Gravity safe, electroweak natural axionic solution to strong CP and SUSY mu problems

Particle physics models with Peccei-Quinn (PQ) symmetry breaking as a consequence of supersymmetry (SUSY) breaking are attractive in that they solve the strong CP problem with a SUSY DFSZ-like axion, link the SUSY breaking and PQ breaking intermediate mass scales and can resolve the SUSY mu problem with a naturalness-required weak scale mu term whilst soft SUSY breaking terms inhabit the multi-TeV regime as required by LHC sparticle mass limits and the Higgs mass measurement. On the negative ledger, models based on global symmetries suffer a generic gravity spoliation problem. We present a model based on the discrete R-symmetry Z_{24}^R-- which may emerge from compactification of 10-d Lorentzian spacetime in string theory-- where the mu term and dangerous proton decay and R-parity violating operators are either suppressed or forbidden while a gravity-safe PQ symmetry emerges as an accidental approximate global symmetry leading to a solution to the strong CP problem and a weak-scale/natural value for the mu term.

∼ 135 GeV [14]. In spite of these impressive successes, a variety of problems arise in SUSY theories-foremost among these being the lack of appearance of the required superpartners at the CERN LHC. Recent analyses of data from LHC run 2 with √ s = 13 TeV pp collisions and ∼ 36 fb −1 of integrated luminosity require the gluino mass mg > ∼ 2 TeV [15] and the top squark mass mt 1 > ∼ 1 TeV [16]. Such large mass limits are far beyond initial expectations from naturalness where for instance Barbieri-Giudice [17] (BG)-requiring no worse than 3% electroweak finetuningexpected that mg and mt 1 are both < ∼ 450 GeV. Before declaring SUSY to be in a finetuning crisis, it was pointed out in Ref's [18] that the BG bounds were computed in multiple soft parameter effective SUSY theories: in this case the BG calculation shows fine-tuning in the effective theory calculation but not in nature herself (as exemplified by more fundamental theories) wherein all soft parameters are interdependent and derived from more fundamental parameters (such as the gravitino mass m 3/2 in gravity-mediated SUSY breaking [19] or the messenger scale Λ in gauge mediation [20]). For correlated soft parameters, then the EW finetuning may be extracted from the MSSM scalar potential minimization conditions which relate the measured Z-boson mass m Z to the weak scale SUSY Lagrangian parameters: Here, m 2 Hu and m 2 H d are squared soft SUSY breaking Lagrangian terms, µ is the superpotential Higgsino mass parameter, tan β = v u /v d is the ratio of Higgs field vacuum-expectation-values and the Σ u u and Σ d d contain an assortment of radiative corrections, the largest of which typically arise from the top squarks. Expressions for the Σ u u and Σ d d are given in the Appendix of Ref. [21]. The fine-tuning measure ∆ EW compares the largest independent contribution on the right-hand-side (RHS) of Eq. (2) to the left-hand-side m 2 Z /2. If the RHS terms in Eq. (2) are individually comparable to m 2 Z /2, then no unnatural fine-tunings are required to generate m Z = 91.2 GeV. The main requirements for low fine-tuning (∆ EW < ∼ 30 1 ) are the following.

