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%% section 3 Radiative Symmetry Breaking [slacpub7050003 in slacpub7050003: slacpub7050004]
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\section{\usemenu{slacpub7050::context::slacpub7050003}{Radiative Symmetry Breaking}}\label{section::slacpub7050003}
Before discussing the scalar sector and radiative symmetry breaking, we
note that, in the allowed window where $\mu, M_1, M_2 \ll M_W$, an
approximate $U(1)$ $R$symmetry exists in the weak gaugino and Higgs
sector, under which $R(H_1)=R(H_2)=0$, and all other chiral matter
fields have $R=1$.\footnote{Under an $R$ transformation the scalar,
fermionic, and auxilliary components of a chiral superfield transform as
$\phi \rightarrow e^{i \alpha R} \phi$, $\psi \rightarrow e^{i \alpha
(R1)} \psi$, and $F \rightarrow e^{i \alpha (R2)} F$, respectively,
where $R$ is the superfield's $R$ charge. The superpotential has $R$
charge $R(W)=2$, and a gauge superfield has $R$ charge
$R(W^{\alpha})=1$.} It is possible to promote this approximate symmetry
to an exact symmetry of the entire MSSM Lagrangian, in which case all
gaugino masses, $A$ terms, and $\mu$ would vanish
\cite{7,11,12,13}. We do not impose such a symmetry by hand,
but simply note that an approximate symmetry exists in the allowed
window. Below we discuss some consequences of extending the approximate
$U(1)_R$ symmetry to other sectors of the MSSM. Such an approximate
symmetry in fact arises accidentally in certain types of hidden sector
SUSY breaking scenarios as discussed below.
The allowed window requires $\tan \beta \approx 1$. Let us therefore
reexamine the lower bound on $\tan \beta$ from the requirement that the
top Yukawa coupling, $h_t$, remain perturbative to high scales. This is
related to the top quark pole mass by the oneloop relation
\cite{14}
\begin{equation}
h_{t}(m_{t}) \simeq \frac{m_{t}^{\mbox{\tiny pole}}}{174{\mbox{ GeV}}}
\frac{\sqrt{1 + \tan^{2}\beta}}{\tan\beta}
\left[ 1 \frac{5}{3}\frac{\alpha_{s}}{\pi} 
\Delta_{\mbox{\tiny SUSY QCD}}  \Delta_{\mbox{\tiny electroweak}}
\right] \lesssim 1.15 \ ,
\label{ht}
\end{equation}
where $1.15$ is our estimate of the quasifixed point value. Neglecting
SUSY, electroweak, and higher loop corrections, one has a $\sim 6\%$
correction to the tree level result, and taking $m_{t}^{\mbox{\tiny
pole}} \gtrsim 160$ GeV, we find the constraint $\tan\beta \gtrsim
1.14$. However, including the oneloop SUSY QCD corrections
\cite{14,15}, we find an additional few percent correction (for a
nonvanishing gluino mass) whose sign depends on the various SUSY
parameters. A $\sim 10\%$ correction is thus possible, which would
lower the $\tan\beta$ bound to $\tan\beta \gtrsim 1.04$. Alternatively,
for a fixed $\tan\beta$ the perturbativity upper bound on
$m_{t}^{\mbox{\tiny pole}}$ could increase. Thus, one can still
consider perturbative values of $h_t$ at Planckian scales for $\tan\beta
\lesssim 1.2$, and we may also consider the possibility of radiative
symmetry breaking (RSB).
Next we consider the scalar Higgs sector. In the allowed window, the
Higgs potential is $V = m_{1}^{2}H_{1}^{2} + m_{2}^{2}H_{2}^{2} 
m_{12}^{2}(H_{1}H_{2} + \mbox{ h.c.})+ D$terms + $\Delta
V^{\mbox{\tiny 1loop}}$, where $m_i$ are in our case simply the soft
SUSY breaking masses, since $\mu$ is generally small. The condition
$m_{2}^{2} < 0$ triggers electroweak symmetry breaking, and $m_{1}^{2}>
0$ is required for the potential to be bounded. At tree level, the
pseudoscalar Higgs mass is $m_{A}^2 = m_{12}^2 (\tan\beta + \cot\beta
)$. Although often assumed, it is not generally true in supergravity
theories that $m_{12}^2$ is proportional to $\mu$. A small $\mu$
parameter does not, therefore, imply the existence of a light
pseudoscalar. Note also that a large $m_{12}^2$ does not violate the
approximate $U(1)_R$ symmetry given above.
