($p_{T}<100$ GeV)-leptonic SUSY phenomenology. The searches for pair-produced
$\tilde{\mu}^{\pm}_{2}\rightarrow\tilde{\chi}^{0}_{1}+\mu^{\pm}$ and
$\tilde{e}^{\pm}_{2}\rightarrow\tilde{\chi}^{0}_{1}+e^{\pm}$ call for
improvement in current LHC SUSY searches of softish leptons plus missing
energy Han:2013gba ; Han:2014nba ; ATLAS:2019lff ; ATLAS:2019lng ; CMS-PAS-
SUS-18-004 . Despite the challenging nature of this channel from the trigger,
signal rate, and SM background, given the multiple motivations for Sleptonic
SUSY the current searches should be improved as much as possible by exploring
the soft-lepton frontier. The experimental program at the LHC has started to
probe such soft signatures, typically with hard initial state radiation (ISR)
ATLAS:2019lng ; CMS-PAS-SUS-18-004 . Further upgrades and development in the
experimental search program could provide definitive information about the
allowed model parameter space.

Further, there are interesting opportunities to search for heavier sleptons
with larger mass splitting with the LSP, e.g., for (heavier) selectron and
smuon in BM1 and BM2, and the (heavier) staus in BM1, BM2, BM3 and BM4. One
can improve the selectron and smuon searches in the non-compressed regime with
higher statistics from the HL-LHC. (S)tau decays and their tagging suffer
lower efficiency and more background. Consequently, if the stau is not too
heavy, such as in BM1 and BM2, the searches for stau plus missing energy,
potentially with ISR, could help probe the model further CMS:2019zmn . The
charged-current production of charged sleptons plus sneutrinos has a larger
rate than neutral-current slepton production. However, the sneutrinos decay
invisibly to LSP plus neutrinos in this scenario. Therefore, while the charged
current typically provides a large production rate, the searches might suffer
a larger background and, hence, be less promising than the neutral current
production of charged sleptons Alvestad:2021sje . Nevertheless, new searches
could be designed to capitalize on the charged current production.

Amongst these new (sub)TeV states, wino pair-production through neutral- and
charged-current processes provide distinctive signatures. The wino
phenomenology is somewhat different from conventional SUSY discussions. Due to
the scale separation between the wino soft mass $M_{2}$ and large Higgsino
mass $\mu$, the dominant decays of the winos are through sleptons, not
directly into electroweak gauge bosons plus the LSP bino. For the wino-
chargino $\tilde{\chi}_{1}^{\pm}$, about half of the decays provide a visible
charged lepton plus missing energy; the rest decay into neutrinos plus
compressed sleptons. For wino-neutralino $\tilde{\chi}_{2}^{0}$, the decays
into sneutrino and neutrino pairs are invisible. The visible decays include
the decays into heavy (light) smuon/selectron and muon/electron pairs that
provide a hard muon/electron and a soft muon/electron and missing energy, and
the decays into staus and taus that provide two visible taus and missing
energy. Overall, the wino pair production has a sizable rate and can give rise
to multi-lepton (3 or 4 leptons) signatures, which is very promising to look
for at the LHC ATLAS:2019wgx ; CMS:2021cox ; CMS:2022nty . In fact, these
multilepton searches provide one of the leading constraints on our model, that
pushes the wino mass to be beyond 900 GeV for our benchmarks. In particular,
our signals typically yield mixed lepton flavors, off-$Z$ multi-leptons, that
would benefit from the improvement in these ongoing searches at the LHC.

###  11.2 $R$-Parity Violating Phenomenology

It is entirely plausible that $R$-parity is violated (RPV) so that all
superpartners are unstable, and dark matter has its origins in some other
sector. This gives rise to new and interesting experimental opportunities. The
LSP could be the light smuon 141414The selectron can be a few GeV heavier and
its major decay will be through the same RPV coupling (assuming it is lepton-
flavor blind). So the phenomenology of smuon also applies to selectrons., such
as in BM5 shown in Table 3. In this case, the phenomenology becomes very rich,
as one is required to turn on a tiny RPV coupling to make the light smuon
unstable so as to be compatible with cosmology. For instance, one can turn on
the $LQd^{c}$ superpotential operator that facilitates decays of the LSP smuon
decays into dijets. So long as one does not simultaneously turn on baryonic
RPV couplings that introduce severe constraints from proton decay, the
$LQd^{c}$ RPV coupling $\lambda^{\prime}$ can be as large as $10^{-3}$
Barger:1989rk ; Allanach:1999ic ; Chamoun:2020aft  without this new source of
flavor violation being constrained by flavor tests. Small R-parity symmetry-
violation is of course a natural possibility.

