Revisiting the SUSY mu problem and its solutions in the LHC era
Kyu Jung Bae Howard Baer Vernon Barger Dibyashree Sengupta





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    OU-HEP-190204 
            
            CTPU-PTC-19-06
            
            
            
             Revisiting the SUSY μ problem
             
            and its solutions in the LHC era
            
            
             Kyu Jung Bae^1[Email: kyujungbae@ibs.re.kr],
            Howard Baer^2[Email: baer@nhn.ou.edu ],
            Vernon Barger^3[Email: barger@pheno.wisc.edu ]
            and Dibyashree Sengupta^2[Email: Dibyashree.Sengupta-1@ou.edu]
            
            
            ^1Center for Theoretical Physics of the Universe,
            
            Institute for Basic Science (IBS), 
            Daejeon 34126, Korea
            ^2Homer L. Dodge Dep't of Physics and Astronomy,
            University of Oklahoma, Norman, OK 73019, USA 
            ^3Dep't of Physics,
            University of Wisconsin, Madison, WI 53706, USA 
            
            
             
            The supersymmetry preserving μ parameter in SUSY theories is naively expected
            to be of order the Planck scale while phenomenology requires it to be 
            of order the weak scale.
            This is the famous SUSY μ problem. 
            Its solution involves two steps: 1. first forbid μ, perhaps via some symmetry, 
            and then 2. re-generate it of order the scale of soft SUSY breaking terms. 
            However, present LHC limits suggest the soft breaking scale m_soft 
            lies in the multi-TeV regime whilst naturalness
            requires μ∼ m_W,Z,h∼ 100 GeV so that a Little Hierarchy
            (LH) appears with μ≪ m_soft. 
            We review twenty previously devised solutions to the SUSY μ problem 
            and re-evaluate them in light of whether they are apt to support the LH. 
            We organize the twenty solutions according to: 
            1. solutions from supergravity/superstring constructions,
            2. extended MSSM solutions,
            3. solutions from an extra local U(1)^' and 
            4. solutions involving Peccei-Quinn (PQ) symmetry and axions.
            Early solutions would invoke a global Peccei-Quinn symmetry
            to forbid the μ term while relating the μ solution to solving the
            strong CP problem via the axion. 
            We discuss the gravity-safety issue pertaining to
            global symmetries and the movement instead toward local gauge symmetries
            or R-symmetries, either continuous or discrete. 
            At present, discrete R-symmetries of order M (ℤ_M^R) 
            which emerge as remnants of Lorentz symmetry
            of compact dimensions seem favored. Even so, a wide variety of regenerative
            mechanisms are possible, some of which relate to other issues such as the strong CP
            problem or the generation of neutrino masses. 
            We also discuss the issue of experimental verification or falsifiability of
            various solutions to the μ problem.
            Almost all solutions seem able to accommodate the LH.
            
            
            
            
            § INTRODUCTION: REFORMULATING THE Μ PROBLEM FOR THE LHC ERA
            
            
            Supersymmetry provides a solution to the Big Hierarchy problem– 
            why does the Higgs mass not blow up to the GUT/Planck scale–
            via a neat cancellation of quadratic divergences which is required by
            extending the Poincare group of spacetime symmetries to its maximal 
            structure<cit.>.
            SUSY is also supported indirectly via the confrontation of data 
            with virtual effects in that 
            1. the measured gauge couplings unify under Minimal Supersymmetric Standard Model (MSSM) 
            renormalization group evolution (RGE) <cit.>, 
            2. the measured value of m_t falls in the range
            required for a radiatively-driven breakdown of electroweak symmetry <cit.>,
            3. the measured value of the Higgs boson mass falls squarely within the
            narrow allowed range required by the MSSM <cit.> and
            4. the measured values of m_W and m_t favor the MSSM with heavy superpartners <cit.>.
            In spite of these successes, so far no direct signal for SUSY has emerged at LHC 
            leading to mass limits m_ 2 TeV and m__1 1 TeV while the rather large
            value of m_h≃ 125 GeV also seemingly requires multi-TeV highly mixed
            top squarks <cit.>. The new LHC Higgs mass measurement and sparticle 
            mass limits seem to have exacerbated the so-called Little Hierarchy problem (LHP) <cit.>:
            why doesn't the Higgs mass blow up to the soft SUSY breaking scale
            m_softseveral TeV, or what stabilizes the apparent hierarchy
            m_h≪ m_soft? The LHP opens up the naturalness question:
            how can it be that the weak scale m_weak∼ m_W,Z,h∼ 100 GeV without 
            unnatural fine-tunings of dimensionful terms in the MSSM Lagrangian?
            
            The most direct link between the magnitude of the weak scale and
            the SUSY Lagrangian comes from minimization of the MSSM Higgs potential
            to determine the Higgs field vevs <cit.>. 
            A straightforward calculation<cit.> reveals that
            
            m_Z^2/2= m_H_d^2+Σ_d^d-(m_H_u^2+Σ_u^u)tan^2β/tan^2β -1-μ^2≃ -m_H_u^2-Σ_u^u(_1,2)-μ^2
            
            where tanβ≡ v_u/v_d is the ratio of Higgs field vevs, 
            μ is the SUSY conserving Higgs/higgsino mass term and m_H_u,d^2 
            are soft SUSY breaking up- and down-Higgs mass terms. 
            The Σ_u^u and Σ_d^d terms contain a large assortment of loop corrections 
            (see the Appendix of Ref. <cit.> for expressions) the largest of which
            are usually the Σ_u^u(_1,2) from the top-squark sector.
            
            We can see immediately from the right-hand-side of Eq. (<ref>)
            that if say one contribution is far larger than m_Z^2/2, then 
            another (unrelated) term will have to be fine-tuned to compensate
            so as to maintain m_Z at its measured value. 
            The  electroweak fine-tuning measure Δ_EW has been introduced <cit.>–
            Δ_EW≡ max |largest term on RHS of Eq. (<ref>)|/(m_Z^2/2)
            
            – to quantify the weak-scale fine-tuning required to maintain m_Z at
            its measured value. 
            While a low value of Δ_ EW seems to be a necessary condition for
            naturalness within the MSSM, the question is: is it also sufficient? It is argued 
            in Ref's <cit.> that for  correlated ( i.e. inter-dependent)
            soft terms as should occur in any more fundamental theory such as SUGRA with a 
            well-specified SUSY breaking sector, or in string theory, then other measures
            such as Δ_ HS≃δ m_h^2/m_h^2 and 
            Δ_ BG≡ max_i|∂log m_Z^2/∂log p_i| 
            (where the p_i are fundamental model parameters) 
            collapse to Δ_ EW so that
            Δ_ EW is sufficient as both an infra-red (IR) and ultra-violet (UV) 
            fine-tuning measure. 
            In contrast, theories with multiple independent soft parameters
            may be susceptible to further fine-tunings which would otherwise cancel in a more 
            fundamental theory. It should be recalled that in the multi-soft-parameter effective theories such
            as CMSSM/mSUGRA, NUHM2 etc., the various soft parameters are introduced to parametrize one's 
            ignorance of the SUSY breaking sector such that some choice of soft parameters will reflect
            the true choice in nature. However, in no sense are the multi-soft-parameter theories
            expected to be fundamental. Thus, in this paper we will adopt Δ_ EW as a 
            measure of naturalness in fundamental theories with the MSSM as the weak scale effective theory.
            In Ref. <cit.>, it is shown that the fine-tuning already turns on 
            for values of Δ_ EW∼ 20-30. We will adopt a value of Δ_ EW<30 as a
            conservative choice for natural models of SUSY.
            
            For a natural theory– 
            where m_W,Z,h∼ 100 GeV because the RHS contributions to Eq. (<ref>) 
            are comparable to or less than the measured value of m_Z^2/2– 
            then evidently
            
              * m_H_u^2(weak) ∼ -(100-300)^2 GeV^2 and
            
              * |μ |∼ 100-300 GeV <cit.>,
            
              *  the largest of the radiative corrections (usually Σ_u^u(_1,2)) 
            are not too large.
            
            
            The first of these conditions pertains to the soft SUSY breaking sector. It can be achieved
            for multi-TeV values of high-scale soft terms (as required by LHC limits) by 
            radiatively driving m_H_u^2 from large, seemingly unnatural high scale values 
            to a natural value at the weak scale. 
            Thus, a high scale value of m_H_u^2(Λ =m_GUT ) must be selected such that
            electroweak symmetry is barely broken. 
            While this may seem to be a tuning in itself, 
            such a selection seems to automatically emerge from SUSY within the 
            string-landscape picture <cit.>. 
            In this scenario, there is a statistical draw towards large soft terms
            which must be balanced by the anthropic requirement that EW symmetry be 
            properly broken and with a weak scale magnitude not too far from its measured value<cit.>.
            The balance between these two tendencies pulls m_H_u^2(m_GUT ) 
            to such large values that EW symmetry is barely broken.
            
            The third of the above conditions– 
            that Σ_u^u(_1,2)∼ 100-300 GeV–
            is achieved for third generation squark soft terms in the several TeV range 
            along with a large trilinear soft term A_t 
            (as is expected in gravity-mediation models). 
            These same conditions which reduce the Σ_u^u(_1,2) values also 
            increase the Higgs mass to its measured value m_h∼ 125 GeV <cit.>.
            
            The second condition– that the superpotential μ parameter
             is of order the weak scale– brings up the famous SUSY μ problem <cit.>:
            since W_ MSSM∋μ H_u H_d is SUSY preserving, naively one expects the 
            dimensionful parameter μ to be of order m_P≃ 2.4× 10^18 GeV
            while phenomenology requires μ∼ m_weak. 
            In this paper, we focus attention on the SUSY μ problem as occurs in 
            gravity-mediation. The SUSY μ problem in gauge-mediated supersymmetry breaking (GMSB) is
            summarized in Ref. <cit.>. In GMSB, since the trilinear soft terms are 
            expected to be tiny, then sparticle masses must become huge with
            highly unnatural contributions to the weak scale in order to accommodate
            a light Higgs boson with m_h≃ 125 GeV <cit.>.[
            We also do not consider SUSY models with non-holonomic soft terms<cit.> or 
            multiple μ terms; 
            it is not clear whether such models have viable UV completions<cit.>.]
            
            There are two parts to solving the SUSY μ problem:
            
              *  First, one must forbid the appearance of μ, usually via
            some symmetry such as Peccei-Quinn (PQ) or better a 
            continuous or discrete gauge or R-symmetry, and then 
            
              *  re-generate μ at the much lower weak scale |μ |∼ 100-300 GeV (the lower the more natural) 
            via some mechanism such as symmetry breaking.
            
            Many solutions to the SUSY μ problem have been proposed, 
            and indeed in Sec. <ref> we will review twenty of these. 
            In most of these solutions, the goal (for gravity-mediation) 
            was to re-generate μ∼ m_3/2 where m_3/2 is the gravitino mass which arises from
            SUGRA breaking and which sets the mass scale for the soft SUSY breaking terms<cit.>. 
            When many of these μ solutions were proposed– well before the LHC era– 
            it was commonly accepted that m_3/2∼ m_weak which would also solve the SUSY 
            naturalness problem. However, in light of the above discussion, the SUSY μ problem
            needs a reformulation for the LHC era: any solution to the SUSY μ problem should first
            forbid the appearance of μ, but then re-generate it at the weak scale, 
             which is now hierarchically smaller than the soft breaking scale:
            
                |μ |∼ m_weak∼ 100-300  GeV≪ m_soft∼ multi-TeV m_3/2 .
            
