Distribution of supersymmetry mu parameter and Peccei-Quinn scale f_a from the landscape

A scan of soft SUSY breaking parameters within the string theory landscape with the MSSM assumed as the low energy effective field theory -- using a power-law draw to large soft terms coupled with an anthropic selection of a derived weak scale to be within a factor four of our measured value -- predicts a peak probability of m_h~125 GeV with sparticles masses typically beyond the reach of LHC Run 2. Such multiverse simulations usually assume a fixed value of the SUSY conserving superpotential mu parameter to be within the assumed anthropic range, mu<~ 350 GeV. However, depending on the assumed solution to the SUSY mu problem, the expected mu term distribution can actually be derived. In this paper, we examine two solutions to the SUSY mu problem. The first is the gravity-safe-Peccei-Quinn (GSPQ) model based on an assumed Z_{24}^R discrete R-symmetry which allows a gravity-safe accidental, approximate Peccei-Quinn global symmetry to emerge which also solves the strong CP problem. The second case is the Giudice-Masiero solution wherein the mu term effectively acts as a soft term and has a linear draw to large values. For the first case, we also present the expected landscape distribution for the PQ scale f_a; in this case, weak scale anthropics limits its range to the cosmological sweet zone of around f_a~ 10^{11} GeV.


Abstract
A scan of soft SUSY breaking parameters within the string theory landscape with the MSSM assumed as the low energy effective field theory-using a power-law draw to large soft terms coupled with an anthropic selection of a derived weak scale to be within a factor four of our measured value-predicts a peak probability of m h 125 GeV with sparticles masses typically beyond the reach of LHC Run 2. Such multiverse simulations usually assume a fixed value of the SUSY conserving superpotential µ parameter to be within the assumed anthropic range, µ < ∼ 350 GeV. However, depending on the assumed solution to the SUSY µ problem, the expected µ term distribution can actually be derived. In this paper, we examine two solutions to the SUSY µ problem. The first is the gravity-safe-Peccei-Quinn (GSPQ) model based on an assumed Z R 24 discrete R-symmetry which allows a gravity-safe accidental, approximate Peccei-Quinn global symmetry to emerge which also solves the strong CP problem. The second case is the Giudice-Masiero solution wherein the µ term effectively acts as a soft term and has a linear draw to large values. For the first case, we also present the expected landscape distribution for the PQ scale f a ; in this case, weak scale anthropics limits its range to the cosmological sweet zone of around f a ∼ 10 11 GeV.

Introduction
One of the curiosities of nature pertains to the origin of mass scales. Naively, one might expect all mass scales to be of order the fundamental Planck mass scale m P l = 1.2 × 10 19 GeV as occurs in quantum mechanics and in its relativistic setting: string theory. For instance, one expects the cosmological constant Λ cc ∼ m 2 P l whereas its measured value is over 120 orders of magnitude smaller. The only plausible explanation so far is by Weinberg [1] in the context of the eternally inflating multiverse wherein each pocket universe has a different value of Λ cc ranging from −m 2 P l to +m 2 P l : if Λ cc were too much larger than its measured value, then the early universe would have expanded so quickly that structure in the form of galaxies, and hence observors, would not occur. This anthropic explanation finds a natural setting in the string theory landscape of vacuum solutions [2] where of order 10 500 [3] (or many, many more [4]) solutions may be expected from string flux compactifications [5].
A further mystery is the origin of the weak scale: why is m weak ∼ m W,Z,h ∼ 100 GeV instead of 10 19 GeV? A similar environmental solution has been advocated by Agrawal, Barr, Donoghue and Seckel (ABDS) [6,7]: if m weak was a factor 2 − 5 greater than its measured value, then quark mass differences would be affected such that complex nuclei, and hence atoms as we know them, could not form (atomic principle).
