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%% section 2 $Z^0$ Width Constraints [slacpub7050002 in slacpub7050002: slacpub7050003]
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\section{\usemenu{slacpub7050::context::slacpub7050002}{$Z^0$ Width Constraints}}\label{section::slacpub7050002}
We first discuss the bounds from $Z^0$ decays. Because the $\mu$
parameter and electroweak gaugino masses enter chargino and neutralino
mass matrices, one might expect that when these parameters are in the
GeV range, charginos and neutralinos are light and in conflict with the
bounds on $Z^0$ decay widths. We will see, however, that in some of
this region of parameter space, charginos are sufficiently massive and
neutralinos are sufficiently decoupled from the $Z^0$ that these bounds
may be satisfied.
First consider the charginos. We assume that the charginos and
neutralinos are the standard mixtures of electroweak gauginos and the
Higgsinos of the two Higgs doublets. We also denote the bino, wino, and
gluino masses by $M_1$, $M_2$, and $M_3$, respectively, and the ratio of
Higgs expectation values by $\tan\beta \equiv \langle H_2 \rangle /
\langle H_1 \rangle $. The chargino mass terms are then $(\psi^)^T
{\bf M}_{\tilde{\chi}^{\pm}} \psi^+ + \mbox{ h.c.}$, where the mass
matrix is
\begin{equation}\label{chamass}
{\bf M}_{\tilde{\chi}^{\pm}} = \left( \begin{array}{cc}
M_2 & \sqrt{2} \, M_W\sin\beta \\
\sqrt{2} \, M_W\cos\beta & \mu \end{array}
\right)
\end{equation}
in the basis $\psi^{\pm} = (i\tilde{W}^{\pm}, \tilde{H}^{\pm})$. The
current bound on chargino masses from LEP measurements is 47 GeV
\cite{1}. This bound requires no additional assumptions,
as charginos remain coupled to the $Z^0$ for all values of the
parameters. We see, however, that for $\mu, M_2 \approx 0$ and
$\tan\beta \approx 1$, both charginos have mass $M_W$ and avoid the
bound. With $\mu \approx 0$ ($M_2 \approx 0$), as $M_2$ ($\mu$)
increases, one chargino mass eigenvalue drops by the seesaw mechanism,
and when $M_2$ ($\mu$) $> 90 \mbox{ GeV}$, the chargino mass limit is
violated for all $\tan\beta$. However, for $1 < \tan\beta \lesssim
2.1$, the parameters $\mu, M_2 \approx 0$ satisfy the chargino mass
bound.
Next we examine the neutralino sector. Unlike charginos, neutralinos
may completely decouple from the $Z^0$, and for this reason, there are
no strict lower bounds on neutralino masses. If one assumes gaugino
mass unification and $\tan\beta > 2$, the lower bound on the lightest
neutralino's mass is 20 GeV \cite{1,2,3}. For $\tan\beta
\lesssim 1.6$, however, this mass bound disappears altogether
\cite{2,3}. It is clear, then, that a discussion of light
neutralinos requires a detailed analysis of their couplings to the $Z^0$
boson. The $Z^0$ width constraints are therefore considerably more
complicated for neutralinos than for charginos, and we will discuss them
in two stages. First, we present a simple discussion that makes clear
the qualitative features of the allowed region. These features are
illustrated in Fig.~\docLink{slacpub7050006.tcx}[fig:1]{1}. We then add a number of refinements to
the analysis and present the resulting allowed region in
Fig.~\docLink{slacpub7050006.tcx}[fig:2]{2}.
