OU-HEP-220301

Mini-review: Expectations for supersymmetry
from the string landscape

Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

Howard Baer1111Email<EMAIL_ADDRESS>, Vernon Barger2222Email:
<EMAIL_ADDRESS>Shadman Salam1333Email<EMAIL_ADDRESS>and
Dibyashree Sengupta3444Email<EMAIL_ADDRESS>
1Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma,
Norman, OK 73019, USA
2Department of Physics, University of Wisconsin, Madison, WI 53706 USA
3Department of Physics, National Taiwan University, Taipei, Taiwan 10617,
R.O.C.

In this mini-review, we summarize a variety of findings pertaining to
consequences of the landscape of string theory for supersymmetry (SUSY)
phenomenology. The idea is to adopt the MSSM as the most parsimonious $4-d$
low energy EFT after string compactification but where the scale of SUSY
breaking is as yet undetermined. A power-law landscape draw to large soft
terms is tempered by the requirement that the derived value of the weak scale
lie within the anthropic window of Agrawal et al. (ABDS). Such a set-up
predicts a light Higgs mass $m_{h}\sim 125$ GeV with sparticles generally
beyond LHC bounds. We discuss consequences for LHC searches: light higgsinos,
highly mixed TeV-scale top squarks, same-sign diboson events and
$m_{\tilde{g}}\sim 2-5$ TeV. We expect dark matter to consist of an
axion/higgsino-like WIMP admixture.

##  1 Introduction and set up

So far, superstring theory provides our only successful unification of quantum
mechanics with general relativity[1, 2]. However, to avoid anomalies, then the
dimensionality of spacetime must be increased to ten (or even eleven in
$M$-theory[3]). To gain accord with the $4-d$ physics of the Standard Model,
one must compactify the extra six dimensions on a suitable compact manifold,
such a Calabi-Yau (CY) manifold, which preserves $N=1$ supersymmetry in the
low energy effective field theory (EFT)[4]. The $4-d$ laws of physics after
compactification then depend on various properties of the compact manifold,
which are parametrized by vacuum expectation values (vevs) of moduli fields:
gravitationally interacting scalar fields with a flat classical potential. To
gain a realistic and potentially predictive theory, the moduli must be
stabilized so as to avoid runaway solutions and to gain well-determined vevs
which in turn determine various low energy properties such as the values of
gauge and Yukawa couplings and soft SUSY breaking terms[5].

At the turn of the 21st century, it was realized that the number of
possibilities for compact CY manifolds was far, far greater than
anticipated[6]. Under flux comactifications[7], the number of distinct vacuum
states might range from $10^{500}$[8] to $10^{272,000}$[9]. These are more
than enough to implement Weinberg’s anthropic solution to the cosmological
constant (CC) problem[10]. It also provides a new understanding of how vastly
different energy scales may emerge from string theory which contains only the
string scale $m_{s}$ at its most fundamental level. In an eternally inflating
multiverse[11, 12], different pocket universes can arise with different
physical constants. Requiring a pocket universe to allow for large scale
structure in the form of galaxies and clusters, then only those with a tiny
cosmological constant are allowed. Indeed, this approach allowed Weinberg to
predict the value of the cosmological constant to a factor of a few a decade
before its tiny yet non-zero value was measured[10, 13].

A similar approach may be applied to the origin of the weak scale
$m_{weak}\sim m_{W,Z,h}$ in models with weak scale supersymmetry[14] (WSS)
(for a recent review of WSS after LHC Run 2, see Ref. [15]). In models such as
the MSSM, the weak scale is determined by the soft SUSY breaking parameters
and the SUSY conserving $\mu$ parameter111Twenty solutions to the SUSY $\mu$
problem are reviewed in Ref. [16].. Minimizing the MSSM scalar (Higgs)
potential, one finds

$m_{Z}^{2}/2=\frac{m_{H_{d}}^{2}+\Sigma_{d}^{d}-(m_{H_{u}}^{2}+\Sigma_{u}^{u})\tan^{2}\beta}{\tan^{2}\beta-1}-\mu^{2}$
(1)

