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\begin{document}
\draft
\preprint{This is Sample Compuscript B}
\begin{title}
$WWZ$, $ZZZ$, and $WW\gamma$ production at $e^+e^-$ colliders
\end{title}
\author{V. Barger and T. Han}
\begin{instit}
Physics Department, University of Wisconsin, Madison, Wisconsin 53706
\end{instit}
\author{R. J. N. Phillips}
\begin{instit}
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, England
\end{instit}
\begin{abstract}
We discuss the prospects for the detection of $WWZ$, $ZZZ$, and
$WW\gamma$ events at a future $e^+e^-$ supercollider with c.m.\
energy 0.35--2.0 TeV.  These modes provide tests of triple and quartic
gauge-boson couplings.  $WWZ$ and $ZZZ$ modes also provide tests of a
neutral Higgs boson $H$ if its mass is in the range 200--600 GeV; there
are especially large enhancements in the $ZZZ$ channel.  For an annual
integrated luminosity of 10 fb$^{-1}$ at optimal incident energies
about a thousand events would be produced per annum in each of the
$WWZ$ and $WW\gamma$ channels, plus about ten $ZZZ$ events in the
absence
of Higgs-boson enhancements.  Of order 20\% of the $WWZ$ and $ZZZ$
final states and 30\% of the $WW\gamma$ final states are in principle
fully reconstructible.  We present dynamical distributions with
and without $H$ effects.
\end{abstract}
\pacs{Ms number DV3663. PACS numbers: 13.10.+q, 12.15.Ji,
14.80.Ef, 14.80.Gt}
 
\narrowtext
\section{INTRODUCTION}
\label{sec:intro}
 
There is great interest in the production of two- \cite{brown79,eich84}
and three- \cite{chan84,han88,niaki} gauge-boson final states at
future high-energy colliders, since they will allow crucial tests of the
electroweak gauge theory.  The predicted $WW\gamma$ and $WWZ$ couplings
will be probed by measurements of $e^+e^-\to W^+W^-$ at CERN LEP II, of
$ep\to eWX$ at DESY HERA, and $pp\to V_1V_2X$ $(V=W$ or $Z$) at the
CERN Large Hadron Collider (LHC) or the Superconducting Super
Collider (SSC) (Refs.~\cite{haiw87,grab86,dicus88,abba88}).
Because of intimate cancellations between
the contributions of gauge-boson exchanges in the $s$ channel and
fermion exchanges in the $t$ and $u$ channels, small deviations from the
standard gauge-theory couplings would be amplified and would give
anomalous magnitude and energy dependence to the cross sections.
Amplitude zeros in radiative subprocesses such as $u\bar{d}\to W^+
\gamma$ are important features of gauge theory \cite{mika79}; they arise
from cancellations between different exchange contributions that
would be upset by the presence of anomalous interactions \cite{haiw87}
that occur in composite models.
 
Attention has mainly focused on the production of two gauge bosons.  In
this paper we address the distributions of three gauge bosons in
$e^+e^-$ collisions, where future machines with high energies and
luminosities may make experimental studies quite feasible.  Such
studies would be interesting because (i) triple gauge-boson production
offers further independent tests of the standard theoretical framework,
(ii) quartic $WWZZ$ and $WWZ\gamma$ couplings enter significantly
for the first time, (iii) the standard neutral Higgs boson $H$
plays an important role in the $WWZ$ and $ZZZ$ channels if its mass
is in the range 200--600 GeV, and (iv) triple gauge-boson production
may be a background to possible future signals from physics beyond
the standard model.  Cross sections for triple gauge-boson production
in $e^+e^-$ collisions have already been presented by two of us
\cite{han88}
using helicity-amplitude techniques; subsequently similar results have
been obtained \cite{niaki} with spinor-inner-product techniques.
Some studies
of the channel $e^+e^-\to WW\gamma$ have also been published
\cite{greco88},
with emphasis on situations where $\gamma$ is not observed and the
process
is a background to Higgs-boson production in $e^+e^-\to e^+e^-W^+W^-$
or $\nu\bar{\nu}W^+W^-$.  In this work we concentrate attention on final
states where all three gauge bosons are observed.
 
Our calculations are performed using the helicity-amplitude techniques
of Ref.~\cite{hagi86}.
In Sec.~\ref{sec:hel} we explain the notation and give helicity
amplitude formulas for $e^+e^-\to V_1V_2$ and $e^+e^-\to V_1V_2V_3$
processes, where the $V_i$ are gauge vector bosons.  In
Sec.~\ref{sec:xsecs}
we present integrated cross sections versus energy for both two- and
three-gauge boson production.  Transverse-momentum and pseudorapidity
cuts $p_{\gamma T}>20$ GeV and $|\eta_\gamma |<2$ are imposed on all
final photons, which are expected to be seen as high-$p_T$
electromagnetic jets.  In the $e^+e^-\to WWZ$ and $ZZZ$ channels, we
show the effects of a Higgs boson on the integrated cross sections
and on the diboson invariant-mass distributions, for c.m.\ energies
$\sqrt s>0.5$, 1.0, 2.0 TeV.  We also illustrate dynamical distributions
in the $WWZ$ and $WW\gamma$ channels versus the boson energies,
transverse momenta, rapidities, polar angles, and pair opening angles.
 
In Sec.~\ref{sec:branchfracs}
we discuss the $W$ and $Z$ decay branching fractions into
identifiable final states, the probabilities of overlap between hadronic
decay jets from different gauge bosons, and the resulting effects on
the final identifiable event rates.  We conclude that for c.m.\ energy
$\sqrt s>0.35$ GeV and annual integrated luminosity of order 10
fb$^{-1}$,
a substantial number of identifiable, fully reconstructible events
would be produced per year in both the $e^+e^-$ $\to WWZ$ and $e^+e^-\to
WW\gamma$ channels.
 
\section{HELICITY AMPLITUDES}
\label{sec:hel}
 
\subsection{Formalism}
\label{subsec:forml}
 
It has been realized that helicity amplitudes provide a convenient means
for Feynman-diagram evaluations.  These amplitude-level techniques
are particularly convenient for calculations involving many Feynman
diagrams, where the usual trace techniques for the amplitude
squared becomes unwieldy.  Our calculations use the helicity techniques
developed in Ref.~\cite{hagi86}, which we briefly summarize below.
 