As long as $\lambda^{\prime}>10^{-7}$, the light sleptons decay promptly, so
there is a sizeable window where this happens and where there are no
experimental flavor constraints. The simplest direct search for pair-produced
smuons give rise to the signatures of dijet resonance pairs, and the limits
are relatively weak at the LHC ATLAS:2017jnp ; CMS:2018mts  due to the large
QCD background, around 120 GeV (from our estimation and reinterpreation of
these searches). Instead, one should look for the more interesting
phenomenology of pair production of the sneutrinos and heavy smuons. This new
channel will generate promising signatures of charged leptons from the
intermediate $Z$ or $W$ boson decay, plus a pair of dijet resonances. Even
better, the pair-produced stau states will give rise to 4 leptons plus dijet
resonance pairs. Searches for these lepton-rich dijet pair channels are
promising at the LHC Brust:2011tb  and could potentially cover the allowed
parameter space of our model.

For $\lambda^{\prime}<10^{-7}$, the slepton LSP is long-lived, producing
displaced decays or even heavy (collider-)stable charged particle (HSCP)
tracks at the LHC. The searches for displaced vertices and HSCP have gained a
lot of attention in the current LHC program, and will improve significantly in
the near future. The displaced vertex search rules out weakly produced
particles below $\sim 900$ GeV Liu:2015bma ; Liu:2018wte ; CMS:2020iwv ,
leaving room only for a longer lifetime of the smuon LSP. Current HSCP search
rules out sleptons below $\sim 460$ GeV Khachatryan:2016sfv ; Aaboud:2019trc ,
leaving BM5 of our model viable, with a RPV coupling
$\lambda^{\prime}<10^{-9}$. Instead of looking for the smuon LSP in future
searches, the HSCP will be produced in a prompt lepton-rich fashion through
the decays of heavier electroweakinos and heavier sleptons. The searches for
HSCP plus leptons provide new LHC opportunities that provide a promising venue
to test our model. Beyond the search for discovery from this channel, the
long-lived signatures will have low background and no ambiguity from the
combinatorics to reconstruct the underlying new state. Hence, one can
immediately access the quantum numbers of the newly discovered states from
this channel through the production rate, decay location distribution, angular
distribution, lepton flavor tagging, etc. One can then identify the SUSY
nature of the underlying physics readily.

##  12 Conclusions

We have argued in this paper that Sleptonic SUSY is an attractive framework
for new physics, from a variety of angles. Most broadly, it is truly
remarkable that the SUSY paradigm for particle physics can be realized in a
relatively elegant and economical manner in terms higher-dimensional
sequestering. This harmoniously results in gaugino-mediated, $D$-mediated and
Higgs-mediated SUSY breaking, in concert with the Giudice-Masiero mechanism
for a comparable $\mu$ parameter. These SUSY-breaking mechanisms automatically
respect and generalize the standard GIM mechanism, and robustly can avoid new
CP-violating phases, thereby satisfying the tightest flavor and EDM bounds
that tightly constrain any paradigm beyond the standard model. The higher-
dimensional framework can also be simply extended to realize orbifold grand
unification, with precision gauge-coupling, $b-\tau$ unification and doublet-
triplet splitting. Therefore, it is well worth asking what more specific
incarnations of this structure are motivated by our current experimental
situation.