Here, $m_{H_{u}}^{2}$ and $m_{H_{d}}^{2}$ are the weak scale values of the
soft SUSY breaking Higgs masses, $\tan\beta\equiv v_{u}/v_{d}$ is the ratio of
Higgs field vevs and the terms $\Sigma_{u}^{u}$ and $\Sigma_{d}^{d}$ contain
over 40 loop corrections to the Higgs potential (for a tabulation, see Ref’s
[17] and [18]). The largest of these typically come from the top squarks:
$\Sigma_{u}^{u}(\tilde{t}_{1,2})$. Equation 1 allows for a definition of the
weak scale finetuning measure $\Delta_{EW}\equiv|largest\ term\ on\ RHS\
|/(m_{Z}^{2}/2)$[19].

If we adopt a so-called fertile patch of the landscape– those vacua whose
$4-d$ low energy EFT is, by parsimony, the MSSM– then we would expect the
magnitude of the weak scale to vary in each pocket universe depending on the
values of the soft breaking terms and $\mu$ parameter in that same pocket
universe[20]. We can write the distribution of vacua versus the soft SUSY
breaking scale $m_{soft}$ (where $m_{soft}\simeq m_{SUSY}^{2}/m_{P}$ in
gravity mediation where $m_{SUSY}\sim 10^{11}$ GeV is the mass scale
associated with hidden sector SUSY breaking) as

$dN_{vac}\sim f_{SUSY}(m_{soft})\cdot f_{EWSB}(m_{soft})\cdot dm_{soft}.$ (2)

Douglas and others[21, 22, 23] originally proposed that

$f_{SUSY}\sim m_{soft}^{2n_{F}+n_{D}-1}$ (3)

where $n_{F}$ is the number of $F$-term SUSY breaking fields and $n_{D}$ is
the number of $D$-term SUSY breaking fields contributing to the overall SUSY
breaking scale. The factor 2 arises since it is expected that the vevs $F_{i}$
are distributed uniformly as complex numbers whilst the $D_{j}$ are
distributed uniformly as real numbers. It was realized shortly thereafter that
this may be too simplistic in that the sources of SUSY breaking may not all be
independent[24]. But even so, for the textbook case of spontaneous SUSY
breaking by a single $F$-term, then already one expects soft terms to be
statistically favored to large values by a linear distribution $f_{SUSY}\sim
m_{soft}^{1}$. An alternative– emphasized by Dine et al.[25, 26]– is that SUSY
is broken non-perturbatively in a hidden sector via e.g. dynamical SUSY
breaking either via gaugino condensation or via instanton effects. In such a
case, then no SUSY breaking scale is favored over any other, which would
result in $f_{SUSY}\sim m_{soft}^{-1}$. These results have some further
support from Broeckel et al.[27] where they investigate the statistics of SUSY
breaking in the landscape including considerations of Kähler moduli ($T_{i}$)
stabilization. For the large volume scenario (LVS)[28] where the $T_{i}$ are
stabilized by a balance between perturbative and non-perturbative effects,
then they find $f_{SUSY}\sim m_{soft}^{-1}$ while for non-perturbative Kähler
moduli stabilization as in KKLT[29] they find $f_{SUSY}\sim m_{soft}^{1}$.
Thus, in the following, we will compare statistical predictions from the
string landscape assuming $f_{SUSY}\sim m_{soft}^{\pm 1}$.

The other relevant distribution contains (anthropic) selection effects in
$f_{EWSB}$. Agrawal, Barr, Donoghue and Seckel (ABDS)[30, 31] showed already
in 1997 that too large a value of the weak scale in various causally
disconnected domains of the multiverse would lead, via the up-down quark mass
difference, to unstable nuclei and lack of atoms which are apparently needed
for life as we know it. They estimate that for pocket universes with
$m_{weak}^{PU}\gtrsim(2-5)m_{weak}^{OU}$, then one is in violation of this so-
called atomic principle (where $m_{weak}^{OU}$ corresponds to the magnitude of
the weak scale in our universe). Without finetuning, then the pocket universe
value of the weak scale corresponds to the maximal term on the RHS of Eq. 1.
Then requiring for definiteness $m_{weak}^{PU}<4m_{weak}^{OU}$ corresponds to
$\Delta_{EW}\lesssim 30$. Allowing for finetuning (where $m_{Z}$ is not
hardwired to 91.2 GeV) then much the same results are obtained in Ref. [32].