A tree-level amplitude in $e^+e^-$ collisions can be expressed in
terms of fermion strings of the form
\begin{equation}
\bar v(p_2,\sigma_2)P_{-\tau}\not\!a_1\not\!a_2\cdots
\not\!a_nu(p_1,\sigma_1)\;,
\end{equation}
where $p$ and $\sigma$ label the initial $e^{\pm}$ four-momenta
and helicities $(\sigma = \pm 1)$, $\not\!a_i=a^\mu_i\gamma_\mu$,
and $P_\tau=\case 1/2(1+\tau\gamma_5)$ is a chirality projection
operator $(\tau = \pm1)$.  The $a^\mu_i$ may be formed from particle
four-momenta, gauge-boson polarization vectors or fermion strings with
an uncontracted Lorentz index associated with final-state fermions.
 
In the chiral representation the $\gamma$ matrices are expressed
in terms of $2\times 2$ Pauli matrices $\sigma$ and the unit matrix 1 as
\begin{eqnarray}
\gamma^\mu \: & = &
\: \left(
\begin{array}{cc}
0 & \sigma^\mu_+ \\
\sigma^\mu_- & 0
\end{array} \; \; \right)\;,
\;\gamma^5= \left(
\begin{array}{cc}
-1 & \; 0\\
0 & \; 1
\end{array} \; \; \right) \;, \nonumber \\ & & \\
\sigma^\mu_{\pm} \:& = & \: ({\bf 1} ,\pm \sigma)\;, \nonumber
\end{eqnarray}
giving
\begin{equation}
\not\!a= \left(
\begin{array}{cc}
0 & (\not\!a)_+\\
(\not\!a)_- & 0
\end{array}\right),\;(\not\!a)_\pm=a_\mu\sigma^\mu_\pm\;.
\end{equation}
The spinors are expressed in terms of two-component Weyl spinors as
\begin{equation}
u=\left(
\begin{array}{c}
(u)_-\\
(u)_+
\end{array}\right),\;v={\bf (}(v)^\dagger_+{\bf ,} \;
(v)^\dagger_-{\bf )}\;.
\end{equation}
The Weyl spinors are given in terms of helicity eigenstates
$\chi_\lambda(p)$ with $\lambda=\pm1$ by
\begin{eqnarray}
u(p,\lambda)_\pm & = & (E\pm\lambda|{\bf p}|)^{1/2}\chi_\lambda(p)\;,
\nonumber \\ & & \\
v(p,\lambda)_\pm & = & \pm\lambda(E\mp\lambda|{\bf p}|)^{1/2}\chi
_{-\lambda}(p)\;, \nonumber
\end{eqnarray}
where
\begin{equation}
\chi_+(p)={\bf [}2|{\bf p}|(|{\bf p}|+p_z){\bf ]}^{-1/2}
\left(
\begin{array}{c}
|{\bf p}|+p_z\\
p_x+ip_y
\end{array}\right)\;,
\end{equation}
\begin{equation}
\chi_-(p)={\bf [}2|{\bf p}|(|{\bf p}|+p_z){\bf ]}^{-1/2}
\left(
\begin{array}{c}
-p_x+ip_y\\
|{\bf p}|+p_z
\end{array}\right)\;.
\end{equation}
For $p_z=-|{\bf p}|$ we follow the convention
\begin{equation}
\chi_+=
\left(
\begin{array}{c}
0\\1
\end{array} \right) , \;\;
\chi_-= \left(
\begin{array}{c}
-1\\0
\end{array}\right)\;.
\end{equation}
 
The fermion strings can be replaced by strings of two-component
spinors and $2\times 2$ matrices.  For example,
\FL
\begin{eqnarray}
\bar v_2P_{-\tau}\not\!a_1\not\!a_2&\cdots& \not\!a_nu_1\nonumber\\
&=&(v_2)^\dagger_\tau [a_1,a_2,\ldots,a_n]_\tau
(u_1)_{\delta_n\tau}\nonumber\\
&=&K\chi^\dagger_{-\sigma_2}(p_2)
[a_1,a_2,\ldots,a_n]_\tau\chi_{\sigma_1}(p_1)\;,\nonumber\\
\end{eqnarray}
where the square brackets with subscript $\tau$ are defined by
\begin{eqnarray}
&&[a_1,a_2,\ldots,a_n]_\tau=
(\not\!a_1)_\tau(\not\!a_2)_{-\tau}\cdots(\not\!a_n)_{\delta_n\tau}\:,
\nonumber  \\ & & \\
&&\delta_n=(-1)^{n+1} \nonumber
\end{eqnarray}
and the overall factor is
\begin{eqnarray}
K&=&\tau\sigma_2(E_2-\tau\sigma_2|{\bf p}_2|)^{1/2}\nonumber\\
&&\times\;[E_1+(-1)^{n+1}\tau\sigma_1|{\bf p}_1|]^{1/2}\;.
\end{eqnarray}
The gauge-boson polarization vectors $\epsilon (Q,\alpha)$ are
expressed in terms of the boson four-momenta as
\FL
\[
\epsilon^\mu(Q,\alpha=1)=(|{\bf Q}|Q_T)^{-1}(0,Q_xQ_z,Q_y
Q_z,-Q_T^2)\;,
\]
\FL
\begin{equation}
\epsilon^\mu(Q,\alpha=2)=Q_T^{-1}(0,-Q_y,Q_z,0)\;,
\end{equation}
\FL
\[
\epsilon_\mu(Q,\alpha=3)=[E_V/(M_V|{\bf Q}|)](|{\bf Q}|^2/E_V,
Q_x,Q_y,Q_z)\;,
\]
with $Q_T=(Q_x^2+Q_y^2)^{1/2}$, $E_V=(|{\bf Q}|^2+M_V^2)^{1/2}$.
At this state it is straightforward to evaluate the helicity amplitudes
numerically and to integrate over the phase space.  In the results
presented below we take the positive $z$ axis to lie along the
$e^-$ beam direction.
 