            
            Our goal in this paper is to review various proposed solutions to the SUSY μ problem
            and confront them with the Little Hierarchy as established by LHC data and as 
            embodied by Eq. <ref>. While many solutions can be  tuned to maintain the 
            Little Hierarchy, others may offer compatibility with or even a mechanism to generate 
            Eq. <ref>. Thus, present LHC data may be pointing to favored solutions to
            the SUSY μ problem which may be reflective of the way nature actually works.
            
            With this end in mind, in Sec. <ref> we will review a variety of 
            mechanisms which have been offered as solutions to the SUSY μ problem. 
            We organize the twenty solutions according to: 
            
              *  solutions from supergravity/superstring constructions,
            
              *  extended MSSM solutions,
            
              *  solutions from an extra local U(1)^' and 
            
              *  solutions involving Peccei-Quinn (PQ) symmetry and axions.
            
            Many of these solutions tend to relate the μ parameter to the scale of soft SUSY breaking 
            which would place the μ parameter
            well above the weak scale and thus require significant EW fine-tuning. 
            One such example is the original Kim-Nilles (KN) <cit.> model (Subsec. <ref>) 
            which generates a μ parameter μ∼ v_PQ^2/m_P and 
            relates v_PQ∼ m_hidden (where m_hidden is a mass scale associated with
            hidden sector SUGRA breaking) and thus obtains  
            μ∼ v_PQ^2/m_P ∼ m_hidden^2/m_P ∼ m_3/2. 
            However, the LHP can also be accomodated by allowing for v_PQ≪ m_hidden 
            so that μ≪ m_3/2. While KN allows this possibility to be implemented
            “by hand”, the later MSY <cit.>, CCK <cit.> and SPM <cit.> models 
            (Subsec. <ref>) implement radiative PQ breaking as a 
            consequence of SUSY breaking  with the result that v_PQ≪ m_hidden 
            and hence μ≪ m_soft <cit.>.
            
            A prominent criticism of the μ solutions based on the existence of a global PQ
            or discrete symmetry is that such symmetries are incompatible with 
            gravity at high scales <cit.>, 
             i.e. that including the presence of gravity 
            could spoil any global or discrete symmetries which may be postulated.
            In Subsec. <ref>, we discuss possible ways around the gravity 
            spoliation of global or discrete symmetries. 
            The MBGW model <cit.> (Subsec. <ref>) adopts a gravity-safe PQ symmetry 
            thanks to a more fundamental discrete gauge symmetry ℤ_22 
            and also generates PQ breaking from SUSY breaking, albeit not radiatively.
            
            An attractive alternative to the discrete or continuous gauge symmetry 
            resides in the possibility of a
            discrete or continuous R symmetry. Several discrete R-symmetries 
            are possible which are anomaly-free (up to a Green-Schwarz term), 
            forbid the μ parameter and other dangerous proton decay operators, and are 
            consistent with an underlying grand unification structure<cit.>.
            Such discrete R-symmetries are expected to arise from compactification of
            extra dimensions in string theory.
            The ℤ_4^R symmetry stands out as a particularly simple approach that also 
            leads to exact R-parity conservation. If one seeks to relate a gravity-safe 
            PQ solution to the strong CP problem with a solution to the μ problem, 
            then two hybrid models based on ℤ_24^R are examined (Subsec. <ref>).
            In this case, the PQ symmetry arises as an accidental approximate global
            symmetry which emerges from the more fundamental discrete R symmetry.
            Here, the PQ breaking is generated through a large negative soft term 
            and not radiatively.  
            
            In Sec. <ref> we discuss the issue of experimental testability 
            and distinguishability of various solutions to the μ problem. 
            In Sec. <ref>, we present a convenient Table <ref> 
            which summarizes our review. 
            Then we draw some final conclusions. 
            Some pedogogical reviews providing an in-depth overview of 
            supersymmetric models of particle physics can be found in Ref's <cit.>. 
            
            
            
            § A REVIEW OF SOME SOLUTIONS TO THE SUSY Μ PROBLEM
            
            
            In this Section, we review some solutions to the SUSY μ problem.
            In the solutions reviewed here, the μ-term is typically generated 
            by breaking the symmetry which originally prohibits the μ-term at the tree-level.
            Depending on the source of such symmetry breaking,
            we categorize the solutions according to 1. those from supergravity/superstring models, 
            2. those from (visible-sector) extensions of the MSSM, 
            3. those including an extra local U(1)^' and 4. 
            those which include also a solution to the strong CP problem 
            with Peccei-Quinn symmetry breaking.
            
            
            
            
             §.§ Solutions in supergravity/string construction
            
            
              §.§.§ Giudice-Masiero (GM)
            
            
            In supergravity models the Kähler function G=K+log |W|^2 is written
            in terms of the real Kähler potential K and the holomorphic 
            superpotential W. 
            If we posit some symmetry (PQ or R-symmetry are suggested in Ref. <cit.>) 
            to forbid the usual MSSM μ term, then one may regenerate it via the
            Higgs fields coupling to hidden sector fields h_m via non-renormalizable 
            terms in K <cit.>:
            
            K∋ H_u^† H_u+H_d^† H_d +(λ_μ/m_PH_u H_d h^† +h.c.) .
            
            If we arrange for SUSY breaking in the hidden sector, then the auxilliary 
            component of h develops a vev ⟨ F_h⟩∼ m_hidden^2 
            so that the gravitino gets a mass m_3/2∼ m_hidden^2/m_P.
            A μ term is generated of order
            μ_ eff=λ_μ⟨ F_h^*⟩/m_P∼λ_μ m_hidden^2/m_P∼λ_μ m_3/2∼ m_soft .
            
            Thus, in the GM case, the μ parameter arises which is 
            typically of order the soft breaking scale unless the coupling 
            λ_μ is suppressed at the ∼ 0.01-0.1 level.
            
            
            
              §.§.§ Casas-Munoz (CM)
            
            
            Casas and Munoz <cit.> propose a string theory inspired solution to the 
            SUSY μ problem.
            In string theory, dimensionful couplings such as μ are already forbidden 
            by the scale invariance of the theory so no new symmetries are needed to forbid it.
            They begin with a superpotential of the form
            
            W=W_0+λ_μW_0H_uH_d/m_P^2
            
            where W_0 is the usual superpotential of the MSSM (but without the μ term) along with the
            hidden sector component which is responsible for SUSY breaking: W_0=W_0^vis(z_i)+W_0^hid(h_m)
            where the z_i comprise visible sector fields while the h_m denote hidden sector fields.
            While the scale-variant μ term is forbidden in W_0^vis, the non-renormalizable
            contribution in Eq. (<ref>) is certainly allowed and, 
            absent any symmetries which could forbid it, probably mandatory.
            Under, for instance, F-term SUSY breaking in the hidden sector, then W_0^hid gains a vev
            ⟨ W_0^hid⟩∼ m_hidden^2m_P (as is easy to see in the simplest Polonyi model 
            for SUSY breaking with W_Polonyi=m_hidden^2(h+β m_P) where β is a dimensionless
            constant). Under these conditions, then a μ term develops with
            μ_ eff∼λ_μm_hidden^2/m_P∼λ_μ m_3/2∼ m_soft.
            
            Ref. <cit.> goes on to show that the CM solution can easily emerge in models 
            of SUSY breaking due to hidden sector gaugino condensation at some intermediate mass scale 
            Λ_h (where then we would associate m_hidden^2≃Λ_h^3/m_P).
            
            A benefit of the CM solution is that it should be consistent with any stringy UV completion <cit.>
            as it avoids the presence of some global (PQ) symmetry. A possible drawback to CM is that the
            μ term is naturally expected to be of order m_soft instead of m_weak 
            unless λ_μ is suppressed (as in GM).
            One way to falsify the CM solution would be to discover a DFSZ-like axion with
            consistent mass and coupling values. Such a discovery would exclude the second term in
            Eq. (<ref>) since it would violate the PQ symmetry.
            
            
            
              §.§.§ μ and a big hierarchy from approximate R-symmetry
            
            
            In string theory models, approximate R-symmetries are expected to
            develop from overall Lorentz symmetry of the 10-dimensional spacetime 
            when compactified to four dimensions. Under a continuous U(1)_R
            symmetry, the superspace co-ordinates transform non-trivially and hence 
            so do the bosonic and fermionic components of superfields. 
            Thus, these symmetries can be linked to overall Lorentz symmetry 
            where also bosons and fermions transform differently. 
            
            Under exact R-symmetry and supersymmetry, then the superpotential
            μ term is forbidden since the gauge-invariant bilinear term of Higgs pair H_uH_d
            carries zero R-charge while the superpotential must have R_W=+2.
            However, H_uH_d may couple to various other superfields
            ϕ_i which carry non-trivial R-charges so that 
            
            W∋ P_μ(ϕ_i )H_uH_d 
            
            where P_μ (ϕ_i ) is a sum over monomials in  the fields ϕ_i^n.
            Unbroken R-symmetry requires a vanishing ⟨ P_μ (ϕ_i )⟩
            but if the R-symmetry is approximate then non-vanishing P_μ (ϕ_i )
            contributions will develop at higher orders in powers of the field vevs
            ⟨ (ϕ_i/m_P)⟩ 1. Thus, a mild hierarchy in the
            field vevs ⟨ϕ_i/m_P⟩ 1, when raised to higher powers 
            ⟨ (ϕ_i/m_P)^n_i⟩≪ 1, can 
            generate a much larger hierarchy of scales <cit.>. 
            In this solution to the μ problem, which is essentially a UV completion of the
            CM solution, then μ∼ m_3/2∼⟨ W⟩ is expected to arise.
            
            
            
              §.§.§ Solution via the discrete R-symmetry ℤ_4^R
            
            
            A particularly attractive way to solve the μ problem in some string
            constructions is via a discrete Abelian R-symmetry ℤ_4^R <cit.>. 
            Such R-symmetries may arise as discrete remnants of the Lorentz
            symmetry of extra dimensional (d=10) models upon compactification to d=4.
            In Ref. <cit.>, the ℤ_4^R symmetry was invoked to forbid the 
            μ term
            as well as dimension-4 baryon- and lepton-number violating operators
            while dangerous dimension-5 operators leading to proton decay 
            are highly suppressed <cit.>.
            The desirable Weinberg neutrino mass operator is allowed. 
            The ℤ_4^R charges
            are assigned so that all anomalies cancel by including Green-Schwarz terms
            (and extra R-charged singlets for gravitational anomalies). 
            The R-charge assignments for the discrete R-symmetry ℤ_4^R 
            are shown in the second row of Table <ref>.
            
            
            
            The charge assignments are consistent with embedding the matter superfields 
            into a single 16 of SO(10) while the split Higgs multiplets would 
            arise from Wilson-line breaking of gauge symmetry. 
            The ℤ_4^R symmetry may be broken via non-perturbative effects such as gaugino 
            condensation breaking of SUGRA in the hidden sector so that 
            a gravitino mass m_3/2 is induced along with soft terms m_soft∼ m_3/2. 
            A μ term may arise via GM (Sec. <ref>) and/or CM (Sec. <ref>) so that
            μ∼⟨ W⟩/m_P^2∼ m_3/2∼ m_soft. 
            Although the discrete ℤ_4^RR-symmetry is broken, the discrete
            matter/R-parity remains unbroken so that the LSP remains absolutely stable.
            This sort of solution to the μ problem is expected to be common in 
            heterotic string models compactified on an orbifold <cit.>.
            Other possibilities for ℤ_N^R with N>4 also occur<cit.> 
            and in fact any N value is possible under anomaly cancellations provided one includes
            additional exotic matter into the visible sector <cit.>.
            
