OU-HEP-220702

Fine-tuned vs. natural supersymmetry:
what does the string landscape predict?

Howard Baer1111Email<EMAIL_ADDRESS>, Vernon Barger2222Email:
<EMAIL_ADDRESS>Dakotah Martinez1444Email<EMAIL_ADDRESS>and Shadman Salam1444Email<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

A vast array of (metastable) vacuum solutions arise from string
compactifications, each leading to different 4-d laws of physics. The space of
these solutions, known as the string landscape, allows for an environmental
solution to the cosmological constant problem. We examine the possibility of
an environmental solution to the gauge hierarchy problem. We argue that the
landscape favors softly broken supersymmetric models over particle physics
models containing quadratic divergences, such as the Standard Model. We
present a scheme for computing relative probabilities for supersymmetric
models to emerge from the landscape. The probabilities are related to the
likelihood that the derived value of the weak scale lies within the Agrawal et
al. (ABDS) allowed window of values leading to atoms as we know them. This
then favors natural SUSY models over unnatural (SUSY and other) models via a
computable probability measure.

##  1 Introduction

Supersymmetry is a key ingredient in superstring theory constructs. An
advantage of compactification of 10-d string theory on a Calabi-Yau
manifold[1] is that it preserves some remnant spacetime supersymmetry in the
4-d theory. Likewise, compactification of 11-d M-theory on a manifold of
$G_{2}$ holonomy preserves some remnant spacetime supersymmetry[2]. Acharya[3]
argues for the proposal that the landscape of all geometric, stable, string/M
theory compactifications to Minkowski spacetime at leading order are
supersymmetric. Non-SUSY preserving compactifications would lead to bubble of
nothing instabilities and presumably lie within the swampland[4].

Making contact with 4-d physics at the TeV scale (which is currently being
explored at the CERN LHC), it is apparent that $N=1$ spacetime SUSY must be
broken.111For a recent review of the status of SUSY after LHC Run 2, see [5].
But the question is: broken at which scale? The gauge hierarchy problem (GHP)
suggests SUSY which is broken at or around the weak scale, thus providing a
“natural” solution to the GHP wherein all quadratically divergent
contributions to the Higgs boson mass cancel. Weak scale supersymmetry is also
supported experimentally via the measured value of gauge couplings, whose
values unify at a scale $m_{GUT}\simeq 2\times 10^{16}$ GeV[6] under
renormalization group evolution[7] within the Minimal Supersymmetric Standard
Model[8] (MSSM) while they do not within the context of the Standard Model
(SM). In addition, the measured value of the Higgs boson mass falls directly
within the narrow range of values allowed by the MSSM[9]. Unfortunately,
superpartners have so far failed to appear at LHC leading to an apparent
naturalness crisis[10].

An apparent alternative to naturalness has emerged from the string
landscape[11]. Under flux compactifications[12], an enormous number of
different compactification possibilities are available[13, 14], each leading
to different 4-d laws of physics. Each of these possibilities can be accessed
within the context of an eternally-inflating multiverse[15]. This scenario
provides the proper setting to realize Weinberg’s anthropic solution to the
cosmological constant problem[16]: we find ourselves in a (pocket) universe
with a tiny cosmological constant $\Lambda_{cc}\sim 10^{-123}m_{P}^{2}$
because if $\Lambda_{cc}$ was much larger, the universe would expand so fast
that structure (galaxies, stars, etc.) would not be able to form and life as
we know it would not arise. Such a solution to the CC problem is known as an
environmental (or anthropic) solution: environmental selection of a tiny
cosmological constant within the plenitude of pocket universes within the
greater multiverse can select a highly fine-tuned value for one (or more) of
our fundamental physical constants. Such a solution may stand in apparent
opposition to a natural solution to the CC problem.

Can the GHP also be explained via environmental/anthropic reasoning rather
than naturalness? Maybe. In the seminal papers by Agrawal, Barr, Donoghue and
Seckel (ABDS)[17, 18], the scenario of the Standard Model emerging from the
multiverse via an anthropic solution to the hierarchy problem is investigated.
The authors consider the SM as the low energy effective field theory (LE-EFT),
but with a variable magnitude for the weak scale. If the weak scale were a
factor $\sim 2-5$ times larger than its actual value, then up-down quark mass
differences would increase, leading to nuclear instability: one enters a
domain of the universe where only protons exist, with no complex nuclei. For
the weak scale reduced by a factor two from its measured value, then protons
become unstable and beta decay to neutrons: there would be no Hydrogen, just
neutron rich matter. In terms of the Higgs vacuum expectation value $v$, one
finds $0.5\lesssim v/v_{0}\lesssim(2-5)$ (where $v_{0}$ is the Higgs vev in
our universe). This narrow range of values for the weak scale has been dubbed
the ABDS window in that values of $v$ outside this range would not lead to a
universe with life as we know it. The anthropic requirement for $v$ to lie
within the ABDS window could allow for a tuning of the weak scale within the
wider multiverse. It also selects out a narrow range of allowed values: namely
$m_{weak}\simeq m_{W,Z,h}\sim 100$ GeV and can explain the magnitude of the
weak scale rather than just accommodate it. The requirement for the magnitude
to lie within the ABDS window is sometimes also referred to as the atomic
principle in that it is required in order for any pocket universe to contain
complex atoms which seem necessary for a rich chemistry and for life as we
know it.