\subsection{$e^+e^-\to VV$ amplitudes}
\label{subsec:VVamps}
 
For the process
\FL
\begin{equation}
e^1(p_1,\sigma_1)+e^+(p_2,\sigma_2)\to W^-(Q_1,\alpha_1)+
W^+(Q_2,\alpha_2)
\end{equation}
there are three contributing Feynman diagrams from neutrino, $Z$,
and $\gamma$ exchanges, which give the amplitude
\begin{equation}
{\cal M}=ig^2\bar{v}(p_2,\sigma_2){\cal O}u(p_1,\sigma_1)\;,
\end{equation}
where
\begin{eqnarray}
{\cal O}=&-&(2l_1^2)^{-1}P+\not\!\epsilon_2\not l_1
\not\!\epsilon_1\nonumber\\
&+&\sum_\tau\;(g_\tau^eD_Z+D_\gamma x_WQ_e)P_\tau\not\! V(2,1)\;.
\end{eqnarray}
Here
\FL
\begin{eqnarray}
l_1  &=&  p_1-Q_1,~~D_a=(p_a^2-M_a^2+iM_a\Gamma_a)^{-1}\;, \nonumber \\
Q_e  &=&  -1,~~g_\tau^e=g_V^e-\tau g_A^3,~~g_V=\case 1/2T_3-x_WQ\;, \\
g_A  &=&  -\case 1/2T_3,~~g^2-8M_W^2G_F/\sqrt{2},~~x_W=\sin^2\theta_W\;.
\nonumber \label{eq:qe} \end{eqnarray}
The triple-gauge boson coupling coefficient is
\begin{eqnarray}
\Gamma(2,1)^\delta=[(&&Q_1-Q_2)^\delta g^{\mu\nu}+(2Q_2+Q_1)^\mu
g^{\delta\nu}\nonumber\\
&&-(2Q_1+Q_2)^\nu g^{\delta\mu}]\epsilon_{2\nu}\epsilon_{l\mu}\;.
\label{eq:triple} \end{eqnarray}
The corresponding helicity amplitudes are
\begin{eqnarray}
{\cal M}&=&ig^2(4E_1E_2)^{1/2}\chi^\dagger_{-\sigma_2}(p_2)\nonumber\\
&&\times \:\{(2l_1^2)^{-1}\delta_{1,\sigma_2}\delta_{-1,\sigma_1}
[\epsilon_2l_1\epsilon_1]_{-1} \nonumber\\
& & \; \; \; {}-  \delta_{\sigma_1,-\sigma_2}(D_Zg_{\sigma_2}^e+
D_\gamma x_W Q_e)[\Gamma(2,1)]_{\sigma_1}\}\nonumber\\
&&\times \:\chi_{\sigma_1}(p_1)\;,
\end{eqnarray}
where $\delta_{a,b}$ is the Kronecker $\delta$ symbol.
 
For the process $e^+e^-\to ZZ$ there are two electron-exchange
diagrams which lead to the helicity amplitudes
\FL
\begin{eqnarray}
{\cal M}=&&ig_Z^2(4E_1E_2)^{1/2}(l_i^2)^{-1}\delta_{\sigma_1,-\sigma_2}
(g_{\sigma_2}^e)^2\chi_{-\sigma_2}^\dagger(p_2)\nonumber\\
&&\times\;[\epsilon_jl_i\epsilon_i]_{\sigma_1}\chi_{\sigma_1}(p_1)
\end{eqnarray}
with an implied summation over $i,j=1,2$ with $j\neq i$; $l_i=p_1
-Q_i,g_Z=g/\cos\theta_W$.
 
For the process $e^+e^- \to Z\gamma$ there are again two
electron-exchange
diagrams which give the helicity amplitudes
\begin{eqnarray}
{\cal M}&=&ig^2{Q_e\sin\theta_W\over\cos\theta_W}(4E_1E_2)^{1/2}
(l_i^2)^{-1}\delta_{\sigma_1,-\sigma_2}g_{\sigma_2}^e\nonumber\\
& &\times\;\chi_{-\sigma_2}^\dagger(p_2)[\epsilon_jl_i\epsilon_i]_
{\sigma_1}\chi_{\sigma_1}(p_1)\;.
\end{eqnarray}
\subsection{$e^+e^-\to VVV$ amplitudes}
\label{subsec:VVVamps}
 
As shown in Fig.~\ref{proc} there are 20 Feynman diagrams
contributing to the process
\begin{eqnarray}
e^-(p_1,\sigma_1)+e^+(p_2,\sigma_2)&\to& Z(Q_1,\alpha_1)+
W^-(Q_2,\alpha_2)\nonumber\\
& &{}+W^+(Q_3,\alpha_3)
\end{eqnarray}
in an arbitrary gauge.  In the following, we present the corresponding
helicity amplitudes in the chiral representation.  For economy of
notation we shall write the helicity amplitudes for $VVV$ production
in the form
\FL
\begin{eqnarray}
{\cal M}(\sigma_1,\sigma_2;\alpha_1,\alpha_2,\alpha_3)&=&i(4E_1
E_2)^{1/2}\chi^\dagger_{-\sigma_2}(p_2)\nonumber\\
& &\times\;{\cal R}(\sigma_1,\sigma_2;\alpha_1,\alpha_2,\alpha_3)
\chi_{\sigma_1}(p_1)\nonumber\\
\end{eqnarray}
and give the reduced amplitude $\cal R$.
 
The three diagrams with double fermion exchanges
in Fig.~\ref{proc}(a) contribute helicity amplitudes
\begin{eqnarray}
{\cal R}^{({\rm a})} \; = & g^2&\!\!g_Z\case 1/2\delta_{1,\sigma_2}
\delta_{- 1\sigma_1}\nonumber\\
&\times\;\biggl(&{g_+^e\over l_1^2h_3^2}[\epsilon_3h_3\epsilon_2
l_1\epsilon_1]_{\sigma_1}+{g^\nu_+\over l_2^2h_3^2}[\epsilon_3
h_3\epsilon_1l_2\epsilon_2]_{\sigma_1}\nonumber\\
& &{}+{g^e_+\over l_2^2h_1^2}[\epsilon_1h_1\epsilon_3l_2\epsilon_2]_
{\sigma_1}\biggr)\;,
\end{eqnarray}
where $l_i=p_1-Q_i$ and $h_i=Q_i-p_2$.
 
