Radiative natural supersymmetry emergent from the string landscape

In string theory with flux compactifications, anthropic selection for structure formation from a discretuum of vacuum energy values provides at present our only understanding of the tiny yet positive value of the cosmological constant. We apply similar reasoning to a toy model of the multiverse restricted to vacua with the MSSM as the low energy effective theory. Here, one expects a statistical selection favoring large soft SUSY breaking terms leading to a derived value of the weak scale in each pocket universe (with appropriate electroweak symmetry breaking) which differs from the weak scale as measured in our universe. In contrast, the SUSY preserving \mu parameter is selected uniformly on a log scale as is consistent with the distribution of SM fermion masses: this favors smaller values of \mu. An anthropic selection of the weak scale to within a factor of a few of our measured value -- in order to produce complex nuclei as we know them (atomic principle) -- provides statistical predictions for Higgs and sparticle masses in accord with LHC measurements. The statistical selection then more often leads to (radiatively-driven) {\it natural} SUSY models over the Standard Model or finely-tuned SUSY models such as mSUGRA/CMSSM, split, mini-split, spread, high scale or PeV SUSY. The predicted Higgs and superparticle spectra might be testable at HL-LHC via higgsino pair production but is certainly testable at higher energy hadron colliders with \sqrt{s}~ 30-100 TeV.


Abstract
In string theory with flux compactifications, anthropic selection for structure formation from a discretuum of vacuum energy values provides at present our only understanding of the tiny yet positive value of the cosmological constant. We apply similar reasoning to a toy model of the multiverse restricted to vacua with the MSSM as the low energy effective theory. Here, one expects a statistical selection favoring large soft SUSY breaking terms leading to a derived value of the weak scale in each pocket universe (with appropriate electroweak symmetry breaking) which differs from the weak scale as measured in our universe. In contrast, the SUSY preserving µ parameter is selected uniformly on a log scale as is consistent with the distribution of SM fermion masses: this favors smaller values of µ. An anthropic selection of the weak scale to within a factor of a few of our measured value-in order to produce complex nuclei as we know them (atomic principle)-provides statistical predictions for Higgs and sparticle masses in accord with LHC measurements. The statistical selection then more often leads to (radiatively-driven) natural SUSY models over the Standard Model or finely-tuned SUSY models such as mSUGRA/CMSSM, split, mini-split, spread, high scale or PeV SUSY. The predicted Higgs and superparticle spectra might be testable at HL-LHC or ILC via higgsino pair production but is certainly testable at higher energy hadron colliders with √ s ∼ 30 − 100 TeV.

Introduction
How can it be that the vacuum energy density ρ vac = Λ cc c 2 /8πG N = Λ cc m 2 P ∼ (0.003 eV) 4 is more than 120 orders of magnitude below its expected value from quantum gravity? Weinberg suggested that in an eternally inflating multiverse [1,2] with each pocket universe (PU) supporting its own non-zero value of the cosmological constant (CC) Λ cc , and with Λ cc being distributed across the decades of allowed values, the value of Λ cc ought to be no larger than the critical value for which large scale structure, which is required for life as we know it to emerge. This allowed Weinberg to predict the value of Λ cc to a factor of several over a decade before it was observed [3,4].
Weinberg's prediction relied on environmental selection of a fundamental constant of nature. His solution to the CC problem found a home in a more nuanced understanding of string theory vacuum states [5]. In compactified string theory, one expects the emergence of a visible sector containing the Standard Model (SM) along with a variety of hidden sectors and a large assortment of moduli fields: gravitationally coupled scalar fields that determine the size and shape of the compactified manifold and whose vacuum expectation values determine most of the parameters of the 4−d low energy effective field theory (EFT). In realistic flux compactifications of type IIB string theories [6], a common estimate for the number of distinct (metastable) vacua can range up to 10 500−1000 [7], and even more for F -theory compactifications [8]. These vacua, each with its own 4 − d EFT and value for Λ cc , are more than enough to support Weinberg's solution to the CC problem.
