactions, which have
to be taken into account necessarily [10] if the BESS model is taken
seriously. These contributions have been calculated to 1-loop order
(and consistently separated from the renormalization terms) in a
series of papers [4,5].
For doing these 1-loop calculations, the tree level
Lagrangian has first to be
completed by gauge-fixing and ghost terms in the
standard way. It was of particular convenience to perform the
calculations in the so-called Landau gauge\footnote{*}
{\ninerm \baselineskip 11 pt The Landau
gauge is distinguished by the fact that ghost-couplings with
scalars ${\scriptstyle (\vec \pi, \vec \sigma)}$ vanish.
Hence, ghost loops
do not contribute to induced interactions (they are completely
absorbed in renormalization). As a consequence, the induced
interaction Lagrangians are fully gauge invariant and not only
BRS-invariant (as they would be in other gauges).}.
It has turned out
thereby, that these expressions can be represented in terms of fully
gauge-invariant Lagrangians as one would expect.
The resulting expressions -- together with the corresponding coupling
strengths -- are presented in ref. [4]
(for fermion interactions) and ref. [5] (for bosonic
self interactions), respectively. Note that -- due to our
parametrization of the NL realization -- the scalar fields
$\pi$ and $\sigma$ emerge
always in the exponents. Consequently, the individual interaction
terms are obtained by expanding in powers of $\vec \pi$ and $\vec
\sigma$ and of the (unmixed) gauge bosons $\vec {\tilde{W^\mu}},
{\tilde Y}^\mu, \vec {\tilde{V^\mu}}$. These terms
are, of course, not individually symmetric -- only their sum is.\par
The main features of the resulting induced interactions can be
described as follows:
\item{-} The strengths of all 1-loop induced couplings are
logarithmically dependent of the cut-off $\Lambda$ (which we
sometimes have identified with a heavy-Higgs-mass $M_H$). Note that
higher loop contributions (though involving higher powers of $\Lambda$)
will not be dominant as long as $\Lambda \lsim 3 TeV$  [10].
\item{-} Quantum corrections to fermionic couplings of vector bosons
[4] are -- although existing -- suppressed by a factor of $({{m_f}
\over {M_W}})^2$, and thus are negligible for light fermions (in
particular for electrons). Therefore, we can safely forget them for
our present purposes.
\item{-} There is no similar suppression for the vector boson self
interactions. In fact, the strengths of these additional interactions
are proportional to polynomials in $\lambda^2$ (of third power in
$\lambda^2$ for cubic self interactions, of fourth power in $\lambda^2$
for quartic ones) and therefore increase with increasing V-mass.
This is the aforementioned
manifestation of the non-decoupling nature of $M_V$
when $\lambda \to \infty$.
As to the specific structure of these interactions, an
interesting difference between cubic and quartic terms emerges
(cf. Tables 5 and 7 of ref. [5]): the cubic self-interactions have
the pure Yang-Mills structure\footnote{*}
{\ninerm \baselineskip 11 pt Apart from the terms
proportional to
${\scriptstyle \partial_\mu G^\mu \scriptstyle (G = \vec
{\tilde W}, \tilde Y,
\vec
{\tilde V})}$.
They do not contribute to physical processes, since we have
to use consistently the Landau gauge, which yields transversal vector
propagators.}, whereas the quartic ones do not. This exceptional role
of the cubic interaction can be traced back to the fact that
nonrenormalizability is inferred to the BESS model via the
NL $\sigma$-model [14].\par
In ref. [5] all vector boson self-interaction Lagrangians
are expressed in terms of unphysical (unmixed) vector fields
$\vec {\tilde W}, \tilde Y, \vec {\tilde V}$. For calculating
physical processes we need the corresponding expressions for the
physical fields $W^\pm, Z, A, V^\pm, V^0$, which can be obtained by
appropriately applying the mixing matrices. The resulting expressions
are quite lengthy. We summarize them in a fairly compact form (for
all interesting vertices) in App. A and B. Similarly, the induced
couplings of physical vector bosons to the (unphysical) scalar
fields $\vec \pi$ and $\vec \sigma$ have been calculated, since
they will be used in computing some amplitudes for
three-boson-production processes (see Fig. 1b), but we won't quote
them here explicitly.
\vfill\eject
\section{Phenomenology}
Present $e^+e^-$-colliders like LEP I  allow only direct tests
of the couplings
between gauge bosons and fermions.
But, as we have shown, the nonrenormalizable
structure of the BESS model shows up most drastically
in the self-couplings of the gauge
bosons due to the new induced couplings.
Future $e^+e^+$-colliders with energies
above the $W^+W^-$ threshold (161 GeV)
will allow direct tests
of these self-couplings.
The first machine
to make the W-pair production process $e^+e^- \to W^+W^-$
possible
will be LEP II ($\sqrt s \sim 190 GeV$),
but due to its very
limited energy range, deviations from the standard model will hardly be
observable [15].
A planned $e^+e^-$ (linear)
collider at $\sqrt s=500 GeV$ (NLC)
will allow much more precise measurements of
the $e^+e^- \to W^+W^-$ cross section and, in particular,
a much better discrimination
of the BESS model
because of the expected higher integrated luminosity
of $20 fb^{-1}$ per year
and because the CM energy of $500 GeV$ is much larger
than the threshold of this reaction,
so that at this energy the violation of the
gauge cancellations due to the induced couplings in the BESS model
yields much higher deviations from the
standard model than at LEP II energies.
Furthermore,
 three gauge boson production processes like
$e^+e^- \to W^+W^-Z$ and $e^+e^- \to W^+W^-\gamma$ [16,17],
which supply a direct test of the quartic vector boson self-couplings
and the non-Yang--Mills structure of these
in the BESS model, will be measurable.
The $e^+e^- \to W^+W^-Z$ threshold is at $250 GeV$ and the
high luminosity will enable even the
measurement of very small cross sections
in the order of $50fb$ as they are
expected for these processes.\par
Future hadron colliders and $\gamma\gamma$-collisions
at NLC will as well
supply tests of vector boson
self-interactions and of the induced couplings in
the BESS model, but these are not considered in the present analysis.
\par
In this paper we present the cross sections for the
two and three gauge boson
production processes at NLC energy of $\sqrt s=500 GeV$.
In addition, we give an outlook
to what happens at an energy of $\sqrt s=2000 GeV$,
which may be interesting for
machines of future generations.
\par
Specifically, we have calculated the following observables:
\item{-} Total cross sections for the
reactions
$e^+e^- \to W^+W^-, e^+e^- \to W^+W^-Z$ and $e^+e^-\to W^+W^-\gamma$
both for polarized and
nonpolarized gauge bosons.
\item{-} The following partial cross sections (distributions):
\itemitem{$\bullet$} $d\sigma/d \cos \theta_G$ for two and
three gauge boson production $(G = W, Z, \gamma)$,
\itemitem{$\bullet$} $d\sigma/dE_G$ for three
gauge boson production,
\itemitem{$\bullet$} $d\sigma/dP_{G,T}$
for three gauge boson production,
\itemitem{$\bullet$} $d\sigma/dy_G$  for three gauge boson production.
\item{-}For the reaction $e^+e^- \to ZZZ$
we have calculated only the total cross section,
since distributions will
presumably not be measurable because of the small size of the cross
sections.\par
To identify the gauge boson production events in experiment, one has
to reconstruct the W and Z bosons from their decay products, while
the photons can be identified directly. As Frank, M\"attig and
Settles [18] have stated, the most significant $e^+e^- \to W^+W^-$
events are those, where one W decays into leptons and the other
one into hadrons. An analysis of the angular distribution of the
decay products of the W and Z bosons yields the polarisation of these
bosons, so that the cross section for the production of polarized
gauge bosons can be measured, too.\par
Figures 1a and b show schematically the tree level Feynman diagrams
which contribute to the two and three gauge boson production processes.
(Since we performed our calculations in the Landau gauge, there is
one diagram with an exchange of an unphysical would-be Goldstone
boson.)