The graphs in Fig.~\ref{proc}(b) with one neutrino and one $W$
exchange give
\FL
\begin{eqnarray}
{\cal R}^{(b)}=&-&g^3\delta_{1,\sigma_2}\delta_{-1,\sigma_1}
\case 1/2\cos\theta_W\nonumber\\
& &\times\;\left({[\Gamma^W(1,3)l_2\epsilon_2]_{\sigma_1}
\over l^2_2}+{[\epsilon_3h_3\Gamma^W(2,1)]_{\sigma_1}
\over h_3^2}\right)\nonumber\\
\end{eqnarray}
with
\FL
\begin{equation}
\Gamma^V(i,j)^\delta={[g^{\delta\delta^\prime}+(1-\xi )Q_{ij}^\delta
Q_{ij}^{\delta^\prime}/(\xi Q_{ij}^2-M_V^2)]\Gamma (i,j)_{\delta^\prime}
\over Q_{ij}^2-M_V^2} \; ,\label{eq:exch}
\end{equation}
where $Q_{ij}=Q_i+Q_j$.  Here $\Gamma(i,j)_\delta$ is defined in
Eq.~(\ref{eq:triple}) and $\xi$ is the gauge parameter of the
$V$ propagator; $\xi =0$ for the unitary gauge, $\xi =1$ for the
Feynman gauge, and $\xi\to \infty$ for the Landau gauge.  For the
photon we choose $\xi =1$ so that $\Gamma^\gamma (i,j)^\delta
=\Gamma (i,j)^\delta /Q_{ij}^2$.
 
The graphs in Fig.~\ref{proc}(c) with one electron and one $Z$
or $\gamma$\ exchange give
\widetext
\begin{eqnarray}
{\cal R}^{\rm c}=-g^2g_Z\delta_{\sigma_1,-\sigma_2}\biggr[&&
(g_{\sigma_2}^2)^2\left({[\Gamma^Z(3,2)l_1\epsilon_1]_{\sigma_1}
\over l_1^2}+{[\epsilon_1h_1\Gamma^Z(3,2)]_{\sigma_1}\over h_1^2}
\right)\nonumber\\
&&+x_WQ_eg_{\sigma_2}^e\left({[\Gamma^\gamma(3,2)l_1\epsilon_1]_
{\sigma_1}\over l_1^2}+{[\epsilon_1h_1\Gamma^\gamma(3,2)]_{\sigma_1}
\over h_1^2}\right)\biggr]\;.
\end{eqnarray}
The graphs in Fig.~\ref{proc}(d) with one $W$ and one $Z$ or $\gamma$
exchange give
\begin{equation}
{\cal R}^{\rm d}=g^3\cos\theta_W\delta_{\sigma_1,-\sigma_2}
\biggr[ g_{\sigma_2}^e\left({[\Gamma^Z(3,21)]_{\sigma_1}\over
Q_{12}^2-M_W^2}+{[\Gamma^Z(13,2)]_{\sigma_1}\over Q_{13}^2-M_W^2}
\right)
+x_WQ_e\left({[\Gamma^\gamma(3,21)]_{\sigma_1}\over
Q_{12}^2-M_W^2}+{[\Gamma^\gamma(13,2)]_{\sigma_1}\over Q_{13}^2-M_W^2}
\right)\biggr]\;,
\end{equation}
\narrowtext
\noindent where $\Gamma^V(i,jk)^\delta$ is defined by
Eqs.~(\ref{eq:triple}) and (\ref{eq:exch}), except that
$\epsilon_{2\nu}$ in Eq.~(\ref{eq:triple}) should be replaced by
$\Gamma (j,k)_\nu$.
 
The graphs of Fig.~\ref{proc}(e) with one $\gamma$ or $Z$ exchange
and a quartic vertex give
\FL
\begin{eqnarray}
{\cal R}^{(e)} =
-g^3\cos\theta_w\delta_{\sigma_1,-\sigma_2}&&\{g^e_{\sigma_2}D_Z
[\tilde \Gamma^Z(1,2,3)]_{\sigma_1}\nonumber\\
&&{} +\:x_WQ_eD_\gamma[\tilde\Gamma^\gamma(1,2,3)]_{\sigma_1}\}\;,
\nonumber \\  \end{eqnarray}
where
\FL
\begin{equation}
\tilde\Gamma^V(i,j,k)^\delta=[g^{\delta\delta '}
+(1-\xi )P^\delta P^{\delta '}/(\xi P^2-M^2_V)
\tilde\Gamma(1,2,3)_{\delta '}
\end{equation}
with
\begin{eqnarray}
\tilde\Gamma(i,j,k)^\delta=&&(2g^{\delta\mu}g^{\nu\lambda}
-g^{\delta\nu}g^{\mu\lambda}-g^{\delta\lambda}g^{\mu\nu})\nonumber\\
&&\times\;\epsilon_\mu(Q_1)\epsilon_\nu(Q_2)\epsilon_\lambda(Q_3)\;,
\end{eqnarray}
and $P=p_1+p_2$ is the total energy momentum.
 
The graphs of Fig.~\ref{proc}(f) involving the exchange of an unphysical
Higgs boson give
\FL
\begin{eqnarray}
{\cal R}^{(f)} & = & -g^3\delta_{\sigma_1,\sigma_2}M^2_Zx^2_W
\left( {g^e_{\sigma_2}D_Z\over\cos\theta_W}-Q_eD_\gamma
\cos\theta_W \right) \nonumber \\
&& \times \left( {[\epsilon_3]_{\sigma_1}\over
Q^2_{12}-M^2_W/\xi}\epsilon_1\cdot\epsilon_2+
{[\epsilon_2]_{\sigma_1}\over
Q^2_{13}-M^2_W/\xi}\epsilon_1\cdot\epsilon_3 \right)\;. \nonumber \\
& &  \end{eqnarray}
These graphs vanish as $\xi\to 0$ (i.e., in the unitary gauge).
 
The graph of Fig.~\ref{proc}(g) involving the exchange of the
physical Higgs boson $H$ gives
\begin{equation}
{\cal R}=-g^2g_Z\delta_{\sigma_1,-\sigma_2}
g^e_{\sigma_2}M^2_W\epsilon_2\cdot\epsilon_3
[\epsilon_1]_{\sigma_1}D_ZD_Z\;.
\end{equation}
For the analogous process,
\FL
\begin{eqnarray}
e^-(p_1,\sigma_1)+e^+(p_2,\sigma_2)\to &&\gamma
(Q_1,\alpha_1)+W^-(Q_2,\alpha_2)\nonumber\\
&&{}+W^+(Q_3,\alpha_3)\;,
\end{eqnarray}
there are 18 Feynman diagrams.  They are similar to those of
Fig.~\ref{proc}, except that the diagram of Fig.~\ref{proc}(a) with
$Z$ coupled to $\nu$ and the real-Higgs-boson diagram 1(g) have no
counterparts here.  The matrix elements ${\cal M}^{\rm (a)}
\cdot{\cal M}^{\rm (f)}$ for $WW\gamma$
production can be recovered from those for $WWZ$ production above
by the following rules.
 