The absence of colored superpartners thus far in LHC searches and the
heaviness of the $125$ GeV Higgs scalar suggest stops at $\sim 10$ TeV or
beyond, but this still leaves an opening for other superpartners to be within
LHC reach. In this way, Sleptonic SUSY, with its sub-TeV uncolored
superpartners and colored and $\sim 10$ TeV Higgs superpartners, stands out,
as both easily realizable within the above theoretical framework and
phenomenologically accessible. Additionally, while some options for realizing
the original hope that the LHC can re-create thermal-relic Dark Matter
particles are now ruled out, Sleptonic SUSY readily incorporates viable co-
annihilating dark matter at the bottom of its spectrum, which is potentially
discoverable at the LHC. It is of course very intriguing that there exists a
potential anomaly between the muon magnetic moment measurement and standard
model theory, and that Sleptonic SUSY again stands out as one of the most
plausible ways of explaining this consistent with an extended GIM
mechanism.151515While a sizeable smuon contribution to the muon $g-2$ is a
robust consequence of Sleptonic SUSY, its magnitude is quite sensitive to the
details of the model parameters. This can be consistent with the nominal size
of the potential anomaly, $\Delta a_{\mu}=259\pm 60\times 10^{-11}$. But if
the experimental anomaly or theoretical computation change in the future, such
changes can still readily be accommodated by modest changes within the
Sleptonic SUSY spectrum.

The Little Hierarchy Problem, of why new physics has not already appeared at
the weak scale to ensure its radiative stability, is an outstanding puzzle in
beyond-standard model theory in general. The Anthropic Principle operating
within a large multiverse with varying effective field theories has been
offered as an alternative paradigm to the naturalness principle underlying the
hierarchy problem. But it is quite possible that neither principle is outright
dominant, but rather anthropic selection “frustrates” naturalness, leaving a
little hierarchy problem. While a detailed understanding of anthropic
selection and multiverse distributions is still lacking, a plausible corollary
of such selection is the Principle of Living Dangerously in which the IR
dynamics should be close to an anthropically-significant phase transition.
From this viewpoint, it is intriguing that Sleptonic SUSY is locally trapped
between two “dangerous” phase transitions: if we de-tune the theory to make it
slightly more natural, in one direction we pass to a phase of unbroken
electroweak symmetry while in the other direction we pass to a phase in which
the photon is massive!

We have generalized UV models of Sleptonic SUSY in a variety of ways, more
fully covering the MSSM parameter space, especially the regime of moderate
$\tan\beta$ as well as the RPV scenario and phenomenology. Furthermore, we
developed and studied a non-minimal model in which the cosmological stability
constraints of the electroweak vacuum are mitigated, thereby allowing richer
phenomenological and cosmological options. For simplicity, we considered the
regime in which UV $A$-terms are subdominant. While we argued they are not
qualitatively important, it would be useful to include them in a more thorough
exploration of the phenomenology, in particular that of the stops.

Experimentally, the general Sleptonic SUSY scenario has many interesting
facets relevant to discovery at the LHC and future colliders. Overall,
sleptons and electroweakinos at weak-TeV scales call for more phenomenological
and experimental explorations. In the bino DM and slepton NLSP case, while
continuing to improve missing energy searches at the LHC, one can also focus
more on the associated heavier sleptons that provide more visible leptons in
the semi-compressed regime. Further, in the RPV case, one can conduct new
searches that significantly enhance the discovery potential of current
searches by requiring additional prompt leptons in the final state, for both
promptly-decaying LSP searches and long-lived LSP searches. These new search
channels could well be the simultaneous discovery channels for LSP and heavier
states. If the sleptonic SUSY spectrum is highly compressed, rendering these
channels very challenging at the LHC, the LHC can still look for the $\sim$
TeV scale Winos decaying into the various weak scale sleptonic states, whose
reach will improve steadily with higher luminosity.

Beyond discovery, one can more ambitiously try to pin down quantum numbers of
the new states to provide evidence that supersymmetry is at work, e.g. to
distinguish from other possible BSM scenarios such as fermionic “lepton
partners” or other weakly-charged new states. In the bino DM and slepton NLSP
case, one needs sizable statistics to extract the angular correlations from
backgrounds, which is not easy to achieve even at HL-LHC given the current
exclusions. Interestingly, in the long-lived RPV scenario, e.g., BM4 with
small RPV coupling, one can pair-produce binos, which provide prompt leptons
to trigger on and the low-background displaced-vertices can extract the spin-
quantum number more easily, providing an exciting program of LHC studies.

