\documentstyle[12pt]{article}

\begin{document}

\title{TESTING THE VECTOR CONDENSATE MODEL OF ELECTROWEAK
INTERACTIONS AT HIGH ENERGY HADRON COLLIDERS}

\author{G.  Cynolter, E.  Lendvai and G.  P\'ocsik \\
 Institute for Theoretical Physics, \\ E\"otv\"os Lorand
University, Budapest }

\date{}
\maketitle

%\date{ITP-Budapest Report No.  520 \\ September 1996 }

\begin{abstract}

In the vector condensate model a doublet of vector fields
plays the role of the Higgs doublet of standard model and
the gauge symmetry is broken dynamically.  This results in
a theory surviving the test of radiative corrections
provided the new charged and neutral vector particles B
have masses of at least several hundred GeV's.  In this
note we show that while at the Tevatron the heavy
B-particle production is too low, at LHC the yield is large
and, for instance, the inclusive cross section of $B^+ B^-$
pairs is 51.5 (15.3) fb at $\sqrt{s}=14$ TeV,
$m_B=400(500)$ GeV.


\end{abstract}

The vector condensate model employs the Lagrangian of the
standard model of electroweak interactions but the usual
scalar doublet is replaced by a doublet of vector fields
$B_{\mu}$ [1] whose neutral component forms a condensate
breaking the gauge symmetry dynamically and providing
nonvanishing $W,Z$ and vanishing photon masses.  A quartic
self-coupling of $B_{\mu}$ gives masses to the B-particles,
as well as the interaction of B-pairs with fermions gives
rise to fermion masses and embeds the Kobayashi-Maskawa
mechanism [1].

Looking at the S parameter it follows that the B-particles
are heavy, at $\Lambda=1$ TeV the threshold is about $m_0
\simeq 400-500 $ GeV for $B^0$ and $m_+ \simeq 200-350$ GeV
for $B^+$ [2].  Higher $\Lambda$ attracts higher minimum
masses.  The allowed regions by S are tightened by the T
parameter [3].  For example at $\Lambda=1 $ TeV, $m_0=400$
GeV the threshold for $m_+$ is increased to 630 GeV.  In
general, to each momentum scale there exists a range of B
masses where the radiative corrections are suitably small
by partial cancellations.  This is why $\Lambda$ cannot be
too large as compared to B masses.

Recently it has been shown that pairs of heavy B--particles
are copiously produced at high energy linear $e^+ e^-$
colliders [4].  In the present note we study the production
of B--particles at the Tevatron and LHC.  We show that
producing heavy B-particles at LHC is very favourable
having a large cross section while at Tevatron energy the
production cross section cannot exceed (0.01-0.02) fb which
is far below the discovery limit.

Since fermions are coupled very weakly to B-pairs [1] in
the vector condensate model, producing B-pairs is expected
to be more considerable from virtual $ \gamma $ and Z
exchanges, that is we consider the Drell-Yan mechanism [5],
$ p ^(\overline{p}^) \rightarrow B \overline{B} +X $ via
quark-antiquark annihilation.

The Drell-Yan cross section for the above hadronic
collisions can be written as [5,6] 
\begin{eqnarray} \sigma(
 p ^(\overline{p}^) \rightarrow B \overline{B} +X)=
\int_{\tau_0}^1 \, d\tau \int_{\tau}^1 \, \frac{dx}{2x}
\sum_i \sigma(q_i \overline{q}_i \rightarrow B
\overline{B}) \cdot \nonumber \\ \left ( f_i^1(x,\hat{s})
f_{\bar i}^2(\tau/x, \hat{s})+ f_{\bar i}^1(x,\hat{s})
f_{i}^2 (\tau/x, \hat{s}) \right )
\end{eqnarray}
where $x$ and $\tau/x$ are the parton momentum fractions,
$\hat{s}=\tau s$ is the square of the centre of mass energy
of $q_i \bar q_i$, s is the same for the hadronic initial
state, $f^1_i (x, \hat{s} )$ means the number distribution
of $i$ quarks in hadron 1 at the scale $\hat{s}$ and the
sum runs over the quark flavours u,d,s,c.  In the
computation the MRS (G) fit program [7] was used for the
parton distributions.