(i) For ${\cal M}^{\rm (a)}$, multiply by $Q_e\sin\theta_W
\cos\theta_W/g^e_+$ and set $g^\nu_+=0$.
 
(ii) For ${\cal M}^{\rm (c)}$, multiply by $Q_e\sin\theta_W
/g^e_{\sigma_2}$.
 
(iii) For ${\cal M}^{\rm (b)},{\cal M}
^{\rm (d)},{\cal M}^
{\rm (e)}$, multiply by $\sin\theta_W/$ $\cos\theta_W$.
 
(iv) For ${\cal M}^{\rm (f)}$, multiply by $-\cos\theta_W/\sin
\theta_W$.
 
Using the helicity amplitude method, it is straightforward to
incorporate
the subsequent decays of the vector bosons.  To obtain the complete
helicity amplitude with fermions as the final states, the polarization
vectors of outgoing vector bosons are replaced by the appropriate
fermion currents to which the vector bosons couple:
\begin{equation}
\epsilon^\ast_\mu(p)\to -c(V)D_V
\sum_\tau c^f_\tau \bar u(f)P_\tau
\gamma_\mu v( \bar f)\;
\end{equation}
where
\begin{eqnarray}
&&c(V)  =  \left\{
\begin{array}{l}
 eQ_f\;\;\mbox{for}\;V=\gamma \;,\\
 V_{ff^\prime}g/\sqrt 2\;\;\mbox{for}\;V=W \;,\\
 g/\cos\theta_W\;\;\mbox{for}\;V=Z \;,
\end{array}\right. \nonumber \\ & & \\
&&c^f_\tau  =  \left\{
\begin{array}{l}
1\;\;\mbox{for}\;V=\gamma\;,\\
\delta_{1\tau}\;\;\mbox{for}\;V=W\;,\\
g^f_V-\tau g^f_A\;\;\mbox{for}\;V=Z\;.
\end{array}\right. \nonumber
\end{eqnarray}
Here $V_{ff'}$ is a fermion mixing matrix element and $D_V$ is
defined in Eq.~(\ref{eq:qe}).
 
The helicity amplitudes for $e^+e^-\to ZZZ$ are given
in Ref.~\cite{han88} and will not be repeated here.
 
\section{CROSS SECTIONS\\ AND GAUGE-BOSON DISTRIBUTIONS}
\label{sec:xsecs}
 
 
We are interested in final states where all three gauge bosons are
identified and measured.  We assume that $W$ and $Z$ will be identified
by their hadronic and charged-lepton decays (Sec.~\ref{sec:branchfracs}
below) and a
final photon will be identified as an electromagnetic jet of neutral
origin; accordingly transverse-momentum and pseudorapidity cuts
\begin{equation}
p_{\gamma T}>20\;\mbox{GeV},\;\;|\eta_\gamma |<2
\label{eq:psubgam} \end{equation}
have been imposed on all final photons
$[\eta=\ln \cot(\case 1/2 \theta )]$.
 
Figure~\ref{xener} shows the integrated cross sections versus
c.m.\ energy for two- and three-gauge-boson final states produced
in $e^+e^-$ collisions. In this figure, possible enhancements from a
physical Higgs boson $H$ have been omitted and the lowest-order QED
``point'' cross section
$\sigma_{\rm pt}=\sigma(e^+e^-\to \gamma^\ast\to\mu^+
\mu^-)=4\pi\alpha^2/(3s)$ has been introduced as a convenient
comparison scale.  The two-gauge-boson cross sections are from 1
to 2 orders of magnitude above those for three gauge bosons,
depending on $\sqrt s$.  Nevertheless, the triple-gauge-boson
cross sections approach 100 fb so that for an $e^+e^-$ collider
with an annual integrated luminosity of 10 fb$^{-1}$ the total
event rates for the $WWZ$ and $WW\gamma$ channels would be about one
thousand per year; see Table~\ref{values}.
 
A standard-model neutral Higgs boson $H$ will enhance the cross
sections for $W^+W^-Z$ and $ZZZ$ as shown in Fig.~\ref{xmass}.
The production of $H$ in $e^+e^-$ collisions comes from its
coupling to the $Z$ boson line in Fig.~\ref{proc}(g).
The enhancement in the $3V$ process is appreciable
only when $2M_V<m_H\sqrt s-M_V$
and is largest when $m_H$ is of order $2M_V$.  The presence of $H$
gives peaks in the distributions versus diboson invariant masses
$M_{WW}$ and $M_{ZZ}$ that are illustrated in Fig.~\ref{contrib}
for $\sqrt s=0.5$, 1.0, 2.0 TeV with a range of $m_H$ values.
 