We formally calculated the cross sections at tree level, but for
each gauge
boson self-interaction vertex we took into account
the 1-loop induced coupling.
All cross sections acquire
contributions from cubic self-couplings, whereas in the three
gauge boson production processes even quartic self-couplings
get involved due
to the last diagram. In the case of $e^+e^- \to ZZZ$ there
is a diagram with a coupling
of four neutral gauge bosons which does not exist for
pure Yang-Mills type
interactions, but exists in the BESS model as a consequence
of the violation of the Yang-Mills structure.
\par
To calculate the cross section for the W pair production we have
proceded completely analytically
using the usual trace techniques. For the calculation of the three
gauge boson production cross sections we had to
procede numerically.
We calculated the amplitudes using
helicity techniques [16,19] and integrated
numerically over the phase space.
In agreement with [16] we imposed the following transverse-momentum and
pseudorapidity cuts on the photon produced in
$e^+e^- \to W^+W^-\gamma$:
$$
P_{T,\gamma}>20 GeV ,~~~~\vert \eta_\gamma \vert < 2 .\eqno(3.1)
$$
In order to obtain numerical values for the
cross sections in the BESS model, we had
to specify the free parameters of the model, i.e.
$g$,$g'$ and $g''$\footnote{*}
{\ninerm \baselineskip 11 pt
Bear in mind, that ${\scriptstyle g}$ and ${\scriptstyle g'}$
are neither
physically nor numerically identical
with the standard model ${\scriptstyle g}$ and ${\scriptstyle g'}$,
since ${\scriptstyle g\cos\varphi}$ and not ${\scriptstyle g}$ describes
the coupling of the ${\scriptstyle W}$ bosons to fermions.},
$f^2$, $\lambda^2$ and the cut-off $\Lambda$. $\Lambda$ was set to
\footnote{$\dagger$}
{\ninerm \baselineskip 11 pt A different choice of
${\scriptstyle \Lambda}$, e.g.
${\scriptstyle \Lambda=2 TeV}$ would change the
values of the deviations from the SM by only some per cent.}
$$
\Lambda = 5 TeV \eqno(3.2)
$$
The other five parameters can be determined from
chosen values of $\alpha_{\rm em}$, $M_W$, $M_Z$, $M_{V_0}$ and $g''/g$.
The first three values are empirically given (the electromagnetic
fine structure
constant $\alpha_{\rm em}$ was taken as 1/127,
which is the value of the running coupling
constant at the $e^+e^- \to W^+W^-Z$
threshold of 250 GeV\footnote{$\ddagger$}{\ninerm \singlespaced
The slight
numerical differences between our predictions and the results of
[16,17]
might be traced back to a slightly
different choice of the coupling parameters.
Different choices for ${\scriptstyle \alpha_{\rm em}}$ would result
approximately only in multiplying in SM and BESS cross sections
by a common factor close to 1, an effect
which would not appreciably change
the deviations of BESS cross sections from those of the SM.\vfill}.)
while the free parameter $g''/g$ was set to
$$
g''/g = 10,\eqno(3.3)
$$
a value suggested by the apparent success of SM in fermionic
processes.
For the unknown mass $M_{V^0}$ we chose the following reference values:
$$
M_{V^0}=400 GeV,~~1000 GeV,~~2000 GeV,
{}~~2500 GeV,\eqno(3.4)
$$
which indicate the presumable range within which $M_{V^0}$
is expected to lie.
Remember that $\lambda^2$ grows with $M_{V^0}$, so a heavy $\rm V^0$
means
strong induced couplings.
\par
As a further input for our calculation
we need the $\rm V^\pm$ and $\rm V^0$
widths. We calculated these widths taking the induced couplings
fully into account. The main
decay channels are two and three gauge bosons and fermion pairs.
This will be discussed in detail in a forthcoming paper [20].
Here we only present
the results for our reference values:\vfill\eject
\MIDTABLE{Table 1}
\Tablebody{5}
& $\lambda^2$ && $M_{V^0} (GeV)$ && $\Gamma_{V^0} (GeV)$ &&
$M_{V^\pm} (GeV)$ && $\Gamma_{V^\pm} (GeV)$ &\CRCR
& 0.241 &&~400&& ~~~~~~~~0.829&&~399.5&& ~~~~~~~~0.748 &\Cr
& 1.505 &&1000&& ~~~~~~31.17&& ~999.0&& ~~~~~~38.54 &\Cr
& 6.012 &&2000&&~~2300&&     1997.3&& ~~1138 &\Cr
& 9.406 &&2500&&31197&&     2496.7&& 23093&\Cr
\endTablebody
\Caption
Widths of the $V$ bosons
\endCaption
\ENDTABLE\medskip
Note that for both $\rm V^\pm$ and $V^0$, the width is dominated by the
two-vector channels and increases strongly with $M_V$,
such that masses of
$M_V \gsim 2200 GeV$ are unlikely.
In this respect
the reference mass
$M_{V^0}= 2500 GeV$ has to be considered as an extreme case.
\par
For comparison we calculated the cross sections for the standard
model as well. Thereby
we neglected all Higgs boson effects, which makes sense if the Higgs
boson is lighter than $2M_W$ or heavier than
$\sqrt s - M_Z$ because then the Higgs boson
yields only a negligible contribution. Else, the
Higgs boson would show up as a
resonance in the $W^+W^-$
channel and can be identified by a Jacobian Peak
 in the energy spectrum of the produced $Z$ bosons.
So Higgs boson effects can
easily be distinguished from
effects of the induced couplings in the BESS model.
\section{Discussion of Numerical Results}
The results for the different observables (cross sections,
distributions, asymmetries) are plotted in Figures 2-21. In all
figures the solid lines represent the results for the higgsless
standard model and the different types of broken lines the
predictions of the BESS model for different choices of $M_{V^0}$, i.e.
for different strengths of the induced couplings.

\subsection{${\bf W^+W^-}$-Production}
In Figs. 2-5 we have depicted the resulting values of the total
cross section for $e^+ e^- \to W^+W^-$ respectively by the BESS
model
(together with the SM-predictions)
for different polarization states of the final vector bosons
(unpolarized, transversally (TT), longitudinally (LL) and mixed
polarized (LT + TL) $W$'s). We plot the cross sections for two
energy intervals:
$$
{\rm (a)} \sqrt s = 150-550 GeV \eqno(4.1)
$$
(which covers the energy region to be reached by the planned NLC)
and
$$
{\rm (b)} \sqrt s = 0 - 2500 GeV \eqno(4.2)
$$
(which roughly represents the total energy region where the BESS
model is reasonably assumed to work).\par
We have used the $M_{V^0}$-values as specified in ch. 3 ($M_{V^0} = 400,
1000, 2000, 2500 GeV$) for the wide energy range (4.2), whereas,
for clarity of presentation, only three $M_{V^0}$-values
($M_{V^0} = 400, 1000, 2500 GeV$) were used for the narrow (NLC-)
energy range a).\par
Differential cross sections $d \sigma/d \cos \theta$
at $\sqrt s = 500 GeV$, for non-polarized, (LT- and TL-polarized
and LL-polarized $W$'s are depicted in Figs. 6-8 for the angular
range $-1 \le \cos \theta \le +1$ (a). In each case, we isolated
in addition the forward direction ($0.9 \le \cos \theta \le 1$)
(b) since, in general, the cross sections are particularly large in
this region and, furthermore, the deviations from the SM show a
substantial variation there. For comparison, we also quoted
$d\sigma /d \cos \theta$ at $\sqrt s = 2000 GeV$ (Fig. 9) for
unpolarized vector bosons and for the same two angular regions.