            A further concern is that a spontaneously broken discrete symmetry 
            may lead to formation of domain walls in the early universe which 
            could dominate the present energy density of the universe <cit.>.
            For the case of gravity mediation, the domain walls would be expected to 
            form around the SUSY breaking scale T∼ 10^12 GeV. However, 
            if inflation persists to lower temperatures, then the domain walls may be 
            inflated away. It is key to observe that many mechanisms of baryogenesis
            are consistent with inflation persisting down to temperatures of
            T∼ 10^6 GeV <cit.>.  
            
            
            
            
            
            
              §.§.§ String instanton solution
            
            
            In string theory models, it is possible for superpotential terms to arise 
            from non-perturbative instanton effects. 
            These are particularly well suited for
            open strings in braneworld scenarios such as IIA and IIB string theory.
            Intriguing applications of stringy instanton effects include the generation
            of Majorana neutrino mass terms, generation of Yukawa couplings and 
            generation of the μ term in the superpotential <cit.>.
            In some D-brane models which include the MSSM at low energy, then the
            superpotential μ term may be forbidden by U(1) symmetries
            but then it is generated non-perturbatively via non-gauge 
            D-brane instanton effects.
            In this case, then a μ term of the form
            
            W∼exp (-S_ cl)M_s H_u H_d 
            
            can be induced where then μ≃exp (-S_ cl)M_s and M_s is the 
            string mass scale. 
            The exponential suppression leads to the possibility of a μ term 
            far below the string scale. Of course, in this case one might expect 
            the μ term to arise at any arbitrary mass scale below 
            the string scale rather than fortuitously at the weak scale. 
            If the μ term does arise at the weak scale from stringy instanton 
            effects, then that value may act as an attractor such that 
            soft terms like m_H_u^2 are pulled statistically to large values by the
            string theory landscape, but not so large that EW symmetry doesn't break.
            Then the weak scale value of m_H_u^2 is of comparable (negative) magnitude 
            to μ (the naturalness condition) to ensure a universe with
            anthropically required electroweak symmetry breaking <cit.>. 
            
            
            
              §.§.§ Mu solution in G_2MSSM
            
            
            In Ref. <cit.> (Acharya  et al.), the authors consider 11-dimensional M-theory
            compactified on a manifold of G_2 holonomy, and derive various 
            phenomenological implications. 
            They consider fields living in multiplets of 
            SU(5) so the doublet-triplet splitting problem is present.
            As opposed to string theory models compactified on orbifolds, in M-theory
            the matter fields live only in four dimensions so a different solution to the
            μ problem is required. Witten suggested the existence of an additional 
            discrete symmetry which forbids the μ term from appearing but which allows 
            the Higgs triplets to gain large enough masses so as to evade proton decay
            constraints <cit.>. In Ref. <cit.>, it is shown that a ℤ_4 symmetry is
            sufficient to forbid the μ term and other dangerous RPV operators while
            allowing massive Higgs triplets. The ℤ_4 discrete symmetry is assumed to 
            be broken via moduli stabilization so that a small μ term develops.
            
            In the G_2MSSM, the gravitino gains mass from non-perturbative effects
            (such as gaugino condensation) in the hidden sector so that 
            m_3/2∼Λ_h^3/m_P^2∼ 10-200 TeV. 
            Matter scalar soft masses are expected at m_ϕ∼ m_3/2 
            so should be very heavy 
            (likely unnatural in the context of Eq. (<ref>)). 
            In contrast, gauginos gain mass from the 
            gauge kinetic function which depends on the vevs of moduli fields 
            so they are expected to be much lighter: m_λ∼TeV scale 
            and in fact these may have dominant AMSB
            contributions <cit.> (with comparable moduli-mediated SUSY breaking contributions) 
            so that the wino may be the lightest of the gauginos. The dominant
            contribution to the μ parameter arises from Kähler contributions
            ala Giudice-Masiero and these are expected to be 
            μ∼ c⟨ S_i⟩/m_pm_3/2∼ 0.1 m_3/2 
            (where c is some constant ∼ 1) 
            and thus is suppressed compared to scalar soft masses, 
            but perhaps comparable to gaugino masses.
            
            
            
            
             §.§ Extended MSSM-type solutions
            
            
              §.§.§ NMSSM: Added singlet with ℤ_3 discrete symmetry
            
            
            The case of adding an additional visible-sector gauge singlet superfield S 
            to the MSSM leads to the next-to-minimal SSM or NMSSM <cit.>. 
            Some motivation for the NMSSM can originate in string
            theory models such as heterotic orbifolds where the μ-term arises as 
            an effective term from couplings of the Higgs pair to a singlet field <cit.>.
            Without imposing any symmetry to forbid singlet couplings,
            we can write a generic NMSSM superpotential as follows:
            
            W_NMSSM=W_MSSM(μ =0 )+λ_μ S H_u H_d+
            ξ_F S +1/2μ_SS^2 + 1/3κ S^3
            
            and corresponding soft terms
             L_soft^NMSSM= L_soft^MSSM-(a_λ S H_uH_d+Bμ H_uH_d
            +1/3a_κ S^3 +1/2b_SS^2+t S+c.c.)-m_S^2|S|^2 .
            
            Here W_MSSM(μ=0) denotes the superpotential for the MSSM but 
            without the μ-term.
            The tadpole t in Eq. (<ref>) may have destabilizing quadratic 
            divergences and must be suppressed <cit.>. 
            A ℤ_3 discrete symmetry is usually imposed 
            wherein chiral superfields transform as ϕ→ e^2π i/3ϕ which 
            sends the dimensionful couplings ξ_F, μ, μ_S, Bμ, b_S and t to zero 
            (only cubic couplings are allowed) at the 
            expense of possibly introducing domain walls into the early universe 
            after the electroweak  phase transition <cit.>.
            (Some means of avoidance of domain walls are proposed in 
            Ref's <cit.>.) 
            By minimizing the scalar potential, now including
            the new singlet scalar S, then vevs v_u, v_d and v_s are induced.
            An effective μ term emerges with
            μ_ eff=λ_μ v_s .
            
               
            An attractive alternative choice for μ-forbidding symmetry 
            than the (perhaps ad-hoc) ℤ_3 would be one of the 
            anomaly-free discrete R-symmetries ℤ_4^R 
            or ℤ_8^R <cit.>. 
            Like the ℤ_3 discrete symmetry, the ℤ_8^R symmetry also forbids 
            the dangerous divergent tadpole term. 
            The  ℤ_4^R symmetry would allow the linear singlet term, but
            it can be argued that in the effective theory the linear term appears 
            when the fields with which the singlet field is coupled acquire VEVs. 
            If these fields belong to the hidden sector, then the coupling will be 
            suppressed by some high mass scale ranging as high as m_P in the 
            case of gravity-mediation. 
            In this case the linear singlet term will be present but it will be 
            highly suppressed <cit.>.
             
            Thus, all the advantages of the ℤ_3 discrete symmetry can be 
            obtained by imposing instead either a ℤ_4^R or ℤ_8^R 
            symmetry: this then  avoids the disadvantages–ad-hocness and introduction 
            of domain walls into the early universe after electroweak phase transition– 
            inherent in the ℤ_3 discrete symmetry.
            
            The added singlet superfield S  in the NMSSM leads to new scalar and 
            pseudoscalar Higgs fields 
            which can mix with the usual MSSM Higgses for v_s∼ v_u,d. So far, 
            LHC Higgs coupling measurements favor a SM-like Higgs so one might expect
            v_s≫ v_u,d which may lead one to an unnatural value of μ_ eff.
            The superfield S also contains a spin-1 2 singlino s̃ 
            which may mix with the usual neutralinos and might even be the 
            LSP <cit.>. 
            In the NMSSM, an additional Higgs quartic potential term is generated from the F-term of the singlet superfield, 
            and thus the SM-like Higgs mass 125 GeV is explained more easily
            without introducing large one-loop corrections.
            This feature can make the NMSSM more attractive to those who are 
            uncomfortable with an MSSM Higgs of mass m_h≃ 125 GeV<cit.>.
            
            
            
              §.§.§ nMSSM
            
            
            An alternative singlet extension of the MSSM is the Nearly-Minimal Supersymmetric Standard 
            Model (nMSSM) (also sometimes called Minimal Nonminimal Supersymmetric Standard Model or 
            MNSSM) <cit.>.
            The nMSSM, like the NMSSM, solves the μ problem via an added singlet superfield S. 
            But in the nMSSM, the model is founded on a discrete R-symmetry either ℤ_5^R or 
            ℤ_7^R. Discrete R-charge assignments for ℤ_5^R are shown in Table <ref>.
            The tree level superpotential is given by
            
            
            W_nMSSM∋λ_μ SH_uH_d+f_uQH_uU^c+f_dQH_d D^c+f_ℓLH_dE^c+f_νLH_uN^c
            +1/2 M_N N^cN^c 
            
            so that unlike the NMSSM with ℤ_3 symmetry, the κ S^3 term is now forbidden. 
            This is why the model is touted as a more minimal extension of the MSSM.
            The discrete R symmetry is broken by SUSY breaking effects in gravity-mediation.
            Then, in addition to the above terms, an effective 
            potential tadpole contribution
            
            
            W_nMSSM^tad∋ξ_F S 
            
            
            is induced at six-loop or higher level where ξ_F∼ m_3/2^2  
            (along with a corresponding soft SUSY breaking term).
            Due to lack of the discrete global ℤ_3 symmetry, the nMSSM then avoids the domain wall
            and weak scale axion problems that might afflict the NMSSM.
            
            
            
            
            Like the NMSSM, the nMSSM will include added scalar and pseudoscalar Higgs particles along with
            a fifth neutralino. However, due to lack of the S self-coupling term and presence of the tadpole term,
            the mass eigenstates and couplings of the added matter states will differ from the 
            NMSSM <cit.>.
            The neutralino in the nMSSM is very light, mostly below 50 GeV,
            but it is hard to get lower than 30 GeV due to the dark matter relic density constraint. 
            Since the neutralinos are so light it is very 
            likely that a chargino will decay into either a MSSM-like χ_2^0 or a singlino χ_1^0, giving rise 
            to a 5 lepton final state. 
            A further decay of the neutralino can give rise to a 7 lepton state. 
            These kinds of multilepton events are more likely in the 
            nMSSM than in the NMSSM. 
            Also, since in the nMSSM the neutralino can be so light, 
            then deviations in Higgs boson h decay branching fractions 
            become more likely than in the case of the 
            NMSSM<cit.>.
            
            
            
            
              §.§.§ Mu-from-nu SSM (μνSSM)
            
            
            The μ-from-νSSM (μνSSM) <cit.> 
            is in a sense a more minimal version of the NMSSM in that it makes use of the 
            gauge singlet right-hand-neutrino 
            superfields N^c_i to generate a μ term.
            The μνSSM first requires a ℤ_3 symmmetry to forbid the usual 
            μ term (and also a usual Majorana neutrino mass term M_iN^cN^c). 
            The superpotential is given by
            
            W    ∋   f_uQH_uU^c+f_dQH_d D^c+f_ℓLH_dE^c+f_νLH_uN^c 
                +   λ_μ i N^c_i H_u H_d+1 3κ_ijkN_i^cN_j^cN_k^c .
            