It is convenient to write the neutralino mass terms
$\frac{1}{2} (\psi ^0)^T {\bf M}_{\tilde{\chi}^0} \psi^0 + \mbox{ h.c.}$
in the basis $(\psi^0)^T = \left(
\frac{1}{\sqrt{2}} ( i \tilde{Z^{0}} + \tilde{H}_A ),
\frac{1}{\sqrt{2}} ( i \tilde{Z^{0}}  \tilde{H}_A ),
i\tilde{\gamma}, \tilde{H}_S \right)$,
where $\tilde{H}_A \equiv \tilde{H}_1 \cos\beta  \tilde{H}_2
\sin\beta$, and $\tilde{H}_S \equiv \tilde{H}_1 \sin\beta + \tilde{H}_2
\cos\beta$. The tree level mass matrix is then
\begin{equation}\label{neutmass}
{\bf M}_{\tilde{\chi}^0} = \left(
\begin{array}{cccc}
M_Z+\frac{1}{2}M+\frac{1}{2}\mu\sin 2\beta &
\frac{1}{2}M\frac{1}{2}\mu\sin 2\beta &
\Delta M & \frac{1}{\sqrt{2}}\mu \cos 2\beta \\
\frac{1}{2}M\frac{1}{2}\mu \sin 2\beta &
M_Z+\frac{1}{2}M+\frac{1}{2}\mu\sin 2\beta &
\Delta M & \frac{1}{\sqrt{2}}\mu \cos 2\beta \\
\Delta M & \Delta M & M_{\tilde{\gamma}} & 0 \\
\frac{1}{\sqrt{2}}\mu\cos 2\beta & \frac{1}{\sqrt{2}}\mu\cos 2\beta
& 0 & \mu \sin 2\beta \end{array}
\right) \ ,
\end{equation}
where $M \equiv M_1 \sin^2 \theta_W + M_2 \cos^2 \theta_W$,
$M_{\tilde{\gamma}} \equiv M_1 \cos^2 \theta_W + M_2 \sin^2 \theta_W$,
$\Delta M \equiv \frac{1}{\sqrt{2}} (M_2M_1)
\cos\theta_W\sin\theta_W$, and $\theta_W$ is the weak mixing angle.
As discussed above, the chargino mass bound is satisfied with $\mu , M_2
\approx 0$. In order to avoid a light neutralino with unsuppressed
coupling to the $Z^0$, it is also necessary that $M_1$ be small. In the
limit $\mu, M_2, M_1 \to 0$, the basis states given above are mass
eigenstates, with masses $M_Z$, $M_Z$, 0, and 0. The light photino,
$\tilde{\gamma}$, does not couple to the $Z^0$, and the light Higgsino,
$\tilde {H}_S$, decouples for $\tan \beta \to 1$. In this case, the
only nonzero coupling of the neutralinos to the $Z^0$ is through
$Z^{0}\tilde{H}_A\tilde{H}_S$, which is suppressed by phase space.
To understand how far one can vary from the limit $\mu, M_2, M_1, \tan
\beta 1 \to 0$ and still satisfy all the constraints, we must discuss
the bounds in greater detail. Let us denote the lightest neutralino,
$\tilde{\chi}^0_1$, by $\chi$, and the heavier neutralinos,
$\tilde{\chi}^0_2$, $\tilde{\chi}^0_3$, and $\tilde{\chi}^0_4$, by
$\chi'$. We will assume that the lightest neutralino $\tilde{\chi}^0_1$
is the lightest supersymmetric particle (LSP) and escapes the detector.
There are then bounds on $\Gamma(Z^0\to \chi\chi)$ from the invisible
$Z^0$ width, and bounds on $\Gamma(Z^0\to \chi\chi')$ and $\Gamma(Z^0\to
\chi'\chi')$ from direct searches for neutralinos.
The current bound on the invisible width of the $Z^0$, in units of the
neutrino width, is $N_{\nu} = 2.988 \pm 0.023$ \cite{1,4}. The
2$\sigma$ upper bound on nonStandard Model invisible decays is then
$\delta N_{\nu} = 0.034$, or a $Z^0$ branching ratio of $B_{\mbox{\tiny
inv}} = 2.3 \times 10^{3}$. We will take this as the bound on the
$\chi\chi$ width.\footnote{Formally, production of $\chi\chi'$ and
$\chi'\chi'$, if followed by $\chi'\to \chi\nu\bar{\nu}$, will also
contribute to the invisible width. However, as we will see, such
processes violate the visible width bounds long before their effect on
the invisible width becomes important, and so may be safely ignored
here.}
The visible width bounds are determined from direct searches for
neutralinos. In Ref.~\cite{2} the L3 Collaboration placed bounds on
neutralinos based on an event sample including 1.8 million hadronic
$Z^0$ events. The decays $\chi'\to \chi {Z^0}^* \to \chi f \bar{f}$,
with $f = q, e, \mu$, and also the radiative decay $\chi' \to \chi
\gamma$ were considered. For given masses $m_{\chi}$ and $m_{\chi'}$,
neutralino events were simulated, and the photonic branching ratio was
chosen to give the weakest bounds. In the regions of most interest to
us, the neutralino masses are $m_{\chi} \approx 0$ and $m_{\chi'} \approx 0,
M_Z$. For these masses, the upper bound on the branching ratio
$B(Z^{0}\to\chi\chi')$ ($B(Z^{0}\to\chi'\chi')$) was found to be at
least $1.2\times 10^{5}$ ($3.5\times 10^{5}$). These bounds
deteriorate rapidly as $m_{\chi} \to m_{\chi'}$, but we will
conservatively assume that they apply for all masses.
