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\preprint{\#HUTP-96/A024\\ 6/96}
\title{
Decays of a Leptophobic Gauge Boson
\thanks{Research
supported in part by the National Science Foundation under Grant
\#.}
}
\author{
Howard Georgi and Sheldon L. Glashow \\
Lyman Laboratory of Physics \\
Harvard University \\
Cambridge, MA 02138 \\
%\vspace{1ex}
%and \\
%\vspace{1ex}
%Other authors
}
\date{}
\abstract{
We discuss the theory and phenomenology of decays of a leptophobic
$U(1)_\X$
gauge boson $X$, such as has been proposed to explain
the alleged deviations of
$R_b$ and $R_c$ from standard model predictions. 
If the scalars
involved in the breaking of the $SU(2)\times U(1)$ symmetry 
are sufficiently light, $X$ will sometimes
decay into a charged (or neutral) scalar along with an oppositely-charged
$W$ (or $Z$). These decay modes
could yield clean signals for
the leptophobic gauge bosons at hadron colliders and provide an
interesting window into the Higgs sector of the theory.
}
\starttext

\section{\label{intro} Introduction}

Several recent papers propose the existence of a
gauge boson $X$ that couples to quarks but not 
leptons~\cite{altarelli,chiappetta,babu,hinchliffe}. Such a boson 
could explain certain deviations from the standard model reported at LEP.
The $X$ boson, even if quite light, may have escaped detection
because its decay into quark-antiquark has a large QCD
background~\cite{cdf}. We focus on the model of reference \cite{hinchliffe}
and suggest the possible detection of the $X$ boson
via its decay into $W$ or $Z$ bosons plus scalars decaying
into heavy quarks. A search for these decay modes could provide evidence
for leptophobic gauge bosons produced at hadron colliders.

\section{\label{model} The Model}

The gauge group is the standard model $\stu$ supplemented by a $U(1)_X$
that does not act on the lepton fields. Left-handed quark doublets carry
$U(1)_\X$ quantum number $q_\Q$, right-handed $U$ quarks $q_\U$, and 
right-handed $D$ quarks $q_\D$. If the $SU(2)$ symmetry breaking 
is done by fundamental Higgs bosons then, in general, three Higgs doublets
are needed to generate quark and lepton masses. Their
Yukawa couplings have the form:
\be
\ol Q\,h_\U\,\tilde H_\U\, U
+\ol Q\,h_\D\, H_\D\, D
+\ol L\,h_\L\, H_{_L}\, E\;,
\label{keterm}
\ee
where the $h_j$ are Yukawa coupling matrices and $\tilde H=-
\sigma_2\,H^{\ds *}$. It follows that the $U(1)_X$ quantum numbers of the 
doublets (which have weak hypercharge is ${1\over2}$) are
\be
q(H_\U) = q_\U-q_\Q\,,
\quad\quad
q(H_\D) = q_\Q-q_\D\,,
\quad\quad
q(H_{_{\!L}}) = 0\,.
%\label{}
\ee
For special values of the $q_j$'s, it is possible to give mass to quarks
and leptons with only two Higgs doublets, but none of these special choices
were favored in the analysis of reference \cite{hinchliffe}. Thus we assume
that\footnote{Note that this rules out some otherwise interesting
possibilities, such as the $\eta$-model of \protect\cite{babu}.}
\be
0\neq q_\U-q_\Q\neq q_\Q-q_\D\neq0\,.
%\label{}
\ee
If electromagnetic gauge invariance is to be left unbroken, the Higgs
doublets
may be written in the form: 
\be
H_j=
\pmatrix{
\xi_j^+\cr
(\xi^0_j+i\,\pi^0_j+v_j)/\sqrt2\cr
}
%\label{}
\ee
for $j=U$, $D$, or $L$. With an appropriate choice of phases for the $H_j$
fields,
their VEVs
$v_j$ may be made real and positive.

For simplicity, and because this is what was assumed in
\cite{hinchliffe}, we assume that the symmetry breaking is done 
primarily by the VEVs
of the $U$ and $D$ doublets, and that the VEV of the $L$ doublet is
negligible, $v_\L\ll v_\U,v_\D$.
The extra $U(1)_\X$ couplings lead to gauge anomalies, so that additional
fermion states must be introduced to cancel them. This is discussed, for
example, in \cite{hinchliffe}. 