While string theory contains only one scale, the string scale m s , Weinberg's solution provides a mechanism for the emergence of a new scale, Λ cc m s via environmental (or anthropic) selection. This result obtains from the expected CC probability distribution where dN vac is the differential distribution of vacua in terms of the cosmological constant. Weinberg assumed the distribution f cc (Λ cc ) was uniformly distributed in the vicinity of 10 −120 m 2 P . Also, f structure (Λ cc ) had the form of a step function f structure ∼ Θ(10 −120 m 2 P − Λ cc ) such that values of Λ cc too much bigger than our (to be) observed value would lead to too rapid cosmological expansion so that structure in the form of condensing galaxies (and hence stars and planets) would not form, and hence observors would not arise.
Can similar reasoning be applied to the origin of other scales such as the weak scale? Indeed, Agrawal et al. [9,10] (ABDS) addressed this question in 1998. They found that-in order to allow the formation of complex nuclei, and hence atoms as we know them which seem essential for life to emerge-the allowed values of the weak scale are located within a rather narrow window of values (the ABDS window). Our measured value of m weak ∼ m W,Z,h seems to be centrally located within the ABDS window which extends roughly from 0.5m OU weak −(2−5)m OU weak , where m OU weak is the measured value of the weak scale in our universe. In the case of the SM, with Higgs potential V Higgs = −µ 2 SM (φ † φ) + λ(φ † φ) 2 , with λ > 0 to ensure stability of the Higgs vev, then one might expect If one assumes all scales of µ SM equally likely, then f µ ∼ 1/µ SM . Meanwhile, if the PU value of the weak scale m P U weak (2 − 5)m OU weak (where OU refers to the measured value in our universe), then the up-down quark mass difference would grow to such an extent that neutrons would no longer be stable within nuclei. Consequently, nuclei consisting of multiple protons would no longer be stable (too much Coulumb repulsion), and the only stable nuclei would consist of single proton states: the universe would be chemically sterile and life as we know it would not arise. This argument has been used as an alternative to the usual naturalness argument in that using anthropic reasoning, then the SM might well be valid all the way up to huge scales Q ∼ m GU T − m s [11] in spite of the presence of quadratic divergences in the Higgs boson mass-squared. Environmental selection can also be applied to supersymmetric models wherein the Higgs mass-squared contains only logarithmic divergences. Indeed it is emphasized in Ref. [12] that in a landscape containing comparable numbers of SM-like and weak scale SUSY-like low energy EFTs, then the anthropically allowed SUSY models should be much more prevalent because there should be a far wider range of natural parameter choices available compared to the finetuned values which are required for the SM. For SUSY models, we expect a distribution of soft term values according to For the soft term distribution f SU SY (m sof t ), positive power law [13][14][15][16] or log [17] distributions pull soft terms to large values and seem favored by LHC SUSY search results [17][18][19][20]. In contrast, negative power law distributions, as expected in dynamical SUSY breaking where all SUSY breaking scales would be equally favored [21][22][23], or large-volume scenario (LVS) compactifications [16]would lead to sparticle masses below LHC limits and light Higgs boson masses much lighter than the measured value m h ∼ 125 GeV [24].
The anthropic selection function f EW SB requires that the derived value of the weak scale in each pocket universe lies within the ABDS window. Thus, one must veto MSSM-like pocket universes wherein m P U Z > (2 − 5)m OU Z where m OU Z = 91.2 GeV is the value of m Z in our universe. Assuming no finetuning of the values entering the right-hand-side of Eq. 4, then conventional sparticle and Higgs mass generators such as Isajet [25] and others [26] can be used to make landscape predictions for sparticle and Higgs boson masses. Without finetuning, then the pocket universe value for the weak scale will typically be the maximal entry on the RHS of Eq. 4. Then, requiring m P U Z 4m OU Z is the same as requiring the electroweak naturalness measure ∆ EW 30 (where ∆ EW ≡ |maximal term on RHS of Eq. 4|/m OU 2 Z /2) [19,27]. Coupling the ABDS requirement with a mild log or power-law draw to large soft terms, then the probability distributions for sparticle and Higgs masses can be computed. It has then been found that the light Higgs mass distribution dP/dm h rises to a peak at m h ∼ 125 GeV whilst sparticle masses are lifted beyond present LHC search limits [17][18][19][20] (see Ref. [28] for a recent review).