Given these bounds, we may now determine the allowed region of parameter
space. As one varies from the point $\mu, M_2, M_1, \tan\beta1 = 0$,
the basis states begin to mix, and have masses given by the diagonal
elements up to corrections of ${\cal O}((M_1, M_2, \mu)^2/M_Z)$. The
mixing angles between the heavy states and between the heavy and light
states are ${\cal O}((M_1, M_2, \mu)/M_Z)$. However, (at tree level)
the mixing between the light states occurs only through the intermediate
heavy states and so is ${\cal O}((M_1, M_2, \mu)/M_Z)^2$. These mass
shifts and mixings may then weaken the various coupling constant and
phase space suppressions. Decays to the following three states
determine the allowed region:
\noindent (a) $\tilde{H}_S \tilde{H}_S$. The ratio $\Gamma(Z^0 \to
\tilde{H}_S \tilde{H}_S)/\Gamma(Z^0 \to \nu\bar{\nu})$ is $\cos^2 2\beta$.
If $\tilde{\chi}^0_2$ has a significant $\tilde{H}_S$ component, the
stringent limits on the visible $Z^0$ width require $\tan \beta < 1.02$,
a range that is in conflict with the perturbativity of the top Yukawa
coupling (see below). However, when $\tilde{\chi}^0_1 \approx
\tilde{H}_S$, the constraint on $\tan\beta$ comes only from the
invisible width bound, which is two orders of magnitude weaker. The LSP
(at tree level) is very nearly pure $\tilde{H}_S$ for
$M_{\tilde{\gamma}}>\mu\sin 2\beta$ and satisfies the invisible
width bound for $\tan \beta < 1.20$. Below, we assume that this
constraint is satisfied and $\tilde{\chi}^0_1 \approx \tilde{H}_S$.
This scenario was previously considered in Ref.~\cite{5}.
\noindent (b) $\tilde{H}_S \tilde{\chi}^0_3$. The decay to $\tilde{H}_S
\tilde{\chi}^0_3$ is suppressed only by phase space. This suppression
is adequate when both $M_Z+\frac{1}{2}M+\frac{1}{2}\mu\sin 2\beta +
\mu \sin 2\beta \gtrsim M_Z$ and $M_Z\frac{1}{2}M 
\frac{1}{2}\mu\sin 2\beta + \mu \sin 2\beta \gtrsim M_Z$. Assuming
$M>0$, this constraint is then $M \lesssim 3\mu\sin 2\beta$ for
$\mu<0$ and $M\lesssim \mu\sin 2\beta$ for $\mu>0$.\footnote{For
$M<0$, the requirements are $M<\mu\sin 2\beta$ for $\mu<0$ and
$M<3\mu\sin 2\beta$ for $\mu>0$. However, we will concentrate on
the case $M>0$, as this holds in most of the allowed region.}
\noindent (c) $\tilde{H}_S \tilde{\chi}^0_2$. For this decay to be
suppressed, the neutralino $\tilde{\chi}^0_2$ must be nearly a pure
photino. The mixing of this eigenstate is controlled by $\Delta M$ and
vanishes when $\Delta M=0$, that is, when $M_1=M_2$.
The allowed regions for $\tan\beta = 1.15$ are presented in
Fig.~\docLink{slacpub7050006.tcx}[fig:1]{1} for three values of the ratio $M_1/M_2$. The allowed
regions are very similar for all $1.02 < \tan\beta < 1.20$. Constraints
(a) and (b) limit the allowed parameter space to a region with
boundaries of definite slope as given above. Constraint (c) provides a
maximum allowed $M_2$, and, as expected, disappears in the limit
$M_1=M_2$.
The analysis above gives a rough picture of what parameter regions may
survive the various constraints. However, several refinements are
necessary. First, as noted above, the photonic branching ratio was
assumed to be unknown in the analysis of Ref.~\cite{2} and was chosen
to give the weakest bounds. However, for specific branching ratios,
additional regions might be excluded. In particular, the radiative
photon decay $\tilde{\chi}^0_2 \to \tilde{\chi}^0_1 \gamma$ has been
studied previously \cite{5} and is expected to be dominant in
our case, where $\tilde{\chi}^0_2 \approx \tilde{\gamma}$ and
$\tilde{\chi}^0_1 \approx \tilde{H}_S$. The production of
$\tilde{\chi}^0_1 \tilde{\chi}^0_2$ then gives a spectacular single
photon signal, and the bound on its rate can be significantly improved.