The related forward-backward asymmetries $A_{FB}$ and centre-edge
asymmetries $A_{CE}$ for unpolarized $W$'s are depicted in Figs. 10
and 11, again for the same two energy ranges as for the total
cross sections.\par
Let us now comment on all these results, in particular on the
differences between SM and BESS model predictions.\par
The deviations from SM stem from three effects:
\item{a)} exchange of heavy boson $V^0$ (including the resulting
resonance effects),
\item{b)} mixing between light and heavy bosons,
\item{c)} deviations of the $Z^0 W^+ W^-$ and $\gamma W^+W^-$
couplings strengths from those of SM due to induced interactions
which result in violation of gauge cancellation between $t$ and
$s$ channel.\par
For $M_{V^0} \lsim 1000 GeV$, the $V^0$-resonance is narrow and
pronounced. For heavy $V^0~\break (M_{V^0} \gsim 2000 GeV)$, the
$V^0$-resonance peak becomes very broad, in fact invisible, but
the induced couplings are then large and yield large deviations from
SM. For instance, at $\sqrt s = 500 GeV$, the relative deviations of
the total cross section (for non-polarized $W$'s) are 3 \%, 5\%, 8\%,
17 \% for $M_{V^0} = 400, 1000, 2000, 2500 GeV$, respectively. Such
deviations should therefore be observable with a $500~GeV~e^+e^-$
collider reaching a luminosity of $20~fb^{-1}$ per year where an
experimental error smaller than 3 \% should be possible (the
statistical error for 90 \% confidence level being very small
$(\sim 0.8 \%)$ for a one year's collection of data\footnote{*}
{\ninerm \baselineskip 11 pt The reduction factor 0.3
was taken into account for the really
reconstructable events, as suggested by Frank, M\"attig and
Settles [18].}. These relative deviations are much more pronounced
for L-T polarized $W$'s (5,7 \%, 11.3 \%, 34 \%, 104 \%, respectively;
statistical error is about 3.5 \%) and especially for L-L
polarized $W$'s (-43 \%, 44 \%, 164 \%, 560 \%,
respectively; statistical
error is about 5.5 \%). On the other hand, for T-T polarized  $W$'s, the
deviations are practically independent of $M_{V^0}$ (about 4 \%;
statistical error being roughly equal to that of the case of
nonpolarized W's).\par
{}From Figs. 10 and 11 we see that the deviations of $A_{FB}$ and
$A_{CE}$ at $\sqrt s = 500 GeV$ are small ($\sim 1 - 2 \%$ for
$M_{V^0} < 1000 GeV$) but will increase drastically with higher
energies.\par
Figs. 6a,b show that the relative deviations of $d \sigma/d \cos
\theta$ from SM values at $\sqrt s = 500 GeV$
for non-polarized $W$'s are substantial for negative values of
$\cos \theta$ (e.g. at $\cos \theta = - 0.5$ they are - 14 \%, + 18 \%,
+ 200 \% for $M_{V^0} = 400, 1000, 2500 GeV$). However, the absolute
values of the cross sections are very small at such angles and the
statistical errors are therefore large ($\sim 14.5 \%$ for
$\Delta (\cos \theta) \approx 0.1$). On the other hand, the
deviations in the forward region $(\cos \theta = 0.9 - 0.99)$ are
approximately 4 \% - 6 \% for any $M_{V^0}$. The statistical error
here is small (about 1 \% for $\Delta (\cos \theta) = 0.09$). Hence,
it appears that it may be more promising to measure $\Delta
(d \sigma/d \cos \theta)_{NP}$ in the
forward directions than in other directions. The relative
deviations are substantially larger for LT + TL channel ($\sim
30 - 45 \%$ for $\cos \theta = -0.5$ and $\sim 1.5 \%$ for $\cos
\theta = 0.9,~M_{V^0} = 400, 1000 GeV$). However, the corresponding
statistical errors under the mentioned conditions are also large
($\sim 45 \%$ and $10 \%$, respectively) due to small absolute values.
These features (large deviations from SM but large small total
cross-sections and large statistical errors) are even more pronounced
in the case of LL polarization. Hence it appears that differential
cross sections for polarized W's are not particularly useful
quantities for discriminating modes, due to large statistical errors.
\par
As seen from Figs. 9 a and b, $(d \sigma/ d (\cos \theta))$
values are drastically  decreased at high energies $(\sqrt s =
2000 GeV)$.
\subsection{Three Gauge Boson Production}
Figs. 12-14 show the total cross sections for the reactions
$e^+e^- \to W^+W^-Z,~e^+e^- \to W^+W^-\gamma$ and $e^+e^- \to ZZZ$
(all vector bosons unpolarized) as functions of the total energy
$\sqrt s$, where $\sqrt s$ varies again in the two ranges (4.1)
and (4.2). Let us first discuss the larger region (4.2) which shows
the global behaviour of the cross sections. In addition to the
resonance peaks at $\sqrt s = M_{V^0}$ (due to the exchange of a
$V^0$ boson in the $s$ channel\footnote{*}
{\ninerm \baselineskip 11 pt In the reaction ${\scriptstyle e^+e^-
\to ZZZ}$ there are no visible resonances, since if the
${\scriptstyle V}$ bosons are
light the induced couplings of four neutral gauge bosons which are
responsible for these resonances are too small and if the
${\scriptstyle V}$ bosons
are heavy the resonances are too wide.}) which also occur in two body
productions, there are $V^0$ resonances in the $W^+W^-$ sub-channel
and $V^\pm$ resonances in the $W^\pm Z$ sub-channel (see Fig. 16)
(for $e^+e^- \to W^+W^-Z$ and similar for $e^+e^- \to W^+W^- \gamma$,
but not for $e^+e^- \to ZZZ$, because there are no diagrams with
trilinear couplings). This means that e.g. the direct $e^+e^- \to
W^+W^-Z$ reaction becomes superimposed by the reactions $e^+e^- \to
V^0Z$ with consequent decay $V^0 \to W^+W^-$ and by the reaction
$e^+e^- \to V^\pm W^\mp$ with the decay $V^\pm \to W^\pm Z$. So above
the threshold of these reactions, i.e. at slightly larger energies
than the $V^0$ resonance, the cross section shows again a maximum. When
$M_{V^0}$ gets larger, all $V$-resonances broaden (see table 1) and
finally dissapear when the $V$-boson width is in the order of or
greater than the $V$ boson mass.\par
An important effect of the induced couplings, as in the case of
$W$ pair production, consists in destroying the gauge cancellations,
i.e. the parts of the amplitudes for the different graphs which grow
with energy do not cancel completely anymore as they would do in
a renormalizable gauge theory. This leads to deviations of the BESS
model cross sections from the standard model ones, which grow both
with energy and with $M_{V^0}$, (since $M_{V^0}$
is proportional to the
strength of the induced couplings). Note that this effect is more
pronounced for triple boson production as compared to $W^+W^-$
production since the induced quartic couplings go with a higher power
of $\lambda^2$.\par
At NLC energies (4.1) the deviations from the SM are not so drastic,
because the non cancelled parts of the amplitudes, which grow with the
energy, are still not so big. Except for the case of a very light
$V$-boson, which causes resonances at low energies, there are only
deviations in the per cent region. (In case of $e^+e^- \to W^+W^-Z$
from 8 \% for medium $M_{V^0}$ up to 20 \% for heavy $M_{V^0}$.)
However, these deviations are large enough to be measurable. The
$e^+e^- \to W^+W^-Z$ cross section is about 50 fb. With an expected
NLC-luminosity of $20 fb^{-1} a^{-1}$ there are 1000 annual events.
Following the analysis of Barger, Han and Phillips [16], 20 \% of
them, that means 200 events per year, will be reconstructable, which
means that the statistical error (for 90 \% confidence level) can
be suppressed to $\sim 5 \%$ after five years of run. The systematical
error is expected to be 2 \% . Thus it should in principle be
possible to distinguish the BESS model with the given parameters from
the standard model empirically\footnote{*}
{\ninerm \baselineskip 11 pt It should be mentioned
that most of the deviations from the standard model for medium
${\scriptstyle M_{V^0}}$ at NLC
energies are tree level effects and not caused by
induced couplings. So an empirical verification of the BESS model
does not neccesarily imply a verification of the induced couplings.}.