This latter solution has been successfully applied in the context of weak scale supersymmetry (WSS) [8] within the string theory landscape. The assumption here is to adopt a fertile patch of landscape vacua where the Minimal Supersymmetric Standard Model forms the correct weak scale effective field theory (EFT), but wherein the soft SUSY breaking terms would scan in the landscape. For perturbative SUSY breaking where no non-zero F -term or D-term is favored over any other in the landscape, then soft terms are expected to scan as a power-law [9,10,11]: where n = 2n F + n D − 1 with n F the number SUSY breaking hidden sector F -terms and n D is the number of SUSY breaking hidden sector D-terms. The factor two comes from the fact that F -terms are distributed as complex values whilst the D-breaking fields are distributed as real numbers. Even for the textbook value n F = 1 and n D = 0, already one expects a statistical draw from the landscape to large soft SUSY breaking terms and one might expect soft terms at the highest possible scale, perhaps at the Planck scale.
However, such huge soft terms would generically result in a Higgs potential with either charge-or-color breaking minima (CCB) or no electroweak symmetry breaking (EWSB) at all. For vacua with appropriate EWSB, then one typically expects the pocket universe value of the weak scale m P U weak m OU weak in violation of the atomic principle (where m OU weak corresponds to the measured value of the weak scale in our universe). Here, for specificity, we will evaluate the expected weak scale value in terms of m P U Z as calculated for each pocket universe via the SUSY EWSB minimization conditions, which read Here, m 2 Hu and m 2 H d are squared soft SUSY breaking Lagrangian terms, µ is the superpotential Higgsino mass parameter, tan β = v u /v d is the ratio of Higgs field vacuum-expectation-values (vevs) and the Σ u u and Σ d d contain an assortment of radiative corrections, the largest of which typically arise from the top squarks. Expressions for the Σ u u and Σ d d are given in the Appendix of Ref. [12].
To remain in accord with the atomic principle according to Ref. [6,7], we will require, for a derived value of µ (so that µ is not available for the usual finetuning in Eq. 3 needed to gain the measured value of m OU Z ), that m P U GeV. This constraint is then the same as requiring the electroweak naturalness parameter [13,12] ∆ EW < ∼ 30. Thus, the anthropic condition is that -for various soft term values selected statistically according to Eq. 1-there must be appropriate EWSB (no CCB or non-EWSB vacua) and that m P U Z < 4m OU Z . These selection requirements have met with success within the framework of gravity-mediation (NUHM2) models [14] and mirage mediation [15] (MM) in that the probability distribution for the Higgs mass m h ends up with a peak around m h ∼ 125 GeV with sparticle masses typically well beyond LHC limits. Such results are obtained for n = 1, 2, 3 and 4 and even for a log(m sof t ) distribution [16,17].
These encouraging results were typically obtained by fixing the SUSY conserving µ parameter at some natural value µ < ∼ 4m OU Z ∼ 350 GeV so that the atomic principle isn't immediately violated. But what sort of distribution of SUSY µ parameter is expected from the landscape? The answer depends on what sort of solution to the SUSY µ problem is assumed in the underlying model (a recent review of 20 solutions to the SUSY µ problem is given in Ref. [18]). Recall that since µ is SUSY conserving and not SUSY breaking, then one might expects its value to be far higher than m weak , perhaps as high as the reduced Planck mass m P . But phenomenologically, its value ought to be at or around the weak scale in order to accommodate appropriate EWSB [19].
In this paper, our goal is to calculate the expected µ parameter probability distribution expected from the string landscape from two compelling solutions to the SUSY µ problem. We will first examine the so-called gravity-safe Peccei-Quinn (GSPQ) model 1 which is based upon a discrete R-symmetry Z R 24 from which the global PQ emerges as an accidental, approximate symmetry; it then solves the SUSY µ problem and the strong CP problem in a gravity-safe manner [20]. The second solution is perhaps most popular: the Giudice-Masiero (GM) mechanism [23] wherein the µ parameter arises from non-renormalizable terms in the Kähler potential.