To estimate this new bound, we reexamine the data of Ref.~\cite{2}. In
that event sample, the dominant Standard Model background, $Z^0 \to
\gamma_{\mbox{\tiny ISR}}\nu\bar{\nu}$, is expected to produce only
$15.7 \pm 1.5$ events with photons passing the cut $p_T > 10 \mbox{
GeV}$. Assuming that $\sqrt{15.7} \simeq 4$ neutralino events could be
hidden in this background, that the efficiency of neutralino detection
in this mode is 50\%, as given in Ref.~\cite{2}, and, for simplicity,
that the neutralino events are uniformly distributed in the range
$0 2 M_2$, the allowed region can be
extended to lower values of $M_2$ in the negative $\mu$ region.)
Qualitatively, the allowed region is very similar to what would be
expected from Fig.~\docLink{slacpub7050006.tcx}[fig:1]{1}, with the exception that points with $\mu
\gtrsim 0$ have been eliminated. These points required the phase space
suppression of $\tilde{H}_S \tilde{\chi}^0_3$ production, and are
eliminated by the $M_Z + 1.8$ GeV data. We see, however, that much of
the $\mu<0$ region still remains.
In the previous figures, radiative corrections have not been included.
Radiative corrections to the diagonal entries of Eq.~(\docLink{slacpub7050002.tcx}[neutmass]{2})
shift the neutralino masses, and thus shift the boundaries slightly.
Corrections to the offdiagonal entries introduce mixings between
states. Mixings between the heavy states and between heavy and light
states are unimportant as similar mixings are already present at tree
level, and all such mixings are highly suppressed because they mix
states whose eigenvalues are split by ${\cal O}(M_Z)$. Offdiagonal
radiative corrections that mix the light states can be important,
however, as they give a Dirac mass that lifts the tree level zero in
Eq.~(\docLink{slacpub7050002.tcx}[neutmass]{2}) \cite{7}. The largest such correction
comes from topstop loops (similar to the ones that induce the radiative
photon decay), is of order 1 GeV, and vanishes when the left and
righthanded stops, $\tilde{t}_L$ and $\tilde{t}_R$, are degenerate
\cite{7}. When this radiative mixing of the light states is
significant with respect to the tree level masses, the LSP is a mixture
of both $\tilde{H}_S$ and $\tilde{\gamma}$. Some regions of the window
that were allowed at tree level are then excluded by the stringent $Z^0$
visible width bound. However, for $\mu$ and $M_2$ sufficiently large,
the radiative mixing becomes negligible, and the LSP can be mostly
$\tilde{H}_S$. For stop masses in the range $100 \mbox{ GeV} <
m_{\tilde{t}_R}, m_{\tilde{t}_L} < 300 \mbox{ GeV}$, we find that the
effect of the radiative Dirac mass is to remove points with $M_2, \mu
\lesssim 24 \mbox{ GeV}$, leaving most of the allowed region displayed
in Fig.~\docLink{slacpub7050006.tcx}[fig:2]{2} intact.
Returning to the chargino mass matrix of Eq.~(\docLink{slacpub7050002.tcx}[chamass]{1}), we see
that, in the allowed Higgsinogaugino window, both charginos are roughly
degenerate with the $W^{\pm}$. Charginos and neutralinos are thus all
within reach of LEP II; if observed there, precision studies may be able
to determine if the SUSY parameters lie in this allowed window
\cite{8}. The chargino mass is lowered by the deviation from $\mu, M_2,
\tan\beta1 = 0$, but remains above 70 GeV. It is important to note
that $m_{\tilde{\chi}^{\pm}_1} + m_{\tilde{\chi}^0_1} > 77 \mbox{ GeV}$
throughout almost all of the allowed region and grows beyond $M_W$ as
$\mu$ and $M_2$ increase in the allowed region, so the branching ratio
for the decay $W \to \tilde{\chi}^{\pm}_1 \tilde{\chi}^0_1$ is highly
suppressed \cite{9}.
Another possible constraint on the light Higgsinogaugino window is from
the relic dark matter density. As the LSP is mostly Higgsino, the
dominant annihilation channel is through $s$channel $Z^0$ to light
fermion pairs. The Higgsino$Z^0$ coupling is necessarily suppressed in
order to avoid the invisible width bound from $Z^0$ decay. This
generally leads to an overproduction of primordial Higgsinos. Using the
nonrelativistic approximation for the freeze out density and the
annihilation cross section for a pure Higgsino state given in
Ref.~\cite{10}, we find that for $\tan \beta \simeq 1.2$, $\Omega h^2
\lesssim 1$ only for $m_{LSP} \gtrsim 20$ GeV. We therefore conclude
that either there is an additional entropy release below the LSP freeze
out temperature to dilute the relic Higgsinos, or that $R$ parity is
broken so that the LSP is not stable, and therefore does not contribute
to the dark matter.
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