The relevant interaction arises via the Higgs mechanism from
the kinetic energy terms of the $H_\U$ and $H_\D$ doublets:
\be
D^\mu H_\U^\dagger\,D_\mu H_\U
+D^\mu H_\D^\dagger\,D_\mu H_\D
\label{ke}
\ee
where
\be
D^\mu=\partial^\mu+ig_2{\vec\tau\over2}\cdot\vec W^\mu
+ig_1{1\over2}\,B^\mu\mp ig_x(q_\Q-q_{_{U,D}})\,X^\mu
\label{covariant}
\ee
and $X^\mu$ is the new $U(1)_X$ gauge field.

To avoid large corrections to the standard model properties of the $Z$ we
must tune the parameters of the model to make the
$X$-$Z$ mixing small. This requires that
\be
g_{_{\!X}}^2\left|
v_\U^2(q_\U-q_\Q)
+v_\D^2(q_\Q-q_\D)
\right|\ll g_2^2v^2\;.
\label{smallmixing}
\ee
Of course, the primary motivation for models of this kind is that a small
amount of mixing can modify the $Z$ couplings slightly and result in a
better
fit to data than the unadorned standard model. However, we are interested
not in these fine details, but in the gross properties of the $X$ boson.
Therefore, we ignore mixing altogether and assume
\be
v_\U^2(q_\U-q_\Q)
+v_\D^2(q_\Q-q_\D)
=0\;.
\label{zeromixing}
\ee
For the same reason, we ignore mixing of the $B$ and the $X$
through the gauge boson kinetic energy terms, assuming that it is
negligible
throughout the range of energies of interest.

For (\ref{zeromixing}) to be satisfied,
$q_\U-q_\Q$ and $q_\D-q_\Q$ must have the same sign, which we take to be
positive by convention. Because $v_\U^2+v_\D^2\approx v^2$ (where $v
\simeq 246$~GeV is
the VEV of the standard model), we can write
\be
v_\U\approx v\,\sqrt{q_\D-q_\Q\over q_\U+q_\D-2q_\Q}\,,\quad\quad
v_\D\approx v\,\sqrt{q_\U-q_\Q\over q_\U+q_\D-2q_\Q}\,.
\label{vuandvd}
\ee
The contribution of the doublet VEVs to the mass of the $X$ boson is
\be
m_\X^{\rm min}\equiv g_\X\,v\,\sqrt{(q_\U-q_\Q)(q_\D-q_\Q)}\,.
\label{xmass}
\ee
Since there may also be $SU(2)$ singlet scalars contributing to the $X$
mass, (\ref{xmass})  should be regarded as a lower bound.

In the Higgs mechanism, one linear combination of the two charged fields,
$\xi_\U^\pm$ and $\xi_\D^\pm$ is transformed into the longitudinal
component of the $W^\pm$, while a similar linear combinations of the two
fields $\pi^0_\U$
and $\pi^0_\D$ becomes the
longitudinal component of the $Z$. If the $U(1)_\X$ breaking comes entirely
from the doublets, the other linear combination becomes
the
longitudinal component of the $X$.\footnote{We will discuss below what
happens if there is additional $U(1)_\X$ symmetry breaking.}
In unitary gauge, we may set
\be
\sum_{j\atop _{U,D}}\,v_j\xi_j^{+}
=\pi^0_\U=\pi^0_\D=0\;.
%\label{}
\ee
In our no-mixing, $v_\L=0$ approximation, we can take
\be
H_\U=
\pmatrix{
v_\D\,\xi^+/v\cr
(\xi^0_\U+v_\U)/\sqrt2\cr
}
\quad\quad
H_\D=
\pmatrix{
-v_\U\,\xi^+/v\cr
(\xi^0_\D+v_\D)/\sqrt2\cr
}
\label{doublets}
\ee
where $\xi^+$ is the surviving combination of $\xi_\U^+$ and
$\xi_\U^+$.
The $\xi^+$, $\xi^0_\U$ and $\xi^0_\D$ fields need not be mass
eigenstates 
(indeed,  we expect some mixing with the components of $H_\L$)
but for now we ignore mixing.