A drawback to the above approach is that it doesn't allow for accidental (finetuned) parameter values conspiring to create m P U Z = m OU Z values which nonetheless end up lying within the ABDS window. This is because the spectrum generators all have the measured value of m Z hardwired into their electroweak symmetry breaking conditions. In the present paper, we build a toy computer code which should provide a better simulation as to what is thought to occur within the multiverse in the case where a subset of vacua containing the minimal supersymmetric standard model (MSSM) is required to be the low energy EFT. This approach allows us to display whether natural SUSY models or finetuned SUSY models are more likely to arise from the landscape. The natural SUSY models 1 are characterized by low ∆ EW 30 while finetuned SUSY models include the constrained MSSM (CMSSM or mSUGRA model) [32], minisplit SUSY [33,34], PeV SUSY [35], high scale SUSY [36], spread SUSY [37] and the G2MSSM [38].
In Sec. 2, we discuss our assumed SUSY model and low energy EFT framework. In Sec. 3, we review expectations for the landscape soft term distribution f SU SY (m sof t ) and why it favors large over small soft SUSY breaking terms. We also discuss the assumed distribution for the SUSY µ parameter. In Sec. 4, we review the crucial anthropic condition that surviving SUSY models lie within the ABDS window, i.e. that the pocket universe value of the weak scale is not-too-far displaced from the value of m weak in our universe. In Sec. 5, we describe our toy model simulation of the multiverse with varying values of m P U weak . In Sec. 6, we present results for natural SUSY models and compare them to results from unnatural SUSY models, and explain why natural SUSY is more likely to emerge from anthropic selection within the string landscape than unnatural SUSY models. The answer is that there exists a substantial hypercube of parameter values leading to m P U weak 4m OU weak for natural SUSY models while the hypercube shrinks to relatively tiny volume for finetuned models: basically, finetuning of parameters implies that only a tiny sliver of parameter choices are likely to be phenomenologically (anthropically) allowed. Some discussion and conclusions are presented in Sec. 7.

SUSY model
In our present discussion, we will adopt the 2-3-4 extra parameter non-universal Higgs model (NUHM2,3,4) for explicit calculations. 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 [39]. In this case, it is really expected that each generation acquires a different soft breaking mass m 0 (1), m 0 (2) and m 0 (3). But for simplicity of presentation, sometimes we will assume 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 [40]. 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 P U weak . This means the 3rd generation is pulled to ∼ 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 [27], whilst their D-term contributions largely cancel under intra-generational universality [41]. 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 [42,43]. 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 [44]. Meanwhile, Higgs multiplets which live in different GUT representations are expected to have independent soft masses m Hu and m H d 2 . Thus, we expect a parameter space of the NUHM models as We assume that only models with appropriate electroweak symmetry breaking (EWSB) are anthropically allowed (thus generating the weak scale), and the scalar potential minimization conditions allow us to trade Bµ for tan β. It is common practise to then finetune either µ or m Hu so as to generate m OU Z = 91.2 GeV. But it is important that here we do not invoke this condition since we wish to allow m P U Z to float to whatever its derived value takes in the multiverse simulation. For instance, the predicted value of m P U Z from scans over the NUHM3 and NUHM4 models is shown in Fig. 1. We also show the ABDS window (shaded green in the Figure). The vast majority of models would be excluded since they lie beyond the ABDS window (much like the vast majority of Λ cc in Weinberg's explanation of the cosmological constant).
is expected where n F is the number of F -breaking fields and n D is the number of D-breaking fields contributing to the overall SUSY breaking scale m sof t ∼ m 2 SU SY /m P where under gravitymediation we also expect the gravitino mass m 3/2 ∼ m 2 SU SY /m P . The above form for f SU SY arises if the SUSY breaking F i terms are distributed independently as complex numbers on the landscape whilst the D-breaking fields D j are distributed as real random numbers. Subsequently, it was then realized that the sources of SUSY breaking should not be all independent which might spoil the above simplistic expectation [46]. However, even under the condition of single F -term source of SUSY breaking, then there is still a linear draw to large soft terms f SU SY ∼ m 1 sof t . Furthermore, in Ref. [16], under considerations of Kähler moduli stabilization, then a linear distribution f SU SY ∼ m 1 sof t would emerge from KKLT [47]-type modulistabilization. In contrast, under dynamical SUSY breaking via e.g. gaugino condensation [48] or instanton effects [49], all SUSY breaking scales are expected to be equally probable leading to a distribution f SU SY ∼ m −1 sof t [21][22][23]. This distribution is also expected to emerge from LVS-type [50] moduli-stabilization [16]. This distribution favors smaller soft terms and leads to sparticle masses below LHC search limits and m h 125 GeV [24] and so we will not consider it further here.