            If the scalar component of one of the RHN superfields 
            ν̃_Ri of N_i^c gains a weak scale vev,
            then an effective μ term develops:
            
            μ_ eff = λ_μ i⟨ν̃_Ri⟩
            along with a weak scale Majorana neutrino mass term 
            M_Njk∼κ_ijk⟨ν̃_Ri⟩.
            By taking small enough neutrino Yukawa couplings, then a weak scale
            see-saw develops which can accommodate the measured neutrino
            masses and mixings.
            
            The μνSSM develops bilinear R-party violating terms via the
            superpotential f_ν LH_uN^c term so that the lightest μνSSM particle
            is not stable and doesn't comprise dark matter: χ̃_1^0→ W^(*)ℓ
            and other modes. 
            As an alternative, a gravitino LSP is suggested with age longer than the 
            age of the universe:
            it could decay as G̃→νγ and possibly yield gamma ray signals
            from the sky <cit.>. 
            The phenomenology of the μνSSM also becomes more
            complex: now the neutrinos inhabit the same mass matrix as neutralinos,
            leptons join charginos in another mass matrix and Higgs scalars and sneutrinos
            inhabit a third mass matrix 
            (albeit with typically small mixing effects).
            Collider signals are strongly modified from usual MSSM 
            expectations <cit.>.
            
            While the μνSSM may be considered the most minimal model to solve 
            the μ problem, it suffers the same ℤ_3 domain wall problem as the NMSSM
            (and perhaps the same routes to avoidance <cit.>). 
            Also, in the context of GUTs, the role that the N_i^c field plays in
            the  16-dimensional spinor of SO(10) woud have to be abandoned.
            
            
            
            
             §.§ μ from an extra local U(1)^'
            
            
            In this class of models <cit.>, 
            a SM singlet superfield S is introduced which is charged under a new 
            U(1)^' gauge interaction, so
            terms with mass dimensions in Eq. (<ref>) are forbidden.
            
            
            
            
            
            
            Due to the U(1)^' gauge charges of S, 
            the cubic coupling S^3 is also absent.
            We will see below three representative realizations of this class of model.
            
            
            
            
              §.§.§ CDEEL model
            
            
            
            Cvetic-Demir-Espinosa-Everett-Langacker <cit.> (CDEEL) 
            propose a U(1)^' extended gauge symmetry model as emblematic of 
            fermionic orbifold string compactifications. 
            While the usual μ term is forbidden by the extended gauge symmetry, 
            the superpotential term
            
            W∋λ_μ S H_u H_d
            
            is allowed and under U(1)^' breaking then S develops a vev ⟨ S⟩∼ m_weak 
            such that a μ term is generated μ_ eff=λ_μ⟨ S⟩
            along with an additional weak scale Z^' gauge boson.
            Forbidding the μ term via a gauge symmetry avoids the gravity spoliation/global symmetry problem. 
            In addition, the μ term is linked to EW
            symmetry breaking and this would be expected to occur at m_weak rather than m_soft. 
            The U(1)^' breaking can occur either via
            large soft SUSY breaking trilinear couplings or via radiative corrections driving
            certain mass-squared terms negative.
            A way to test this class of models, in the exotica decoupling limit,
            is to search for new Z^' gauge bosons with exotic decays to
            light higgsinos <cit.>.
            
            To maintain anomaly cancellation, a variety of (intermediate scale) exotic quark and lepton fields
            must be introduced along with extra SM gauge singlets.
            If these new states come in GUT representations,
            then gauge coupling unification can be maintained.
            A set of possible U(1)^' gauge charges are listed in Table <ref>.
            
            
            
            
            
              §.§.§ sMSSM model
            
            
            An alternative U(1)^'-extended MSSM (abbreviated as sMSSM)<cit.> 
            also solves the μ problem by invoking multiple SM singlet superfields charged under U(1)^' symmetry.
            
            In this model, a visible-sector singlet field S directly couples to Higgs doublets
            but avoids stringent constraints on having an 
            additional weak scale Z^' gauge boson by introducing as well a 
             secluded sector containing three additional singlets S_1, S_2, S_3 charged under U(1)^'.
            The superpotential is given by
            
            W_sMSSM∋λ_μ S H_u H_d+λ_s S_1S_2S_3
            
            so that the secluded sector has a nearly F- and D-flat scalar potential. 
            The U(1)^' and electroweak symmetry breaking then occurs as a result of SUSY breaking
            A-terms. 
            Then the secluded sector scalars can obtain vevs much larger than the weak scale; if also the trilinear singlet coupling λ_s is small, 
            then the additional Z^' essentially decouples. 
            Nonetheless, additional Higgs and singlinos appear in the weak scale effective theory 
            so that this model phenomenologically resembles the nMSSM 
            (described in Subsec. <ref>) which has very different
            manifestations from what is expected from the CDEEL U(1)^' model.
            
            
            
            
              §.§.§ HPT model
            
            
            
            The Hundi-Pakvasa-Tata (HPT) model <cit.> also solves the SUSY μ problem
            by positing an additional U(1)^' gauge symmetry in a supergravity 
            context. 
            The U(1)^' charges of the multiplets in the HPT scheme are shown
            in Table <ref>. 
            With these U(1)^' charge assignments, the μ term
            is forbidden in the superpotential but (unlike the CDEEL model) a dim-4 term 
            as μ solution à la Kim-Nilles is allowed:
            
            W∋λ_μ S^2 H_u H_d/M_p .
            
            The U(1)^' gauge symmetry also forbids trilinear RPV couplings
            and dangerous p-decay operators.
            When the U(1)^' breaks (at an intermediate scale Q∼ 10^11 GeV), 
            the S field acquires a vev to yield an effective μ parameter of the 
            required magnitude.
            
            A distinctive feature of the HPT model is that a bilinear RPV (bRPV) term, LH_u is allowed at
            the right magnitude so as to generate phenomenologically-allowed neutrino masses <cit.>.
            The desired pattern of neutrino masses and mixing angles
            are also accommodated through radiative corrections.
            The bRPV leads to an unstable lightest neutralino which decays via 
            χ̃_1^0→ℓ W^(*) or ν Z^(*) and may lead to displaced vertices
            in collider events. Dark matter must be comprised of some other particles ( e.g. axions).
            Also, the U(1)^' is broken at the intermediate scale Q∼ 10^11 GeV so that
            the additional Z^' has a mass far beyond any collider reach.
            
            Since solving the μ problem as well as generating the neutrino mass scale of suitable order
            requires introduction of a new gauge group U(1)^', care must be taken so that
            associated anomalies are cancelled. 
            Anomaly cancellation requires introducing various additional exotic fields
            including color triplets K_i and K_i^' states. The lightest of these 
            leads to stable weak-scale exotic hadrons which may also yield 
            highly-ionizing tracks at collider experiments. 
            In the HPT scheme, gauge coupling unification may be upset. 
            
            
             §.§ Solutions related to Peccei-Quinn symmetry breaking
            
            
            In this Subsection, we examine natural μ-term solutions related to 
            the PQ symmetry used to solve the strong CP problem.
            In this class of models, the μ-term is forbidden by the PQ symmetry, 
            but generated once the PQ symmetry is spontaneously broken.
            Then the model also provides a solution to the strong CP problem and 
            generates axion dark matter.
            In Subsec. <ref>, <ref>, and <ref>,
            we review μ-term generation models with various sources of PQ breaking.
            
            Meanwhile, imposing a global symmetry causes the `quality' 
            issues of the symmetry which may spoil the PQ solution to the strong CP problem,
            since global symetries are not protected from quantum gravity effects.
            In Subsec. <ref>, we discuss a criterion for protecting the PQ solution to the strong CP problem,
            and in Subsec. <ref> we present examples based on 
            discrete R-symmetries which satisfy the gravity-safety criterion
            and can be considered as generating an accidental, approximate PQ symmetry.
            Also, we review the natural Higgs-flavor-democracy (HFD) solution 
            which contains an approximate PQ symmetry
            from a discrete symmetry in Subsec. <ref>.
            
            Finally, we review μ-term generation by breaking of PQ symmetry from SUSY breaking:
            radiative breaking of PQ symmetry (Subsec. <ref>), breaking of an accidental approximate PQ symmetry from a gauged U(1)_R symmetry (Subsec. <ref>) and a ℤ_22 discrete gauge symmetry (Subsec. <ref>) by a large negative trilinear term. 
            
            
            
            
              §.§.§ Kim-Nilles solution
            
            
            Kim and Nilles (KN) <cit.> presented the first formulation 
            of the SUSY μ problem along with a proposed solution. 
            In KN, it is proposed that there exists a global Peccei-Quinn (PQ) symmetry U(1)_PQ 
            which is needed at first as a solution to the strong CP problem. 
            The PQ symmetry is implemented in the context of the supersymmetrized version
            of the DFSZ <cit.> axion model[In the DFSZ axion model <cit.>, the SM is extended to include two Higgs doublets which then couple to singlets
            which contain the axion.] wherein the Higgs multiplets carry PQ
            charges  e.g.Q_PQ(H_u)=Q_PQ(H_d)=-1 so that the μ term is 
            forbidden by the global U(1)_PQ. 
            Next, the Higgs multiplets are coupled 
            via a non-renormalizable interaction to a SM gauge singlet field X
            which carries a PQ charge Q_PQ(X)=+2/(n+1):
            
                W_μ∋λ_μ/m_P^nX^n+1H_uH_d
            
            for n≥ 1. 
            
            It is arranged to spontaneously break PQ by giving the X field a vev ⟨ X⟩
            which also generates a (nearly) massless axion a which solves the strong CP problem.
            To obtain cosmologically viable axions– with 
            ⟨ X⟩∼ 10^11 GeV and with m_p ≃ 2.4× 10^18 GeV, we can obtain 
            the μ parameter of the order of m_3/2 only if n = 1 (for which Q_PQ(X) = +1).
            The matter superfields also carry appropriate PQ charge so as to 
            allow the MSSM trilinear superpotential terms: see Table <ref>.
            
            
            
            
            The intermediate PQ breaking scale can be gained from a PQ superpotential of the form:
            
            W_PQ=λ_PQZ( XY-v_PQ^2).
            
            The scalar components of X and Y develop vevs 
            ⟨ X⟩ =⟨ Y⟩ = v_PQ
            such that a μ term is generated:
            μ = λ_μ⟨ X⟩^2/m_P .
            