##  2 Sparticle and Higgs mass distributions from the landscape: $n=\pm 1$

In our present discussion, we will adopt the 3 extra parameter non-universal
Higgs model (NUHM3) for explicit calculations[33, 34, 35]. In this model, the
matter scalars of the first two generations are assumed to live in the
16-dimensional spinor of $SO(10)$ as is expected in string models exhibiting
local grand unification, where different gauge groups are present at different
locales on the compactified manifold[36]. In this case, it is really expected
that each generation acquires different $m_{0}(1)$, $m_{0}(2)$ and $m_{0}(3)$
soft breaking masses. But for simplicity of presentation, we will assume
first/second generational degeneracy. At first glance, one might expect that
the generational non-degeneracy would lead to violation of flavor-changing-
neutral-current (FCNC) bounds. The FCNC bounds mainly apply to first-second
generation nonuniversality[37]. However, the landscape itself allows a
solution to the SUSY flavor problem in that it statistically pulls all
generations to large values provided they do not contribute too much to
$m_{weak}^{PU}$. This means the 3rd generation is pulled to $\sim$ several TeV
values whilst first and second generation scalars are pulled to values in the
$10-50$ TeV range. The first and second generation scalar contributions to the
weak scale are suppressed by their small Yukawa couplings[17], whilst their
$D$-term contributions largely cancel under intra-generational
universality[38]. Their main influence on the weak scale then comes from two-
loop RGE contributions which, when large, suppress third generation soft term
running leading to tachyonic stop soft terms and possible charge-or-color
breaking (CCB) vacua which we anthropically veto[39, 40]. These latter bounds
are flavor independent so that first/second generation soft terms are pulled
to common upper bounds leading to a quasi-degeneracy/decoupling solution to
both the SUSY flavor and CP problems[41]. Meanwhile, Higgs multiplets which
live in different GUT representations are expected to have independent soft
masses $m_{H_{u}}$ and $m_{H_{d}}$222 In models of local grand unification,
the matter multiplets can live in the the $SO(10)$ spinor representations
while the Higgs and gauge fields live in split multiplets due to their
geography on the compactified manifold[42].. Thus, we expect a parameter space
of the NUHM3 models as

$m_{0}(1)\simeq m_{0}(2),\ m_{0}(3),\ m_{H_{u}},\ m_{H_{d}},\ m_{1/2},\
A_{0},\ \mu,\ {\rm and}\ m_{A},$ (4)

where we used the EW minimization conditions allow for the more convenient
weak scale variables $\mu$ and $m_{A}$. For simplicity, sometimes we will
adopt $m_{0}(1)=m_{0}(2)=m_{0}(3)$ in which case we have the NUHM2 model.

Since $n=1$ is expected for KKLT moduli stabilization, then it is perhaps more
warranted to use the generalized mixed moduli anomaly (mirage) mediation model
GMM[43].333Also of interest is the more general anomaly-mediation model with
separate bulk contributions to Higgs soft terms and a bulk $A$-term
contribution[44]. Then one can obtain naturalness within the AMSB framework;
in this case, while winos are the lightest gauginos, the higgsinos are the
lightest EWinos. Applying landscape statistics to the GMM model, then the
magnitude of the mirage unification scale can be predicted. For results, see
Ref. [45].