Distribution of µ parameter and PQ scale for the GSPQ model
The first µ term solution we will examine is the so-called gravity-safe PQ (GSPQ) model which was specified in Ref. [20]. The GSPQ model is based upon a discrete Z R 24 R-symmetry to at first forbid the µ parameter. The set of discrete R symmetries that allow for all anomalycancellations in the MSSM (up to Green-Schwarz terms) and are consistent with SO(10) or SU (5) GUT matter assignments were catalogued by Lee et al. in Ref. [24] and found to consist of Z R 4 , Z R 6 , Z R 8 , Z R 12 and Z R 24 . These discrete R-symmetries 1. forbid the SUSY µ term, 2. forbid all R-parity-violating operators, 3. suppress dimension-5 proton decay operators while 4. allowing for the usual superpotential Yukawa and neutrino mass terms.
The superpotential for the GSPQ model introduces two additional PQ sector fields X and Y and is given by where f u,d, ,ν are the usual MSSM+right-hand-neutrino (RHN) Yukawa couplings and M N is a Majorana neutrino mass term which is essential for the SUSY neutrino see-saw mechanism.
Since the µ term arises from the PQ sector of the superpotential (second line of Eq. 4), this is an example of the Kim-Nilles solution to the SUSY µ problem [25]. The GSPQ model is a hybrid between the Choi-Chun-Kim [21] (CCK) radiative PQ breaking model and the Babu-Gogoladze-Wang model [22] (BGW) based on discrete gauge symmetries. For the case of Z R 24 symmetry applied to the GSPQ model, then it was also found that all further non-renormalizable contributions to W GSP Q are suppressed by powers up to 1/m 7 P : terms such as X 8 Y 2 /m 7 P and X 4 Y 6 /m 7 P being allowed. These terms contribute to the scalar potential with terms suppressed by powers of 1/m 8 P . The wonderful result is that the Peccei-Quinn symmetry needed to resolve the strong CP problem emerges as an accidental, approximate symmetry much like baryon-and lepton-number emerge in the SM as a result of the SM gauge symmetries. The Z R 24 symmetry is strong enough to sufficiently suppress PQ breaking terms in W GSP Q such that a very sharp PQ symmetry emerges: enough to guarantee that PQ-violating contributions to the strong CP violatingθ parameter keep its value belowθ < ∼ 10 −10 in accord with neutron EDM measurements. Thus, the GSPQ model based on Z R 24 discrete R-symmetry yields a gravity-safe global PQ symmetry! The PQ symmetry ends up being violated when SUSY breaking also breaks the Z R 24 discrete R-symmetry, leading to the emergence of the µ parameter with value µ ∼ λ µ v 2 X /m P . In the GSPQ model, the F -term part of the scalar potential is augmented by SUSY breaking soft term contributions SUSY breaking with a large value of trilinear soft term −A f leads to Z R 24 breaking (allowing a µ term to develop) and consequent breaking of the approximate, accidental PQ symmetry, leading to the pseudo-Goldstone boson axion a (a combination of the X and Y fields).
The GSPQ scalar potential minimization conditions are [26] (neglecting the Higgs field contributions which lead to vevs at far lower mass scales) To simplify, we will take A f and f to be real so that the vevs v X and v Y are real as well. Then, the first of these may be solved for v Y and substituted into the second equation to yield a cubic polynomial in v 4 X which can be solved for either analytically or numerically. Viable solutions can be found for |A f | ≥ √ 12m 0 3.46m 0 (where for simplicity, we assume a common scalar mass m X = m Y = m 3/2 ≡ m 0 ). Then, for typical soft terms of order m sof t ∼ 10 TeV and f = 1, we develop vevs v X ∼ v Y ∼ 10 11 GeV. For instance, for m X = m Y = 10 TeV, f = 1 and A f = −35.5 TeV, then v X = 10 11 GeV, v Y = 5.8 × 10 10 GeV, v P Q ≡ v 2 X + v 2 Y = 1.15 × 10 11 GeV and the PQ scale f a = v 2 X + 9v 2 Y = 2 × 10 11 GeV. The µ parameter for λ µ = 0.1 is given as µ = λ µ v 2 X /m P 417 GeV. In Fig. 1, we plot contours of the derived value of µ in the m 3/2 vs. −A f parameter space for λ µ = 0.1. The gray-shaded region does not yield admissible vacuum solutions while the right-hand region obeys the above bound | − A f | > ∼ √ 12m 3/2 . From the plot we see that, for any fixed value of gravitino mass m 3/2 , low values of µ occur for the lower allowed range of |A f |. There is even a tiny region with µ < 100 GeV in the lower-left which may be ruled out by negative search results for pair production of higgsino-like charginos at LEP2. As |A f | increases, then the derived value of µ increases beyond the anthropic limit of µ < ∼ 350 GeV and would likely lead to too large a value of the weak scale unless an unnatural finetuning is invoked in m P U Z .