Building upon the SM and ABDS, Arkani-Hamed and Dimopoulos[19, 20, 21]
proposed a model known as Split Supersymmetry wherein the natural SUSY
solution to the GHP is eschewed in exchange for an environmental solution.
This then allows the possibility of a highly fine-tuned supersymmetric model.
The authors then investigate the consequences of scalar masses $\tilde{m}$ far
beyond the naturalness limit, taking $\tilde{m}$ as high as $\sim 10^{9}$ GeV.
SUSY fermions, higgsinos and gauginos, may be protected by a chiral or
$R$-symmetry and may still live around the EW scale. This set-up maintains the
successful gauge coupling unification and WIMP dark matter of SUSY models, but
enlists the vast number of landscape solutions to effectively tune the weak
scale to lie within the ABDS window as required by the atomic principle. The
advantage of very heavy scalars (especially first/second generation matter
scalars), as noted much earlier by Dine et al.[22] and others[23, 24] is that
they provide a decoupling solution to the SUSY flavor and CP problems and may
also suppress proton decay. In addition, under gravity mediation wherein
scalars get mass of order the gravitino mass, this provides a solution to the
cosmological gravitino and moduli problems.222For a recent overview of the
cosmological moduli problem, see e.g. [25].

Thus, Split SUSY and a variety of successor models[26, 27] have been
considered as legitimate expressions of what sort of SUSY models are expected
to emerge from the string landscape. In the literature, it is sometimes
claimed that a rather heavy Higgs mass and no sign of SUSY scalars at LHC
might be construed as evidence for finetuning within the multiverse as opposed
to a natural solution to the GHP, wherein there is no finetuning. Split SUSY,
and the other high-scale SUSY models considered here, are motivated by the
expectation that the soft SUSY breaking terms are statistically favored to
occur at large as opposed to small values on the landscape via a power law
relation $P(m_{soft})\sim m_{soft}^{2n_{F}+n_{D}-1}$ which obtains if the
complex-valued SUSY breaking $F$-term fields and real-valued SUSY breaking
$D$-term fields are distributed uniformly on the landscape[28, 29, 30]. (Here,
$n_{F}$ is the number of hidden sector $F$-term fields and $n_{D}$ is the
number of hidden sector $D$-term fields contributing to the overall SUSY
breaking scale.) This landscape draw to large soft terms must be balanced by
the anthropic/environmental condition that the derived value of the weak scale
in each pocket universe lies within the ABDS window of allowed values[31, 32].

In this paper we survey a variety of finetuned models (both the SM and SUSY),
and compare these to natural SUSY models, all within the context of the string
landscape. What we find is somewhat at odds with the literature: natural SUSY
models are more likely to emerge from the string landscape than finetuned
models. We advance a particular probability measure $P_{\mu}$ which quantifies
these probabilities. By taking ratios, we are able to evaluate the relative
probabilities for different models to emerge from the landscape.

In our deliberations, weak scale naturalness plays a key role, and we must
define what we mean by naturalness. We adopt the definition of so-called
practical naturalness[33]: an observable ${\cal O}$ is natural provided that
all independent contributions to ${\cal O}$ are comparable to or less than
${\cal O}$. For the case of the SM, where the Higgs potential is given by

$V=-\mu_{SM}^{2}\phi^{\dagger}\phi+\lambda(\phi^{\dagger}\phi)^{2}$ (1)

a vacuum expectation value $v=\sqrt{\mu_{SM}^{2}/\lambda}$ develops and the
tree-level Higgs boson mass is given by $m_{h}^{2}=2\mu_{SM}^{2}$. The loop-
corrected Higgs mass is quadratically divergent up to some cutoff scale
$\Lambda_{SM}$ where

$m_{h}^{2}=2\mu_{SM}^{2}+\delta m_{h}^{2}$ (2)

where at one loop

$\delta
m_{h}^{2}\simeq\frac{3}{4\pi^{2}}\left(-\sum_{i}\lambda_{i}^{2}+\frac{g^{2}}{4}+\frac{g^{2}}{8\cos^{2}\theta_{W}}+\lambda\right)\Lambda_{SM}^{2}$
(3)

where the $\lambda_{i}$ are Yukawa couplings for the $i$th fermion, $g$ is the
$SU(2)_{L}$ gauge coupling and $\lambda$ is the Higgs quartic coupling[34].
Requiring practical naturalness then leads to $\Lambda_{SM}\lesssim 1$ TeV
whilst finetuning is required for much higher values of $\Lambda_{SM}\gg 1$
TeV.