In Fig. 1[46], we show the $A_{0}$ vs. $m_{0}$ plane for the NUHM2 model with
$m_{1/2}$ fixed at 1 TeV, $\tan\beta=10$ and $m_{H_{d}}=1$ TeV. We take
$m_{H_{u}}=1.3m_{0}$. The plane is qualitatively similar for different
reasonable parameter choices. We expect $A_{0}$ and $m_{0}$ statistically to
be drawn as large as possible while also being anthropically drawn towards
$m_{weak}\sim 100-200$ GeV, labelled as the red region where $m_{weak}<500$
GeV. The blue region has $m_{weak}>1.9$ TeV and the green contour labels
$m_{weak}=1$ TeV. The arrows denote the combined statistical/anthropic pull on
the soft terms: towards large soft terms but low $m_{weak}$. The black contour
denotes $m_{h}=123$ GeV with the regions to the upper left (or upper right,
barely visible) containing larger values of $m_{h}$. We see that the combined
pull on soft terms brings us to the region where $m_{h}\sim 125$ GeV is
generated. This region is characterized by highly mixed TeV-scale top
squarks[47, 48]. If instead $A_{0}$ is pulled too large, then the stop soft
term $m_{U_{3}}^{2}$ is driven tachyonic resulting in charge and color
breaking minima in the scalar potential (labelled CCB). If $m_{0}$ is pulled
too high for fixed $A_{0}$, then electroweak symmetry isn’t even broken.

Figure 1: Contours of $m_{weak}$ in the $A_{0}$ vs. $m_{0}$ plane for
$m_{1/2}=1$ TeV, $m_{H_{u}}=1.3m_{0}$, $\tan\beta=10$ and $m_{H_{d}}=1$ TeV.
The arrows show the direction of statistical/anthropic pull on soft SUSY
breaking terms. Within the black contour is where $m_{h}>123$ GeV. There is
also a slight black contour in the upper-right horn as well.

Next, we scan over parameter values

$\displaystyle m_{0}(1,2)$ $\displaystyle:$ $\displaystyle 0-60\ {\rm TeV}$
(5) $\displaystyle m_{0}(3)$ $\displaystyle:$ $\displaystyle 0.1-10\ {\rm
TeV}$ (6) $\displaystyle m_{H_{u}}$ $\displaystyle:$ $\displaystyle
m_{0}(3)-2m_{0}(3)$ (7) $\displaystyle m_{H_{d}}(\sim m_{A})$ $\displaystyle:$
$\displaystyle 0.3-10\ {\rm TeV}$ (8) $\displaystyle m_{1/2}$ $\displaystyle:$
$\displaystyle 0.5-3\ {\rm TeV}$ (9) $\displaystyle-A_{0}$ $\displaystyle:$
$\displaystyle 0-50\ {\rm TeV}$ (10) $\displaystyle\mu_{GUT}$ $\displaystyle:$
$\displaystyle\ fixed$ (11) $\displaystyle\tan\beta$ $\displaystyle:$
$\displaystyle 3-60$ (12)

The soft terms are all scanned according to $f_{SUSY}\sim m_{soft}^{\pm 1}$
while $\mu$ is fixed at a natural value $\mu=150$ GeV. For $\tan\beta$, we
scan uniformly. The goal is to take scan upper limits beyond those imposed by
$f_{EWSB}$ so the plot upper bounds do not depend on scan limits. The lower
limits for the $n=-1$ case are selected in accord with previous scans for
$n=1$ with a draw to large soft terms just for consistency. If we lower the
lower bound scan limits, then the $n=-1$ histograms will migrate to what
becomes even worse discord with experimental limits.

In Fig. 2, we show putative landscape distributions for various NUHM3
parameters. In frame a), we show the distribution for first/second generation
scalar masses $m_{0}(1,2)$. For $n=1$, then we see the probability
distributions peaks around $m_{0}(1,2)\sim 20$ TeV but extends as high as
$\sim 45$ TeV. Such large first/second generation scalar masses provide the
decoupling/quasi-degeneracy soluton to the SUSY flavor and CP problems. In
contrast, for $n=-1$ then the distribution is sharply peaked near 0 as
expected. In frame b), we show the distribution in third generation scalar
soft mass $m_{0}(3)$. Here, the $n=1$ distribution peaks at 5 TeV but runs as
high as 10 TeV. The $n=-1$ distribution again peaks at zero, which will lead
to very light third generation squarks. The distribution in $m_{1/2}$ shown in
frame c) for $n=1$ peaks around $m_{1/2}\sim 1.5$ TeV leading to gaugino
masses typically beyond the present LHC limits. For $n=-1$, the distribution
peaks at low $m_{1/2}$ leading to gauginos that are typically excluded. And in
frame d) we see the distribution in trilinear soft term $-A_{0}$. For $n=1$,
the distribution has a double peak structure with most values in the multi-TeV
range leading to large stop mixing and consequently cancellations in the
$\Sigma_{u}^{u}(\tilde{t}_{1,2})$ and upift of $m_{h}$ to $\sim 125$ GeV. For
$n=-1$, then $A_{0}$ peaks around zero, and we expect little stop mixing and
lighter values of $m_{h}$.