The same is true for the reaction $e^+e^- \to W^+W^-\gamma$. On the
other hand, the cross section for the reaction $e^+e^- \to ZZZ$ is
only 1fb, which means the statistical error is probably too large
to get precise results, in reasonable time, although this reaction
would be of largest importance because of its singular nature.\par
Figure 15 shows the cross sections for the production of polarized
gauge bosons in the reaction $e^+e^- \to W^+W^-Z$. The differences
of the BESS model to the standard model are small if no or only one
longitudinally polarized gauge boson is produced and they are large
if two or three longitudinally polarized gauge bosons are produced.
This is because the amplitudes of the single Feynman graphs grow with
higher powers of $\sqrt s$ the more longitudinal bosons are in the
final state, so the effect of non-cancellation of the leading powers
of $\sqrt s$ is especially strong if mainly longitudinal gauge bosons
are produced. Unfortunately, in those cases the total cross sections
are very small and so the statistics are very bad, while if
transversal gauge bosons are produced, the statistics are better
because of the larger cross sections but the deviations are small.\par
Figures 16 to 21 show different partial cross sections at the NLC
energy of 500 GeV. Except for resonance effects (in case of a light
$V^0$) like Jacobian peaks\footnote{$\dagger$}
{\ninerm \baselineskip 11 pt There are only very small
effects of a ${\scriptstyle V^\pm \to W^\pm \gamma}$
resonance in the ${\scriptstyle e^+e^-
\to W^+W^- \gamma}$ process since the responsible coupling vanishes
on tree level and the induced coupling is very small.} the
deviation of the BESS model from the SM results are distributed
regularly over the angular, energy etc. region. Thus, a measurement
of only those processes where one of the produced gauge bosons is
emitted in a certain part of its phase space would not improve
the expected deviation from the standard model, but it would make
statistics worse because there are less events.
\section{Conclusions}
The discussion of the last chapter has shown that the specific
structures of the BESS model (existence of heavier vector bosons,
mixing between heavy and light ones, new induced couplings) will
become effective in boson production by $e^+e^-$-collisions at
energies of about 500 GeV with sufficient magnitude, such that an
identification of these effects (and, consequently discrimination
of BESS and SM) would be possible with the help of the planned New
Linear Collider (NLC). The most promising observables in this respect
are total cross sections for production of unpolarized and
longitudinally polarized gauge bosons. Further valuable information
can also be obtained by measuring asymmetries and partial cross
sections although the results will be less conclusive due to limited
statistics.\par
In general, quantities connected with two boson production will be
measurable to much greater accuracy and yield more distinctive
bounds. But results for three boson production processes will be
of particular theoretical interest because they are determined
partially by the four-boson self-interactions which are much more
sensitive to the specific model than the three-vector vertices.\par
If no deviations from the SM will be found in future measurements
of the above-mentioned process the results will nevertheless allow
to restrict the parameter ranges of the BESS model parameters due to
finite experimental accuracy. A careful investigation of the
corresponding expectations will be published in a forthcoming paper
[7].
\vfill\eject
\noindent{\bf Appendix A}\medskip
\centerline{\underbar{Trilinear gauge boson self-interactions}}
\medskip
The 1-loop induced interaction Lagrangians containing three
unphysical (unmixed) vector boson fields\footnote{*}
{\ninerm see footnote
on page 7} are written down in Table 5 of ref. [5]. Here we
consistently can forget about the terms proportional to $\partial_\mu
G^\mu~~~(G = \vec W, Y, \vec V)$ since we work in the Landau gauge.
Decomposing the remaining expressions into charge-eigenstates,
applying the mixing matrices ${\cal C}$ (for charged vector bosons)
and ${\cal N}$ (for neutral ones) (cf. 2.2 and 2.3) and adding the
corresponding tree level interactions we get the total Lagrangian
for the cubic self interactions of physical vector bosons (tree level
and 1-loop induced ones). It can be written in a compact form by
using the notation
\widenspacing
$$
{W^\pm \choose V^\pm} \equiv  {c_1^\pm
\choose c_2^\pm}~~~~~~~~~~~~~~
\left( \matrix{
A \cr
Z \cr
V^0\cr} \right) \equiv \left( \matrix{
n_1 \cr
n_2 \cr
n_3 \cr}\right)\eqno(A.1a,b)
$$
One obtains
$$
\eqalign{
{\cal L}_{\rm 3 gauge~bosons} &= i \sum^2_{a,b =1} \sum^2_{i=0}
K_{a b, i} \cdot \{ (c_a^+)^\mu (c_b^+)^\nu (n_i)_{\mu\nu} +\cr
&+ [(c_a^-)_{\mu \nu} (c_b^+)^\mu - (c_b^+)_{\mu\nu} (c_a^-)^\mu]
(n_i)^\nu\}}\eqno(A.2)
$$
where
$$
\eqalign{
K_{a b, i} &= 2 {\cal C}_{1a} {\cal C}_{1b} [\alpha_1 {\cal N}_{oi}
+ \alpha_6 {\cal N}_{1i} + \alpha_2 {\cal N}_{2i}] + \cr
&+ 2 {\cal C}_{2a} {\cal C}_{2b} [\alpha_3 {\cal N}_{oi} + \alpha_4
{\cal N}_{1i} + \alpha_7 {\cal N}_{2i}] + \cr
&+ ({\cal C}_{1a} {\cal C}_{2b} + {\cal C}_{2a} {\cal C}_{1b})
[2 \alpha_2 {\cal N}_{1i} + 2 \alpha_4 {\cal N}_{2i} + \alpha_5
{\cal N}_{oi}]}\eqno(A.3)
$$
and
\vfill\eject
$$
\eqalign{
\alpha_1 &= - t {1 \over 4} (\lambda^2 - 1) (\lambda^2 - 3)^2\cr
\alpha_2 &= t {1\over 2} (\lambda^6 - 3 \lambda^4 + 5 \lambda^2 + 1)\cr
\alpha_3 &= - t (\lambda^6 - 2 \lambda^4 - 1)\cr
\alpha_4 &= \alpha_3\cr
\alpha_5 &= t (\lambda^6 - 5 \lambda^4 + 3 \lambda^2 + 1)\cr
\alpha_6 &= - t {1 \over 4} (\lambda^6 + \lambda^4 - 5 \lambda^2 + 11)
+ {1 \over {2 g^2}}\cr
\alpha_7 &= t \cdot 2 (\lambda^6 - 2 \lambda^4 - 1) + {2 \over
{g^{\prime\prime 2}}}}\eqno(A.4)
$$
$$
(t \equiv {1 \over {48 (4 \pi)^2}}~~\ln~~{\Lambda \over {M_W}})
$$
\vfill\eject
\noindent{\bf Appendix B}\medskip
\centerline{\underbar{Quadrilinear gauge boson self-interactions}}
\medskip
The 1-loop induced interaction Lagrangians connecting four (unmixed)
vector boson fields can be found in Table 7 of ref. [5]. They can be
inverted into expressions for quartic interactions of the physical
bosons by appropriately using the mixing matrices ${\cal C}$ and
${\cal N}$ (cf. (2.2) and (2.3)). Here, we refrain from quoting the
full Lagrangian but we list only quartic (physical) vector boson
self-couplings (tree level + 1-loop induced ones) which contribute to
the processes $e^+e^- \to W^+W^-Z,~~e^+e^- \to W^+W^-\gamma$ and
$e^+e^- \to ZZZ$.