In SUSY models with the MSSM as the LE-EFT, then the weak scale is actually
predicted in terms of the weak scale soft SUSY breaking terms and
superpotential $\mu$ parameter. Minimization of the Higgs potential in the
MSSM leads to

$m_{Z}^{2}/2=\frac{m_{H_{d}}^{2}+\Sigma_{d}^{d}-(m_{H_{u}}^{2}+\Sigma_{u}^{u})\tan^{2}\beta}{\tan^{2}\beta-1}-\mu^{2}\simeq-
m_{H_{u}}^{2}-\mu^{2}-\Sigma_{u}^{u}(\tilde{t}_{1,2})$ (4)

where the right-hand-side approximation is obtained for moderate-to-large
$\tan\beta\gtrsim 5$. Here, $m_{H_{u}}^{2}$ and $m_{H_{d}}^{2}$ are the Higgs
soft breaking masses and $\tan\beta$ is the usual ratio of Higgs field vacuum
expectation values $v_{u}/v_{d}$. The $\Sigma$ terms contain over 40 loop
corrections (explicit formulae for the $\Sigma$ terms may be found in Ref’s
[35] and [36] and leading two-loop terms may be found in Ref. [37]). Requiring
the MSSM weak scale as given by the measured value of $m_{Z}$ to be natural
then requires $|\mu|\lesssim 350$ GeV while $m_{H_{u}}^{2}$ is driven
radiatively to small negative values at the weak scale (electroweak symmetry
is barely broken). Also, the leading loop corrections
$\Sigma_{u}^{u}(\tilde{t}_{1,2})$ are minimized for TeV scale top squarks with
large, negative $A_{t}$ trilinear soft terms[38] which also give rise to
nearly maximal stop mixing and large values of $m_{h}\sim 125$ GeV. The
finetuning measure[38]

$\Delta_{EW}\equiv|maximal\ term\ on\ RHS\ of\ Eq.\
\ref{eq:mzs}|/(m_{Z}^{2}/2)$ (5)

is then a measure of practical naturalness in the MSSM, where for natural
models usually $\Delta_{EW}\lesssim 30$ is required. Notice that $\Delta_{EW}$
is closely related to the ABDS anthropic window in that requiring
$\Delta_{EW}\lesssim 30$ then requires all independent contributions to the
MSSM weak scale to be within the ABDS window.

##  2 A survey of some natural and unnatural SUSY models

###  2.1 CMSSM

For a long time, the mSUGRA[39] or CMSSM[40] model served as a sort of
paradigm model for SUSY phenomenology. This model posits gravity-mediated SUSY
breaking which induces a common scalar mass $m_{0}$, a common gaugino mass
$m_{1/2}$ and a common trilinear soft term $A_{0}$ all prescribed at the GUT
scale $m_{G}\simeq 2\times 10^{16}$ GeV. The weak scale soft terms are
determined by RGE running to the weak scale, where electroweak symmetry is
radiatively broken via a large top quark Yukawa coupling. The $\mu$ term is
tuned via Eq. 4 to give the measured value of $m_{Z}$. In pre-LHC days, it was
possible within the CMSSM model to gain accord with naturalness (low
$\Delta_{EW}$) and with an acceptable thermal relic abundance of the LSP.
After LHC Run 2– while respecting the LHC measured Higgs mass and also LHC
sparticle search limits– natural CMSSM spectra are no longer possible[41, 42,
65].

For illustrative purposes, we compute the mSUGRA/CMSSM spectra using the
Isasugra spectrum generator[43, 44] for a mSUGRA/CMSSM benchmark point with
($m_{0},\ m_{1/2},\ A_{0},\ \tan\beta=5000\ {\rm GeV},1200\ {\rm GeV},\ -8000\
{\rm GeV},10$) which yields a gluino mass $m_{\tilde{g}}=2.8$ TeV (well above
current LHC bounds) with $m_{h}=124.3$ GeV and with $\Delta_{EW}=2641$ (highly
EW finetuned). The thermal bino LSP abundance is $\Omega_{\chi}h^{2}\simeq
249$ so non-thermal processes would need to be invoked to bring the relic
density into alignment with the measured dark matter abundance[45].

###  2.2 PeV SUSY

PeV-scale supersymmetry[46, 47] is motivated by the possibility of SUSY
breaking via “charged” SUSY breaking fields $S$. For charged SUSY breaking,
scalar partners gain mass via Kähler potential terms
$K\ni\frac{S^{\dagger}S}{m_{P}^{2}}Q^{\dagger}Q$ where the $Q$ are visible
sector fields and $S$ are hidden sector fields which carry some charge,
perhaps $R$ charge. Thus, scalar fields gain a mass $m_{Q}^{2}\sim
F_{S}^{\dagger}F_{S}/m_{P}^{2}\sim m_{3/2}^{2}$ whilst gaugino masses, which
ordinarily gain mass via the gauge kinetic function $f\ni kS$ are forbidden.
Hence, the leading contribution to gaugino masses (and also $A$-terms) are the
loop-suppressed anomaly-mediated contributions
$m_{\lambda}=\frac{\beta(g_{\lambda})}{g_{\lambda}}m_{3/2}$ and we expect
$M_{1}\simeq m_{3/2}/120$, $M_{2}\simeq m_{3/2}/360$ and $M_{3}\simeq
m_{3/2}/40$. The wino is then the LSP and can make up the dark matter.
Thermally produced relic winos can make up all the missing dark matter for
$m_{wino}\sim 3$ TeV. Then, with a 3 TeV wino, one expects scalar masses
$\tilde{m}\sim 1000$ TeV, i.e. close to the PeV scale (1 PeV=1000 TeV). The
PeV scale scalar masses provide a decoupling solution to the SUSY flavor and
CP problems[22]. The $\mu$ parameter may range anywhere between $m_{wino}$ and
$\tilde{m}$. The resultant light Higgs mass is expected in the range $125\
{\rm GeV}<m_{h}<155\ {\rm GeV}$[20].