A final consideration is whether all soft terms should have common probability distributions on the landscape [51]. For instance, gaugino masses arise from the SUGRA gauge kinetic function f AB which is typically of the form ∼ k · Sδ AB in string models where S is the dilaton superfield and k is some constant. Under Eq. 6, then this would give a linear draw to large m 1/2 while scalar masses which arise from the Kähler function might have a stronger draw to large values. In addition, the other soft terms such as A 0 have very different dependencies on SUSY breaking fields and hence are expected to scan independently on the landscape [51]. For the bulk of this work, we will generally assume a single source of SUSY breaking so all soft terms scan linearly in m sof t to large values.

µ term
Since in this work we do not finetune the µ parameter to gain the measured value of m OU Z , we must also be concerned with the expected distribution f µ (µ). In solutions to the SUSY µ problem [52], it is usually expected that the µ term arises from SUSY breaking via Kähler potential terms such as Giudice-Masiero [53] (GM) operators (wherein µ is expected to scan as do the soft terms) or via superpotential terms such as W λ µ φ n+1 H u H d /m n P where n = 0 gives the next-to-minimal MSSM [54] (NMSSM) and n > 0 gives the Kim-Nilles [55] (KN) solution. Since Eq. 4 strongly favors µ ∼ m weak m sof t (the Little Hierarchy), then we expect the GM solution disfavored as well. Likewise, we expect the NMSSM solution to be disfavored since there is no evidence for visible sector singlet fields which can re-introduce the gauge hierarchy [56] or lead to domain wall issues [57]. Thus, we expect the KN mechanism, which may also be mixed with the SUSY solution to the strong CP problem, as the most likely avenue towards generating a superpotential µ term. In Ref. [58], the distribution of µ terms under SUSY breaking in the landscape was derived for fixed λ µ values, where the µ term was generated from the Z R 24 discrete-R-symmetry model [59] which generates the SUSY µ-term of order m weak while also generating a gravity-safe accidental, approximate global Peccei-Quinn (PQ) symmetry needed to solve the strong CP problem in a stringy setting where no global symmetries are allowed. Instead of fixing λ µ as in that work, we expect λ µ to scan as would the other Yukawa couplings in the superpotential. In Ref. [60], Donoghue et al. showed that the distribution of fermion masses are distributed uniformly on a log scale as expected from the landscape. Since in our case the µ term also arises as a superpotential Yukawa coupling, we will expect it to be scale-invariant and hence distributed as on the landscape (thus favoring small values of µ P U ).

The ABDS window
Agrawal et al. [9,10] (ABDS) explored the plausible range of the weak scale for pocket universes within the multiverse already in 1998. They found that-in order to allow the formation of complex nuclei, and hence atoms as we know them which seem essential for life to emergethe measured value of the weak scale is located within a rather narrow window of values (the ABDS window). Let us characterize the weak scale according to the oft-used Z-boson mass, with m OU Z = 91.2 GeV being the Z mass in our universe while m P U Z = m OU Z is the Z-boson mass in each different pocket universe. ABDS assumed an ensemble of pocket universes with the SM as the 4 − d EFT but with variable values of m P U Z . Then, ABDS found m P U Z (2 − 5)m OU Z as an upper bound, while (for us) the less essential lower bound is m P U Z m OU Z /2. The ABDS pocket universe window is depicted in Fig. 2.
While the exact upper bound on m P U Z is not certain, what is important is that there is some distinct upper bound: for too large of values of m P U Z , then complex stable nuclei will not form, and hence the complex chemistry of our own universe will also not form. The explicit ABDS upper bound is quite distinct from many previous approaches which would penalize too large values of m P U Z by a factor (m OU weak /m SU SY ) 2 , a putative finetuning factor which penalizes but does not disallow a large mass gap (a Little Hierarchy) between the SUSY breaking scale and the measured value of the weak scale.