Bosons from $H\to V_1V_2$ decay are dominantly longitudinally
polarized in the $H$ rest frame \cite{dunc86}.  This gives rise to
$V_1\to ab$ decay angular distributions of the form $d\sigma
/d\cos \theta^*\sim\sin^2\theta^*$, where $\theta^*$ is the
polar angle in the $V_1$ rest frame defined with respect to the
$V_2$ momentum vector.  Transverse polarization, on the other hand,
gives $d\sigma /d\cos\theta^\ast\sim(1+\cos^2\theta^\ast)$ distributions
(averaging the two decay products $a$ and $b$).  Polarization therefore
offers another way to confirm the Higgs-boson signal, if the
non-Higgs-boson contributions are mainly transverse.  In
$e^+e^-\to W^+W^-Z$ events, for example, the $\cos \theta^\ast$
distribution in the decay of $W^+$ or $W^-$ in each bin of invariant
mass $M_{WW}$ is determined by the fraction of events $P^W_L$ with
longitudinal polarization:  namely,
\begin{equation}
\sigma^{-1}d\sigma/d\cos\theta^\ast =\case 3/8 [1+P_L
+(1-3P_L)\cos^2\theta^\ast]\;.
\end{equation}
In a mass bin corresponding to a strong $H$ signal, $P_L$ approaches
1 and the distribution approaches a sin$^2\theta^\ast$ shape.
Calculations show that the non-Higgs-boson contributions are indeed
mainly transverse.  Figure~\ref{depend}(a) illustrates
the cos$\theta^\ast$ dependence of $W^-$ decays in
$e^+e^-$ $\to W^+W^-Z$ for $\sqrt s=0.5$, $m_H=0.2$ TeV;
summing over all events we have
$P^W_L\simeq 0.3$ and a concave-upward shape, but in the bin
$180<M_{WW}<220$ GeV we have $P^W_L\simeq 0.5$ and a markedly
convex shape.  Figure~\ref{depend}(b) shows the corresponding
dependence of $P^W_L$ on $M_{WW}$; it is clear that $P^W_L$
would be even more enhanced if it were practicable to select a
narrower bin at the Higgs-boson peak.
If we increase $m_H$ above 0.2 TeV the
longitudinal-polarization fraction $P_L$ of the Higgs-boson
signal will increase toward 1.0, since $\Gamma(H\to W_L
W_L)\sim m^3_H/M^2_W$ whereas $\Gamma (H\to W_TW_T)\sim
M^2_W/m_H$.
 
Figure~\ref{momen} shows the transverse-momentum
distributions of $W$ and $Z$ in the $WWZ$ channel at $\sqrt s=0.5$,
1.0, 2.0, TeV with $m_H=0.2$ or 0.4 TeV; the $W^+$ and $W^-$
curves coincide. The Jacobian peak in $p_T(Z)$ at the value
\begin{eqnarray}
p_T(Z,{\rm peak})=&&[s^2-2s(M^2_Z+M^2_H)\nonumber\\
&&{}+(m^2_H-M^2_Z)^2]^{1/2}/(2\sqrt s)
\end{eqnarray}
is due to the two-body kinematics of the Higgs-boson contribution
in Fig.~\ref{proc}(g).  Figure~\ref{distrib} shows the $W$
and $Z$ c.m.\ energy distributions for the same choices of
energy and Higgs-boson mass.  The spike in $E_Z$ at the value
\begin{equation}
E_Z({\rm peak}=(s+M^2_Z-m^2_H)/(2\sqrt s)
\end{equation}
is again due to the two-body kinematics of the Higgs-boson
contribution (smeared by the widths of $H$ and $Z$); it is hard to
distinguish in the case of Fig.~\ref{distrib}(c).
 
Figure~\ref{radid} shows the distributions versus
$W^+$, $W^-$, and $Z$
c.m.\ rapidities in the $WWZ$ channel, defining the $z$ axis
to lie along the $e^-$ beam direction.  The three curves peak
in different regions and the peaks broaden systematically as
energy increases.  There are no distinctive Higgs-boson effects
here and the case of no Higgs-boson contribution is shown.
Figure~\ref{chan} illustrates the dependence on gauge-boson polar
angles $\theta_V$
and diboson opening angles $\theta_{VV^\prime}$ in the $WWZ$
c.m.\ frame.  The spike in $\theta_{WW}$ comes from the Higgs-boson
contribution; its position is determined by $m_H$ and its width
is related to the $H$ width.
 
Similar distributions for $e^+e^-\to WW\gamma$ events are shown
in Figs.~\ref{result} and~\ref{trans}.  Here there is no
Higgs effect, but
otherwise the features broadly resemble those of the $WWZ$ case.
 
\section{BRANCHING FRACTIONS \\ AND EVENT RATES}
\label{sec:branchfracs}
 
The identification of three gauge-boson events depends on
the channels into which the $W^{\pm}$ and $Z$ decay.  For the
branching fractions of the individual $W$ and $Z$ bosons we
assume $m_t=45$ GeV with three neutrino species and use the
calculated values
\begin{eqnarray}
Z&\rightarrow &q\bar q,\;\;0.70\;,\nonumber\\
&&e\bar e,\;\;0.03\;,\nonumber\\
&&\mu\bar\mu ,\;\;0.03\;,\nonumber\\
&&\tau\bar\tau ,\;\;0.3\;,\nonumber\\
&&\nu\bar\nu ,\;\;0.20\;;\nonumber\\
& & \\
W&\rightarrow &q\bar q^\prime ,\;\;0.73\;,\nonumber\\
&&e\nu ,\;\;0.09\;,\nonumber\\
&&\mu\nu ,\;\;0.09\;,\nonumber\\
&&\tau\nu ,\;\;0.09\;.\nonumber \label{eq:ztoq}
\end{eqnarray}
Here, the $q\bar q$ and $q\bar q^\prime$ final states are summed over
all contributing quark flavors and lead primarily to two-jet states.
For a two-jet or two-lepton state to be identified, its invariant
mass must reconstruct the mass of the parent $Z$ or $W$.  This
can be achieved in principle for $e^+e^-,\mu^+\mu^-$, and pairs of
purely hadronic jets.  However, in decays to heavy-quark jets
(initiated by $c$, $b$, or $t$) there are appreciable probabilities
of semileptonic decays with unmeasureable neutrinos; the latter will
only be identified when the neutrinos are soft and subtract little
from the visible invariant mass (or when vertex detectors identify
the weakly decaying heavy flavors.)  Including semileptonic modes
where the visible invariant mass exceeds 95\% of the parent $W$
or $Z$ mass, the {\it reconstructible} branching fractions are
\begin{eqnarray}
Z&\to &q\bar q ,\;\;0.60\;,\nonumber\\
&&e\bar e ,\;\;0.03\;,\nonumber\\
&&\mu\bar\mu ;\;\;0.03\; ;\\ && \nonumber \\
W&\to &q\bar q^\prime ,\;\;0.61\;. \nonumber
\end{eqnarray}
In practice these values will be reduced a little by detector
resolution and by cutting the tails of the $Z$ and $W$ resonance
mass distributions, but we take them as the basis for the present
discussion.  A final-state photon can be identified as an
electromagnetic shower with a neutral initiator (i.e., no charged
track in a central detector), with high efficiency in principle.
 