$$
\eqalignno{
{\cal L}_{W^\pm W^\pm \gamma\gamma} &= \sigma_1 (A^\mu A_\mu)
(W^{+\nu}W^-_\nu) &~\cr
&+ 2 \sigma_2 (A^\mu A^\nu) (W^+_\mu W^-_\nu) &(B.1a)\cr
{\cal L}_{W^\pm W^\pm \gamma Z} &= \sigma_3 (A^\mu Z_\mu)
(W^{+\nu}W^-_\nu) &~\cr
&+ \sigma_4 (A^\mu Z^\nu) (W^+_\mu W^-_\nu + W^+_\nu W^-_\mu) &(B.1b)\cr
{\cal L}_{W^\pm W^\pm \gamma V^0} &= \sigma_5 (A^\mu V^0_\mu)
(W^{+\nu}W^-_\nu) &~\cr
&+ \sigma_6 (A^\mu V^{0 \nu})
(W^+_\mu W^-_\nu + W^+_\nu W^-_\mu) &(B.1c)\cr
{\cal L}_{W^\pm W^\pm ZZ} &= \sigma_7 (Z^\mu Z_\mu)
(W^{+\nu}W^-_\nu) &~\cr
&+ 2 \sigma_8 (Z^\mu Z_\mu) (W^+_\mu W^-_\nu) &(B.1d)\cr
{\cal L}_{W^\pm W^\pm ZV^0} &= \sigma_9 (Z^\mu V^0_\mu)
(W^{+\nu}W^-_\nu) &~\cr
&+ \sigma_{10} (Z^\mu V^{0 \nu})
(W^+_\mu W^-_\nu + W^+_\nu W^-_\mu)&(B.1e)\cr
{\cal L}_{ZZZZ} &= \sigma_{11} (Z^\mu Z_\mu) (Z^\nu Z_\nu)&(B.1f)\cr
{\cal L}_{ZZZV^0} &= \sigma_{12} (V^{0\mu} Z_\mu) (Z^\nu Z_\nu)
&(B.1g)\cr}
$$
It turns out that all couplings of four neutral gauge bosons where
at least one of these is a photon are zero. Note also that there are
(non-vanishing) interaction terms involving photon fields which
individually are not invariant under $U(1)_{em}$. However, since
they are obtained from fully invariant expressions (by expansion in
power of fields), electromagnetic gauge invariance is established
if the appropriate terms (including in general also cubic boson
interaction terms) are added.\par
The coupling constants are given by:
$$
\eqalignno{
\sigma_1 = &2
\{ \lbrack
\beta_2 {\cal N}_{00} {\cal N}_{10} +
\beta_4 {\cal N}_{10} {\cal N}_{20} &~\cr
&+ \beta_{11} {\cal N}^2_{00} + \beta_{15} {\cal N}_{20}^2 &~\cr
&+ \beta_{24} {\cal N}_{00} {\cal N}_{20} + 2 \beta_{40}
{\cal N}_{10}^2
\rbrack
{\cal C}^2_{11} &~\cr
&+
\lbrack
\beta_4 {\cal N}^2_{10} + \beta_8 {\cal N}^2_{20} &~\cr
&+ 2 \beta_{18} {\cal N}_{10} {\cal N}_{20} + \beta_{26} {\cal N}_{00}
{\cal N}_{10} &~\cr
&+ \beta_{29} {\cal N}_{00}^2 + \beta_{35} {\cal N}_{00}
{\cal N}_{20}
\rbrack
{\cal C}_{11} {\cal C}_{21} &~\cr
&+
\lbrack
\beta_8 {\cal N}_{10} {\cal N}_{20} + \beta_{10} {\cal N}_{00}
{\cal N}_{20} &~\cr
&+ \beta_{15} {\cal N}^2_{10} + \beta_{20} {\cal N}^2_{00} &~\cr
&+ \beta_{33} {\cal N}_{00} {\cal N}_{10} + 2 \beta_{42}
{\cal N}^2_{20}
\rbrack
{\cal C}^2_{21}
\}
&(B.2a)\cr
\sigma_2 = &
\lbrack
\beta_1 {\cal N}_{00} {\cal N}_{10} +
\beta_3 {\cal N}_{10} {\cal N}_{20} &~\cr
&+ \beta_{12} {\cal N}^2_{00} + \beta_{16} {\cal N}_{20}^2 &~\cr
&+ \beta_{25} {\cal N}_{00} {\cal N}_{20} + 2 \beta_{39}
{\cal N}_{10}^2
\rbrack
{\cal C}^2_{11} &~\cr
&+
\lbrack
\beta_3 {\cal N}^2_{10} + \beta_7 {\cal N}^2_{20} &~\cr
&+ 2 (\beta_{17} + \beta_{19}) {\cal N}_{10}
{\cal N}_{20} + (\beta_{27} + \beta_{28})  {\cal N}_{00}
{\cal N}_{10} &~\cr
&+ \beta_{30} {\cal N}_{00}^2 + (\beta_{36} + \beta_{37})
{\cal N}_{00} {\cal N}_{20}
\rbrack
{\cal C}_{11} {\cal C}_{21} &~\cr
&+
\lbrack
\beta_7 {\cal N}_{10} {\cal N}_{20} + \beta_9 {\cal N}_{00}
{\cal N}_{20} &~\cr
&+ \beta_{16} {\cal N}^2_{10} + \beta_{21} {\cal N}^2_{00} &~\cr
&+ \beta_{34} {\cal N}_{00} {\cal N}_{10} + 2 \beta_{41}
{\cal N}^2_{20}
\rbrack
{\cal C}^2_{21}  &(B.2b)\cr
\sigma_3 = &2
\{ \lbrack
\beta_2 ({\cal N}_{01} {\cal N}_{10} +
{\cal N}_{00} {\cal N}_{11}) +
\beta_4 ({\cal N}_{11} {\cal N}_{20} + {\cal N}_{10} {\cal N}_{21})&~\cr
&+ 2 \beta_{11} {\cal N}_{00} {\cal N}_{01} +
2 \beta_{15} {\cal N}_{20} {\cal N}_{21} &~\cr
&+ \beta_{24} ({\cal N}_{01} {\cal N}_{20} + {\cal N}_{00}
{\cal N}_{21}) + 4 \beta_{40}
{\cal N}_{10} {\cal N}_{11}
\rbrack
{\cal C}^2_{11} &~\cr
&+
\lbrack
2 \beta_4 {\cal N}_{10}{\cal N}_{11} + 2 \beta_8
{\cal N}_{20}{\cal N}_{21} &~\cr
&+ 2 \beta_{18} ({\cal N}_{11}
{\cal N}_{20} + {\cal N}_{10} {\cal N}_{21}) +
\beta_{26} ({\cal N}_{01} {\cal N}_{10} + {\cal N}_{00}
{\cal N}_{11}) &~\cr
&+ \beta_{29} {\cal N}_{00} {\cal N}_{01} + \beta_{35} ({\cal N}_{01}
{\cal N}_{20} + {\cal N}_{00} {\cal N}_{21})
\rbrack
{\cal C}_{11} {\cal C}_{21} &~\cr
&+
\lbrack
\beta_8 ({\cal N}_{11} {\cal N}_{20} + {\cal N}_{10}
{\cal N}_{21})
\beta_{10} ({\cal N}_{01} {\cal N}_{20} + {\cal N}_{00}
{\cal N}_{21}) &~\cr
&+ 2 \beta_{15} {\cal N}_{10} {\cal N}_{11} +
2 \beta_{20} {\cal N}_{00} {\cal N}_{01} &~\cr
&+ \beta_{33} ({\cal N}_{01} {\cal N}_{10} + {\cal N}_{00}
{\cal N}_{11}) +
4 \beta_{42}
{\cal N}_{20} {\cal N}_{21}
\rbrack
{\cal C}^2_{21}
\}
&(B.2c)\cr
\sigma_4 = &
\lbrack
\beta_1 ({\cal N}_{01} {\cal N}_{10} +
{\cal N}_{00} {\cal N}_{11}) +
\beta_3 ({\cal N}_{11} {\cal N}_{20} + {\cal N}_{10} {\cal N}_{21})&~\cr
&+ 2 \beta_{12} {\cal N}_{00} {\cal N}_{01} +
2 \beta_{16} {\cal N}_{20} {\cal N}_{21} &~\cr
&+ \beta_{25} ({\cal N}_{01} {\cal N}_{20} + {\cal N}_{00}