###  2.3 Split SUSY

In split SUSY[19, 20, 21], the motivation is that the string landscape may
provide a selection mechanism for the finetuning of the electroweak scale in
that the weak scale must lie within the ABDS window in order to have a
universe with complex atoms as we know them, which seem necessary for life.
However, SUSY may still be needed for consistency with string theory, but the
SUSY breaking scale may now be far higher than that which is usually required
by naturalness. One may then allow masses of squarks and sleptons (which occur
in multiplets of $SU(5)$) to be as high as $m_{\phi}\sim 10^{9}$ GeV while
fermion masses, which are protected by chiral symmetry, can lie near the weak
scale. This model then preserves the SUSY success stories of gauge coupling
unification and WIMP dark matter while appealing to vacuum selection from the
string landscape to “tune” the EW scale to its value as required by the atomic
principle. Thus, in split SUSY, one expects both gauginos and higgsinos around
the weak scale whilst squarks and sleptons decouple at some intermediate scale
(e.g. $10^{9}$ GeV). Such a split hierarchy of masses can arise from $D$-term
SUSY breaking which maintains an approximate, accidental $R$-symmetry[21]. The
very high scalar mass scale $\tilde{m}$ provides a decoupling solution to the
SUSY flavor and CP problems and also alleviates the cosmological gravitino and
moduli problems by making these particles sufficiently heavy and thus
shortlived in the early universe. The striking signature of split SUSY models
is long lived gluinos which may decay with displaced vertices or even outside
of the collider detector. For scalar masses as high as $\sim 10^{9}$ GeV, then
the lightest Higgs scalar is expected to have mass $m_{h}\sim 130-145$
GeV[48].

###  2.4 High-scale SUSY

In high-scale SUSY (HS-SUSY)[49, 50, 51], it is assumed that the underlying
4-d theory is indeed SUSY, but with a much higher SUSY breaking scale than
that which is usually assumed to solve the gauge hierarchy problem. Thus, in
HS-SUSY, the superpartners are typically clustered at some very high mass
scale $\tilde{m}\sim 10-10^{13}$ TeV. In HS-SUSY, the SM is the LE-EFT and
only the light Higgs particle is expected to be produced at LHC. Indeed, by
requiring the model to yield the measured Higgs mass $m_{h}\sim 125$ GeV, then
$\tilde{m}\sim 10^{1}-10^{7}$ TeV[48, 52].

###  2.5 Mini-Split

Mini-Split[27] SUSY is a version of split SUSY wherein the scalar mass
$\tilde{m}$ is lowered to the $\sim 10^{2-4}$ TeV range in order to
accommodate the measured Higgs mass $m_{h}\simeq 125$ GeV while gauginos
remain near the TeV scale. Several scenarios are envisaged in [27] including
non-sequestered AMSB and $U(1)^{\prime}$ mediation. These scenarios include a
small $A$ parameter while $\mu$ may be either at the gaugino scale (light) or
at the scalar scale (heavy).

###  2.6 Simply unnatural SUSY

In simply unnatural SUSY[53] (SUN-SUSY), the scalar mass scale $\tilde{m}$ is
determined by the measured value of the Higgs mass $m_{h}\simeq 125$ GeV to be
$\tilde{m}\sim 10^{2}-10^{3}$ TeV where the trilinear soft terms $A_{i}$ are
assumed to be tiny (little mixing in the stop sector). The SUSY $\mu$ term is
also expected to be $\mu\sim\tilde{m}$ while the gaugino masses, which require
an $R$-symmetry breaking to gain mass, are expected to be at the TeV scale.
Minimally, the gaugino masses are expected to obtain the AMSB form, but the
presence of heavy vector-like states could alter those relations leading to a
more compressed gaugino spectrum. Typically, the wino is expected to be the
LSP, and the relic abundance may be produced either thermally or non-thermally
due to late-decaying TeV-scale moduli fields.

###  2.7 Spread SUSY

In Ref. [26], it is emphasized that there may exist a forbidden region on the
scale of SUSY breaking $\tilde{m}$ such that if $\tilde{m}\gtrsim{\cal O}(1)$
TeV, then LSP dark matter will be overproduced which can violate the anthropic
bounds which disfavor/forbid DM overproduction in that the baryon-to-DM ratio
may be insufficient for baryonic structure formation in the universe[54]. This
forbidden region should persist up to $\tilde{m}\sim T_{R}$ where $T_{R}$ is
the reheat temperature of the universe at the end of inflation. Higher values
of $\tilde{m}>T_{R}$ are allowed in that SUSY particles wouldn’t be produced
during the reheat process. Taking $\tilde{m}>T_{R}$ then leads to a very heavy
SUSY spectrum (High Scale SUSY) whilst taking $\tilde{m}\sim 1$ TeV leads to
Spread SUSY in the case of SUSY breaking via “charged” hidden sector fields
(where scalars gain mass $\tilde{m}$ but gauginos and $A$ terms do not) or via
uncharged hidden sector fields (which leads to all sparticles at
$\tilde{m}\sim 10$ TeV, dubbed the “environmental MSSM”). The spread SUSY
spectrum divides into two possibilities: 1. scalar masses $\tilde{m}\sim
10^{5}$ TeV with gauginos at $10^{2}$ TeV and higgsinos at $\sim 1$ TeV and 2.
scalars around $10^{3}$ TeV with higgsinos and gravitinos $\sim 10^{2}$ TeV
and gauginos $\sim 1$ TeV. Thus, the spread SUSY models typically have SUSY
mass spectra spread across three mass scales.