Using conventional SUSY spectra generators where m Z is fixed to its measured value, then one may estimate the value of m P U Z from the finetuning measure ∆ EW in the limit of no finetuning (where the weak scale is determined by the maximal value on the RHS of Eq. 4) as which gives for ∆ EW ∼ 30 a value m P U Z ∼ 360 GeV, about four times its value m OU Z . We can then plot out the allowed range of MSSM weak scale parameters from Eq. 4 while setting the radiative corrections Σ u u and Σ d d to zero. The result is shown in Fig. 3 where we plot allowed values of µ P U vs. −m 2 Hu (weak). To fulfill the requirement that m P U Z 4m OU Z , then one must live in between the red and green curves. For parameter choices above the red curve then one obtains m P U Z m OU Z and the pocket universe value of the weak scale is too big. For points below the green curve, then m P U 2 Z goes negative signalling inappropriate EWSB. One immediately notices that there is a large range of parameter values in the lower-left corner of the plot that land in the ABDS window. Alternatively, for large values of µ P U 360 GeV and −m 2 Hu (weak) 360 GeV, then in order to gain m P U Z in the ABDS window one must land within the tiny gap between the red and green curves. Unnatural or finetuned SUSY models (such as High-Scale SUSY [36,61], Split SUSY [62,63] and Minisplit SUSY [33,34], labelled in upper right) thus must fall in the narrow gap whilst natural SUSY models characterized by low ∆ EW would lie in the substantial lower-left allowed region. If µ P U and −m 2 Hu (weak) were fundamental parameters (as in pMSSM) that were distributed in a scale invariant fashion (uniform on a log scale), then it would be easy to see why natural SUSY is more likely to emerge than finetuned SUSY from the landscape: for a random distribution of parameters (on a log scale), one is more likely to land in the large lower-left region than in the narrow gap between the red and green curves in the upper right.

−m 2
Hu (weak) parameter space in a toy model ignoring radiative corrections to the Higgs potential. The region between red and green curves leads to m P U weak < 4m OU weak so that the atomic principle is satisfied.
Of course, the value of m 2 Hu (weak) is highly distorted from its high scale value due to RG running and the requirement of radiatively-induced EWSB (REWSB). However, the value of µ P U roughly tracks the high scale value of µ since µ does not run that much between high and low scales. In addition, the Σ u,d u,d terms are not zero and can often be the dominant contributions to the RHS of Eq. 4. Our goal in this paper is to explore these connections numerically via a toy simulation of the multiverse.

The hypercube of ABDS-allowed parameter values in the NUHM2 model
Before we turn to our toy multiverse simulation, let us illustrate the hypercube of ABDS-allowed parameters for the simple case of the NUHM2 model.  In Fig. 4, we plot the value of m P U Z versus variation in several soft SUSY breaking terms for a NUHM2 benchmark model. We take m P U Z = ∆ EW /2m OU Z . In frame a), we vary the parameter m 0 . The red dots correspond to m P U Z > 4m OU Z while green points have m P U Z < 4m OU Z . From the plot, we see range of m 0 : 4.5−5.2 TeV which leads to ABDS-allowed pocket universes. For larger or smaller values, then the Σ u u (t 1,2 ) values which enter Eq. 4 become too large and then finetuning is required to lie within the ABDS window. This range of allowed m 0 values is correlated with the window of A 0 values in that large cancellations can occur in both Σ u u (t 1 ) and Σ u u (t 2 ) for large A 0 at nearly maximal stop mixing which is where m h is lifted to ∼ 125 GeV [29]. For lower or higher m 0 values, this cancellation is destroyed and top-squark loop contributions to the weak scale become too large.