GSPQ model in the multiverse
To begin our calculation of the expected distribution of the µ parameter from the landscape, we adopt the two-extra-parameter non-universal Higgs SUSY model NUHM2 [27,28,29,30,31,32] where matter scalar soft masses are unified to m 0 whilst Higgs soft masses m Hu and m H d are independent. 2 The latter soft Higgs masses are usually traded for weak scale parameters µ and m A so the parameter space is given by We will scan soft SUSY breaking terms with the n = 1 landscape power-law draw, with an independent draw for each category of soft term [33]. The scan must be made with parameter space limits beyond those which are anthropically imposed. Our p-space limits are given by A crucial assumption is that the matter scalar masses in the PQ sector are universal with the matter scalar masses in the visible sector: hence, we adopt that m 0 = m X = m Y ≡ m 3/2 . We also assume correlated trilinear soft terms: A f = 2.5A 0 . This latter requirement is forced upon us by requiring |A f | ≥ √ 12m 0 to gain a solution in the PQ scalar potential while in the MSSM sector if |A 0 | is too large, then top squark soft-squared masses are driven tachyonic leading to CCB vacua. We also adopt f = 1 throughout.
For our anthropic requirement, we will adopt the atomic principle from Agrawal et al. [7] where m P U weak < ∼ (2 − 5)m OU weak . To be specific, we will require m P U Z < 4m OU Z (which corresponds to the finetuning measure ∆ EW < 30 [13,12]). The finetuned solutions are possible but occur rarely compared to non-finetuned solutions in the landscape [34]. The anthropic requirement results in upper bounds on soft terms such as to maintain a pocket-universe weak scale value not-too-displaced from its measured value in our universe. We also must require no chargeor-color-breaking (CCB) minima and also an appropriate breakdown in electroweak symmetry (i.e. that m 2 Hu is actually driven negative such that EW symmetry is indeed broken). Given this procedure, then the value of µ can be calculated from the GSPQ model scalar potential minimization conditions and then the entire SUSY spectrum can be calculated using the Isajet/Isasugra package [35]. The resulting spectra can then be accepted or rejected according to the above anthropic requirements.  In this subsection, we restrict our results to parameter scans with λ µ = 0.1. In Fig. 2, we show the distribution of scan points in the A 0 vs. µ plane for a) all derived weak scale values m P U weak and b) for only points with m P U weak < 4m OU weak . From frame a), we see that only the colored portion of parameter space yields appropriate EWSB, albeit mostly with a huge value of m P U weak well beyond the ABDS anthropic window. The points with too low a value of −A 0 do not yield viable GSPQ vacua (unless compensated for with an appropriately small value of m 0 ) while points with too large a value of −A 0 typically yield CCB minima in the MSSM scalar potential. The surviving points are color coded according to the value of m P U weak with the dark blue points yielding the lowest values of m P U weak , which occur in the lower-right corner. In frame b)-which is a blow-up of the red-bounded region from frame a)-we add the anthropic condition m P U weak < 4m OU weak . In this case, the range of −A 0 and µ values becomes greatly restricted since the large µ points require large values of m 0 and m 1/2 , leading to too large values of Σ u u (t 1,2 ). This can be seen from Fig. 3, where we plot the color-coded µ values in the m 0 vs. m 1/2 plane for λ µ = 0.1. From the right-hand scale, the dark purple dots have µ < ∼ 100 GeV (and so would be excluded by LEP2 chargino pair searches which require µ > ∼ 100 GeV). The green and yellow points all have large values of µ ∼ 300 − 350 GeV, but these occur at the largest values of m 0 and m 1/2 . For even larger m 0 and m 1/2 values, the derived µ value exceeds 365 GeV; and absent fine-tuning, such points would lead to m P U weak lying beyond the ABDS window, in violation of the atomic principle.