###  2.8 G2MSSM

The G2MSSM labels the sort of SUSY spectra expected to emerge from
11-dimensional $M$-theory compactified on a manifold of $G_{2}$ holonomy[55,
56] which preserves $N=1$ SUSY in the low energy 4-d effective field theory
(LE-EFT). The LE-EFT then consists of the usual MSSM fields plus an assortment
of moduli fields which are string remnants from the compactification. Scalar
masses $\tilde{m}$ and the lightest modulus field are expected to gain masses
of order the gravitino mass $m_{3/2}$ and in order to solve the cosmological
moduli/gravitino problems then $\tilde{m}\sim 30-100$ TeV. Gaugino masses are
suppressed relative to scalars by a factor $\log(m_{P}/m_{3/2})\sim 30$ so
gauginos (and higgsinos) are expected at the 1-3 TeV range and may have
comparable moduli/anomaly-mediated contributions. The LSP may be bino or wino-
like but the relic density is seriously affected by non-thermal production via
the late-decaying lightest modulus field[57]. In later renditions, the
possibility of a hidden sector LSP is entertained[58].

###  2.9 Radiatively-driven/Stringy natural SUSY

In radiatively-driven natural SUSY (RNS)[38, 35], large high scale soft terms
can be radiatively driven to small weak scale values. Then all weak scale
contributions to the weak scale are of order the weak scale. This corresponds
to $\Delta_{EW}\lesssim 30$.333A computer code DEW4SLHA is available which
computes the value of $\Delta_{EW}$ for any MSSM model listed in SUSY Les
Houches Accord format[36]. The RNS models can be generated from NUHM2 or NUHM3
models[38, 35], from generalized mirage mediation[59] and from natural
anomaly-mediation[60]. As an example, we take a simple NUHM2 model with
first/second/third generation GUT scale scalar masses
$m_{0}(1,2)=m_{0}(3)=4.5$ TeV, $m_{1/2}=1$ TeV, $A_{0}=-7.2$ TeV,
$\tan\beta=10$ with $\mu=200$ GeV and $m_{A}=2$ TeV. The model has
$m_{\tilde{g}}\sim 2.4$ TeV (LHC safe) with $\Delta_{EW}=12.8$ and
$m_{h}=124.3$ GeV. The higgsino-like LSP is $m_{\chi}=195.3$ GeV with
$\Omega_{\chi}h^{2}=0.011$ (so room for additional axion dark matter).

While RNS models are typically slightly more natural for lower $m_{0}$ and
$m_{1/2}$ values, we expect from the string landscape, under spontaneous SUSY
breaking via a single $F$-term field distributed uniformly as a complex number
throughout the landscape, a linear statistical draw to large soft terms[31].
For more SUSY breaking fields, the statistical draw goes as $f_{SUSY}\sim
m_{soft}^{2n_{F}+n_{D}-1}$ where $n_{F}$ is the number of hidden sector $F$
breaking fields and $n_{D}$ is the number of hidden sector $D$-breaking fields
(the latter distributed as real numbers)[28, 29, 30].444In [61], it is found
that a linear $n=1$ soft term draw is obtained for KKLT[62] moduli-
stabilization models. Convolution of the statistical draw to large soft terms
with the anthropic requirement that the derived weak scale lies within the
ABDS window then leads to a probability distribution for $m_{h}$ that rises to
a peak around $m_{h}\sim 125$ GeV[32] (in part because $A_{0}$ is also drawn
to large (negative) values giving maximal stop mixing[63]) with sparticles
typically beyond LHC reach. In this rendition, naturalness is replaced by what
Douglas calls stringy naturalness[64], where a mode is more stringy natural if
more landscape vacua lead to such a result. In stringy natural SUSY, a 3 TeV
gluino is more (stringy) natural than a 300 GeV gluino[65]. The RNS benchmark
given above is thus highly stringy natural. Thus, under stringy naturalness,
RNS models with LHC-compatible sparticle masses most commonly emerge from the
landscape[66].

##  3 A scheme for computing relative probabilities from the landscape

The central question we wish to address is: how likely are various SUSY models
(and the SM) to arise from the landscape? To answer this, we will restrict
ourselves to string vacua containing the MSSM as the low energy EFT, and where
SUSY breaking is mediated by gravity, i.e. spontaneous SUSY breaking in a
$N=1$ supergravity framework.[67] In such a SUGRA framework, scalar masses are
generically non-universal[68, 69, 70, 71] unless protected by some symmetry:
e.g. the matter scalars of each generation fill the 16-dimensional spinor rep
of $SO(10)$ so one might expect these to have a common mass $m_{0}(i)$,
$i=1-3$ a generation index.555We regard the AMSB soft terms as included in the
gravity-mediated soft terms. Since the Higgs scalars come in split multiplets,
there is no reason to expect $m_{0}(i)=m_{H_{u,d}}$ and thus we expect the LE-
EFT to be a non-universal Higgs model (NUHM). This framework accommodates all
of the high-scale and natural SUSY models under consideration here.666 For
instance, in this framework, there is no known reason to favor the CMSSM model
over any of the NUHM models. While an absolute probability for any particular
LE-EFT (including those not within the realm of the MSSM) is not possible to
calculate (at least at this time), we can make estimates of relative
probabilities amongst gravity-mediated MSSM models based on certain reasonable
assumptions.