In frame b), we show m P U Hu (weak) is driven to large negative values (see e.g. Fig. 4 of Ref. [12]), and hence gives a large contribution to m P U Z via Eq. 4. As m Hu (GU T ) increases, then −m 2 Hu (weak) decreases, until at m Hu (GU T ) ∼ 6.5 TeV, then electroweak symmetry is no longer broken and we do not generate a weak scale. Such pocket universes must be vetoed since they lack massive SM fermions and gauge bosons. The landscape pull on m Hu is to large values such that electroweak symmetry (EWS) is barely broken. While this viable portion of the hypercube looks small, it must be remembered that there is a landscape pull to large values stopping just short of the no EWSB limit (dashed vertical line). We can view this differently in Fig. 5 where instead we plot m Hu (GU T ) on the y-axis and m Hu (weak) on the x-axis. In this case, the more substantial allowed range of m Hu (weak) values required by Fig. 3 is apparent as the green region on the right side of the curve.

A toy model of vacuum selection within the multiverse
Instead of using any of the publicly available SUSY spectra codes for which m Z is fixed at its measured value in our universe, we will construct a toy program with variable weak scale where both µ and m Hu are input parameters and m P U Z = m OU Z is an output parameter. We begin by creating a code which solves the 26 coupled renormalization group equations (RGEs) of the MSSM via Runge-Kutta method starting with GUT scale inputs of parameters m 0 (1, 2, 3), m Hu , m H d , m 1/2 , A 0 , tan β, and µ (9) where we have used the EWSB minimization conditions to trade the bilinear soft term b = Bµ for tan β, but where we have not imposed the relation between m Hu (weak) and µ(weak) in terms of the measured value of m Z . We use the one-loop RGEs but augmented by the twoloop terms from Eq. 11.22 of Ref. [64] which set the upper limits on first/second generation scalar masses. We run the set of soft terms, gauge and Yukawa couplings and µ term from Q = m GU T 2 × 10 16 GeV down to the weak scale Q weak which we define as that scale at which m 2 Hu first runs negative so long as Q weak < m 0 (3). Otherwise, we set Q weak = m 0 (3). This method implements the condition of barely broken EWSB [65]. Then, we use Eq. 4 to calculate m P U Z to see if it lies within the ABDS window. We veto vacua with no EWSB or color-or-charge-breaking (CCB) minima (where charged or colored scalar squared masses run negative) as these would presumably lead to unlivable vacua.
In our toy simulation of this fertile patch (those vacua leading to the MSSM as the 4 − d low energy EFT) of the string landscape, we will scan over parameters as such: The soft terms are all scanned according to f SU SY ∼ m 1 sof t (as expected for SUSY breaking from a single F -term field) while µ is scanned according to f µ ∼ 1/µ. For tan β, we scan uniformly.
6 Numerical results 6.1 Results for m H u vs. µ plane We are now ready to present results from our toy model simulation of vacuum selection from the multiverse, where for simplicity we restrict ourselves to those vacua with the MSSM as the low energy EFT, but where soft terms and the µ parameter vary from vacuum to vacuum, and with a linear draw to large soft terms (as expected in models with spontaneous supersymmetry breaking from a single F -term field where all field values are equally likely). In this case, since F is distributed randomly as a complex number, then the overall SUSY breaking scale m SU SY has a linear draw to large soft terms [18]. We couple this with the MSSM prediction for the magnitude of the weak scale as given by Eq. 4. This is one of the most important Figure 6: The value of m Hu (weak) vs. µ P U The green points denote vacua with appropriate EWSB and with m P U weak < 4m OU weak so that the atomic principle is satisfied. Blue points have m P U weak > 4m OU weak .