            This value of the μ term μ∼λ_μv_PQ^2/m_P 
            is to be compared to the soft breaking scale
            in models of gravity-mediation: m_soft∼ m_3/2∼ m_hidden^2/m_P.
            Here, v_PQ is identified as v_PQ∼ m_hidden and thus μ is 
            obtained as μ∼ m_3/2.  
            But, a value μ∼ m_weak≪ m_soft∼ m_3/2 can be accomodated
            for v_PQ< m_hidden,  i.e. if the scale of PQ breaking lies somewhat 
            below the mass scale associated with hidden sector SUSY breaking.[
            In models with SUSY breaking arising from  e.g. gaugino condensation
            at an intermediate scale Λ_h, 
            then m_3/2∼Λ_h^3/m_P^2
            in which case we would define m_hidden^2 ∼Λ_h^3/m_p.][
            The model <cit.> shows a more complete ultraviolet theory which includes 
            a mechanism to get v_PQ in the intermediate scale through the introduction
            of a chiral superfield in the hidden brane, yielding an 
            ultraviolet suppressed term
            in the hidden brane which gives rise to μ∼ m_weak when SUSY is broken 
            in the hidden brane through the shining mechanism <cit.>.]
            A virtue of the KN solution is that it combines a solution to the strong CP
            problem with a solution to the SUSY μ problem which also allows for a 
            Little Hierarchy. A further benefit is that it provides an additional
            dark matter particle– namely the DFSZ <cit.> axion– to co-exist with the
            (thermally under-produced) higgsino-like WIMP from natural SUSY.
            Thus, dark matter is then expected to be comprised of a 
            WIMP/axion admixture <cit.>.
            For the lower range of PQ scale v_PQ, then
            the dark matter tends to be axion dominated with typically 10-20% WIMPs
            by mass density <cit.>. For larger v_PQ values, then non-thermal processes 
            such as saxion and axino <cit.> decay augment the WIMP abundance while for even larger
            values of v_PQ then the higgsino-like WIMPs are overproduced and one typically
            runs into BBN constraints from late-decaying neutral particles 
            (saxions and axinos) or overproduction of relativistic axions from saxion decay which contribute to the effective number of neutrino species
            N_eff 
            (which is found to be N_eff=3.13± 0.32 from the recent Particle Data Group tabulation <cit.>).
            In the context of the DFSZ model embedded within the MSSM, 
            then the presence of higgsinos in the aγγ triangle diagram 
            is expected to reduce the axion-photon-photon coupling to levels
            below present sensitivity making the SUSY DFSZ axion very challenging to 
            detect <cit.>.
            
            
            
            
              §.§.§ Chun-Kim-Nilles
            
            
            In the CKN model <cit.>, it is assumed that SUSY is broken in the hidden sector
            due to gaugino condensation 
            ⟨λλ⟩∼Λ_h^3∼ (10^13 GeV)^3 
            in the presence of a hidden SU(N)_h gauge group. Furthermore, 
            there may be vector-like hidden sector quark chiral superfields present
            Q and Q^c which transform as N and N^* under SU(N)_h. 
            The Higgs and hidden quark superfields carry PQ charges as in Table <ref>:
            
            
            This allows for the presence of a superpotential term
            
            W_CKN∋λ_μ/m_PQQ^cH_uH_d .
            
            Along with gauginos condensing at a scale Λ_h to break SUGRA with
            m_3/2∼Λ_h^3/m_P^2, 
            the hidden sector scalar squarks condense at a scale Λ <Λ_h
            to break the PQ symmetry and to generate a μ term
            μ_ eff∼λ_μΛ^2/m_P .
            
            Thus, this model provides a framework for μ <m_soft. 
            It also generates a DFSZ axion to solve the strong CP problem
            along with a string model-independent (MI) axion which could provide 
            a quintessence solution for the cosmological constant (CC) <cit.>.
            The CC arises from the very low mass MI axion field slowly settling to the 
            minimum of its potential.
            
            
            
            
            
              §.§.§ BK/EWK solution linked to inflation and strong CP
            
            
            In Ref's <cit.>, a model is proposed with superpotential
            
            W_EWK∋λ_μϕ H_u H_d+κϕ N^2
            
            where the ϕ field plays the role of inflaton and the N field is a
            waterfall field leading to hybrid inflation in the early universe <cit.>.
            Although the model appears similar to the NMSSM, it is based on a 
            PQ rather than ℤ_3 symmetry with charges as in Table <ref>. 
            Thus, it avoids the NMSSM domain wall problems which arise
            from a postulated global ℤ_3 symmetry. 
            Augmenting the scalar potential with soft breaking terms, then the ϕ and
            N fields gain vevs of order some  intermediate scale Q∼ 10^12 GeV
            so that Yukawa couplings λ_μ and κ are of order 
            10^-10. Such tiny Yukawa couplings might arise from type-I 
            string theory constructs <cit.>.
            To fulfill the inflationary slow-roll conditions, then the field
            ϕ must gain a mass of less than 5-10 MeV and a reheat temperature
            of 1-10 GeV. Domain walls from breaking of the PQ symmetry are inflated away.
            
            
            
            
              §.§.§ Global symmetries and gravity
            
            
            It is well known that gravitational effects violate global symmetries, as has been
            considered via black hole “no hair” theorems <cit.> and 
            wormhole effects <cit.>.
            In such cases, it has been questioned whether the PQ mechanism can be realistic 
            once one includes gravity or embeds the SUSY PQ theory into a 
            UV complete string framework <cit.>.
            Indeed, Kamionkowski and March-Russell <cit.> (KMR) considered the effect of 
            gravitational operators such as 
            
            
            V(ϕ )∋g/m_P^2m+n-4|ϕ|^2mϕ^n+h.c. +c
            
            involving PQ charged fields ϕ in the scalar potential upon the axion potential. In the case of 2m+n=5, 
             i.e. a term suppressed by a single power of m_P, then these gravitational terms
            would displace the minimum of the PQ axion potential such that the QCD 
            CP violating term G_μν AG̃_A^μν settles to a non-zero minimum 
            thus destroying the PQ solution to the strong CP problem. To maintain 
            θ̅ 10^-10, KMR calculated that all gravitational 
            operators contributing to the axion potential should be suppressed by at least
            powers of (1/m_P)^8. This is indeed a formidable constraint! 
            
            
            
            
            To avoid such terms, additional symmetries are required <cit.>. 
            In string theory, it is known that discrete symmetries arising from gauge symmetries 
            are gravity-safe, as are other discrete symmetries or R-symmetries arising from
            string compactification. 
            In Fig. <ref> the Kim diagram is shown <cit.>. 
            The red/lavender column denotes an infinite set of Lagrangian terms in the model 
            under consideration which obey some exact, gravity-safe, discrete symmetry. 
            Of this set of terms, the few 
            lower order terms, denoted by the lavender region, obey an exact global symmetry, 
            understood here to be the PQ symmetry whose breaking yields the QCD axion.
            The red-shaded terms obey the discrete symmetry but violate any global
            symmetry.
            The green/lavender row denotes the full, infinite set of global symmetry terms, of which the
            green-shaded terms are not gravity-safe. If the discrete symmetry is strong enough, 
            then the gravity-unsafe terms will be sufficiently suppressed. 
            The global PQ symmetry is expected to be approximate. The question then is: 
            is it sufficiently strong so as to be gravity-safe? 
            Some additional gravity-safe symmetry is required to ensure the PQ mechanism is robust. 
            The lavender region represents gravity-safe terms which obey the global symmetry.
            
            
            
              §.§.§ Gravity-safe symmetries : gauge symmetries or R-symmetries: 
            continuous or discrete 
            
            
            Given that global symmetries are not gravity-safe (and hence not fundamental),
            it is common to turn to gauge symmetries as a means to forbid the μ term.
            Some models based on an extra local U(1)^' were examined in Subsec. <ref>.
            Some problems with this approach emerge in that one has to suitably hide any new
            gauge bosons associated with the extra gauge symmetry and one must also typically 
            introduce (and hide) extra exotic matter which may be needed to ensure 
            anomaly cancellation. In addition, such exotic matter may destroy the desireable feature
            of gauge coupling unification should the new exotica not appear in complete GUT multiplets.
            
            An alternative approach is to introduce  discrete gauge 
            symmetries <cit.>. 
            Such ℤ_M symmetries may emerge from a local U(1)^' when a charge 
            M object (charged under the new U(1)^') condenses at very high
            energy leaving a discrete ℤ_M gauge symmetry in the low energy effective theory.
            Since the ℤ_M emerges from a local gauge theory, it remains gravity-safe.
            In Subsec. <ref>, the MBGW model <cit.> which is based on a 
            ℤ_22 discrete gauge symmetry is examined. 
            The model under ℤ_22 is found to be anomaly-free and is used to not only
            forbid the μ term but to generate a PQ symmetry needed to solve the strong CP problem.
            The lowest order PQ violating term allowed by the ℤ_22 
            is sufficiently suppressed so that PQ arises as an  
            accidental approximate global symmetry thereby rendering the model to be gravity-safe. 
            The ℤ_22 discrete gauge charges of the multiplets turn out to be 
            not consistent with GUTs which should be manifested at some level in the high energy theory. 
            Also, the presence of a charge 22 object which  condenses
            at some high energy scale may not be very plausible and might be inconsistent 
            with the UV completion of the theory ( i.e. lie in the swampland). 
            
            Continuous or discrete R-symmetries offer a further choice for gravity-safe symmetries.
            A solution using a continuous U(1)_R symmetry was examined in Subsec. <ref>.[See also Ref. <cit.>.]
            In the interest of minimality, it is noted that continuous R symmetries
            are not consistent with the particle content of just the MSSM <cit.>.
            Then it is also of interest to examine the possibility of discrete 
            remnant R-symmetries ℤ_N^R which arise upon compactification
            of the full Lorentz symmetry of 10-d string theories.
            R-symmetries are characterized by the fact that superspace co-ordinates 
            θ carry non-trivial R-charge: 
            in the simplest case, Q_R(θ )=+1 so that Q_R( d^2θ ) =-2. 
            For the Lagrangian L∋∫ d^2θ W to be invariant under 
            ℤ_N^R-symmetry, the superpotential W must carry Q_R(W)= 2 +integer 
            multiples of N. 
            
            
            
             
            These remnant discrete R-symmetries ℤ_N^R– if sufficiently strong–
            can forbid lower order operators in powers of 1/m_P which would violate
            putative global symmetries such as PQ.
            Such a built-in mechanism from string theory may enable the PQ symmetry 
            to be strong enough to support the axion solution
            to the strong CP problem. Since the R-symmetry is necessarily supersymmetric 
            (it acts on superspace co-ordinates), this is another instance in how the implementation of 
            the axion solution to the strong CP problem is enhanced and made more plausible by the 
            presence of supersymmetry. 
            However, not all possible R-symmetries are a suitable candidate for a fundamental 
            symmetry. 
            Table <ref> (as derived in Ref's <cit.>) 
            shows the R-symmetries along with the R-charges of the multiplets 
            which are consistent with either SU(5) or 
            SO(10) unification, anomaly-free (allowing for a Green-Schwarz term),
            forbid the μ term and also forbid the 
            R-parity violating and dimension-five proton decay operators and 
            hence can serve the purpose of being a fundamental symmetry.
            In fact, the  ℤ_N^R symmetries of Table <ref> have been shown to be the
             only anomaly-free symmetries which allow for fermion masses and suppress the
            μ term while maintaining consistency with GUTs. As a bonus, they allow for
            neutrino masses while forbidding R-parity and dangerous proton decay operators. 
            Implementation of the discrete R-symmetries is only possible in extra-dimensional
            GUTs, making their implementation in string compactifications very natural <cit.>.
            
            
            
            
            
              §.§.§ Natural Higgs-Flavor-Democracy (HFD) solution to μ problem
            
            
            
            
            
            In Ref. <cit.>, the μ problem is solved by introducing additional identical 
            Higgs doublet superfields to those of the MSSM. 
            The theory then contains a direct product of discrete interchange symmetries 
            S_2(H_u)× S_2(H_d). 
            This is  Higgs flavor democracy (HFD).
            Besides solving the μ problem, this mechanism also gives rise to an approximate PQ symmetry and 
            hence a light QCD axion, thereby solving the strong CP problem whilst avoiding the 
            gravity spoliation problem. 
            The HFD discrete symmetry can be found in several string theory models.
            