In Table 1, we list a variety of supersymmetric models, along with the SM, and
the proposed range for various first/second $m_{0}(1,2)$ and third generation
$m_{0}(3)$ scalar masses, along with the expected range for gaugino and
higgsino masses and the range of the light Higgs mass. In the last column we
list the relative probability measure $P_{\mu}$ to be explained below. For the
two SUSY models CMSSM and RNS, we have approximate supersymmetry extending all
the way down to the weak scale. For the remainder of SUSY models, which
include rather high mass scales $\tilde{m}$, we assume the heavy SUSY states
are integrated out at scale $Q\simeq\tilde{m}$ which then destroys softly
broken SUSY below the $\tilde{m}$ scale, so that quadratic divergences arise
which are proportional to $\Lambda=\tilde{m}$ as in Eq. 3. A pictorial
comparison of the spectra from the various models is given in Fig. 1.

model | $\tilde{m}(1,2)$ | $\tilde{m}(3)$ | gauginos | higgsinos | $m_{h}$ | $P_{\mu}$
---|---|---|---|---|---|---
SM | - | - | - | - | - | $7\cdot 10^{-27}$
CMSSM ($\Delta_{EW}=2641$) | $\sim 1$ | $\sim 1$ | $\sim 1$ | $\sim 1$ | $0.1-0.13$ | $5\cdot 10^{-3}$
PeV SUSY | $\sim 10^{3}$ | $\sim 10^{3}$ | $\sim 1$ | $1-10^{3}$ | $0.125-0.155$ | $5\cdot 10^{-6}$
Split SUSY | $\sim 10^{6}$ | $\sim 10^{6}$ | $\sim 1$ | $\sim 1$ | $0.13-0.155$ | $7\cdot 10^{-12}$
HS-SUSY | $\gtrsim 10^{2}$ | $\gtrsim 10^{2}$ | $\gtrsim 10^{2}$ | $\gtrsim 10^{2}$ | $0.125-0.16$ | $6\cdot 10^{-4}$
Spread ($\tilde{h}$LSP) | $10^{5}$ | $10^{5}$ | $10^{2}$ | $\sim 1$ | $0.125-0.15$ | $9\cdot 10^{-10}$
Spread ($\tilde{w}$LSP) | $10^{3}$ | $10^{3}$ | $\sim 1$ | $\sim 10^{2}$ | $0.125-0.14$ | $5\cdot 10^{-6}$
Mini-Split ($\tilde{h}$LSP) | $\sim 10^{4}$ | $\sim 10^{4}$ | $\sim 10^{2}$ | $\sim 1$ | $0.125-0.14$ | $8\cdot 10^{-8}$
Mini-Split ($\tilde{w}$LSP) | $\sim 10^{2}$ | $\sim 10^{2}$ | $\sim 1$ | $\sim 10^{2}$ | $0.11-0.13$ | $4\cdot 10^{-4}$
SUN-SUSY | $\sim 10^{2}$ | $\sim 10^{2}$ | $\sim 1$ | $\sim 10^{2}$ | $0.125$ | $4\cdot 10^{-4}$
G2MSSM | $30-100$ | $30-100$ | $\sim 1$ | $\sim 1$ | $0.11-0.13$ | $2\cdot 10^{-3}$
RNS/landscape | $5-40$ | $0.5-3$ | $\sim 1$ | $0.1-0.35$ | $0.123-0.126$ | $1.4$
Table 1: A survey of some unnatural and natural SUSY models along with general
expectations for sparticle and Higgs mass spectra in TeV units. We also show
relative probability measure $P_{\mu}$ for the model to emerge from the
landscape. For RNS, we take $\mu_{min}=10$ GeV.  Figure 1: Mass spectra from
various unnatural and natural SUSY models as depicted in Table 1.

For the two SUSY models RNS and CMSSM, the dominant contribution to the weak
scale can be extracted from the value of $\Delta_{EW}$. Then the pocket
universe value of $m_{Z}^{PU}$ can be computed using Eq. 4 as

$\frac{(m_{Z}^{PU})^{2}}{2}=\frac{(m_{Z}^{OU}\sqrt{\Delta_{EW}})^{2}}{2}-\mu_{PU}^{2}$
(6)

(assuming the dominant contribution dominates all other contributions to
$(m_{Z}^{PU})^{2}$, which is usually the case.) Here, $m_{Z}^{OU}=91.2$ GeV,
the value of $m_{Z}$ in our universe (OU). In most SUSY spectrum calculations,
the value of the $\mu$ parameter is finetuned to ensure that $m_{Z}$ gains its
measured value in our universe. However, in the multiverse, each pocket
universe containing the MSSM as the LE-EFT will have a different value of
$\mu_{PU}$ which will in general lead to a value for the weak scale which is
very different from the one in our universe: $m_{Z}^{PU}\neq m_{Z}^{OU}$. In
fact, frequently $m_{Z}^{PU}$ may differ from $m_{Z}^{OU}$ by many orders of
magnitude. If it does, then one will have a pocket universe with $m_{weak}$
outside the anthropic ABDS window, thus violating the atomic principle.