\section{\label{xdecays}$X$ Decays}

The couplings responsible for the decay of $X$ into $W$ or $Z$ plus a
scalar are obtained by
putting (\ref{doublets}) into (\ref{keterm}): 

\be\ba{c}
g_\X\,g_2\,{v_\U v_\D\over v}(q_\U+q_\D-2q_\Q)
\biggl(\xi^+\,X^\mu\,W^-_\mu
+\xi^-\,X^\mu\,W^+_\mu\biggr)\cra
+{g_\X\,g_2\over\cos\theta}
\biggl((q_\D-q_\Q)v_\D\,\xi^0_\U-(q_\U-q_\Q)v_\U\,\xi^0_\D\biggr)
X^\mu\,Z_\mu\;.\ea
%\label{}
\ee
Using (\ref{vuandvd}) and $m_\W=g_2\,v/2=m_\Z\cos\theta$, we find:
\be
2\,g_\X\,\sqrt{(q_\U-q_\Q)(q_\D-q_\Q)}\,
\biggl(m_\W\,\Bigr(\xi^+\,X^\mu\,W^-_\mu
+\xi^-\,X^\mu\,W^+_\mu\Bigr)
+m_\Z\,\xi^0\,X^\mu\,Z_\mu
\biggr)
\label{result}
\ee
where
\be
\xi^0\equiv\xi^0_\D\,\sqrt{q_\D-q_\Q\over q_\U+q_\D-2q_\Q}-
\xi^0_\U\,\sqrt{q_\U-q_\Q\over q_\U+q_\D-2q_\Q}\;.
%\label{}
\ee

Equation (\ref{result}), our central result, is valid provided that
all $SU(2)$ breaking is done by the VEVs of $H_\U$ and $H_\D$.  Our result
is unaffected by additional $U(1)_\X$ breaking due to the VEVs of
$SU(2)$ {\it singlet\/} fields. These would contribute to $m_\X$  and lead
to the survival of a linear combination of the various $\pi$ fields, but
one which
does not appear in (\ref{result}). 


Note that $\xi^\pm$ and $\xi^0$ form a triplet under the custodial $SU(2)$
symmetry~\cite{custodial}. The orthogonal linear combination of the two
neutral states is a custodial $SU(2)$ singlet, which is the analog in this
model of the standard model Higgs boson. The custodial $SU(2)$ symmetry may
be broken by the mass mixing between the neutral states or by the masses and
mixing of all of the states with other spinless bosons in the model, but it
remains manifest in the couplings.

The dominant decay mode of $X$ is into quark-antiquark pairs, where the QCD
background may obscure the resonance. For this reason,
we are interested in the branching ratio for the decay of the $X$
into $W\xi$ and $Z\xi$ due to interaction
(\ref{result}). If the $\xi$s are sufficiently
light, these decays may offer clearer signatures of a leptophobic gauge
boson.

For each family, the rate
$\Gamma$ for $X$ to decay into a $Q={2\over3}$ quark-antiquark pair is:
\be\ba{l}
\Gamma(X\rightarrow U\ol U)\cra\approx
{1\over4\pi}\,g_\X^2\,(q_\Q^2+q_\U^2)\,(1-m_q^2/m_\X^2)\,p(m_\X,m_q,m_q)\cra
={1\over8\pi}\,g_\X^2\,(q_\Q^2+q_\U^2)\,(1-m_q^2/m_\X^2)\,\sqrt{m_\X^2-
4m_q^2}\;.\ea
\label{slg1}
\ee
where $p$ is the final particle momentum in the rest frame,
\be
p(m_\X,m_a,m_b)
={m_\X\over2}\,\sqrt{\left(1-{(m_a+m_b)^2\over m_\X^2}\right)
\left(1-{(m_a+m_b)^2\over m_\X^2}\right)}\,.
%\label{}
\ee
For each family, the rate
$\Gamma$ for $X$ to decay into a $Q=-{1\over3}$ quark-antiquark pair is:
\be\ba{l}
\Gamma(X\rightarrow D\ol D)\cra\approx
{1\over4\pi}\,g_\X^2\,(q_\Q^2+q_\D^2)\,(1-m_q^2/m_\X^2)\,p(m_\X,m_q,m_q)\cra
={1\over8\pi}\,g_\X^2\,(q_\Q^2+q_\D^2)\,(1-m_q^2/m_\X^2)\,\sqrt{m_\X^2-
4m_q^2}\;.\ea
\label{slg2}
\ee


One might expect the gauge boson decays to be
suppressed by powers of $m_\W/m_\X$ because of the explicit factors of
$m_\W$ and $m_\Z$ in (\ref{result}). However,
these factors are compensated
by the enhancement from the longitudinal $W$ and $Z$. Thus the partial
widths for $W\xi$, $Z\xi$ and $q\bar q$ decays are of the same order,
differing only by kinematic and counting factors.

