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%% section 4 The $CP$ Problem, \rb, and Proton Decay [slacpub7050004 in slacpub7050004: slacpub7050005]
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\section{\usemenu{slacpub7050::context::slacpub7050004}{The $CP$ Problem, \rb, and Proton Decay}}\label{section::slacpub7050004}
The light Higgsinogaugino window has a number of interesting
consequences. With conventional weak scale SUSY breaking parameters,
the present bound on the electric dipole moments (EDMs) of atoms,
molecules, and the neutron limit the $CP$ violating phases in the
dimensionful parameters of the MSSM to be less than $10^{2}  10^{3}$
over much of the parameter space \cite{18}. This is generally
referred to as the SUSY $CP$ problem. As shown in Ref. \cite{19},
all flavorconserving $CP$ odd observables are proportional to the
phases of $M_{\lambda} \mu (m_{12}^2)^*$, $A^* M_{\lambda}$, or $A \mu
(m_{12}^2)^*$, where $M_{\lambda}$ is any one of the three gaugino
masses. The phase of $M_3$ does not enter the electron EDM at oneloop.
It follows that the electron EDM is proportional at lowest order to at
least one insertion of $\mu$, $M_1$, or $M_2$. For $\mu, M_1, M_2
\sim {\cal O}({\rm GeV})$, one see that the electron EDM is suppressed by
${\cal O}(M/M_{\mbox{\tiny SUSY}})$, where $M \sim {\cal O}({\rm GeV})$.
This largely eliminates the SUSY $CP$ problem for atoms with unpaired
electrons which are sensitive to the electron EDM. In addition, if
leptonic $A$ terms are small, as would be the case if the approximate
$U(1)_R$ symmetry discussed above were extended to the leptonic sector,
the electron EDM would be suppressed by ${\cal O}(M/M_{\mbox{\tiny
SUSY}})^2$.
$CP$ violation in the strongly interacting sector depends on the gluino
mass $M_3$, and so is not necessarily suppressed in the
phenomenologically allowed window. However, if the approximate $U(1)_R$
is extended to the entire Lagrangian, then all gaugino masses, $\mu$,
and all $A$ terms are suppressed. The EDMs of the neutron and atoms
with paired electrons (which are senstive to strong sector $CP$
violation) are then suppressed by ${\cal O}(M/M_{\mbox{\tiny
SUSY}})^2$.\footnote{The full $U(1)_R$ symmetry imposed in
Refs.~\cite{7,11,12,13} is not required to solve the SUSY $CP$
problem. If any two types of the four classes of dimensionful
parameters $\{M_{\lambda}, \mu, A, m_{12} \}$ are ${\cal O}(M) \ll
M_{\mbox{\tiny SUSY}}$ at the high scale, then all $CP$ odd observables
are suppressed by ${\cal O}(M/M_{\mbox{\tiny SUSY}})^2$. This may be
verified by noting that $U(1)_{PQ}$ and $U(1)_{RPQ}$ field
redefinitions \cite{19} may be used to isolate the phases on the
small parameters.} Even in the RSB scheme given in the previous section
with a large gluino mass and small $A$ terms at the high scale, the
strong sector $CP$ violation is still suppressed by ${\cal
O}(M/M_{\mbox{\tiny SUSY}})$. This is apparent for the first and third
type of combinations of $CP$ violating parameters given above, as both
involve an insertion of $\mu$. For the second type this follows since,
even though a sizeable $A$ term can be induced by the gluino from
running to the low scale, the phase is then aligned with that of the
gluino mass, {\it i.e.}, ${\rm Arg}(A) \simeq {\rm Arg} (M_3)$.
Supersymmetric models can in principle give large enough oneloop
corrections to the $Z^0 b \bar{b}$ vertex to explain the $\sim 3\sigma$
discrepancy between the experimental \cite{20} and Standard Model
values of $R_b$ \cite{21,22,23,24}. The most important
contributions are from vertex corrections involving a top quark Yukawa
coupling $b_L \tilde{H}_2 \tilde{t}_R$. A sizeable effect requires a
light chargino with a substantial Higgsino component, $\tan
\beta \simeq 1$, and a light $\tilde{t}_R$
\cite{21,22,23,24}. The first two of these requirements
are met in the light Higgsinogaugino window. In addition, as
demonstrated earlier, it is also possible to arrange for a light
$\tilde{t}_R$ consistent with RSB. This is highly nontrivial, as it is
generally difficult to obtain solutions that explain the $R_b$
discrepancy consistent with RSB. (See, however, Ref.~\cite{23} for
a solution with conventional weakscale SUSY parameters.) The effect in
the light Higgsinogaugino window (see, for example, Ref.~\cite{24}),
may be determined from the figures of Ref.~\cite{21}. We find that
for $\tan \beta = 1$, $\mu = M_2 = 0$, and
$m_{\tilde{t}_R} = 100$ GeV, the SUSY shift in $R_b$ is $\delta R_b
\approx 0.002$, or roughly equal to what can be achieved in the Higgsino
region with similar chargino and stop masses.\footnote{Here we have
ignored leftright stop mixing angle suppressions. In principle, the
mixing, which in our case is due to weak scale $A$ parameters, can be
small for $m_{\tilde{t}_R} \lesssim m_t$ if the soft mass squared
$m_{\tilde{t}_R}^2$ is slightly negative at the weak scale (when
permitted by the stability of the potential).} This effect is, of
course, greatly increased for smaller $m_{\tilde{t}_R}$, and may
therefore significantly reduce (but not eliminate) the current
discrepancy of 0.006 between experiment and the Standard Model
\cite{20}.
Proton decay at oneloop is also suppressed in the allowed window. The
supersymmetric baryon violating coupling $QQQL$ must be dressed with an
offshell gaugino in order to obtain a fourFermi interaction. With
degenerate squarks, and ignoring any flavor changing, the gluino
contribution vanishes, so the largest dressing typically comes from
charginos \cite{25}. A chiral insertion is necessary on the
chargino line to obtain the fourFermi interaction. In order to avoid a
light quark Yukawa coupling to the Higgsino component of the chargino,
an $M_2$ insertion is required. The proton decay rate is then
suppressed in this limit at oneloop by ${\cal O} (M_2/M_{\mbox{\tiny
SUSY}})^2 \sim 10^{4}$ in the allowed window. If the gluino is
massive, gluino exchange could then dominate the decay rate if there are
flavor changing squark masses.
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