            HFD: One starts by introducing two pairs of Higgs doublets 
            at the GUT scale m_G namely : {H_u^(1), H_d^(1)} and {H_u^(2), H_d^(2)}. 
            However, the weak scale MSSM requires only one pair of Higgs doublets: {H_u, H_d}. 
            If, at the GUT scale, the two pairs of Higgs doublets : 
            H_u = {H_u^(1), H_u^(2)} and H_d = {H_d^(1), H_d^(2)} 
            are indistinguishable then there must exist the permutation
            symmetries S_2(H_u)× S_2(H_d) . 
            Then the Higgsino mass matrix has a democratic form given by: 
            
                {m_G/2    m_G/2  
                m_G/2    m_G/2 
                } .
             
            The Higgs mass eigenvalues are m_G and 0. 
            Hence, the Higgs pair in the weak scale MSSM is obtained to be massless. 
            Still, the model construction of the MSSM requires a massive Higgs pair at the weak scale
            with mass value μ. 
            In order to fulfill this criteria, the HFD must be broken and this mechanism 
            results in μ≈𝒪(TeV).
            
            Generation of μ: The minimal Kahler potential is considered as K = Φ_i Φ_i^† 
            where Φ_i ( i =1, 2) is a doublet under the gauge group such as the Higgs superfield and 
            X_i and X̅_̅i̅ (i=1,2) are singlets under the gauge group. 
            Both Φ_i and X_i and the corresponding barred fields obey the S_2× S_2 symmetry. 
            X^(0) and X̅^(0) are SM singlet fields containing a very light QCD axion for 
            10^9 GeV ≤v_PQ≤10^12 GeV. 
            With this construct, the S_2(L) ×  S_2(R) symmetric nonrenormalizable term is:
            
            
                W^(nonrenormalizable) = ∑_i,j=1,2( X^(i)X̅^(j)/m_P) H_u^(i) H_d^(j)
                    + ∑_ij∑_kl( X^(i)X̅^(j)/m_P) H_u^(k) H_d^(l)
            
            With the HFD breaking minimum at ⟨ X_1 ⟩ = ⟨X̅_̅1̅⟩ = v_PQ 
            and ⟨ X_2 ⟩ = ⟨X̅_̅2̅⟩ = 0, Eq. (<ref>) becomes 
            
                W^(nonrenormalizable) = λ_μ  v_PQ^2/2m_P(H_u^(0)+H_u^(M_G))(H_d^(0)+H_d^(M_G))
            
            This choice of HFD breaking minimum is spontaneous.
            Thus we obtain μ = λ_μ  v_PQ^2/2m_P. 
            With 10^10 GeV ≤v_PQ≤10^12 GeV and λ_μ≈𝒪(1), 
            we obtain μ≈𝒪(0.1-10^3  TeV). 
            The LH can be accomodated for the lower range of v_PQ or if λ_μ <1.
            
            Light QCD Axion - Integrating out the heavy fields in Eq. (<ref>), one obtains
            
                W = λ_μ  X^(0)X̅^(0)/2m_PH_u^(0)H_d^(0) .
            
            The PQ charges of Higgs multiplets are obtained from their interaction with the 
            quarks and PQ charges of X^(0) and X̅^(0) are defined by Eq. (<ref>). 
            Thus, a term m_3/2λ^2/4m_P^21/M_GH_uH_d(XX^c)^2 is 
            obtained which violates PQ and hence adds a tiny correction to μ. 
            Here, M_G is the GUT scale higgsino mass. 
            Hence, PQ symmetry emerges as an approximate symmetry, thereby giving rise to a light QCD axion 
            which does not suffer from the gravity-spoliation problem.
            
            
            
            
            
              §.§.§ Radiative PQ breaking from SUSY breaking
            
            
            The above models are particularly compelling in that
            they include supersymmetry which solves the gauge hierarchy problem, 
            but also include the axion solution to the strong CP problem of QCD. 
            In addition, they allow for the required Little Hierarchy of μ≪ m_soft.
            A drawback to the KN model is that it inputs the PQ scale “by hand” 
            via the superpotential Eq. (<ref>). It is desireable if the PQ scale can 
            be generated via some mechanism and furthermore, the emergence of
            three  intermediate mass scales in nature– the hidden sector
            SUSY breaking scale, the PQ scale and the Majorana neutrino scale– 
            begs for some common origin. 
            A model which accomplishes this was first proposed
            by Murayama, Suzuki and Yanagida (MSY) <cit.>.
            
            In radiative PQ breaking models, the MSSM superpotential is
            
            W_MSSM=∑_i,j=1^3[ ( f_u )_ijQ_iH_u U_j^c+ 
            ( f_d )_ijQ_iH_d D_j^c+( f_e )_ijL_iH_d E_j^c+( f_ν )_ij
            L_iH_u N_j^c ]
            
            where we explicitly include the right hand neutrino superfields N_i and the
            generation indices i,j run from 1-3. 
            To this, we add a PQ superpotential containing new PQ-charged fields X and Y of
            the form
            
            W_PQ∋1/2h_ijXN_i^cN_j^c+f/m_PX^3Y+W_μ
            and where 
            
            W_μ^MSY =g_MSY/m_PXYH_uH_d ,
            
            
            
            where the PQ charges Q_PQ(matter)=1/2, Q_PQ(Higgs)=-1, Q_PQ(X)=-1
            and Q_PQ(Y)=3. 
            Along with the MSY superpotential terms, we include the corresponding 
            soft SUSY breaking terms
            
            V_MSY   ∋    m_X^2|ϕ_X|^2+m_Y^2|ϕ_Y|^2+m_N_i^2|ϕ_N_i|^2
               +   ( 1/2h_iA_iϕ_N_i^2ϕ_X+f/m_PA_fϕ_X^3ϕ_Y+
            g_MSY/m_PA_gH_uH_dϕ_Xϕ_Y +h.c.).
            
            For simplicity, we assume a diagonal coupling h_ij= h_iδ_ij.
            The model may be defined as applicable at the reduced 
            Planck scale m_P≃ 2.4× 10^18 GeV and the corresponding 
            Renormalization Group Equations (RGEs) can be found in Ref. <cit.>
            at 1-loop and Ref. <cit.> at 2-loop order.
            Under RG evolution, the large Yukawa coupling(s) h_i push the 
            soft mass m_X^2 to negative values at some intermediate mass scale resulting
            in the radiatively-induced breakdown of PQ symmetry as a consequence of 
            SUSY breaking. 
            The scalar potential consists of the terms V=V_F+V_D+V_ soft.
            The Higgs field directions can be ignored since these develop vevs at much lower
            energy scales. Then the relevant part of the scalar potential is just
            
            V_F∋|f|^2/m_P^2|ϕ_X^3|^2+9|f|^2/m_P^2|ϕ_X^2ϕ_Y|^2 .
            
            Augmenting this with V_ soft, we minimize V at a scale Q=v_PQ 
            to find the vevs of ϕ_X and ϕ_Y (v_X and v_Y):
            
            0   =   9|f|^2/m_P^2|v_X^2|^2v_Y +f^*A_f^*/m_Pv_X^*3+m_Y^2v_Y 
            
            
            0   =   3|f|^2/m_P^2|v_X^2|^2v_X+18|f|^2/m_P^2|v_X|^2|v_Y|^2v_X
            +3f^*A_f^*/m_Pv_X^*2v_Y^*+m_X^2v_X .
            
            The first of these may be solved for v_Y. Substituting into the second, we find a 
            polynomial for v_X which may be solved for numerically. 
            The potential has two minima in the v_X and v_Y plane 
            symmetrically located with respect to the origin. 
            For practical purposes, we use the notation v_X=|v_X| and v_Y=|v_Y|.
            
            The fields ϕ_X and ϕ_Y obtains vevs v_X and v_Y
            at the intermediate mass scale, taken here to be v_PQ=√(v_X^2+9v_Y^2).
            The corresponding axion decay constant is given by f_a = √(2)v_PQ.[For axion interactions,
            the actual decay constant is f_A≡ f_a/N_DW where N_DW is the domain wall number.]
            
            
            
            A DFSZ-like axion a arises as the pseudo-Goldstone boson of spontaneous
            PQ breaking, thus solving the strong CP problem. A μ parameter, which is
            originally forbidden by PQ symmetry, is generated with a value
            μ_ eff=g_MSYv_Xv_Y/m_P
            and a Majorana neutrino mass, also initially forbidden by PQ symmetry, 
            is generated at 
            
            M_N_i=h_i|_Q=v_x v_X .
            
            Since the μ term depends on an arbitrary coupling g_MSY, one may obtain any desired
            value of μ for particular v_X and v_Y vevs by suitably adjusting g_MSY. 
            However, if the required values of g_MSY are very different from unity, 
             i.e.g_MSY≫ 1 or g_MSY≪ 1, 
            we might need to introduce an additional physical scale to explain the μ term.
            To generate a value of μ =150 GeV, then values of g_MSY as shown in Fig. <ref> 
            are required depending on the values of m_3/2 and h(M_P) which are assumed. 
            
            
            The virtues of this model then include:
            
              *  it is supersymmetric, thus stabilizing the Higgs sector and allowing for
            a gauge hierarchy,
            
              *  it solves the strong CP problem via a DFSZ-like axion a,
            
              *  it presents a unified treatment of the three intermediate mass scale
            where the PQ and Majorana neutrino scales arise as a consequence of 
            SUSY breaking and
            
              *  it allows for a Little Hierarchy μ≪ m_soft for the case where
            v_PQ< m_hidden.
            
            
            
            Detailed numerical calculations in the MSY model have been carried out in Ref. <cit.>. 
            There, it is found that for generic W_μ^MSY couplings
            g_MSY∼ 0.1-1, then a μ parameter μ∼ 100-200 GeV can easily be 
            generated from TeV-scale soft breaking terms.
            Furthermore, since the μ term sets the mass scale for the W,Z,h 
            boson masses and is determined itself by the PQ vevs v_X and v_Y, then
            the axion mass 
            m_a≃ 0.48 f_πm_π/f_a=6.25× 10^-3  GeV/f_a 
            is related to the Higgs mass m_h and the higgsino masses 
            m__1,_1,2∼μ.
            
            
            
            
            The required PQ charges for the MSY model are listed in Table <ref>.
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            Other closely related models make different choices for which fields enter into W_μ.
            We can also have:
            
            W_μ^CCK   =   g_CCK/m_PX^2 H_uH_d    or 
            
            W_μ^SPM   =   g_SPM/m_PY^2 H_uH_d .
            
            The above three possibilities for W_μ correspond to Ref's <cit.> (MSY), 
            <cit.> (CCK) and <cit.> (SPM).
            The corresponding PQ charges for the three radiative PQ breaking models are listed in 
            Table <ref>.
            
            We list in Fig's <ref> and <ref> also the values of g_CCK and g_SPM which are needed
            to generate a value of μ≃ 150 GeV. For a given value of h(m_P) and m_3/2, 
            then typically g_CCK< g_MSY<g_SPM. 
            The MSY model has the interesting
            feature that the PQ charge assignments are consistent with SO(10) unification. 
            We also remark that all three models can easily generate
            weak scale values of μ from multi-TeV values of m_3/2: 
             i.e.μ≪ m_3/2 so that a Little Hierarchy is naturally generated.
            