The angle integrated, colour averaged annihilation cross section
$ \sigma(q_i \overline{q}_i \rightarrow B \overline{B} ) $
is calculated to lowest order in the gauge couplings, and
QCD corrections are neglected.  We hope this approximation
shows the order of magnitude of the cross section.
 For $B^0 \overline{B}^0$
final state the Z exchange is working, and $B^+ B^- $
 pairs appear via $\gamma+Z $ exchange.
This is because in the model the following Lagrangians
relevant to the annihilation process emerge [1]:

\begin{eqnarray} 
L \left( B^{0} \right)&=&{ig \over 2cos
 \theta_W } \partial^{\mu} B^{(0)\nu +} \left( Z_{ \mu}
 B_{\nu}^{(0)} - Z_{\nu} B^{(0)}_{\mu} \right) + h.c.,
 \nonumber \\ L \left( B^+ {B^-} Z
 \right)&=&-cos2\theta_W \cdot L \left(B^{(0)}
 \rightarrow B^{(+)} \right),
\end{eqnarray} 
where $B^{(0)}_{\mu} \left( B^{(+)}_{\mu} \right ) $ denotes the
field of neutral (charged) B--particles. 
At $q_j \overline{q}_jZ$-vertex the usual coupling 
$ig \gamma_{\mu} (g_{Vj}+g_{Aj} \gamma_5 )$ acts, here

\begin{eqnarray} \left.  \matrix{
g_{Vj}& = & \frac{1}{2}&-\frac{4}{3} sin^2 \theta_W, \cr & & & \cr
g_{Aj}& = &{ \frac{1}{2} }& } \right \} j=u,c \nonumber \\ \\ \left.
\matrix{ g_{Vj}& = &-\frac{1}{2}&+\frac{2}{3} sin^2 \theta_W,
\cr & & & \cr g_{Aj}&=&-\frac{1}{2}& } \right \} j=d,s  \ .\nonumber
\end{eqnarray}


From (2) we get for $B^0 \overline{B}^0$ final states
\begin{eqnarray} 
\sigma(q_i \overline{q}_i \rightarrow B^0
\overline{B}^0)= \frac{1}{3} \frac{1}{256 \pi} \left (
\frac{g}{cos \theta_W} \right )^4 ( g_{Vi}^2+g_{Ai}^2)
\left( 1-\frac{4 m_0^2}{\hat{s}} \right)^{3/2} 
\frac{\hat{s} +8 m_0^2 }{4 m_0^4} .  
\end{eqnarray}

This is decreasing at high, increasing $m_0$ and for
$\hat{s} \gg 4 m_0^2$ it is proportional to
$\hat{s}/m_0^4$ reflecting that the Lagrangian (2) is
coming form a nonrenormalisable, effective model.  The
cross section of $B^+ B^-$ pairs can be expressed by (4) in
the following way 
\begin{eqnarray} 
\sigma(q_i
\overline{q}_i \rightarrow B^+ B^-)= \sigma(q_i
\overline{q}_i \rightarrow B^0 \overline{B}^0; m_0
\rightarrow m_+) \cdot \frac{1}{g_{Vi}^2+g_{Ai}^2} \cdot
\nonumber \\ \left [ (g_{Vi}^2+g_{Ai}^2 )cos^2 2 \theta_W+ 2
Q_{q_i} g_{Vi} sin^2 2 \theta_W cos 2 \theta_W + 4
Q_{q_i}^2 sin^2 2\theta_W \right].
\end{eqnarray}
The individual terms are due to $Z$ exchange, $\gamma-Z$
interference and $\gamma$ exchange. 