In Fig. 4  However, once the anthropic constraint is applied, then we obtain the red distribution which varies between µ ∼ 50 − 365 GeV with a peak at µ P U ∼ 200 GeV followed by a fall-off to larger values.
In Fig. 5, we plot the derived value of the PQ scale f a from all models with appropriate EWSB (blue) and those models with m P U weak < 4m OU weak (red). In this case, the PQ scale comes out in the cosmological sweet spot where there are comparable relic abundances of SUSY DFSZ axions and higgsino-like WIMP dark matter [36]. The unrestricted histogram ranges up to values of f a ∼ (2 − 4) × 10 11 GeV. This differs from an earlier work which sought to derive the PQ scale from the landscape by imposing anthropic conditions using constraints on an overabundance of mixed axion-neutralino dark matter [36]. In the present case, the GSPQ soft terms are correlated with the visible sector soft terms and the latter are restricted by requiring the derived weak scale to lie within the ABDS window. The fact that the present results lie within the cosmological sweet zone then resolves a string theory quandary as to why the PQ scale isn't up around the GUT/Planck scale [37]. By including the weak scale ABDS anthropic requirement, the red histogram becomes rather tightly restricted to lie in the range f a : (1 − 2) × 10 11 GeV.
In Fig. 6, we show the expected distribution in light Higgs mass m h without (blue) and with (red) the anthropic constraint. For the blue histogram, the upper bound on soft terms is set by a combination of our scan limits but also the requirement of getting an appropriate breakdown of PQ symmetry (as in lying outside the gray-shaded region of Fig. 1). In this case, the distribution peaks around m h ∼ 128 GeV with only small probability down to m h ∼ 125   GeV. When the anthropic constraint m P U weak < 4m OU weak is imposed, then we gain instead the red histogram which features a prominent peak around m h ∼ 125 GeV, which is supported by the ATLAS/CMS measured value of m h [38].
In Fig. 7, we show the expected distribution in gluino mass mg. For the blue curve, without the anthropic constraint, we have a strong statistical draw from the landscape for large gluino masses which is only cut off by our artificial upper scan limits along with the requirement of appropriate PQ breaking. Once the anthropic condition is imposed, then the mg distribution peaks around mg ∼ 3 TeV with a tail extending up to about 5 TeV. The ATLAS/CMS requirement that mg > ∼ 2.2 TeV only restricts the lowest portion of the derived mg probability distribution.

Results for other values of λ µ
We have repeated our calculations to include other choices of λ µ = 0.02, 0.05, 0.1 and 0.2. By lowering the value of λ µ , then correspondingly larger GSPQ soft term values (and hence NUHM2 soft term values) may lead to acceptable vacua. In Fig. 8, we show the derived µ parameter distribution for three choices of λ µ after the anthropic weak scale condition is applied. A fourth histogram for λ µ = 0.02 actually peaks below ∼ 100 GeV and so the bulk of this distribution would be ruled out by LEP2 limits which require µ > ∼ 100 GeV due to negative searches for chargino pair production. As λ µ increases, then the µ distribution becomes correspondingly harder: for λ µ = 0.2, then the distribution actually peaks around µ ∼ 250 − 300 GeV. This could offer an explanation as to why ATLAS and CMS have not yet seen the soft dilepton plus jets plus E T signature which arises from higgsino pair production [39,40,41,42,43] at LHC [44,45]. Current limits on this process from ATLAS extend out to µ ∼ 200 GeV for mχ0 2 − mχ0 1 mass gaps of ∼ 10 GeV [44,45]. In Fig. 9, we show the distribution in f a for the three different values of λ µ . Here the model is rather predictive with the PQ scale lying at f a ∼ (0.5 − 2.5) × 10 −11 GeV, corresponding to an axion mass of m a ∼ 144 − 720 µeV. Unfortunately, in the PQMSSM, the axion coupling g aγγ is highly suppressed compared to the non-SUSY DFSZ model due to cancelling contributions from higgsino states circulating in the aγγ axion coupling triangle diagram [46]. Thus, axion detection at experiments like ADMX may require new advances in sensitivity in order to eek out a signal.