             Gravity safety of radiative PQ breaking models: 
            An important issue for the radiative PQ breaking models is whether the 
            required PQ symmetry is actually gravity-safe and whether it may emerge from
            any of the aforementioned ℤ_N^R symmetries.
            We have examined whether or not the three radiative PQ breaking models of 
            Table <ref> (CCK, MSY and SPM) can be derived from any of the more
            fundamental ℤ_N^R symmetries in Table <ref> <cit.>.
            In almost all cases, the hXN^cN^c operator is disallowed: then there is no
            large Yukawa coupling present to drive the PQ soft term m_X^2 negative 
            so that PQ symmetry is broken. And since the PQ symmetry does not allow for
            a Majorana mass term 1/2M_NN^cN^c, then no see-saw scale can be developed.
            One exception is the MSY model under ℤ_4^R symmetry with charge
            assignments Q_R(X)=0 and Q_R(Y)=2: then a YH_uH_d term is allowed
            which would generate a μ term of order the intermediate scale.
            Also, without considering any specific R-charges for the fields X and Y, 
            we can see that the R-charges for X and Y should be such that the term 
            XYH_uH_d is allowed and since the R-charges of H_u and H_d are 0, 
            then a term  MXY would always be allowed: this term breaks PQ 
            at high order and is not gravity safe.
            A second exception is SPM under the ℤ_6^R symmetry with
            charges Q_R(X)=0 and Q_R(Y)=2: then operators like Y^4/m_p are allowed
            which break PQ but are not sufficiently suppressed so as to be gravity-safe. 
            Furthermore, we can see that in this model that the R-charge of Y 
            is such that terms like  M^2Y which break PQ are
            always allowed but are not gravity safe.
            Thus, we conclude that while the radiative PQ breaking models are indeed compelling and
            can address all three intermediate scales in a unified framework, the 
            required PQ symmetry does not appear gravity-safe.
            
            
            
              §.§.§ CCL model from gauged U(1)_R symmetry
            
            
            In the model of Choi, Chun and Lee <cit.> (CCL), 
            the μ term is generated in a manner similar to the SPM model <cit.>, 
            but with the difference that the fundamental symmetry is a 
            gauged U(1)_R symmetry out of which the PQ symmetry arises to be an 
            accidental approximate symmetry. 
            The superpotential for CCL is
            
            W_CCL   =   f_uQH_uU^c+f_dQH_dD^c+f_eLH_dE^c+f_νLH_uN^c+
               +   λ_μY^2H_uH_d/m_p+κ X^3Y/m_P+λ_NX^nN^cN^c/2m_P^n-1 ,
            
            with U(1)_R and PQ charges for the n=2 case given in Table <ref>.
            
            
            
            The singlets X and Y get their VEVs at the intermediate scale 
            when the PQ symmetry is broken via a large (relative to m_3/2) 
            negative trilinear soft term contribution to the scalar potential, 
            thereby giving rise to μ∼ m_soft. 
            The U(1)_R gauge boson has mass of order the compactification scale
            so the low energy theory is that of the MSSM.
            Because the fundamental symmetry of CCL is a gauged U(1)_R symmetry, 
            the phenomenology of this model is dictated by a hierarchy of soft terms 
            m_1/2≫ m_scalars>m_3/2 (m_1/2: gaugino mass). 
            Scalar soft masses are fixed in terms of U(1)_RD-terms
            and typically lead to large negative m_H_u^2 at the weak scale
            which then requires a large, unnatural μ term which would violate 
            the μ≪ m_soft Little Hierarchy.
            The gravitino or the RH sneutrino turns out to be the LSP and hence 
            end up as cold dark matter candidates. 
            If the neutrino is Majorana type then the gravitino is the LSP and 
            if the neutrino is Dirac type then the RH sneutrino is the LSP.
            
            
            
              §.§.§ MBGW model of PQ breaking from SUSY breaking
            
            
            The Martin-Babu-Gogoladze-Wang (MBGW) model <cit.> begins with a 
            superpotential
            
            W   =   f_uQH_uU^c+f_dQH_dD^c+f_eLH_dE^c+f_νLH_uN^c
                  +1/2M_RN^cN^c+λ_μX^2H_uH_d/m_p+λ_2(XY)^2/m_P
            which is augmented by soft SUSY breaking terms
            
            V_soft∋ m_X^2|ϕ_X|^2+m_Y^2|ϕ_Y|^2+(λ_2C(ϕ_Xϕ_Y)^2/m_P
            +h.c.)
            
            so that the scalar potential is
            
            V_MBGW=V_F+V_soft
            with 
            
            V_F∋ 4λ_2^2/m_P|ϕ_Xϕ_Y|^2( |ϕ_X|^2+|ϕ_Y|^2).
            
            The scalar potential admits non-zero minima in the fields ϕ_X and ϕ_Y for
            C<0. The scalar potential for the case of m_X=m_Y≡ m_s=10^4 GeV 
            and C=-3.5× 10^4 GeV is shown in Fig. <ref>.
            
            
            
            It is found in Ref. <cit.> that the model admits a remnant ℤ_22 
            discrete gauge symmetry which is anomaly free up to Green-Schwarz terms and forbids 
            lower order operators which would lead to gravitational instability. 
            Beside the terms in Eq. (<ref>), the lowest order 
            PQ-violating term in the superpotential is (Y)^11/m_P^8: 
            thus this model is gravity safe according to the KMR criterion.
            An approximate PQ symmetry emerges as an accidental consequence of the discrete 
            ℤ_22 gauge symmetry.
            The ℤ_22 and PQ charges are listed in Table <ref>.
            
            
            
            By taking ⟨ϕ_X⟩≡ v_x and ⟨ϕ_Y⟩≡ v_Y, 
            then the scalar potential minimization conditions read
            
            0    =    
            2λ_2/m_PC^*v_xv_Y^2+m_X^2v_X+4λ_2^2/m_P^2(
            v_Xv_Y^2(v_X^2+v_Y^2)+v_X^3v_Y^2)
            
            0    =    
            2λ_2/m_PC^*v_x^2v_Y+m_Y^2v_Y+4λ_2^2/m_P^2(
            v_X^2v_Y(v_X^2+v_Y^2)+v_X^2v_Y^3).
            
            A simplifying assumption of m_X^2=m_Y^2≡ m_s^2 and v_X=v_Y≡ v_s leads to
            
            v_s^2=-C±√(C^2-12m_s^2)/12λ_2m_P
            
            so that the μ term is
            μ_MBGW≃λ_μv_s^2/m_P
            with v_s^2≃|C|/12λ_2m_P.
            Taking m_s≃ m_3/2=10^4 GeV with μ =150 GeV and 
            C=-3.5× 10^4 GeV
            leads to v_s≃ v_PQ≃ 10^11 GeV for λ_2=0.7 and 
            λ_μ≃0.036. Thus, the MBGW model admits a Little Hierarchy 
            μ≪ m_3/2 whilst generating the PQ scale v_PQ∼ 10^11 GeV 
            (which generates mainly axion dark matter with a smaller portion of higgsino-like WIMPs <cit.>).
            The allowed range of MBGW model parameter space is shown in Fig. <ref>
            where we show contours of λ_μ values which lead to μ =150 GeV.
            
            
            
            As mentioned previously, the MBGW model appears gravity-safe under the ℤ_22 
            discrete gauge symmetry,
            The discrete gauge symmetry ℤ_M might arise if a charge Me 
            field condenses and is integrated out of the low energy theory while
            charge e fields survive (see Krauss and Wilczek, Ref. <cit.>). 
            While the ensuing low energy theory should be gravity safe, 
            for the case at hand one might wonder at the 
            plausibility of a condensation of a charge 22 object and whether it might 
            occupy the so-called  swampland <cit.> of theories not 
            consistent with a UV completion in string theory. 
            In addition, the charge assignments <cit.> are not consistent with  
            SU(5) or SO(10) grand unification which may be expected at some
            level in a more ultimate theory.
            
            Alternatively, it is worth checking whether MBGW is gravity-safe under any of the 
            discrete R-symmetries listed in Table <ref>.
            To check gravity safety, we note that additional superpotential terms of the
            form λ_3X^pY^q may be allowed for given ℤ_N^R charge 
            assignments and powers p and q. Such terms will typically break the PQ
            symmetry and render the model not gravity safe if the scalar potential V(ϕ )
            includes terms which are not suppressed by at least eight 
            powers of 1/m_P <cit.>. 
            The largest dangerous scalar potential terms develop
            from interference between λ_2 (XY)^2/m_P and 
            λ_3 X^pY^q/m_P^p+q-3 when constructing the scalar potential
            V_F=∑_ϕ̂|∂ W/∂ϕ̂|_ϕ̂→ϕ^2
            (here, the ϕ̂ label chiral superfields with ϕ being their leading components).
            We find the MBGW model to be not gravity safe under any of the ℤ_N^R
            discrete R-symmetries of Table <ref>.
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
             §.§ Hybrid models of PQ breaking from SUSY breaking
            
            
            
            
            In this Subsection, we review three models which combine approaches 
            where PQ symmetry breaking is triggered by SUSY breaking and 
            where a gravity-safe accidental approximate PQ symmetry 
            might emerge from a discrete R-symmetry.
            
             
            
              *  These models are obtained by adopting a hybrid approach <cit.> 
            between the radiative breaking models and the MBGW model.
            
              *  In the radiative breaking models, a Majorana neutrino scale is 
            generated as the PQ field X gets VEV. However, in the hybrid models, 
            the Majorana mass term MN^cN^c/2 is allowed but it is not generated through PQ breaking– 
            similar to MBGW model.
            
              *  In the radiative breaking models, intermediate PQ and Majorana neutrino scales
             develop as a consequence of intermediate scale SUSY breaking and the running of 
            soft SUSY breaking mass term to negative squared values. In contrast, in the MBGW model
             and in the hybrid models, PQ breaking is triggered by large negative soft terms instead
             of radiative breaking.
            
            
            Three  hybrid models as listed below : 
             
            
            
              §.§.§ Hybrid CCK Model
            
            
            The superpotential for the hybrid CCK model (hyCCK) is given by <cit.>:
            
            W_hyCCK   ∋   f_uQH_uU^c+f_dQH_d D^c+f_ℓLH_dE^c+f_νLH_uN^c+
            M_N N^cN^c/2
                +    fX^3Y/m_P+λ_μ X^2 H_uH_d/m_P .
            
            Thus when the PQ symmetry breaks, the μ parameter is obtained as 
            
            μ_ eff = λ_μ⟨ X⟩^2/m_P .
            
            
            We have checked that the hyCCK model is not gravity-safe under the ℤ_N^R 
            symmetries for N=4,6,8 or 12. 
            However, it does turns out to be gravity-safe under ℤ_24^R symmetry
            with the ℤ_24^R charge and PQ charge assignments as shown in Table <ref>.
            
            
            
            The scalar potential for hyCCK is found to be
            
            
                V = [fA_fϕ_X^3ϕ_Y/m_P+h.c.] + m_X^2|ϕ_X|^2 + m_Y^2|ϕ_Y|^2 + f^2/m_P^2[9|ϕ_X|^4|ϕ_Y|^2 + |ϕ_X|^6]
            
            and is shown in Fig. <ref> vs. scalar field values ϕ_X and ϕ_Y. For large
            negative values of soft term A_f, then a ℤ_24^R and PQ breaking minimum develops.
            