What is the likely distribution of SUSY $\mu_{PU}$ parameters in the
multiverse? Here, we assume the $\mu$ parameter arises in the superpotential
as in the Kim-Nilles (KN) solution to the SUSY $\mu$ problem[72],777 For a
recent review of twenty solutions to the SUSY $\mu$ problem, see Ref. [73].
where we expect $W\ni\lambda_{\mu}S^{2}H_{u}H_{d}/m_{P}$. The PQ charged field
$S$ acquires a vev of order $f_{a}\sim 10^{11}$ GeV under PQ breaking so that
a $\mu$ parameter arises:

$\mu(KN)\sim\lambda_{\mu}f_{a}^{2}/m_{P}\sim m_{weak}.$ (7)

Thus, the KN $\mu$ parameter has the form of a (Planck-suppressed) Yukawa
coupling, in accord with the other Yukawa couplings which occur in the
superpotential. But the question is: what sort of distribution for $\mu$ would
we expect on the landscape? For fixed $\lambda_{\mu}$, this has been computed
in a particular well-motivated KN solution based on an anomaly-free discrete
$R$-symmetry ${\bf Z}_{24}^{R}$[74]. However, for non-fixed $\lambda_{\mu}$,
this may well be different. In fact, Donoghue, Dutta and Ross[75] make a
convincing case that Yukawa couplings are distributed uniformly across the
decades of possible values, which appears to match well with the measured
fermion mass values. We will adopt the Donoghue et al. ansatz for the KN $\mu$
parameter as well: that no particular scale for the $\mu_{PU}$ value is
favored over any other from the string landscape. This seems reasonable in
that the only scale inherent in string theory is the string scale, and all
other scales likely arise dynamically: i.e. there is no preferred scale for
$\mu_{PU}$. This corresponds to a landscape distribution $f_{\mu}\sim 1/\mu$
so that the integrated distribution is indeed scale invariant.

In Fig. 2, we show on the $x$-axis over 15 decades of possible values for
$\mu_{PU}$. For the RNS model, where the maximal contribution to the RHS of
Eq. 4 is bounded to lie within a factor a few of our measured value of the
weak scale, then there is a substantial range of $\mu_{PU}$ values leading to
$m_{Z}^{PU}$ lying within the (blue-shaded) ABDS window. We will take (quite
arbitrarily) the lower limit of $\mu_{PU}$ to be $\sim 10$ GeV. Values of
$\mu_{PU}(min)$ higher or lower by an order of magnitude from this value lead
to differences in $P_{\mu}$ of a factor $\sim 2$ which is inconsequential for
our purposes. The probability for a random value of $\mu_{PU}$ to give rise to
$m_{Z}^{PU}$ within the ABDS window is then

$P_{\mu}\equiv\log_{10}(\mu_{PU}(max)/\mu_{PU}(min))$ (8)

i.e. the length of the interval of logarithmically distributed $\mu_{PU}$
values. Using this interval, we find $P_{\mu}(RNS)\sim 1.4$.

Figure 2: Values of $m_{Z}^{PU}$ vs. $\mu_{PU}$ or $\mu_{SM}$ for various
natural (RNS) and unnatural SUSY models and the SM. The ABDS window extends
here from $m_{Z}^{PU}\sim 50-500$ GeV.

For the CMSSM benchmark model with $\Delta_{EW}=2641$, then the maximal
contribution to the RHS of Eq. 4 is well beyond the ABDS window. Thus, a
finely-tuned value of $\mu_{PU}$ will be needed in order for $m_{Z}^{PU}$ to
lie within the ABDS window, in accord with the atomic principle. One will have
to live in the nearly vertical portion of the red CMSSM curve, for which the
interval length is $P_{\mu}(CMSSM)\sim 0.005$. While the absolute values of
$P_{\mu}$ don’t have a particular meaning (we don’t know the overall
normalization), the ratios of probabilities do. In this case, we would expect
the RNS model to be $P_{\mu}(RNS)/P_{\mu}(CMSSM)\sim 260$ times more probable
on the landscape than the CMSSM benchmark model. In this case, the “natural”
value for the weak scale in the case of the CMSSM benchmark model would be
$m_{weak}\sim m_{Z}\sqrt{\Delta_{EW}}\sim 5$ TeV.

We can also calculate a value of $P_{\mu}$ for the SM, assuming the SM is
valid all the way up to a scale $Q\sim m_{GUT}$ as is assumed in estimates of
the SM vacuum stability[76]. Here, we will also assume that $\mu_{SM}$ has a
scale invariant distribution so that the $x$-axis of Fig. 2 pertains to
$\mu_{SM}$ of Eq. 2 as well as to $\mu_{PU}$. Taking the value of
$m_{Z}^{PU}\sim m_{h}^{PU}$, we can use Eq. 2 to plot the value of the weak
scale in the SM. The plot is shown in Fig. 2 as the SM curve. Here, we see a
value of $\mu_{SM}\sim 10^{15}$ GeV is needed for $m_{Z}^{PU}(SM)$ to lie
within the ABDS window while the natural value of $m_{Z}^{PU}(SM)$ is $\sim
10^{15}$ GeV. This shows the extreme finetuning needed by the SM in order to
ensure the weak scale lies within the ABDS window. We can compute
$P_{\mu}(SM)$ and find it to be $\sim 7\cdot 10^{-27}$, that is the RNS model
about $10^{26}$ times more likely than the SM to emerge from the landscape.