We have seen above that $\tan\beta \approx 1$ in the allowed window. At
tree level, the light $CP$ even Higgs mass satisfies the bound
$m_{h^{0}} < M_Z\cos 2\beta$ and so vanishes in the limit $\tan \beta
\to 1$. However, $m_{h^{0}} \propto h_{t}m_{t}$ is generated by
topstop contributions to $\Delta V^{\mbox{\tiny 1loop}}$, and, in
principle, a large Higgs mass can be obtained to satisfy the current
experimental bound of $m_{h^{0}} \gtrsim 60$ GeV
\cite{1}. (The lower bound on the Standard Model Higgs boson
mass is the relevant one in the limit $\tan\beta \rightarrow 1$.) If
the approximate $U(1)_R$ symmetry is extended to the entire Lagrangian,
though, achieving $m_{h^{0}} \gtrsim 60$ GeV is not trivial. In this
case the mixing between the stops $\tilde{t}_L$ and $\tilde{t}_R$, which
can significantly enhance the loop contributions to $m_{h^0}$ \cite{16},
is small since $\mu, A \approx 0$. In addition, the stop masses
$m_{\tilde{t}_{L,\,R}}$ are constrained from above if RSB with minimal
particle content is required. This may be seen by recalling the
minimization condition $m_2^2 = \left(m_1^2 + \frac{1}{2}M_Z^2
(1\tan^2\beta)\right)/\tan^2\beta$. The constraint $m_1^2>0$ then
implies a lower bound on $m_{2}^{2}$ of $
\frac{1}{2}M_Z^2 (\tan^2\beta  1)/\tan^2\beta$. In the $U(1)_R$
symmetric case, where all gaugino masses are small, the RGE equation for
$m^2_2$ is $\partial{m_{2}^{2}}/\partial{\ln{Q}} \simeq
\frac{3}{8\pi^{2}} h_{t}^{2}[m_{2}^{2} + m^2_{{\tilde{t}}_{L}}+
m^2_{{\tilde{t}}_{R}}]$. The requirement that $m_2^2$ not be driven too
negative then places an upper bound of typically $\lesssim M_Z$ on the
boundary condition for the stop masses at the grand scale. One then
finds that the stop masses at the weak scale are not large enough to
push $m_{h^0}$ above its lower bound. Thus, unless the $U(1)_R$
symmetry is explicitly broken by a gluino mass, RSB and $m_{h^{0}}
\gtrsim 60$ GeV cannot be achieved simultaneously with minimal particle
content. (Note that in Ref.~\cite{11} the authors assume a global
$U(1)_R$ symmetry in the whole Lagrangian, but do not require
satisfactory RSB.)
Let us elaborate on the above observations. We have seen that to have
satisfactory RSB with minimal particle content, the combination
$[m_{2}^{2} + m_{\tilde{t}_{L}}^{2} +m_{\tilde{t}_{R}}^{2}]$ that
controls $m_{2}^{2}$ renormalization is constrained to be approximately
zero at the grand scale. If we assume also a common scalar mass $m_{0}$
at the grand scale and vanishing gaugino masses and $A$ parameters
($i.e.$, the $U(1)_R$ symmetric limit), one finds $m_{0} \lesssim
\frac{1}{3} M_{Z}$ (for $\tan\beta \lesssim 1.15$ and $h_{t}$ at its
quasifixed point) and an unacceptable spectrum. (Stronger constraints
apply for nonvanishing dimensionthree terms, and the symmetry limit is
preferred.) However, if we relax the universality assumption, we are
led to consider the following soft parameter boundary conditions at the
grand scale: $m_{2}^{2}(0) \approx [m_{\tilde{t}_{L}}^{2}(0)
+m_{\tilde{t}_{R}}^{2}(0)] > 0$ and $m_{2}^{2}(0) \neq m_{1}^{2}(0)$.
If we add a nonvanishing gluino mass, thereby explicitly breaking the
$U(1)_R$ symmetry in the colored sector, $m^2_{\tilde{t}_{L}}$ and
$m^2_{\tilde{t}_{R}}$ both turn positive and possibly large in the
course of renormalization, and the radiatively induced $m_{h^{0}}$ is
sufficiently large. (Such boundary conditions can be realized, $e.g.$,
in certain stringy schemes \cite{17}.) As long as the scalar
potential in the full and effective theories is bounded from below at
all scales, these boundary conditions are acceptable. In particular, we
find solutions with righthanded stops ranging in mass from 45 GeV to
many hundreds of GeV, Higgs bosons in the 6070 GeV range, and the two
charginos between 7090 GeV (as is favored by $R_b$ (see below)). We
present typical spectra in Fig.~\docLink{slacpub7050006.tcx}[fig:3]{3}, assuming the above pattern
for boundary conditions. Only those masses that are constrained by RSB
and $m_{h^0}$ are presented. Note that because the trilinear terms in
the scalar potential are small, dangerous color breaking directions of
the potential, which are generic in the limit $\tan\beta \to 1$
\cite{16}, are eliminated. (However, if $m_{\tilde{t}}\lesssim m_t$,
dangerous directions may persist.)
The above scheme is an example of boundary conditions that can
successfully generate RSB. (Note that all other boundary conditions are
only negligibly constrained by RSB and $m_{h^{0}}$.) The tuning
required in order to achieve RSB is a reflection of the fact that we did
not have at our disposal an arbitrary $\mu$, which typically absorbs the
tuning. (For example, generically one expects $\mu \gtrsim 1$ TeV for
$\tan\beta \to 1$ \cite{16}.) Instead, the tuning is now in the soft
parameters. Alternatively, $\partial m_{2}^{2}/\partial
\ln{Q}$ can be adjusted by introducing a righthanded neutrino
superfield at an intermediate scale with a soft mass
$m_{\tilde{\nu}_{R}}^{2} < 0$. A new neutrino Yukawa term
$h_{\nu}^{2}m_{\tilde{\nu}_{R}}^{2}$ then enters the RGE for $m_2^2$,
and may be used to balance the RGE. The additional freedom results from
the fact that $m_{\tilde{\nu}_{R}}^{2}$ is unconstrained, as the
physical mass of the scalar neutrino is determined essentially by the
intermediate scale.
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