            
            
            The lowest order PQ violating terms in the superpotential are 
            X^8Y^2/ m_P^7, X^4Y^6/m_P^7 and Y^10/ m_P^7 which implies that the lowest order 
            PQ breaking term in the scalar potential is suppressed by 1/ m_P^8. 
            Therefore, this model satisfies the KMR condition for being gravity-safe.
            
            The allowed range of hyCCK model parameter space is shown in Fig. <ref>
            where we show contours of λ_μ values which lead to μ =200 GeV
            in the m_3/2 vs. -A_f plane for f=1. 
            We also show several representative contours of v_PQ values.
            Values of λ_μ∼ 0.015-0.2 are generally sufficient for a natural
            μ term and are easily consistent with soft mass 
            m_soft∼ m_3/2∼ 2-30 TeV as indicated by LHC searches.
            We also note that for m_3/2∼ 5-20 TeV, then v_PQ∼ 10^11 GeV which corresponds 
            to the sweet spot for axion cold dark matter. 
            
            
            
              §.§.§ Hybrid SPM Model
            
            
            The superpotential for the hybrid SPM model (hySPM) is given by <cit.>
            W_hySPM   ∋   f_uQH_uU^c+f_dQH_d D^c+f_ℓLH_dE^c+f_νLH_uN^c+M_NN^cN^c/2
                +    fX^3Y/m_P+λ_μ Y^2 H_uH_d/m_P  .
            
            In this case, when PQ symmetry breaks, the μ parameter is generated to be
            
            μ_ eff = λ_μ⟨ Y⟩^2/m_P .
            
            This model also turns out to be not gravity-safe under ℤ_N^R symmetries for 
            N=4,6,8 and 12 but is gravity-safe for ℤ_24^R symmetry.
            The gravity-safe ℤ_24^R charge 
            and PQ charge assignments are shown in Table <ref>.
            
             
            
            The scalar potential is obtained similar to that in the hyCCK model with the 
            only difference being that now the lowest order  PQ violating terms in the superpotential are 
            Y^8X^2/m_P^7, Y^4X^6/ m_P^7 and X^10/m_P^7 which means that the lowest order PQ breaking terms 
            in the scalar potential are suppressed by 1/ m_P^8 so that the hySPM model also satisfies
            the KMR condition for being gravity-safe.  
            
            The allowed range of hySPM model parameter space is shown in Fig. <ref>
            where we show contours of λ_μ values which lead to μ =150 GeV
            in the m_3/2 vs. -A_f plane for f=1. 
            We also show several representative contours of v_PQ values.
            
            
            
              §.§.§ Hybrid MSY model
            
            
            The superpotential in the hybrid MSY model (hyMSY) is given as <cit.>:
            
            W_hyMSY   ∋   f_uQH_uU^c+f_dQH_d D^c+f_ℓLH_dE^c+f_νLH_uN^c+M_NN^cN^c/2
                +    fX^3Y/m_P+λ_μ XY H_uH_d/m_P .
            
            However, we have checked that the hyMSY model does not satisfy
            the KMR condition for being gravity-safe 
            under any of the R-symmetries listed in Table <ref>.
            
               
            
            
            § ARE THE VARIOUS Μ SOLUTIONS EXPERIMENTALLY 
            DISTINGUISHABLE?
            
            
            An important question arises: are the various solutions to the SUSY μ problem 
            experimentally testable and experimentally distinguishable from one another? 
            
            Obviously, one important consequence is the existence of weak scale SUSY (WSS) 
            so that if WSS is disproved, then the whole discussion on the origin of the μ term is moot.
            The main  raison d'etre for SUSY is to stabilize the weak scale under the presence 
            of quantum corrections. 
            In addition, WSS provides a natural mechanism for electroweak symmetry breaking.
            This means no severe fine-tuning of parameters involved in determining
            the magnitude of the weak scale, which we take to be no  fine-tuning in Eq. (<ref>).
            Upper limits have been derived on sparticle masses within the context of unified SUSY 
            models with no fine-tuning <cit.> ( i.e.Δ_EW 30). 
            These imply typically m_ 6 TeV and m__1 3 TeV and |μ | 360 GeV.
            To explore such high sparticle masses, then about 15 ab^-1 of pp collisions
            at √(s) 27 TeV is required for a hadron collider <cit.> or
            √(s) 720 GeV is needed for an e^+e^- collider <cit.>. 
            If no sparticles are seen at such colliders, 
            then SUSY as we understand it would no longer be a viable hypothesis for 
            stabilization of the weak scale.
            
            Some of the μ solutions are expected to give rise to the MSSM-only as the weak scale
            effective theory. In this case, it may be difficult to distinguish for instance a
            GM solution from a CM solution. In the case of the G2MSSM solution, distinctive 
            mass relations amongst sparticles are expected to occur which could support
            or deter such explanations <cit.>.
            
            In addition to weak scale SUSY, several models– KN, CKN, EWK, HFD,
            radiative PQ models (MSY, CCK, SPM),
            MBGW and hybrid models predict a SUSY DFSZ axion. 
            Recent searches for axions at axion haloscope experiments <cit.> have reached the 
            non-SUSY DFSZ coupling strengths for a narrow range of m_a possibilities.
            However, the SUSY DFSZ axion– by virtue of including higgsinos in the aγγ triangle
            vertex– has a much smaller coupling <cit.>. 
            It is not clear whether present technology has the 
            capability to probe such tiny aγγ couplings. In the event that a thorough
            search can be made for SUSY DFSZ axions over their allowed range of masses and couplings 
            strengths, then (non)observation of axions could rule out or verify this class of μ
            problem solutions. A related test could be the determination of a diminished abundance
            of higgsino-like WIMPs such that the presence (or not) of additional dark matter particles 
            such as axions is required.
            
            Several of the μ solutions require as well additional distinctive particles.
            The NMSSM solution requires the presence of additional scalar and pseudoscalar Higgs bosons
            and a fifth neutralino arising from the NMSSM singlino. For many NMSSM parameter choices,
            some deviations in the h boson coupling strengths are expected <cit.>.
            
            The U(1)^'μ solutions also include distinctive new particle predictions. 
            The CDEEL model <cit.> requires the presence of an additional weak scale Z^' boson
            which could decay to higgsinos as well as SM particles <cit.>. 
            For the HPT model <cit.>, the Z^' is expected to be far beyond any collider reach projections. 
            Instead, for HPT, 
            one expects bilinear RPV leading to distinctive collider signatures and altered
            expectations for dark matter. Also, in these models one may expect the presence of
            stable weak scale exotic hadrons or other exotica which arise from the requirement for
            anomaly cancellation.
            
            
            
            § CONCLUSIONS
            
            
            In this paper, we have re-examined the SUSY μ problem with
            perspective gained from experimental results from LHC through Run 2
            with 150 fb^-1 of data. The two parts to the SUSY μ solutions are 
            1. first forbid the μ term, perhaps via some symmetry and then 
            2. regenerate it, perhaps via symmetry breaking. The new perspective from
            LHC and the naturalness issue is that μ should be generated
            of order m_weak∼ m_W,Z,h∼ 100-300 GeV whilst the soft SUSY breaking
            terms likely inhabit the multi-TeV regime. Thus, a Little Hierarchy (LH)
            should now be included in SUSY μ solutions where |μ |≪ m_soft.
            This is different from pre-LHC expectations where solutions
            sought to generate |μ |≃ m_soft.
            
            To gain an updated perspective on the SUSY μ problem, we examined
            twenty solutions. 
            These solutions are summarized in Table <ref> where we list 
            each solution and how it may admit a LH, whether it also addresses 
            the strong CP problem, whether it is gravity-safe, 
            its relation to neutrino masses
            (Standard see-saw or other) and any distinctive experimental consequences.
            While all solutions have the capacity to be consistent with the LH (usually
            by adjusting some arbitrary constant λ_μ), some actually
            generate μ∼ m_weak≪ m_soft with λ_μ∼ 1 (such as the
            radiative PQ breaking models MSY, CCK and SPM). 
            
            Also, early attempts to solve the SUSY μ problem could appeal to an
            underlying global symmetry such as PQ to suppress the μ term. 
            It soon became clear that such global symmetries are not consistent with
            an ultra-violet completion which includes gravity effects since gravitational
            interactions don't respect global symmetries. Continuous (U(1)^') or
            discrete gauge symmetries are gravity-safe but usually require the addition
            of perhaps unwanted exotica in order to preserve anomaly-freedom. 
            The more recent emergence of discrete R-symmetries <cit.>, 
            which can arise 
            from compactification of extra dimensions in string theory, seems to 
            provide the cleanest suppression symmetry for the μ term.
            A delineation of anomaly-free (including a GS term) 
            ℤ_N^R symmetries which are consistent with SO(10) or SU(5)
            unification (thus preserving gauge coupling unification) offers perhaps
            the most compelling solutions for the first half of the SUSY μ problem.
            For N=4,6,8,12 and 24, these symmetries forbid μ along with RPV
            trilinear terms and dimension-5 p-decay operators whilst allowing
            the required Yukawa couplings and neutrino mass operators. Of these, 
            the ℤ_4^R stands out as both simple and compelling.
            It should probably now replace R-parity as a standard pillar 
            upon which the MSSM is constructed. 
            
            If one also seeks to simultaneously solve the strong CP problem, 
            then the ℤ_24^R symmetry works in the hybrid models to suppress
            unwanted superpotential terms while providing the underlying fundamental
            symmetry from which a global PQ can emerge as an accidental, approximate
            symmetry which is gravity-safe.
            Several other solutions also have their roots in stringy behavior
            (CM, U(1)^', instanton, G2MSSM).
            
            If the naturalness edict is followed– which requires |μ | not too far
            from m_weak∼ 100 GeV– then one expects thermally-underproduced
            higgsino-like WIMPs as (part of) dark matter. If the natural WIMP
            abundance is enhanced by non-thermal processes to make up the 
            entirety of dark matter, then they become excluded by a combination
            of direct and indirect WIMP detection experiments <cit.>. 
            Thus, additional dark matter beyond WIMPs then seems to be required. 
            The axion is a highly motivated candidate to make up the remaining 
            bulk of dark matter. To gain accord with the requirements of 
            cold dark matter, a gravity-safe solution to the strong CP problem
            and a solution to the SUSY μ problem (while also suppressing dangerous 
            p-decay operators and allowing for see-saw neutrino masses), then the
            hybrid models based on ℤ_24^R discrete R-symmetry stand out
            as a rather complete answer.
            
            Overall, the SUSY μ problem has generated a rich panoply of 
            solutions over the past 35 years. To begin the process of selecting amongst 
            them or building others, it is of the essence to first discover SUSY
            and then to proceed with precision measurements of the SUSY spectra along with
            any exotica to gain insight into which if any of the solutions 
            best describes nature.
            Future collider and dark matter experiments should go a long way towards
            selecting amongst or ruling out these various solutions and 
            other solutions perhaps yet to come.
            
            
            
            
            § ACKNOWLEDGMENTS
            
            
            We thank H. Serce for help in the early stages of this project.
            This work was supported in part by the US Department of Energy, Office of High Energy Physics.
            The work of KJB was supported by IBS under the project code, IBS-R018-D1.
            
            
            
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