{\cal N}_{21}) + 4 \beta_{39}
{\cal N}_{10} {\cal N}_{11}
\rbrack
{\cal C}^2_{11} &~\cr
&+
\lbrack
2 \beta_3 {\cal N}_{10}{\cal N}_{11} + 2 \beta_7
{\cal N}_{20}{\cal N}_{21} &~\cr
&+ 2 (\beta_{17} + \beta_{19}) ({\cal N}_{11}
{\cal N}_{20} + {\cal N}_{10} {\cal N}_{21}) +
(\beta_{27} + \beta_{28}) ({\cal N}_{01}
{\cal N}_{10} + {\cal N}_{00}
{\cal N}_{11}) &~\cr
&+ 2 \beta_{30} {\cal N}_{00} {\cal N}_{01} +
(\beta_{36} + \beta_{37}) ({\cal N}_{01}
{\cal N}_{20} + {\cal N}_{00} {\cal N}_{21})
\rbrack
{\cal C}_{11} {\cal C}_{21} &~\cr
&+
\lbrack
\beta_7 ({\cal N}_{11} {\cal N}_{20} + {\cal N}_{10}
{\cal N}_{21}) +
\beta_{9} ({\cal N}_{01} {\cal N}_{20} + {\cal N}_{00}
{\cal N}_{21}) &~\cr
&+ 2 \beta_{16} {\cal N}_{10} {\cal N}_{11} +
2 \beta_{21} {\cal N}_{00} {\cal N}_{01} &~\cr
&+ \beta_{34} ({\cal N}_{01} {\cal N}_{10} + {\cal N}_{00}
{\cal N}_{11}) +
4 \beta_{41}
{\cal N}_{20} {\cal N}_{21}
\rbrack
{\cal C}^2_{21}  &(B.2d)\cr
\sigma_{11} = &~~(\beta_1 + \beta_2) {\cal N}_{01} {\cal N}_{11}^3 &~\cr
&+ (\beta_3 + \beta_4)  {\cal N}_{21} {\cal N}^3_{11} &~\cr
&+ \beta_5 {\cal N}_{11} {\cal N}^3_{01} &~\cr
&+ \beta_6 {\cal N}_{21} {\cal N}^3_{01} &~\cr
&+ (\beta_7 + \beta_8) {\cal N}_{11}
{\cal N}^3_{21} &~\cr
&+ (\beta_9 + \beta_{10}) {\cal N}_{01} {\cal N}^3_{21} &~\cr
&+ (\beta_{11} + \beta_{12} + \beta_{13} + \beta_{14})
{\cal N}^2_{01} {\cal N}^2_{11} &~\cr
&+ (\beta_{15} + \beta_{16} + \beta_{17} + \beta_{18} + \beta_{19})
{\cal N}^2_{11} {\cal N}^2_{21} &~\cr
&+ (\beta_{20} + \beta_{21} + \beta_{22} + \beta_{23})
{\cal N}^2_{01} {\cal N}^2_{21} &~\cr
&+ (\beta_{24} + \beta_{25} + \beta_{26} + \beta_{27} + \beta_{28})
{\cal N}_{01} {\cal N}_{21} {\cal N}^2_{11} &~\cr
&+ (\beta_{29} + \beta_{30} + \beta_{31} + \beta_{32})
{\cal N}_{11} {\cal N}_{21} {\cal N}^2_{01} &~\cr
&+ (\beta_{33} + \beta_{34} + \beta_{35} + \beta_{36} + \beta_{37})
{\cal N}_{01} {\cal N}_{11} {\cal N}^2_{21} &~\cr
&+ \beta_{38} {\cal N}^4_{01} &~\cr
&+ (\beta_{39} + \beta_{40}) {\cal N}^4_{11} &~\cr
&+ (\beta_{41} + \beta_{42}) {\cal N}^4_{21} &(B.2e)\cr
\sigma_{12} = &~~(\beta_1 + \beta_2)
({\cal N}_{02} {\cal N}_{11}^3 + 3 {\cal N}_{01} {\cal N}_{12}
{\cal N}^2_{11}) &~\cr
&+ (\beta_3 + \beta_4) ({\cal N}_{22} {\cal N}^3_{11}
+ 3 {\cal N}_{21} {\cal N}_{12} {\cal N}_{11})
&~\cr
&+ \beta_5 ({\cal N}_{12} {\cal N}^3_{01} + 3 {\cal N}_{11}
{\cal N}_{02} {\cal N}^2_{01})
&~\cr
&+ \beta_6 ({\cal N}_{22} {\cal N}^3_{01}  + 3 {\cal N}_{21}
{\cal N}_{02} {\cal N}^2_{01})
&~\cr
&+ (\beta_7 + \beta_8) ({\cal N}_{12}
{\cal N}^3_{21} + 3 {\cal N}_{11} {\cal N}_{22} {\cal N}^2_{21})
&~\cr
&+ (\beta_9 + \beta_{10}) ({\cal N}_{02} {\cal N}^3_{21} +
3 {\cal N}_{21} {\cal N}_{22} {\cal N}^2_{21}) &~\cr
&+ (\beta_{11} + \beta_{12} + \beta_{13} + \beta_{14})
({\cal N}_{02} {\cal N}_{01} {\cal N}^2_{11} + {\cal N}_{12}
{\cal N}_{11} {\cal N}^2_{01}) &~\cr
&+ 2 (\beta_{15} + \beta_{16} + \beta_{17} + \beta_{18} + \beta_{19})
({\cal N}_{12} {\cal N}_{11} {\cal N}^2_{21} + {\cal N}_{22}
{\cal N}_{21} {\cal N}^2_{11})&~\cr
&+ 2 (\beta_{20} + \beta_{21} + \beta_{22} + \beta_{23})
({\cal N}_{02} {\cal N}_{01} {\cal N}^2_{21} + {\cal N}_{22}
{\cal N}_{21} {\cal N}^2_{01})&~\cr
&+ (\beta_{24} + \beta_{25} + \beta_{26} + \beta_{27} + \beta_{28})
({\cal N}_{02} {\cal N}_{21} {\cal N}^2_{11} + {\cal N}_{01}
{\cal N}_{22} {\cal N}^2_{11} + 2 {\cal N}_{01} {\cal N}_{21}
{\cal N}_{12} {\cal N}_{11}) &~\cr
&+ (\beta_{29} + \beta_{30} + \beta_{31} + \beta_{32})
({\cal N}_{12} {\cal N}_{21} {\cal N}^2_{01} + {\cal N}_{11}
{\cal N}_{22} {\cal N}^2_{01} + 2 {\cal N}_{11} {\cal N}_{21}
{\cal N}_{02} {\cal N}_{01})&~\cr
&+ (\beta_{33} + \beta_{34} + \beta_{35} + \beta_{36} + \beta_{37})
({\cal N}_{02} {\cal N}_{11} {\cal N}^2_{21} + {\cal N}_{01}
{\cal N}_{12} {\cal N}^2_{21} + 2 {\cal N}_{01} {\cal N}_{11}
{\cal N}_{22} {\cal N}_{21})&~\cr
&+ 4 \beta_{38} {\cal N}_{02} {\cal N}^3_{01} &~\cr
&+ 4 (\beta_{39} + \beta_{40}) {\cal N}_{12} {\cal N}^3_{11} &~\cr
&+ 4 (\beta_{41} + \beta_{42}) {\cal N}_{22} {\cal N}^3_{21}.&(B.2f)}
$$
$\sigma_7$ can be obtained from the formula for $\sigma_1$ and
$\sigma_8$ from the formula for $\sigma_2$ by the substitution
${\cal N}_{i0} \to {\cal N}_{i1}$. $\sigma_5$ and $\sigma_6$ are
constructed from $\sigma_3$ and $\sigma_4$ respectively replacing
${\cal N}_{i1} \to {\cal N}_{i2}$, to find $\sigma_9$ and $\sigma_{10}$
one has to replace ${\cal N}_{i0} \to {\cal N}_{i2}$ in $\sigma_3$
and $\sigma_4$.\par
The ${\cal C}_{ij}$ and ${\cal N}_{ij}$ are again the elements of the
mixing matrix (2.2) and (2.3) and the $\beta_i$ are the couplings
of the unmixed gauge bosons.\vfill\eject
$$
\eqalign{