Even if a modestly compressed Sleptonic SUSY spectrum evades discovery at the
LHC, it would be guaranteed discovery at future colliders. Future lepton
colliders, such as ILC, CLIC or muon colliders, can readily achieve center-of-
mass energy greater than twice the $\sim$ weak-scale slepton mass. We will
then be able to make definitive discoveries through slepton pair production in
all scenarios, and determine the quantum numbers through production rates,
differential observables, etc. Even at the low-energy stage of future lepton
colliders, operating at the $Z$-pole, it may be possible to detect the shift
in the $Z\overline{\tau}\tau$ coupling loop-mediated by weak-scale staus, due
to the large mixing that connects to the large shifts in $g-2$ (of muon and
$\tau$), the near-instability of the EW vacuum, and frustrated naturalness.
Staus this light, consistent with cosmologically stable EW vacuum, are most
straightforwardly achieved within our non-minimal model.

In this paper, we have focused on the limit in which the GIM mechanism is
perfectly satisfied, with negligible new sources of BSM flavor violation.
However, it is possible that there are observably large deviations from such a
limit. This can obviously happen if there are sizeable RPV couplings. But even
with R-parity, as we reviewed, extra-dimensional sequestering can be
imperfect, introducing small flavor-violating SUSY-breaking effects. Either
case can lead to opportunities for new signals in flavor physics, in
particular associated to the third generation.

At future hadron colliders, such as FCC-hh and SPPC, we will be able to make a
full discovery by observing the 10 TeV scale squarks and gluinos. Even at a
lower center-of-mass energy below the production of the colored states, we
will likely make the first discovery by observing the Winos and their decays
into lepton-rich final states.

We have seen that Sleptonic SUSY, especially in the regime with sizeable
corrections to the muon magnetic moment, can readily have a deeper vacuum than
the standard EW vacuum, in which staus condense and Higgs electromagnetism.
While the EW vacuum can have cosmological lifetime at low temperatures, it
will be important to study a fuller cosmological history to assess whether the
EW vacuum is stable enough when thermal effects are included starting from
inflationary reheating. It will also be important to improve upon the vacuum
lifetime estimates presented here. It is however possible to find regions of
parameter space in which the EW vacuum is absolutely stable, in particular in
our non-minimal model.

###### Acknowledgements.

We would like to thank Z. Chacko, Junwu Huang, Keith Olive, Riccardo Rattazzi,
and Tuhin Roy for helpful discussions. We would also like to thank Tsutomu
Yanagida for bringing ref. Inoue:1991rk  to our attention. KA, ME and RS were
supported in part by the NSF grant PHY-1914731 and by the Maryland Center for
Fundamental Physics. ME was also supported by the Swiss National Science
Foundation under contract 200020-188671 and through the National Center of
Competence in Research SwissMAP. ZL was supported in part by the U.S.
Department of Energy (DOE) under grant No. DE-SC0022345. ZL would like to
thank the Aspen Center for Physics, supported by National Science Foundation
(NSF) grant PHY-1607611, where part of this work was completed.

##  Appendix A A Toy Model for Vacuum (Meta-)stability Considerations

This appendix is not necessary for the numerical estimates of the EW
metastable vacuum lifetime, which we performed as outlined in the main text,
but it does provide a useful analytic toy model with only one scalar field
which makes the essential/parametric considerations of the vacuum stability
analysis involving $H_{u}$ and stau fields transparent. Consider a single real
scalar field $\phi$ with the following Lagrangian:

$\mathcal{L}=\frac{1}{2}\left(\partial\phi\right)^{2}-\frac{1}{2}m^{2}\phi^{2}+\frac{1}{3}\delta\,\phi^{3}-\frac{1}{4}\lambda\,\phi^{4}.$
(39)

To connect with the potential of eq. (9.1), we can think of $\phi$ as the
distance in the field space from the standard EM preserving vacuum along the
direction connecting the standard vacuum to the possible EM breaking vacuum.
$m^{2}$ then corresponds to a combination of the stau and the SM Higgs mass
terms, which if stau masses are larger, is dominantly set by $m^{2}_{L_{3}}$
and $m^{2}_{\bar{e}_{3}}$. The cubic coupling $\delta$ corresponds
parametrically to $y_{\tau}\mu$ and $\lambda$ to a combination of quartic
couplings set by the gauge couplings, $y_{\tau}$, and the one loop Higgs
quartic $\delta\lambda_{H}$ induced by top and stop loops.