predictions of supersymmetric models. However, it is often hidden in phenomenological work since parameters are tuned in the computer codes so that the value of m Z has its numerical value as given in our universe. In Fig. 6, we show the results of our toy model where m P U Z = m OU Z . We show results in the µ P U vs. sign(−m 2 Hu ) | − m 2 Hu |(weak) parameter plane as in Fig. 3. We adopt parameter choices m Hu = 1.3m 0 and A 0 = −1.6m 0 with tan β = 10 while allowing m 0 and m 1/2 to be statistically determined as in Sec. 5. The soft term m H d is scanned over the range given above with n = 1. The light blue points all have m P U Z > 4m OU Z and so lie beyond the ABDS window: the weak scale is too large to allow for formation of complex nuclei and hence atoms as we know them: these points are anthropically vetoed. The green points have values of m P U Z < 4m OU Z and hence fall within the ABDS window: these points should allow for formation of complex nuclei and obey the atomic principle [15]. We see that the bulk of allowed points live within the parameter hypercube as shown in Fig. 3. However, in our toy model, a small number of green points now do live beyond the Fig. 3 parameter hypercube. The latter points are generated with accidental finetuning of parameters such that m P U Z is still less than 4m OU Z in spite of large contributions to the weak scale. Nonetheless, we do see that the natural SUSY models with µ P U and m Hu (weak) 360 GeV are much more numerous than the finetuned solutions. Let us compare the above results to the more common methodology of simply requiring m P U Z = ∆ EW /2m OU Z < 4m OU Z 360 GeV (corresponding to ∆ EW < 30) as can be computed in available SUSY spectrum generators 3 . We show these results as a scan in the same parameter space as in Fig. 3 but now as shown in Fig. 7. In this case, the weak scale is taken to be the largest of the elements contributing to the RHS of Eq. 4, so it assumes no finetuning of parameters within the landscape vacuum states. Again, the blue points lie beyond the ABDS window whilst the green points are anthropically allowed. In this case, the green points fill out the parameter hypercube of Fig. 3, albeit including the Σ u,d u,d radiative corrections. Since no allowance for finetuning is made, then no green points extend along the finetuned diagonal in Fig. 7. The plot does show why the event generator runs with f EW SB = Θ(30 − ∆ EW ) gives a good representation of expected superparticle and Higgs mass spectra in scans over the landscape of string vacua [17,19,20].

Distribution of µ parameter
As a byproduct of our toy model of the string landscape, we are able to plot out the expected distribution of the superpotential µ parameter. This has also been done in Ref. [58] but in that Figure 8: Distribution of µ parameter from scan over NUHM2 model with P µ ∼ 1/µ and with m P U Z < 4m OU Z and with other parameters fixed as in Fig. 6.
case a fixed µ-term Yukawa coupling λ µ is adopted for a particularly well-motivated solution to the µ problem wherein the global U (1) P Q symmetry needed to solve the strong CP problem emerges as an accidental, approximate, gravity-safe global symmetry from a discrete anomalyfree R-symmetry Z R 24 which also solves the SUSY µ problem and provides a basis for R-parity [59]. In the present case, we allow λ µ to also scan in the landscape so that µ is distributed uniformly across the decades of possible values, as may be expected for other superpotential terms (the matter Yukawa couplings) as shown by Donoghue et al. [60].
In Fig. 8, we show the distribution of the weak scale value of the SUSY µ parameter as expected from our toy landscape model where µ P U is distributed as P µ ∼ 1/µ and where we also require appropriate EWSB and m P U Z < 4m OU Z . Other parameters are fixed as in Fig. 6. From the plot, we see that the µ P U distribution is peaked at low values and falls off at higher µ P U values of several hundred GeV, with a steep drop beyond the non-finetuned ABDS window which ends at µ ∼ 360 GeV. For our toy model, there is still some probability to gain µ P U 360 GeV due to the possibility of finetuning in our toy model. It should be noted that the lower range of µ values ∼ 100 − 200 GeV is now under pressure from LHC soft dilepton plus jets plus MET searches [66,67] via higgsino pair production [68][69][70][71]. The plot also emphasizes that there would be a good chance that ILC would turn out to be a higgsino factory in addition to a Higgs factory for √ s 2m(higgsino) [72,73].
6.3 Why finetuned SUSY models are scarce on the landscape compared to natural SUSY In Fig. 6, we see that finetuned SUSY models that lie with parameters µ P U and m Hu (weak) 4m OU Z are relatively scarce in the multiverse compared to natural SUSY models with low ∆ EW . While the finetuned models are logically possible, selection of their parameters is restricted to a hypercube of tiny-volume compared to natural SUSY models, and so we expect natural SUSY as the more likely expression of anthropically selected pocket universes.