Figure 2: Probability distributions for NUHM3 soft terms a) $m_{0}(1,2)$, b)
$m_{0}(3)$, c) $m_{1/2}$ and d) $A_{0}$ from the $f_{SUSY}=m_{soft}^{\pm 1}$
distributions of soft terms in the string landscape with $\mu=150$ GeV.

In Fig. 3, we plot the landscape distributions for light and heavy SUSY Higgs
boson masses. In frame a), for $n=1$ we see a distribution with a strong peak
around $m_{h}\sim 124-126$ GeV in accord with data. The distribution cuts off
for $m_{h}\gtrsim 127$ GeV because otherwise the
$\Sigma_{u}^{u}(\tilde{t}_{1,2})$ contributions become too large leading to
too large a value of $m_{weak}$ beyond the ABDS window. For $n=-1$, the
distribution peaks at $m_{h}\sim 118$ GeV with really no significant
probability beyond $m_{h}\sim 124$ GeV. This essentially rules out the $n=-1$
case. In frame b), the distribution in heavy pseudoscalar mass $m_{A}$ is
shown. For $n=+1$, the distribution peaks at $m_{A}\sim 2.5$ TeV with a
distribution extending as high as $m_{A}\sim 8$ TeV. These values are well
beyond recent ATLAS search limits[49] from $H,\ A\rightarrow\tau\bar{\tau}$,
which are plotted in the $m_{A}$ vs. $\tan\beta$ plane. For $n=-1$, then we
expect rather light $m_{A}$, possibly at a few hundred GeV, leading to large
light-heavy Higgs mixing. This also seems in contradiction with LHC results
which favor a very SM-like light Higgs as expected in the decoupling limit.

Figure 3: Probability distributions for light Higgs scalar mass a) $m_{h}$ and
pseudoscalar Higgs mass b) $m_{A}$ from the $f_{SUSY}=m_{soft}^{\pm 1}$
distributions of soft terms in the string landscape with $\mu=150$ GeV.

In Fig. 4, we plot the expected strongly interacting sparticle mass
distributions from the landscape. In frame a), we see for $n=1$ that
$m_{\tilde{g}}$ peaks around $m_{\tilde{g}}\sim 2.5-4$ TeV which is well
beyond current LHC limits which require $m_{\tilde{g}}\gtrsim 2.1$ TeV. The
upper distribution edge extends as far as $m_{\tilde{g}}\sim 6$ TeV. In
contrast, for the $n=-1$ distribution, then the bulk of probability is below
2.1 TeV, although a tail does extend somewhat above present LHC bounds. In
frame b), we show the distribution in first generation squark mass
$m_{\tilde{u}_{L}}$. For $n=1$, the distribution peaks around
$m_{\tilde{q}}\sim 20$ TeV but extends to beyond 40 TeV. For $n=-1$, then
squarks are typically expected at $m_{\tilde{q}}\lesssim 1-2$ TeV and one
would have expected squark discovery at LHC (although a tail extends into the
multi-TeV range). In frame c), we show the light top squark mass distribution
$m_{\tilde{t}_{1}}$. Here, the $n=1$ distribution lies mainly between
$1<m_{\tilde{t}_{1}}<\sim 2.5$ TeV whereas LHC searches require
$m_{\tilde{t}_{1}}\gtrsim 1.1$ TeV. For $n=-1$, then somewhat lighter stops
are expected although there still is about a 50% probability to lie beyond LHC
bounds on $m_{\tilde{t}_{1}}$. In frame d), we show the distribution in
$m_{\tilde{t}_{2}}$. For $n=1$, we expect $m_{\tilde{t}_{2}}\sim 2-6$ TeV
whilst for $n=-1$ then we expect instead that $m_{\tilde{t}_{2}}\sim 1-3$ TeV.