The potential for this field (assuming $m^{2}>0$) has a minimum at $\phi=0$
(corresponding to the standard vacuum). The condition for this to be the
absolute minimum of the potential is same as the condition for the potential
being non-negative for all $\phi$ which is

$\delta^{2}<\frac{9}{2}m^{2}\lambda.$ (40)

This corresponds parametrically to smallness of the ratio $\mu
y_{\tau}/(g\,m_{\tilde{\tau}})$.

Now let us assume that this condition is not satisfied and there is another
deeper minimum for the potential. We can then look for the action of the
bounce solution for tunneling out of the $\phi=0$ vacuum. By a field and
coordinate rescaling, the Euclidean action for the $O(4)$-symmetric bounce can
be put in the following form:

$S=2\pi^{2}\frac{m^{2}}{\delta^{2}}\int
d\tilde{r}\tilde{r}^{3}\left[\frac{1}{2}\left(\frac{d\tilde{\phi}}{d\tilde{r}}\right)^{2}+\frac{1}{2}\tilde{\phi}^{2}-\frac{1}{3}\tilde{\phi}^{3}+\frac{1}{4}\frac{\lambda
m^{2}}{\delta^{2}}\tilde{\phi}^{4}\right].$ (41)

The bounce is a solution to the equation of motion following from the action
above with the boundary conditions $\tilde{\phi}\rightarrow 0$ as
$\tilde{r}\rightarrow\infty$ and $\frac{d\tilde{\phi}}{d\tilde{r}}=0$ at
$\tilde{r}=0$ Coleman:1977py . Since the boundary conditions do not introduce
any further dependence on the parameters of the potential, the bounce action
has the following parametric form:

$S_{B}\approx 204\frac{m^{2}}{\delta^{2}}f_{B}\left(\frac{\lambda
m^{2}}{\delta^{2}}\right)$ (42)

The prefactor has been chosen such that $f_{B}(0)=1$ and it has been fixed by
numerically solving the bounce for $\lambda=0$. For small $\frac{\lambda
m^{2}}{\delta^{2}}$, $S_{B}$ can be perturbatively corrected by expanding in
$\frac{\lambda m^{2}}{\delta^{2}}$. We obtain numerically, for small $x$,

$f_{B}(x)\approx
1+1.4\frac{9x}{2}+1.9\left(\frac{9x}{2}\right)^{2}+2\left(\frac{9x}{2}\right)^{3}+\mathcal{O}(x^{4}).$
(43)

Also we know from eq. (40) that $\lim_{x\rightarrow\frac{2}{9}}f(x)=\infty$,
i.e. the lifetime should become infinite as the $\phi\neq 0$ minimum becomes
degenerate with the $\phi=0$ one. In terms of the parameters of the potential
of eq. (9.1), for large $\mu$ and thus large $\delta$, we can set $f\approx 1$
in eq. (43) so that
$S_{B}\propto\frac{m^{2}_{\tilde{\tau}}}{(\mu\,y_{\tau})^{2}}$ using eq. (42).
As we decrease $\mu$ (keeping other parameters fixed), bounce action
increases: at some point, $\delta$ becomes small enough that the full
expression for $f$ in eq. (43) has to be used. The bounce action continues to
increase as we reduce $\mu$ further until as $\mu$ decreases, it diverges when
the EM conserving and the EM breaking vacua become degenerate. In that limit
the bounce action can be computed analytically using the thin-wall
approximation giving

$S_{B}\approx\frac{3\pi^{2}m^{2}}{4\delta^{2}\left(1-\frac{9\lambda
m^{2}}{2\delta^{2}}\right)^{3}}\approx
7.4\frac{m^{2}}{\delta^{2}\left(1-\frac{9\lambda
m^{2}}{2\delta^{2}}\right)^{3}}.$ (44)