We can show this in a different way in this subsection. In Fig. 9a), we adopt as an example our natural SUSY benchmark model as before, but with variable µ P U . We then plot the value of m P U Z as obtained with our natural SUSY parameter choice but with varying µ P U . Recall, µ P U is distributed uniformly across the decades of values using f µ ∼ 1/µ P U . From frame a), we see a rather large window of µ P U values from 100 − 210 GeV which gives values of m P U Z within the ABDS window (green portion of curve). For larger values of µ P U , m P U Z drops below the ABDS window lower bound and then hits the boundary where electroweak symmetry is not properly broken.
In contrast, in Fig. 9b) we instead adopt a value of m Hu (GU T ) < m 0 so that m 2 Hu is driven to large (unnatural) negative values at the weak scale. In this case, when we plot the µ P U values needed to gain m P U Z within the ABDS window, we find a tiny range of parameters around µ P U ∼ 4 TeV which is anthropically allowed. Thus, compared to frame a), we see that, given a uniform distribution of µ parameter on the landscape, the unnatural model is logically possible-but highly improbable-compared to natural SUSY models.
Another example of a finetuned model occurring on the landscape comes from the CMSSM/mSUGRA model (frame c)) where all scalar masses are unified to m 0 . In this case, with m 0 = 5 TeV, m 1/2 = 1.2 TeV, A 0 = 0 and tan β = 10 with µ > 0, one can see that a very thin range of µ P U values around 2 TeV allow for m P U Z lying within the ABDS window. Thus, we would expect the CMSSM to also be rare on the landscape as compared to natural SUSY models which instead have non-universal scalar masses with m Hu (GU T ) ∼ 1.3m 0 so that m 2 Hu runs barley negative at the weak scale.

Conclusions
Weak scale supersymmetry (WSS) provides a well-known solution to the gauge hierarchy problem of the Standard Model via cancellation of all quadratic divergences in scalar field masses. WSS is also suppported by a variety of virtual effects, especially gauge coupling unification and the numerical value of the Higgs boson mass. N = 1 SUSY is expected as a byproduct of string theory compactified on a Calabi-Yau manifold. In models of string flux compactification, enormous numbers of vacuum states are possible which allows for Weinberg's anthropic solution to the CC problem. The question then arises: what sort of soft terms arise statistically on the landscape, and what are the landscape predictions for WSS?
We addressed this question here via construction of a toy landscape model wherein the low energy EFT below the string scale was the MSSM but where each vacuum solution contained different soft term and µ term values. Under such conditions, it is expected there is a power-law draw to large soft terms favoring models with high scale SUSY breaking. However, since the soft Figure 9: The value of m P U weak vs. µ P U The green points denote vacua with appropriate EWSB and with m P U weak < 4m OU weak so that the atomic principle is satisfied. Red points have m P U weak > 4m OU weak .
terms and SUSY µ parameter determine the scale of EWSB, then anthropics provides an upper limit on the various soft terms: if they lead to too large a value of the pocket-universe weak scale, m P U weak (2−5)m OU weak (the ABDS window), then complex nuclei and chemistry that seems necessary for life would not arise. This scenario has been used to motivate unnatural models like the SM valid to some high scale Q m weak , or other unnatural models of supersymmetry such as split, minisplit, PeV, spread SUSY or high scale SUSY.
Our toy simulation gives a counterexample in that models with low EW finetuning (radiativelydriven naturalness) have a comparatively large hypercube of parameter values on the landscape leading to a livable universe. For finetuned models, then the hypercube of anthropically-allowed parameters shrinks to a tiny volume relative to natural models. Thus, for a landscape populated with vacua including the MSSM as the low energy EFT, the unnatural models, while logically possible, are expected to be selected with much lower probability compared to natural SUSY models characterized by low ∆ EW . This result is simply a byproduct of the finetuning needed for unnatural models which shrinks their hypercube of allowed parameter space to tiny volumes. As a result, we expect weak scale SUSY to ultimately emerge at sufficiently high energy colliders with m h ∼ 125 GeV but with sparticles typically beyond present LHC search limits. In the natural models, higgsinos must lie in the 100 − 350 GeV range while top-squarks typically lie within the range mt 1 ∼ 1 − 2.5 TeV with near maximal mixing (due to the landscape pull to large A 0 values) which easily distinguishes them from unnatural models which would have either far heavier top squarks or else top squarks with very low mixing.