Figure 4: Probability distributions for a) $m_{\tilde{g}}$, b)
$m_{\tilde{u}_{L}}$, c) $m_{\tilde{t}_{1}}$ and d) $m_{\tilde{t}_{2}}$ from
the $f_{SUSY}=m_{soft}^{\pm 1}$ distributions of soft terms in the string
landscape with $\mu=150$ GeV.

##  3 Stringy naturalness

For the case of the string theory landscape, in Ref. [50] Douglas has
introduced the concept of stringy naturalness:

> Stringy naturalness: the value of an observable ${\cal O}_{2}$ is more
> natural than a value ${\cal O}_{1}$ if more phenomenologically viable vacua
> lead to ${\cal O}_{2}$ than to ${\cal O}_{1}$.

We can compare the usual naturalness measure $\Delta_{BG}$ to what is expected
from stringy naturalness in the $m_{0}$ vs. $m_{1/2}$ plane[51]. We generate
SUSY soft parameters in accord with Eq. 2 for values of $n=2n_{F}+n_{D}-1=1$
and 4. The more stringy natural regions of parameter space are denoted by the
higher density of sampled points.

Figure 5: The $m_{0}$ vs. $m_{1/2}$ plane of the NUHM2 model with
$A_{0}=-1.6m_{0}$, $\mu=200$ GeV and $m_{A}=2$ TeV and an $n=1$ draw on soft
terms, The higher density of points denotes greater stringy naturalness. The
LHC Run 2 limit on $m_{\tilde{g}}>2.25$ TeV is shown by the magenta curve. The
lower yellow band is excluded by LEP2 chargino pair search limits. The green
points are LHC-allowed while black are LHC-excluded.

In Fig. 5, we show the stringy natural regions for the case of $n=1$. Of
course, no dots lie below the CCB boundary since such minima must be vetoed as
they likely lead to an unlivable pocket universe. Beyond the CCB contour, the
solutions are in accord with livable vacua. But now the density of points
increases with increasing $m_{0}$ and $m_{1/2}$ (linearly, for $n=1$), showing
that the more stringy natural regions lie at the highest $m_{0}$ and $m_{1/2}$
values which are consistent with generating a weak scale within the ABDS
bounds. Beyond these bounds, the density of points of course drops to zero
since contributions to the weak scale exceed its measured value by at least a
factor of 4. There is some fluidity of this latter bound so that values of
$\Delta_{EW}\sim 20-40$ might also be entertained. The result that stringy
naturalness for $n\geq 1$ favors the largest soft terms (subject to
$m_{Z}^{PU}$ not ranging too far from our measured value) stands in stark
contrast to conventional naturalness which favors instead the lower values of
soft terms. Needless to say, the stringy natural favored region of parameter
space is in close accord with LHC results in that LHC find $m_{h}=125$ GeV
with no sign yet of sparticles.

In Fig. 6, we show the same plane under an $n=4$ draw on soft terms. In this
case, the density of dots is clearly highest (corresponding to most stringy
natural) at the largest values of $m_{0}$ and $m_{1/2}$ as opposed to naive
expectations where the most natural regions are at low $m_{0}$ and $m_{1/2}$.
In this sense, under stringy naturalness, a 3 TeV gluino is more natural than
a 300 GeV gluino!

Figure 6: The $m_{0}$ vs. $m_{1/2}$ plane of the NUHM2 model with
$A_{0}=-1.6m_{0}$, $\mu=200$ GeV and $m_{A}=2$ TeV and an $n=4$ draw. The
higher density of points denotes greater stringy naturalness. The LHC Run 2
limit on $m_{\tilde{g}}>2.25$ TeV is shown by the magenta curve. The lower
yellow band is excluded by LEP2 chargino pair search limits. The green points
are LHC-allowed while black points are LHC-excluded.

##  4 Consequences of string landscape for SUSY collider searches

A figurative depiction of the expected sparticle and Higgs mass spectra from
the landscape is shown in Fig. 7.

Figure 7: Sparticle and Higgs mass spectra for a natural SUSY benchmark point.