For the $W^+$ decay, the square of the
invariant matrix element is
\be
{4\over3}
g_\X^2(q_\U-q_\Q)(q_\D-q_\Q)\,m_\W^2\,
\left(-g_{\mu\nu}+{{p_\W}^\mu {p_\W}^\nu\over m_\W^2}\right)
\left(-g_{\mu\nu}+{{p_\X}_{\!\mu} {p_\X}_{\!\nu}\over m_\X^2}\right)\;.
%\label{}
\ee
Using $(p_\X p_\W)=(m_\X^2+m_\W^2-m_\xi^2)/2$. we find
\be
{4\over3}g_\X^2\,(q_\U-q_\Q)(q_\D-q_\Q)\,m_\W^2\,
\left(2+{(m_\X^2+m_\W^2-m_\xi^2)^2\over4m_\X^2m_\W^2}\right)\;.
%\label{}
\ee
Thus the partial width
into $W^\pm\xi^\mp$ is
\be\ba{l}
\Gamma(X\rightarrow W^\pm\xi^\mp)\cra\approx
{1\over3\pi}g_\X^2\,(q_\U-q_\Q)(q_\D-q_\Q)\cra\cdot
\left(2+{(m_\X^2+m_\W^2-m_{\xi^\pm}^2)^2\over4m_\X^2m_\W^2}\right)
\,{m_\W^2\over m_\X^2}\,p(m_\X,m_\W,m_{\xi^\pm})\;.\ea
\label{gammaw}
\ee
For the decay into $Z\,\xi^0$, because of custodial symmetry,
the partial
width is given by half of this result, with
$m_\W\rightarrow
m_\Z$ and $m_{\xi^\pm}\rightarrow m_{\xi^0}$:
\be\ba{l}
\Gamma(X\rightarrow Z\xi^0)\cra\approx
{1\over6\pi}g_\X^2\,(q_\U-q_\Q)(q_\D-q_\Q)\cra\cdot
\left(2+{(m_\X^2+m_\Z^2-m_{\xi^0}^2)^2\over4m_\X^2m_\Z^2}\right)
\,{m_\Z^2\over m_\X^2}\,p(m_\X,m_\Z,m_{\xi^0})\;.\ea
\label{gammaz}
\ee

Branching ratios for the decay modes
$X\rightarrow W^\pm\xi^\mp$, $X\rightarrow Z\xi^0$ 
 are determined by equations
(\ref{slg1}), (\ref{slg2}), (\ref{gammaw}) and (\ref{gammaz}).
Figures \ref{fig121}, \ref{fig1212} and \ref{fig011} show these branching
ratios (and that of  $X\rightarrow t\bar t$)
as a function of $m_\X$
for two representative leptophobic models and two values for the $\xi$
masses. One model is that discussed in
reference \cite{hinchliffe}; in the other
we choose $q_\Q=0$, for which 
anomaly cancellation is more straightforward. The branching ratios for
these signature modes of decay of a leptophobic gauge boson are large
enough to be of experimental interest.


\section{\label{phenomenology}Phenomenology}

The production cross section for $X$ depends on $g_\X$, which otherwise
does not enter into our analysis except to determine
the minimum value of the $X$
mass, $m_\X^{\rm min}$. 
With $g_\X=0.15$, one of the values
discussed in reference \cite{hinchliffe}, this cross section is given
approximately in figure \ref{figsigma}. It is well below the published limit
from CDF for all values of $m_\X$. For this value of $g_\X$, $m_\X^{\rm
min}\approx 90$~GeV. 

The $\xi$'s produced in $X$ decays
decay primarily into heavy quarks, giving rise to the processes: 
{\renewcommand{\arraystretch}{.9}
\be
\begin{array}{r@{}l}
X\rightarrow W^- & \xi^+ \\
&\,\hookrightarrow
c\bar s
\ea
\ee
\be
\begin{array}{r@{}l}
X\rightarrow Z & \xi^0 \\
&\,\hookrightarrow
b\bar b
\end{array}
%\label{decayschemes}
\ee
If, as expected,  the $\xi$'s mix slightly with the 
states in the $H_L$ doublet, 
there will also be decay modes in which $\tau$'s are
produced:
\be
\begin{array}{r@{}l}
X\rightarrow W^- & \xi^+ \\
&\,\hookrightarrow
\tau^+\nu_\tau
\ea\ee
\be
\begin{array}{r@{}l}
X\rightarrow Z & \xi^0 \\
&\,\hookrightarrow
\tau^+\tau^-
\end{array}
%\label{taudecayschemes}
\ee
}
Note also that if the $X$ is above $t\bar t$ threshold, its decay into
$t\bar t$ could be a significant source of $t$s.