\beta_1 &= {1 \over 2} t (\lambda^8 - 2 \lambda^6 - 2 \lambda^4 +
10 \lambda^2 - 7)\cr
\beta_2 &= {1 \over 4} t (\lambda^8 + 4 \lambda^6 - 26 \lambda^4 +
52 \lambda^2 - 31)\cr
\beta_3 &= - {1 \over 2} t (2 \lambda^8 - 5 \lambda^6 + \lambda^4 +
9 \lambda^2 + 1)\cr
\beta_4 &= - {1 \over 2} t (\lambda^8 - \lambda^6 - 7 \lambda^4 +
21 \lambda^2 + 2)\cr
\beta_5 &= {3 \over 4} t (\lambda^8  - 10 \lambda^4 +
24 \lambda^2 - 15)\cr
\beta_6 &= - {3 \over 2} t (\lambda^8  - 2 \lambda^6 - 2 \lambda^4 +
10 \lambda^2 + 1)\cr
\beta_7 &= - 2 t (2 \lambda^8 - \lambda^6 + 1)\cr
\beta_8 &= - 2 t (\lambda^8 + \lambda^6 + 2)\cr
\beta_9 &= \beta_7\cr
\beta_{10} &= \beta_8\cr
\beta_{11} &= {1 \over 8} t (\lambda^8 - 8 \lambda^6 + 30 \lambda^4 -
40 \lambda^2 + 17)\cr
\beta_{12} &= {1 \over 4} t (\lambda^8 + 4 \lambda^6 + 6 \lambda^4 -
28 \lambda^2 + 17)\cr
\beta_{13} &= {1 \over 4} t (\lambda^8 - 10 \lambda^6 + 34 \lambda^4 -
42 \lambda^2 + 17)\cr
\beta_{14} &= {1 \over 2} t (\lambda^8 + 8 \lambda^6 - 2 \lambda^4 -
24 \lambda^2 + 17)\cr
\beta_{15} &= {1 \over 2} t (\lambda^8 - 3 \lambda^6 + 4 \lambda^4 +
\lambda^2 + 1)\cr
\beta_{16} &= t (\lambda^8 +  \lambda^4 +
7 \lambda^2 + 1)\cr
\beta_{17} &= {1 \over 2} t (2 \lambda^8 - 7 \lambda^6 + 10 \lambda^4 +
\lambda^2 + 2)\cr
\beta_{18} &= {1 \over 2} t (2 \lambda^8 + 5 \lambda^6 - 14 \lambda^4 +
13 \lambda^2 + 2)\cr
\beta_{19} &= {1 \over 2} t (2 \lambda^8 - \lambda^6 - 8 \lambda^4 +
\lambda^2 + 2)\cr
\beta_{20} &= \beta_{15}\cr
\beta_{21} &= \beta_{16}\cr
\beta_{22} &= \beta_{17}\cr
\beta_{23} &= \beta_{18}\cr}
$$
\vfill\eject
$$
\eqalign{
\beta_{24} &= - {1 \over 2} t (\lambda^8 - 3 \lambda^6 + 9 \lambda^4 -
9 \lambda^2 + 2)\cr
\beta_{25} &= - {1 \over 2} t (2 \lambda^8 + 3 \lambda^6 + 3 \lambda^4 -
9 \lambda^2 + 1)\cr
\beta_{26} &= - t (\lambda^8 + 8 \lambda^6 - 13 \lambda^4 +
2 \lambda^2 + 2)\cr
\beta_{27} &= - {1 \over 2} t (2 \lambda^8 + \lambda^6 - 5 \lambda^4 +
\lambda^2 + 1)\cr
\beta_{28} &= - {1 \over 2} t (2 \lambda^8 - 11 \lambda^6 + 25 \lambda^4
- 17 \lambda^2 + 1)\cr
\beta_{29} &= \beta_{24}\cr
\beta_{30} &= \beta_{25}\cr
\beta_{31} &= - {1 \over 2} t (4 \lambda^8 + 17 \lambda^6 - 31 \lambda^4
+ 5 \lambda^2 + 5)\cr
\beta_{32} &= \beta_{28}\cr
\beta_{33} &= t (\lambda^8 + 7 \lambda^6 - 8 \lambda^4
- \lambda^2 + 1)\cr
\beta_{34} &= 2 t (\lambda^8 - 2 \lambda^6 +  \lambda^4
- 7 \lambda^2 + 1)\cr
\beta_{35} &= t (2 \lambda^8 + 3 \lambda^6 +  6 \lambda^4
- 13 \lambda^2 + 2)\cr
\beta_{36} &= t (2 \lambda^8 - 3 \lambda^6 + 12 \lambda^4
- \lambda^2 + 2)\cr
\beta_{37} &= t (2 \lambda^8 + 3 \lambda^6 - 6 \lambda^4
- \lambda^2 + 2)\cr
\beta_{38} &= {3 \over {16}} t (\lambda^8 - 4 \lambda^6 + 6 \lambda^4
- 4 \lambda^2 + 17)\cr
\beta_{39} &= {1 \over 8} t (\lambda^8 - 4 \lambda^6 + 2 \lambda^4
+ 4 \lambda^2 - 3) + {1 \over {4g^2}}\cr
\beta_{40} &= {1 \over {16}} t (\lambda^8 - 4 \lambda^6 + 14 \lambda^4
- 20 \lambda^2 + 57) - {1 \over {4g^2}}\cr
\beta_{41} &= 2 t (\lambda^8 - \lambda^4)
+ {1 \over {g^{\prime\prime 2}}}\cr
\beta_{42} &=  t (\lambda^8 + 2 \lambda^4 + 3)
- {1 \over {g^{\prime\prime 2}}}\cr}\eqno(B.3)
$$
$$
(t = {1 \over {48}} {1 \over {(4 \pi)^2}}~\ln {\Lambda \over {M_W}})
$$
\vfill\eject
\doublespaced
\noindent{\bf References}\medskip
\item{[1]} See, f.i., ``$e^+e^-$ Collisions at 500 GeV, the Physics
Potential'', ed. P. Zerwas, DESY, Hamburg 1992, to be published;
Proceedings of the ``Workshop on Physics and Experiments with Linear
Colliders'', Saariselk\"a, Finnland, 1991, to be published.
\item{[2]} For cubic selfinteractions a fairly model dependent
analysis based on symmetry considerations has been performed in\hfill
\break
G. Gounaris, J.L. Kneur, J. Layssac, G. Moultaka, F.M. Renard,
D. Schildknecht (Bielefeld-Montpellier-Thessaloniki-Collaboration),
Bielefeld preprint BI-TP 91/40\break (1991)
\item{[3]} R. Casalbuoni, S. de Curtis, D. Dominici and R. Gatto,
Nucl. Phys. \underbar{B282} (1987) 235; Phys. Lett. \underbar{B155}
(1985) 95
\item{[4]} G. Cveti\v c and R. K\"ogerler, Nucl. Phys. \underbar{B328}
(1989) 342; Z. Phys. \underbar{C48} (1990) 109
\item{[5]} G. Cveti\v c and R. K\"ogerler, Nucl. Phys. \underbar{B363}
(1991) 401
\item{[6]} G. Cveti\v c, C. Grosse-Knetter and R. K\"ogerler,
Bielefeld preprint BI-TP 91/37 (1991), to be published in ref. [1]
``$e^+e^-$ Collisions at 500 GeV, the Physics Potential'', ed.
P. Zerwas, Hamburg
\item{[7]} R. B\"onisch, C. Grosse-Knetter and R. K\"ogerler,
to be published
\item{[8]} A. Hosoya and K. Kikkawa, Nucl. Phys. \underbar{B101}
(1975) 271;\hfill\break
J. Alfaro and P.H. Damgaard, Ann. Phys. (N.Y.) \underbar{202} (1990)
398
\item{[9]} T. Kunimasa and T. Goto, Progr. Theor. Phys.