We can now also compute $P_{\mu}$ values for the various high-scale SUSY
models listed in Table 1. The key point here is that quadratic divergences
still cancel out at energy scales $Q>\tilde{m}$. But once $Q$ drops below
$\tilde{m}$, then we must integrate out the heavy sparticles in the LE-EFT and
the quadratic divergences no longer cancel. Then we may use the uncanceled
terms in Eq. 3 to compute corrections to the Higgs mass, again with
$m_{Z}^{PU}\simeq m_{h}$. For most of these models, we take
$\Lambda\sim\tilde{m}=m_{0}(3)$ to compute the curves of $m_{Z}^{PU}$ vs.
$\mu_{SM}$, where now the Higgs potential is that of the SM for
$Q<\tilde{m}$.888For the SM parameter values entering Eq. 3 in the case of
high scale SUSY models with scale boundary $\tilde{m}$, we use FlexibleSUSY
and FlexibleEFTHiggs to extract the appropriate values [77, 78, 79].

The various curves are shown in Fig. 2 for the assorted high scale SUSY models
of Table 1. We can then extract the values of $P_{\mu}$ for each case. As an
example, Split SUSY with $m_{0}(3)\sim 10^{6}$ TeV gives $P_{\mu}\sim 7\cdot
10^{-12}$ so that RNS is $\sim 10^{12}$ times more likely than Split SUSY to
emerge from the landscape. Lest one be dismayed by the low relative
probability for Split SUSY to emerge from the landscape, it is worth noting
that the Split SUSY benchmark is $\sim 10^{15}$ times more likely to emerge
from the landscape than the SM (when the SM is valid up to $Q=m_{GUT}$).
Scaling $\tilde{m}$ to lower values in order to accommodate the measured value
of $m_{h}$ as in mini-split helps matters somewhat: in this case, mini-split
with a wino LSP and $\tilde{m}\sim 10^{2}$ TeV has $P_{\mu}\sim 4\cdot
10^{-4}$, so the RNS benchmark is more likely to emerge than the mini-split
benchmark by a factor $\sim 3000$.

##  4 Conclusions

In this paper we examined the question: are finetuned or natural models of
particle physics more likely to emerge from selection of string vacua within
the multiverse? We required as the anthropic condition that the atomic
principle be fulfilled in that the derived value of the weak scale lies within
the ABDS window. We also assumed a scale invariant distribution for the
superpotential $\mu$ parameter in that for string theory with only a single
mass scale, the string scale, all other scales are equally likely. We assumed
the same distribution for $\mu_{SM}$. For natural models with $\Delta_{EW}$
below say 30, then all contributions to the weak scale lie within the ABDS
window. In this case, then a wide range of $\mu_{PU}$ values still lead to
$m_{Z}^{PU}$ within the ABDS window (since no finetuning of $\mu_{PU}$ is
required). For unnatural models with large contributions to the weak scale,
then only tiny ranges of $\mu_{PU}$ or $\mu_{SM}$ are allowed in order for
$m_{Z}^{PU}$ to lie within the ABDS window.

Basically, particle physics models which require electroweak finetuning may be
possible on the landscape, but for a uniform distribution of the tuning
parameters, they are likely to be rare because the finetuned solutions should
be rare on the landscape relative to natural models. This seems like common
sense, but apparently contradicts the common wisdom in the literature which
asserts that the string landscape provides motivation to take finetuned models
as a plausible possibility since the cosmological constant also appears to be
finetuned. The origin of the discrepancy may be traced to how the anthropic
condition is implemented. In many early works (e.g. [19, 28]), the anthropic
penalty (finetuning factor) is given as a factor $(m_{weak}/m_{soft})^{2}$
which favors $m_{soft}\sim m_{weak}$ but also allows for $m_{soft}\gg
m_{weak}$. This finetuning factor can be overwhelmed by a landscape draw to
large soft terms $m_{soft}^{n}$ with $n\geq 3$, thus favoring high scale SUSY
breaking. In contrast, we require as the anthropic condition that the derived
value of the weak scale lies within the ABDS window. In our method, if any
contribution to the weak scale in a given model is far beyond the measured
value of $m_{weak}$, then only a teensy range of $\mu_{PU}$ values are allowed
to regain the ABDS window. This is in contrast to natural models where a wide
range of $\mu_{PU}$ values lead to $m_{weak}$ within the ABDS window (since no
finetuning is needed).

Figure 3: Values of relative probabilities $P_{\mu}(model)/P_{\mu}(RNS)$ for
the various benchmark scenarios considered in the text.

We examined the relative probabilities of various natural and finetuned SUSY
models, and the SM, to emerge from the landscape via a computable measure of
relative probabilities. A summary of our results is shown in Fig. 3 where we
present a bar chart of the relative probabilities of the various benchmark
models considered here in relation to the radiatively-driven natural SUSY
benchmark where a wide range of $\mu_{PU}$ values lie within the ABDS window.
Basically, the more finetuned a model is, the less likely it is to emerge from
the landscape in comparison to particle physics models with electroweak
naturalness.

## Acknowledgements

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Basic Energy Sciences Energy Frontier Research
Centers program under Award Number DE-SC-0009956 and U.S. Department of Energy
Grant DE-SC-0017647.

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