Given such a spectra and the above distributions, we briefly describe
expectations for SUSY at future collider options. The big picture is that for
a positive power-law draw to large soft terms from the string landscape, then
we expect a Higgs mass $m_{h}\sim 125$ GeV with sparticles beyond present LHC
search limits: exactly what LHC is seeing so far with $\sqrt{s}=13$ TeV and
139 fb-1 of integrated luminosity.

###  4.1 LHC

####  4.1.1 Light higgsinos

Since the SUSY $\mu$ parameter is SUSY conserving rather than SUSY breaking,
it feeds mass to $W,\ Z$ and $h$ and also higgsinos (which mix with gauginos).
Gaugino masses are SUSY breaking and we expect the lightest EWinos to be
mainly higgsino-like, but not pure higgsino. We expect the higgsino-like
lightest EWinos $\tilde{\chi}_{1}^{\pm}$ and $\tilde{\chi}_{1,2}^{0}$ to have
mass in the range $\sim 100-350$ GeV. Since the higgsino-like EWinos have very
compressed spectra with mass gaps $\sim 5-10$ GeV, then their visible decay
products are expected to be very soft and difficult to detect. LHC searches
for pair production of higgsino-like EWinos was suggested in Ref. [52] as a
probe of low $\mu$ via a soft dimuon trigger. At present, the best search
strategy seems to be to look for
$pp\rightarrow\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{0}+jet$ with
$\tilde{\chi}_{2}^{0}\rightarrow\ell^{+}\ell^{-}\tilde{\chi}_{1}^{0}$[53, 54,
55]. For Snowmass 2022, the landscape parameter space has been mapped out in
Ref. [56] and improved angular cuts for LHC searches have been proposed in
Ref. [57].

####  4.1.2 Gluino searches

In the landscape, gluinos may range from $\sim 2-6$ TeV while top squarks are
in the $1-2.5$ TeV range. This means gluinos should decay to top+stop or else
three-body modes to top and bottom quarks[58, 59]. The HL-LHC $5\sigma$ reach
assuming 3000 fb-1 is found to be $m_{\tilde{g}}\sim 2.7$ TeV so there is some
possibility these will be discovered at LHC but more likely a higher energy
hadron collider with $\sqrt{s}\gtrsim 30$ TeV will be needed[60, 61, 62].

####  4.1.3 Top squark pair searches

In landscape SUSY, we expect light top squarks with mass
$m_{\tilde{t}_{1}}\sim 1-2.5$ TeV whilst the current LHC limits require
$m_{\tilde{t}_{1}}>1.1$ TeV. The reach of HL-LHC with 3000 fb-1 extends to
$m_{\tilde{t}_{1}}\sim 1.3-1.7$ TeV. Thus, a higher energy hadron collider
will be needed to probe the entire expected light stop mass range[60, 61, 62].
An important feature of landscape SUSY is that top squarks should be nearly
maximally mixed due to the required large weak scale value of the trilinear
soft term $A_{t}$.

####  4.1.4 Same-sign diboson signature

A qualitatively new signature for SUSY arises in natural models when $\mu$ is
small. The reaction $pp\rightarrow\tilde{\chi}_{2}^{\pm}\tilde{\chi}_{4}^{0}$
whre the EWinos are mainly wino-like can occur at high rates followed by decay
to same-sign $W$ bosons. This gives a unique SS dilepton plus MET signature
with minimal jet activity (just that from ISR) in distinction to SS dileptons
from gluino and squark production where substantial jet activity is expected.
Signal and background and LHC reach have been plotted out in Ref’s [63, 64,
65].

###  4.2 Linear $e^{+}e^{-}$ collider

####  4.2.1 Direct production of light higgsinos

Since light higgsinos are expeted in landscape SUSY with radiatively-driven
naturalness, then it makes sense to build something like the International
Linear $e^{+}e^{-}$ Collider (ILC). The ILC is touted as a Higgs factory, but
if $\sqrt{s}>2m(higgsino)\sim 250-700$ GeV, then it may turn out to be a
higgsino factory as well[66]. The soft dileptons arising from higgsino pair
production ($e^{+}e^{-}\rightarrow\tilde{\chi}_{1}^{+}\tilde{\chi}_{1}^{-}$
and $\tilde{\chi}_{1}^{0}\tilde{\chi}_{2}^{0}$) should be easily seen at ILC
and their invariant mass and energy spectra will allow precision determination
of their masses and mixings. One may even test gaugino mass unification[67].