Ideally, with a highly hermetic detector, one could measure the
total missing energy and missing three-momentum for the events
that occur at $\sqrt {s}=2E_{\rm beam}$.  Then those modes
where just one of the $W$ or $Z$ decayed into neutrinos could
also be reconstructed.  Because of the strong beamstrahlung
effects \cite{feld88} expected at future $e^+e^-$ high-energy linear
colliders, many events will actually occur at lower energies; it
is likely that such measurements will be restricted here to the
missing transverse momentum $\not\! p_T$.  Nevertheless, it should
still be possible to reconstruct a substantial fraction of the $W
\to l\nu (l=e$ or $\mu )$ decays where the transverse mass \cite{barg83}
\begin{equation}
m_T(l\not\! p_{lT})=[(|{\bf p}_{lT}|+|{\bf \not\! p}_T|)^2-
(\not\! p_{lT}+\not\! p_T)^2]^{1/2}
\end{equation}
lies close to $M_W$ while the other two gauge bosons appear as
hadronic jets.  The fraction of events reconstructible in this
way will depend on the experimental $\not\! p_T$ resolution.
We therefore regard Eq.~(\ref{eq:ztoq}) as providing only a
conservative lower bound on the eventually reconstructible decays.
 
Another factor affecting the identification of $VVV$ events is
jet overlap.  If two jets from two different gauge bosons
overlap substantially, it will not be possible to separate them
and reconstruct the masses accurately.  Also if an electron from
$Z\to e^+e^-$ or $W\to e\nu$ decay overlaps with a jet, it may
well escape identification and confuse the kinematics.  To
study this question we have calculated the cross-section dependence
on a cut $\Delta_0$, requiring all opening angles $\Delta\theta$
between pairs of leptons or jets (from two different gauge-boson
parents) to exceed $\Delta_0$:
\FL
\begin{eqnarray}
\Delta_{\min}\equiv\min\Delta\theta (&&\mbox{jet+jet or
lepton}\nonumber\\
&&{}+\mbox{jet from different }V)\geq\Delta_0\;.\label{eq:dmin}
\end{eqnarray}
Figure~\ref{open} shows the distributions versus
$\Delta_{\min}$ for $e^+e^- \to WWZ$ and $WW\gamma$,
for typical energies $\sqrt{s}=0.5$, 1.0,
and 2.0 TeV.
 
Yet another factor is the beam pipe.  There is inevitably a loss
of information about final particles emerging near the beam axis,
so for practical purposes we require all final leptons and jets to lie
outside the extreme forward and backward cone defined by an angle
$\theta_0:$
\begin{equation}
\theta_0\leq\theta(\mbox{lepton or jet})\leq\pi -\theta_0\;.
\end{equation}
Note that our photon cuts Eq.~(\ref{eq:psubgam}) already include
a minimum angle requirement with $\theta_0=15.4^\circ$
on the photon direction.
 
The $\Delta_0$ and $\theta_0$ cuts are approximately independent.
If realistically we require both $\Delta_0 =15^\circ$ and $\theta_0
=10^\circ$ simultaneously, the fractions of events satisfying these
cuts will be
\narrowtext
\begin{quasitable}
\begin{tabular}{ccc}
$\sqrt{s}$ & & \\
(TeV) &$WWZ$ &$WW\gamma$ \\
\tableline
0.5&0.84&0.88 \\
1.0&0.70&0.75 \\
2.0&0.53&0.63
\end{tabular}
\end{quasitable}
\begin{equation}
\label{tev}
\end{equation}
Putting together the results of Eqs.~(\ref{eq:ztoq}) and (\ref{tev}),
and remembering the possibility of identifying some other decays
in addition, we conclude that at least 20\% of
$e^+e^-\to W^+W^-Z$ and 32\% of
$e^+e^-\to W^+W^-\gamma$ events should
be fully reconstructible
at $\sqrt{s}=0.5$ TeV (falling to 13\% and 23\%, respectively, at
$\sqrt{s}=2.0$ TeV).
 
Since the product of branching fraction and geometrical efficiency
for reconstructible final states is substantial, the crucial factor
for event rates will be the production cross section.  We summarize in
Table~\ref{values} the numbers of $WWZ$, $ZZZ$, and $WW\gamma$ events
that would be produced at $e^+e^-$ colliders of various
c.m.\ energies for an
estimated \cite{weis88} integrated annual luminosity of 10
fb$^{-1}=10^{40}$
cm$^{-2}$.  The optimal energy for $WW\gamma$ production is about
$\sqrt{s}=0.35$ TeV; the optimal energy for $WWZ$ production is 1
TeV.  For detection efficiencies of order 20--30\%, hundreds of events
would be obtained per year with this luminosity to test the
standard-model gauge couplings.
 
Reconstructible events in the $WWZ$ and $WW\gamma$ channels should
be essentially free of backgrounds, due to the simultaneous mass-shell
constraints.  Continuum processes such as $e^+e^-\to W^+W^-$,
$t \bar{t}$, $q\bar{q}gg$, etc., may have integrated cross sections
that are an order of magnitude or more larger than the $3V$ signals,
but the requirement that three gauge bosons be simultaneously
reconstructed will strongly suppress such backgrounds (presuming
reasonable detector resolution).
 