 We have calculated various distributions of
$B \overline{B}$ pairs for $p \overline{p}$
collisions at $\sqrt{s}=1.8 $ TeV and for $pp$ collisions
at $\sqrt{s}=14$ TeV, assuming $m_B=400,500,600$ GeV.
Typically, at the Tevatron no notable result can be
presented, however, increasing the energy to LHC, we get
sizable cross sections.  Also the yield of $B^+ B^-$ is
larger than that of $B^0 \overline{B}^0$.  As an example we
show in Fig.1 the $\tau$-distribution of $B^+ B^-$ pairs
at LHC, $m_+=400$ GeV.  $\frac{d \sigma}{d \tau}$ is
sharply peaked after threshold ($4m_+^2/s$) and
insignificant from about $\tau=0.1$, that is for higher
invariant masses of $B^+B^-.$ Fig.2 shows
$\frac{\partial^2 \sigma}{\partial p_T \partial y }
\vert_{y=0}$ as the function of the transverse momentum
$p_T$ of $B^+$ at vanishing rapidity y of $B^+ B^-$ for
LHC, $m_+=400$ GeV.

For the total cross section (1) we obtain 
\begin{eqnarray}
& \sigma_{\rm Tev}&=0.020 (0.0145) fb \, \hbox{ for } B^+B^-
(B^0 \overline{B}^0), m_B=400 \hbox{GeV}, \nonumber \\
&\sigma_{LHC}&=51.46; 15.30; 5.65 fb \; (44.35; 13.54; 4.90
fb) \\ & & \hbox{ for } B^+B^- (B^0 \overline{B}^0),
m_B=0.4; 0.5; 0.6 \hbox{TeV}.  \nonumber 
\end{eqnarray}


For instance, at an expected integrated luminosity of $10^5
{\rm pb}^{-1}$ one gets about 5100 $B^+ B^-$ pairs of
$m_+=0.4 $ TeV at LHC per annum.

In conclusion, we have shown that heavy B--particle pairs
have a large inclusive cross section due to $q
\overline{q}$ annihilation at LHC in a suitable mass range
making the detection of $B^{+,0}$ at LHC possible.
\vspace*{.5cm}

This work is supported in part by OTKA I/7, No.  16248.
\pagebreak

\begin{thebibliography}{9}

\bibitem{1} G.  P\'ocsik, E.  Lendvai and G.  Cynolter,
Acta Phys.  Pol.  {\bf B24}, 1495 (1993); G.Cynolter, E.
Lendvai and G.  P\'ocsik, in Electroweak Symmetry Breaking,
World Scientific, 1995 (Ed.  F.  Csikor and G.  P\'ocsik)
p.  85.  
\bibitem{2} G.  Cynolter, E.  Lendvai and G.
P\'ocsik, Mod.  Phys.  Lett. {\bf A10}, 2193 (1995).
\bibitem{3} G.  Cynolter, E.  Lendvai and G.  P\'ocsik,
Mod.  Phys.  Lett.  {\bf A9 }, 1701 (1994) \bibitem{4} G.
Cynolter, E.  Lendvai and G.  P\'ocsik, Acta Phys.  Pol.
{\bf B26}, 921 (1995). 
\bibitem{5} S.D.  Drell and T.M.
Yan, Phys.  Rev.  Lett.  {\bf 25}, 316 (1970); Ann.  Phys.
(N.Y) {\bf 66}, 578 (1971).
\bibitem{6} E.  Eichten, I.
Hinchliffe, K.  Lane and C.  Quigg, Rev.  Mod.  Phys.  {\bf
56}, 579 (1984). 
\bibitem{7} A.D.  Martin, W.J.  Stirling
and R.G.  Roberts, Phys.  Lett {\bf B 354}, 155 (1995).
\end{thebibliography}

\pagebreak

{\Large \bf Figure Captions}
\vspace*{1cm}

Fig.  1:  The differential $\tau$-distribution of $B^+B^-$
pairs at LHC,$m_+=400$ GeV.

\vspace*{.5cm}

Fig.  2:  $\frac{\partial^2 \sigma}{\partial p_T \partial y
 } \vert_{y=0}$ at LHC, $m_+=400$ GeV.  $p_T$ denotes the
 transverse momentum of $B^+$ and $y$ is the rapidity of
 $B^+B^-$.

\pagebreak

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