####  4.2.2 Precision Higgs measurements at a Higgs factory

A primary goal of an $e^{+}e^{-}$ machine operating with
$\sqrt{s}>m_{Z}+m_{h}$ is that it can precisely measure Higgs boson
properties, especially coupling strengths $\kappa_{i}$ which could show
deviations from SM predictions. In landscape SUSY, since the soft terms are
pulled to large values, one gets decoupling and the expected Higgs couplings
should look very SM-like[68].

##  5 Consequences for WIMP and axion searches

In natural SUSY with light higgsinos, the lightest neutralino is higgsino-like
and typically thermally underproduced by about a factor $5-15$. If the
underdensity of neutralinos is augmented by non-thermal higgsino production in
the early universe, then higgsino-only dark matter seems excluded by direct
and indirect DM detection experiments[69]. However, since axions are needed to
solve the strong CP problem, then a neutralino/axion dark matter mixture is to
be expected[70]. In SUSY context with two Higgs doublets, the SUSY DFSZ axion
model is naturally expected[71, 72], and axions tend to make up the bulk of
the DM abundance. The lower neutralino DM abundance allows light higgsinos to
escape DD and IDD DM bounds[73]. The full mixed axion/higgsino DM abundance
reqires solution of eight coupled Boltzmann equations which include the
effects of axinos, axions, saxions and gravitinos[74, 75]. The axions are more
difficult to detect than otherwise projected since now higgsinos circulate in
the axion-$\gamma$-$\gamma$ loop and reduce the axion-photon coupling to tiny
levels[76].

##  6 Consequences for $(g-2)_{\mu}$

The $a_{\mu}\equiv(g-2)_{\mu}/2$ anomaly has recently been reinforced by first
data from the Fermilab E989 experiment. To match the anomaly, SUSY theories
typically need light smuons and mu-sneutrinos. The landscape tends to pull
first/second generation sfermions into the 10-40 TeV range so that $a_{\mu}$
should look very SM-like. In this case, we would expect little or no
anomaly[77].

##  7 Conclusions

The emergence of the string landscape of vacua has exciting consequences for
SUSY phenomenology (in addition to providing a solution to the CC problem).
With of order $10^{500}$ vacua to explore, statistical methods can be brought
to bear, and may even place string theory on a long-awaited predictive
footing. We present here a mini-review of our work on the topic of stringy
naturalness. We examined two main scenarios: a power-law draw on soft terms to
large ($n=1$) or small ($n=-1$) soft terms. The former is motivated by the
expectation of SUSY breaking by a single $F$ term which is distributed
uniformly as a complex number on the landscape, and by KKLT moduli
stabilization. The latter is motivated by an expectation that the SUSY
breaking scale is distributed uniformly over the decades of possibilities and
arises in LVS moduli stabilization. These statistical expectations must be
tempered by the anthropic requirement that the derived value for the weak
scale in each pocket universe must lie within the ABDS window of values. The
LHC data clearly are in accord with the $n\geq 1$ statistics, as they predict
$m_{h}\sim 125$ GeV with sparticles typically beyond present LHC search
limits. We also discussed these implications for LHC SUSY searches and for
WIMP and axion dark matter searches, since we expect dark matter to consist of
a WIMP/axion admixture.

Acknowledgements:

This work is a Snowmass 2022 whitepaper. This material is based upon work
supported by the U.S. Department of Energy, Office of Science, Office of High
Energy Physics under Award Number DE-SC-0009956 and DE-SC-001764. Some of the
computing for this project was performed at the OU Supercomputing Center for
Education and Research (OSCER) at the University of Oklahoma (OU). The work of
DS was supported by the Ministry of Science and Technology (MOST) of Taiwan
under Grant No. 110-2811-M-002-574.

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