\section*{Acknowledgements}

We are grateful for interesting conversations with Melissa Franklin, Paolo
Giromini and Ken Lane. We are particularly grateful to Tom Baumann for help
with structure functions.


\begin{thebibliography}{99}

\bibitem{altarelli} Guido Altarelli, Nicola Di Bartolomeo, Ferruccio
  Feruglio, Raoul Gatto, Michelangelo L. Mangano, R(B), R(C)
  AND JET DISTRIBUTIONS AT THE TEVATRON IN A MODEL WITH AN
  EXTRA VECTOR BOSON, CERN-TH-96-20, Jan 1996.
e-Print Archive:  .

\bibitem{chiappetta}
  P. Chiappetta, J. Layssac, F.M. Renard, C. Verzegnassi,
  HADROPHILIC Z-PRIME: A BRIDGE FROM LEP-I, SLC AND CDF TO
  LEP-II ANOMALIES. PM-96-05, Jan 1996. e-Print Archive:  
  .

\bibitem{babu}
By K.S. Babu, Chris Kolda, John March-Russell, LEPTOPHOBIC U(1)S AND THE
R(B) - R(C) CRISIS, IASSNS-HEP-96-20, Feb 1996.
e-Print Archive: 

\bibitem{hinchliffe} K. Agashe, M. Graesser, Ian Hinchliffe, M. Suzuki,
A CONSISTENT MODEL OF ELECTROWEAK DATA INCLUDING Z ---> B ANTI-B AND Z
---> C ANTI-C,
LBL-38569, Apr 1996.
e-Print Archive: .

\bibitem{cdf}
  F. Abe et al., SEARCH FOR NEW PARTICLES DECAYING TO DIJETS
  IN P ANTI-P COLLISIONS AT S**(1/2) = 1.8-TEV.
  Phys. Rev. Lett. 74 (1999) 3538-3543.
e-Print Archive: .

\bibitem{custodial}
 P. Sikivie, L. Susskind, M. Voloshin, V. Zakharov,
Nucl. Phys. {\bf B173} (1980) 189.
See also M. Weinstein, Phys. Rev. {\bf D8} (1973) 2511.

\end{thebibliography}

\newpage

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\caption{\sf\label{fig121} Branching ratios in $X$ decay for
$(q_\Q,q_\U,q_\D)=(-
1,2,1)$ and $m_{\xi^\pm}=m_{\xi^0}=100$~GeV. The solid line is
$B(X\rightarrow W^\pm\,\xi^\mp)$. The dashed line is
$B(X\rightarrow Z\,\xi^0)$. The dotted line is
$B(X\rightarrow t\bar t)$.
}\end{figure}}

\newpage

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\caption{\sf\label{fig1212} Branching ratios in $X$ decay for
$(q_\Q,q_\U,q_\D)=(-
1,2,1)$ and $m_{\xi^\pm}=m_{\xi^0}=200$~GeV. The solid line is 
$B(X\rightarrow W^\pm\,\xi^\mp)$. The dashed line is
$B(X\rightarrow Z\,\xi^0)$. The dotted line is
$B(X\rightarrow t\bar t)$.
}\end{figure}}

\newpage

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\caption{\sf\label{fig011} Branching ratios in $X$ decay for
$(q_\Q,q_\U,q_\D)=(
0,1,1)$ and $m_{\xi^\pm}=m_{\xi^0}=100$~GeV. The solid line is
$B(X\rightarrow W^\pm\,\xi^\mp)$. The dashed line is
$B(X\rightarrow Z\,\xi^0)$. The dotted line is
$B(X\rightarrow t\bar t)$.
}\end{figure}}

\newpage

{\figsize\begin{figure}[htb]
$$\beginpicture
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\endpicture$$
\caption{\sf\label{figsigma}An estimate of the $X$ production cross section
in
picobarns at center of mass energy 1800 GeV for $(q_\Q,q_\U,q_\D)=(-1,2,1)$
and $g_\X=0.15$.}\end{figure}}

\end{document}