Distribution of µ parameter in Giudice-Masiero model
For GM, one assumes first that the µ parameter is forbidden by some symmetry (R-symmetry or Peccei-Quinn (PQ) symmetry?). Then one assumes that in the SUSY Kähler potential K, there is a Planck suppressed coupling of the Higgs bilinear to some hidden sector field h m which gains a SUSY-breaking vev:  where λ GM is some Yukawa couping of order ∼ 1. When h m develops a SUSY breaking vev F h ∼ m 2 hidden with the hidden sector mass scale m hidden ∼ 10 11 GeV, then a weak scale value of would ensue, where m P is the reduced Planck mass m P = m P l / √ 8π 2.4 × 10 18 GeV. In the GM model, since µ ∝ F h (a single F -term), then one would expect also that µ GM would scale as m 1 sof t in the landscape. Nowadays, models invoking the µ-forbidding PQ global symmetry are expected to lie within the swampland of string-inconsistent theories since quantum gravity admits no global symmetries [47,48,49]. Discrete or continuous R-symmetries or gauge symmetries may still be acceptable; the former are expected to emerge from compactification of manifolds with higher dimensional spacetime symmetries.
In Fig. 10, we show the expected distribution of the µ GM parameter (µ in the GM model) without (blue) and with (red) the anthropic constraint that m P U weak < 4m OU weak . The blue histogram is just a linear expectation of the µ parameter up to the upper scan limit. Thus, for the GM model in the landscape, one expects a huge µ parameter. Varying the coupling λ GM just rescales the µ GM distribution. And since the µ GM sector effectively decouples from the visible sector (unlike for the GSPQ model), we do not find that varying λ GM has any effect on the expected µ GM distribution from the landscape.
Next, the µ GM distribution must be tempered by the anthropic constraint which then places an upper limit of µ < ∼ 365 GeV, but also excludes some parameter space with too large Σ u u values. Here, for λ GM = 1, we see the expected µ parameter distribution peaks around ∼ 250 followed by a drop-off to ∼ 360 GeV.

Summary and conclusions
In this paper we have explored the origin of several mass scale mysteries within the MSSM as expected from the string landscape. Soft SUSY breaking terms are expected to be distributed as a power-law or log distribution (although in dynamical SUSY breaking they are expected to scale as 1/m sof t [50]). But other mass scales arise in supersymmetric models: the SUSY conserving µ parameter, the PQ scale f a (if a solution to the strong CP problem is to be included) and the Majorana neutrino scale M N . Here, we have examined the expected distribution of the SUSY µ parameter from the well-motivated GSPQ model which invokes a discrete Z R 24 symmetry to forbid the µ term (along with R-parity violating terms and while suppressing dangerous p-decay operators). It also generates an accidental, approximate global PQ symmetry which is strong enough to allow for the theta parameterθ < ∼ 10 −10 (hence it is gravity-safe [51,52,53,54]). The breaking of SUSY in the PQ sector then generates a weak scale value for the µ parameter and generates a gravity-safe PQ solution to the strong CP problem. For the GSPQ model, we expect the PQ sector soft terms to be correlated with visible sector soft terms which scan on the landscape and are susceptible to the anthropic condition that m P U weak < 4m OU weak in accord with the ABDS window. Thus, a landscape distribution for both the µ parameter and the PQ scale f a are generated. For small values of Yukawa coupling λ µ , then the µ distribution is stilted towards low values µ ∼ 100 GeV which now seems ruled out by recent ATLAS/CMS searches for the soft-dilepton plus jets plus E T signature which arises The PQ scale f a also ends up lying in the cosmological sweet zone so that dark matter would be comprised of an axion/higgsino-like WIMP admixture [55,56,57,46]. We also examined the µ distribution expected from the Giudice-Masiero solution. In this case, the µ parameter is expected to scan as m 1 sof t with a distribution peaking around µ ∼ 200 − 300 GeV.