\acknowledgments
 
This research was supported in part by the University of Wisconsin
Research Committee with funds granted by the Wisconsin Alumni
Research Foundation and in part by the U.S. Department of Energy
under Contract No.\ DE-AC02-76ER00881.
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\figure{Feynman diagrams for the process $e^+e^- \to$ $W^+W^-Z$.\label
{proc}}
\figure{Integrated cross sections for the production of two and
three gauge bosons in $e^+e^-$ collisions, vs c.m.\ energy
$\sqrt s$, omitting Higgs-boson effects.  The QED point cross section
$\sigma_{\rm pt}$ is shown for comparison.\label{xener}}
\figure{Integrated cross sections for (a) $e^+e^-\to WWZ$ and
(b) $e^+e^-\to ZZZ$ vs Higgs-boson mass showing the enhancement from
the possible presence of a heavy standard-model Higgs boson,
for $\sqrt s=0.5,1.0,2.0$ TeV.\label{xmass}}
\widetext
\figure{Effects of Higgs-boson contribution on diboson invariant-mass
distributions.  The $M_{WW}$ distribution in $e^+e^-\to WWZ$ is
shown for (a) $\sqrt s=0.5$ TeV, (b) $\sqrt s=1.0$ TeV, and
(c) $\sqrt s=2.0$ TeV.  The $M_{ZZ}$ distribution in $e^+e^-\to
ZZZ$ is shown for (d) $\sqrt s=0.5$ TeV, (e) $\sqrt s=1.0$ TeV,
(f) $\sqrt s=2.0$ TeV, averaged over the three $ZZ$ pairings.
\label{contrib}}
\narrowtext
\figure{(a) Dependence of $e^+e^-\to W^+W^-Z$ cross section on
cos$\theta^\ast$ of $W^-$ decay at $\sqrt s=0.5$, $m_H=0.2$ TeV;
$W^+$ results are the same; the dashed curve refers to all events,
the solid curve refers to the mass bin 0.18$<M_{WW}<0.22$ TeV.
(b) Corresponding dependence of the longitudinal and transverse
polarization fractions $P^W_L$ and $P^W_T$ on $WW$ invariant
mass, denoted by solid and dashed curves, respectively.\label{depend}}
\widetext
\figure{$W$ and $Z$ momentum distributions transverse to the beam
axis in the $WWZ$ channel for (a) $\sqrt s=0.5$, $m_H$ $=0.2$ TeV,
(b) $\sqrt s=1.0$, $m_H=0.4$ TeV, and (c) $\sqrt s=2.0$, $m_H=0.4$
TeV.\label{momen}}
\figure{$W$ and $Z$ c.m.\ energy distributions in the $WWZ$ channel for
(a) $\sqrt s=0.5$, $m_H=0.2$ TeV, (b) $\sqrt s=1.0$, $m_H=0.4$ TeV,
and (c) $\sqrt s=2.0$, $m_H=0.4$ TeV.\label{distrib}}
\figure{$W^+$, $W^-$, and $Z$ c.m.\ rapidity distributions in the $WWZ$
channel for (a) $\sqrt s=0.5$ TeV, (b) $\sqrt s=1.0$ TeV, and
(c) $\sqrt s=2.0$ TeV.  There is no distinctive Higgs effect here
and we show the case of no Higgs-boson contribution.\label{radid}}
\narrowtext
\figure{Distributions in the $WWZ$ channel vs (a) gauge-boson
polar angles $\theta_V$ and (b) diboson opening angles
$\theta_{VV^\prime}$ in the c.m.\ frame, for $\sqrt s=0.5$,
$m_H=0.2$ TeV.\label{chan}}
\widetext
\figure{Calculation results in the $e^+e^-\to WW\gamma$ channel:
distributions vs gauge-boson c.m.\ energies $E_V$ at
(a) $\sqrt s=0.5$ TeV, (b) $\sqrt s=1.0$ TeV, (c) $\sqrt s=2.0$
TeV $(W^+$ and $W^-$ curves coincide); distributions vs diboson
invariant masses $M_{VV^{\prime}}$ at (d) $\sqrt s=0.5$ TeV,
(e) $\sqrt s=1.0$ TeV,
and (f) $\sqrt s=2.0$ TeV $(W^+\gamma$ and $W^-\gamma$ curves
coincide).\label{result}}
\figure{Distributions in the $WW_\gamma$ channel vs gauge-boson
transverse momenta $p_T(V)$ at (a) $\sqrt s=0.5$ TeV, (b) $\sqrt s=1.0$
TeV, and (c) $\sqrt s=2.0$ TeV; distribution vs gauge-boson rapidities
$y_V$ at (d) $\sqrt s=0.5$ TeV, (e) $\sqrt s=1.0$ TeV, and
(f) $\sqrt s=2.0$ TeV.\label{trans}}
\narrowtext
\figure{Dependence on the minimum opening angle $\Delta_{\min}$ of
Eq.~(\ref{eq:dmin}) for (a) $e^+e^-\to WWZ$ and
(b) $e^+e^-$ $\to WW\gamma$.
\label{open}}
 
\newpage
\mediumtext
\begin{table}
\caption{Values of $R=\sigma(VVV)\sigma _{{\rm pt}}$ and corresponding
annual event rates for $e^+e^- \to WWZ$, $ZZZ$, and $WW\gamma$ assuming
a luminosity 10 fb$^{-1}$ per year.}\label{values}
\begin{tabular}{clccrcc}
\multicolumn{1}{c}{$\sqrt s$ (TeV)} &\multicolumn{1}{c}{$R(W^+W^-Z)$}
&\multicolumn{1}{c}{$N(W^+W^-Z)$} &\multicolumn{1}{c}{$R(ZZZ)$}
&\multicolumn{1}{c}{$N(ZZZ)$} &\multicolumn{1}{c}{$R(W^+W^-\gamma)$}
&\multicolumn{1}{c}{$N(W^+W^-\gamma)$}\\
\tableline
\multicolumn{7}{c}{$m_H<2M_W$ or $m_H\geq$1 TeV
(negligible Higgs-boson contribution)}\\
\dec 0.35 &1.5$\times 10^{-2}$ &1.2$\times 10^2$ &7.1$\times
10^{-4}$ &6 &\dec 0.20 &1.6$\times 10^3$ \\
\dec 0.5 &9.7$\times 10^{-2}$ &3.9$\times 10^2$ &2.9$\times
10^{-3}$ &12 &\dec 0.35 &1.4$\times 10^3$ \\
\dec 1.0 &5.8$\times 10^{-1}$ &5.8$\times 10^2$ &1.0$\times
10^{-2}$ &10 &\dec 0.73 &7.3$\times 10^2$ \\
\dec 2.0 &1.8 &4.4$\times 10^2$ &2.4$\times 10^{-2}$ &6
&\dec 1.4 &3.4$\times 10^2$ \\
 \\
\multicolumn{7}{c}{$m_H=0.2$ TeV} \\
\dec 0.35 &2.9$\times 10^{-2}$ &2.4$\times 10^2$ &
9.5$\times 10^{-3}$ &77 & & \\
\dec 0.5 &1.2$\times 10^{-1}$ &4.6$\times 10^2$ &
1.5$\times 10^{-2}$ &58 & & \\
\dec 1.0 &0.6 &6.0$\times 10^2$ &2.3$\times 10^{-2}$ &22 & & \\
\dec 2.0 &1.8 &4.5$\times 10^2$ &3.2$\times 10^{-2}$ &8 & & \\
\end{tabular}
\end{table}
\end{document}
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