In the models considered here, unless $\tan\beta$ is very large, $\lambda$ is
set primarily by the gauge couplings and is not sensitive to the other
parameters. Then to maximize the lifetime of the vacuum (or to achieve
absolute stability), one can maximize the ratio
$\frac{m^{2}}{\delta^{2}}\sim\frac{m^{2}_{\tilde{\tau}}}{(\mu\,y_{\tau})^{2}}$.
In the MSSM realization of Sleptonic SUSY, without the help from
$y_{\tau}$-mediated contribution to the stau soft mass-squareds proportional
to large and negative $m^{2}_{H_{d}}$, eq. (16), this ratio is too small to
lead to a cosmologically meta-stable vacuum, once the $125$ GeV Higgs, EWSB
scale and the muon $g-2$ are fit, and it is essential to use the yukawa/Higgs
mediation RG effects to increase the stau masses. Moreover, importantly,
increasing this effect does not indefinitely increase the vacuum lifetime, and
is not enough to remove the EM breaking minimum. To see this assume that
$|m^{2}_{H_{d}}|$ and/or $\tan\beta$ is large enough so that the stau masses
are dominated by the $y_{\tau}$-mediation effect, that is
$m^{2}_{\tilde{\tau}}\propto-m^{2}_{H_{d}}\tan^{2}\beta$. In this regime,
$\mu$ also needs to be large enough ($|\mu|^{2}+m^{2}_{H_{d}}>0$) to prevent
large $H_{d}$ condensation. Therefore, we see that in this regime the ratio
$\frac{m^{2}}{\delta^{2}}$ is at most of order one, and it turns out that it
is numerically not large enough to lead to an absolutely stable vacuum. In
fact in this limit, the bounce action turns out to be close to the limit for
the vacuum to have a lifetime of the order of the age of the universe today.

However, in the non-minimal model, $\mu$ is not constrained to be large by
$m^{2}_{H_{d,u}}$ anymore, as the non-condensation of $H_{d}$ can be achieved
using other generalized $\mu$-terms. Since $\mu$ can then be chosen to be
smaller in the non-minimal model and $\delta\propto\mu$, we see that $\delta$,
in turn, is allowed to be smaller, enough for having an absolutely stable
standard EWSB vacuum.

##  Appendix B Other Charge- and/or Color- breaking Vacua

As it was mentioned in section 9, there may be minima of the potential with
nonzero VEVs for fields that are not included in the potential of eq. (9.1).
In this appendix we briefly discuss the regions in field space where such
charge- and/or color-breaking vacua my lie.

In the absence of SUSY-breaking, there exist paths in the field space along
which the potential is (approximately) flat, that is the potential arising
from the $D$-terms and $F$-terms vanish. The soft SUSY breaking effects can
either lift the potential along these paths, or lower it which can lead to new
minima of the potential, or even possibly to potentials that are unbounded
from below Komatsu:1988mt ; Casas:1995pd ; Strumia:1996pr . For the models
studied in this paper, we have considered scenarios where the soft mass
squared terms for $m^{2}_{H_{u,d}}$ are negative. So the potentially
problematic flat paths involve the associated Higgs fields. In the following,
we first consider the MSSM and then later briefly comment on the corrections
in the non-minimal model. In particular the following three paths can lead to
charge- and/or color- breaking vacua:

(i) $H_{u},\tilde{\tau}_{L},\tilde{\tau}_{R},\tilde{\nu}_{i\neq 3}\neq 0$ and
other fields set to zero. Only one of the sneutrinos of the first two
generations need to be non-zero. The $F$-term for $H_{d}$ can be set to zero
relating the staus and $H_{u}$,
$\tilde{\tau}_{L}\tilde{\tau}_{R}^{*}=-\frac{y_{\tau}}{\mu}H_{u}$, and the
$D$-term potential vanishes if

$|\tilde{\nu}|^{2}=|H_{u}|^{2}+|\tilde{\tau}_{L}|^{2}=|H_{u}|^{2}+|\tilde{\tau}_{R}|^{2}$
(45)

(ii) $H_{u},\tilde{b}_{L},\tilde{b}_{R},\tilde{\nu}\neq 0$ and other fields
set to zero, similar to path (i), with sbottoms instead of staus cancelling
the $F$-term for $H_{u}$. The $SU(3)$ $D$-term also vanishes if the two
sbottom fields have equal magnitudes.