\underbar{37} (1967) 452;\hfill\break
T. Sonoda and S.Y. Tsai, Progr. Theor. Phys. \underbar{71} (1984)
878
\item{[10]} T. Appelquist, C. Bernard, Phys. Rev. \underbar{D22}
(1980) 200; J. van der Bij and M. Veltman, Nucl. Phys. \underbar{B231}
(1984) 205
\item{[11]} M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida,
Phys. Rev. Lett. \underbar{54} (1985) 1215; see also A.P.
Balachandran, A. Stern and G. Trahern, Phys. Rev. \underbar{D19}
(1979) 2416
\item{[12]} for 2-dimensional theories:\hfill\break
V. Golo and A.M. Perelomov, Phys. Lett. \underbar{B79} (1978) 112;
\hfill\break
A. D'Adda, P. Di Vecchia and M. L\"uscher, Nucl. Phys. \underbar{B146}
(1978) 63; Nucl. Phys. \underbar{B152} (1979) 125;\hfill\break
for 3-dimensional theories:\hfill\break
I. Ya. Aref'eva and S.I. Azakov, Nucl. Phys. \underbar{B162} (1980) 298
\hfill\break
for 4-dimensional theories:\hfill\break
R. K\"ogerler, W. Lucha, H. Neufeld and H. Stremnitzer, Phys. Lett.
\underbar{201B} (1988) 335;\hfill\break
T. Kugo, Soryushiron Kenkyu \underbar{71} (1985) E78;\hfill\break
T. Kugo, H. Terao and S. Uehara, Prog. Theor. Phys. Suppl.
\underbar{85} (1985) 122
\item{[13]} R. Casalbuoni, D. Dominici, F. Feruglio and R. Gatto,
Nucl. Phys. \underbar{B310} (1988) 181; Phys. Lett. \underbar{200B}
(1988) 495
\item{[14]} K. Shizuja, Nucl. Phys. \underbar{B121} (1977) 125
\item{[15]} A. de Rujula, M.B. Gavela, P. Hernandez, E. Masso,
CERN preprint CERN-TH 6272/91 (1991);\hfill\break
for a critics see: M. Bilenki; J.L. Kneur, F.M. Renard and
D. Schildknecht, Bielefeld-preprint
BI-TP 92/25
\item{[16]} V. Barger, T. Han and R.J.N. Phillips, Phys. Rev.
\underbar{D39} (1989) 146
\item{[17]} A. Tofighi-Niaki and J.F. Gunion, Phys. Rev.
\underbar{D39} (1989) 720
\item{[18]} M. Frank, P. M\"attig, R. Settles, W. Zeuner, to be
published in ref. [1] ``$e^+e^-$ Collisions at $500~GeV$, the Physics
Potential'', ed. P. Zerwas, Hamburg
\item{[19]} K. Hagiwara and D. Zeppenfeld, Nucl. Phys. \underbar{B274}
(1986) 1
\item{[20]} For first (incomplete) calculation see: G. Cveti\v c,
R. K\"ogerler and J. Trampeti\'c, Phys. Lett. \underbar{248B} (1990)
128
\vfill\eject
\noindent{\bf Figure Captions}\medskip
\def\litem{\par\noindent
               \hangindent=\parindent\ltextindent}
\def\litemitem{\par\noindent
               \hangindent=2\parindent\ltextindent}
\def\ltextindent#1{\hbox to \hangindent{#1\hss}\ignorespaces}
{\parindent=2.5cm
\litem{Fig. 1}
Feynman diagrams
\hfill\break a) for $e^+e^- \to W^+W^-$
\hfill\break b) for three gauge boson production
\litem{Fig. 2}
Energy dependence of $\sigma_{tot} (e^+e^- \to W^+W^-)$ (production
of unpolarized\break gauge bosons)
in SM and BESS for different values of
$M_{V^0}$\hfill\break
(full line: SM prediction; dotted line: $M_{V^0} = 400 GeV$;
dashed line: $M_{V^0} = 1000 GeV$; chaindotted line:
$M_{V^0} = 2000 GeV$;
chaindashed line: $M_{V^0} = 2500 GeV$)
\hfill\break a) for $\sqrt s = 150 - 550 GeV$
\hfill\break b) for $\sqrt s = 0 - 2500 GeV$
\litem{Fig. 3}
Same as Fig. 2 for $\sigma_{tot} (e^+e^- \to W^+_T W^-_T)$
(production of polarized gauge bosons)
\litem{Fig. 4}
Same as Fig. 2 for $\sigma_{tot}
(e^+e^- \to W^+_L W^-_T + W^+_T W^-_L)$
\litem{Fig. 5}
Same as Fig. 2 for $\sigma_{tot}
(e^+e^- \to W^+_L W^-_L)$
\litem{Fig. 6}
Partial cross section $d \sigma/d \cos \theta (e^+e^- \to W^+W^-)$
as a function of $\cos \theta$ for $\sqrt s = 500 GeV$
\hfill\break a) $- 1 \le \cos \theta \le 1$
\hfill\break b) $0.9 \le \cos \theta \le 1$
\litem{Fig. 7}
Same as Fig. 6 for $d \sigma/d \cos \theta (e^+e^- \to W^+_L W^-_T +
W^+_T W^-_T)$ (polarized gauge bosons)
\litem{Fig. 8}
Same as Fig. 6 for $d \sigma/d \cos \theta (e^+e^- \to W^+_L W^-_L)$
\litem{Fig. 9}
Same as Fig. 6 for $d \sigma/d \cos \theta (e^+e^- \to W^+ W^-)$
at an energy of $\sqrt s = 2000 GeV$
\litem{Fig. 10}
Same as Fig. 2 for $A_{FB} (e^+e^- \to W^+W^-)$
\litem{Fig. 11}
Same as Fig. 2 for $A_{CE} (e^+e^- \to W^+W^-)$
\litem{Fig. 12}
Same as Fig. 2 for $\sigma_{tot} (e^+e^- \to W^+W^-Z)$
\litem{Fig. 13}
Same as Fig. 2 for $\sigma_{tot} (e^+e^- \to W^+W^-\gamma)$
\litem{Fig. 14}
Same as Fig. 2 for $\sigma_{tot} (e^+e^- \to ZZZ)$
\litem{Fig. 15}
Energy dependence of $\sigma_{tot}$ for production of polarized
gauge bosons in\break
$e^+e^- \to W^+W^- Z$ for $\sqrt s = 350 GeV - 250 GeV$
\hfill\break a) $e^+e^- \to W^+_T W^-_T Z^{~}_T$
\hfill\break b) $e^+e^- \to W^+_L W^-_T Z^{~}_T
+ W^+_T W^-_L Z^{~}_T$
\hfill\break c) $e^+e^- \to W^+_L W^-_L Z^{~}_T$
\hfill\break d) $e^+e^- \to W^+_T W^-_T Z^{~}_L$
\hfill\break e) $e^+e^- \to W^+_L W^-_T Z^{~}_L
+ W^+_T W^-_L Z^{~}_L$
\hfill\break f) $e^+e^- \to W^+_L W^-_L Z^{~}_L$
\litem{Fig. 16}
Angular distribution $d \sigma/d \cos \theta_B (e^+e^- \to W^+W^-Z)$
as a function of $\cos \theta_B\break
(B = W^+, Z)$ for $\sqrt s = 500 GeV$
\hfill\break a) distribution of $\cos \theta_Z$
\hfill\break b) distribution of $\cos \theta_W$
\litem{Fig. 17}
Same as Fig. 16 for $d \sigma/d \cos \theta_B
(e^+e^- \to W^+W^-\gamma)$
\litem{Fig. 18}
Same as Fig. 16 for the energy distribution $d \sigma/dE_B
(e^+e^- \to W^+W^- Z)$
\litem{Fig. 19}
Same as Fig. 16 for $d \sigma/dE_B
(e^+e^- \to W^+W^- \gamma)$
\litem{Fig. 20}
Same as Fig. 16 for the transverse momentum distribution $d \sigma/dP_T$
($e^+e^-$\break $\to W^+W^- Z$)
\litem{Fig. 21}
Same as Fig. 16 for the rapidity distribution $d \sigma/dy
(e^+e^- \to W^+W^- Z)$}
\end