(iii) $H_{d}^{0},\tilde{t}_{L},\tilde{t}_{R},\tilde{e}_{L},\tilde{e}_{R}\neq
0$: and all other scalars set to zero. Setting
$|\tilde{t}_{L}|=|\tilde{t}_{R}|\equiv\tilde{t}$, the $SU(3)$ $D$-term
vanishes. Then setting $|\tilde{t}^{2}|=|\frac{\mu}{y_{t}}H_{d}^{0}|$, and
choosing appropriate phases/signs, the $F$-term for $H^{0}_{u}$ vanishes. The
charged component of the doublet selectron $\tilde{e}_{L}$ can be used to set
the $SU(2)$ $D$-term to zero, and finally the hypercharge $D$-term can be set
to zero using the $SU(2)$-singlet selectron,

$|\tilde{e}_{R}|^{2}=|\tilde{e}_{L}|^{2}=|H_{d}^{0}|^{2}+|\tilde{t}^{2}|.$
(46)

The supersymmetric potential along this path is however not exactly flat since
there are quartic couplings $\propto y_{e}^{2}$ from $F$-terms corresponding
to $H_{d}$ and slectrons, but these quartic couplings are very small,
$y_{e}^{2}\sim 10^{-11}(\tan\beta)^{2}$.

To see the possibility of the potential charge and/or color breaking vacua
more clearly, one can write the potential in terms of only one of the nonzero
fields, and including the soft SUSY breaking terms. For example for path
(iii), in terms of $H_{d}^{0}\equiv\varphi$,

$V(\varphi)=(m^{2}_{H_{d}}+m^{2}_{L_{1}}+m^{2}_{\bar{e}_{1}})|\varphi|^{2}+\frac{|\mu|}{y_{t}}(m^{2}_{Q_{3}}+m^{2}_{\bar{u}_{3}}+m^{2}_{L_{1}}+m^{2}_{\bar{e}_{1}})|\varphi|+\delta\lambda_{H_{d}}|\varphi|^{4}+\mathcal{O}(y_{e}^{4}),$
(47)

where $\delta\lambda_{H_{d}}$ is the one-loop quartic coupling for $H_{d}$
dominantly induced by bottom/sbottom loops. This shows that if
$m^{2}_{H_{d}}+m^{2}_{L_{1}}+m^{2}_{\bar{e}_{1}}<0$, the EWSB vacuum can be
destabilized.

Having outlined the different directions in which deeper vacua than the EWSB
vacuum can be encountered, we of course need the potential energy to be
bounded from below in order for the theory to be physically sensible. In the
renormalizable limit of the MSSM, in which we also include Dirac neutrinos,
the charged lepton Yukawa coupling effects (such as the last term in eq. (47)
above) and neutrino Yukawa coupling effects give rise to (small) quartic
couplings that bound the potential at very large field values. Further, there
may be other stabilizing effects in the very far UV where the MSSM is
completed into a unified or string theory. None of these effects would be
relevant at the moderate field values which dominate the tunneling processes
determining the EWSB vacuum lifetime, and are therefore dropped in the related
estimates.

In the non-minimal model, the $F$-terms for $H^{\prime}$ fields are nonzero
along the above paths. In order to set those $F$-terms to zero, one of the
$H^{\prime}$ scalars should be non-zero, but then the large positive
$m^{2}_{H^{\prime}}$ stabilize the potential. Also with vanishing
$H^{\prime}$, the positive contribution of $H^{\prime}$ F-terms, lead to a
more stable standard vacuum. In particular, for path (iii), the
$H^{\prime}_{u}$ F-term adds a contribution $|\mu_{d}|^{2}|\varphi|^{2}$ to
the potential in eq. (47). Therefore a choice of
$|\mu_{d}|^{2}>-m^{2}_{H_{d}}$ would be sufficient to avoid a minimum along
this path.

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