• m 2
Hu is driven radiatively to small, and not large, negative values at the weak scale [28,21].
• The top squark contributions to the radiative corrections Σ u u (t 1,2 ) are minimized for TeVscale highly mixed top squarks [28]. This latter condition also lifts the Higgs mass to m h ∼ 125 GeV. For ∆ EW < ∼ 30, the lighter top squarks are bounded by mt 1 < ∼ 3 TeV.
• The gluino mass, which feeds into the Σ u u (t 1,2 ) via renormalization group contributions to the stop masses [27], is required to be mg < ∼ 6 TeV, possibly beyond the reach of the √ s = 13 − 14 TeV LHC. 2 • First and second generation squark and slepton masses may range as high as 5-30 TeV with little cost to naturalness [29,21,30,22].
In light of the finetuning clarification, it is perhaps not surprising that SUSY has yet to emerge at the LHC. Indeed, simple statistical arguments from the string theory landscape [31] suggest a pull to large values of soft terms albeit tempered by the (anthropic) requirement that the weak scale not stray too far from its value m weak ≡ m W,Z,h ∼ 100 GeV. In this latter case, then present LHC mass limits are just beginning to probe natural SUSY parameter space and indeed it may require an energy upgrade of LHC to √ s ∼ 27 TeV for a full exploration [33]. Along with non-appearance of superpartners, the MSSM suffers several structural problems arising from the superpotential. Including non-renormalizable terms (up to m −1 P ) as should be 1 The onset of fine-tuning for ∆ EW > ∼ 30 is visually displayed in Ref. [22]. 2 The upper bound on mg increases to 9 TeV for the natural anomaly-mediated SUSY breaking model [32,33]. present in supergravity, then the gauge invariant MSSM superpotential reads: The first term on line 1 of Eq. 3, if unsuppressed, should lead to Planck-scale values of µ while phenomenology (Eq. 2) requires µ of order the weak scale ∼ 100 − 350 GeV. This is the famous SUSY µ problem albeit modified for the LHC era: why is µ ∼ 100 − 350 GeV whilst LHC Higgs mass and sparticle limits require soft terms m sof t ∼multi-TeV? The κ i , λ ijk , λ ijk and λ ijk terms violate either baryon number B or lepton number L or both and can, if unsuppressed, lead to rapid proton decay and an unstable lightest SUSY particle (LSP). The f ij u,d,e are the quark and lepton Yukawa couplings and must be allowed to give the SM fermions mass via the Higgs mechanism. The κ (1,2) ijkl terms lead to dimension-five proton decay operators and are required to be either highly suppressed or forbidden. It is common to implement discrete symmetries to forbid the offending terms and allow the required terms in 3. For instance, the Z M 2 matter parity (or R-parity) forbids the κ i and λ ( , ) ijk terms but allows for µ and the κ (1,2) ijkl terms: thus, the ad-hoc R-parity conservation all by itself is insufficient to cure all of the ills of Eq. 3.
A promising approach, which addresses both the strong CP problem and many of the offending terms in Eq. 3, is to implement models where PQ symmetry breaking [34,35,36,37] occurs as a consequence of SUSY breaking. In this approach, one posits the presence of a global PQ symmetry with PQ charges as listed in Table 1 along with a superpotential which includes the Yukawa couplings (second line) of Eq. 3 but then introduces additional chiral superfields X and Y and augments the superpotential with terms along with three possibilities for the bilinear H u H d couplings: The model is postulated to hold just below the reduced Planck scale m P . The global U (1) P Q forbids the µ term, the RPV terms, the Majorana neutrino mass term and the (last line) dangerous proton decay terms of Eq. 3. But when one augments the radiative PQ Lagrangian with soft SUSY breaking terms and allows for running of the PQ parameters m 2 X , m 2 Y , h ij and trilinears (for simplicity, we will here adopt h ij = h i δ ij as diagonal in generation space and assume h 1 = h 2 = h 3 ≡ h), then it is found-for large PQ Yukawa coupling h ∼ 1.5 − 4-that m 2 X is driven radiatively to negative values at an intermediate scale resulting in spontaneous PQ symmetry breaking wherein the X and Y fields develop vevs v X and v Y respectively. The spontaneously broken global PQ symmetry generates a Goldstone boson-the axion which solves the strong CP problem-but then also generates a superpotential mu term Y and the axion decay constant (given by f a = i q 2 P Q v 2 i ) depend on the SUSY PQ parameters via the scalar potential minimization conditions (listed e.g. in Ref. [40]).
An attractive feature of the models-similar to the Kim-Nilles model [38,39]-is that µ ∼ f 2 a /m P as compared to the soft SUSY breaking scale in gravity mediation m sof t ∼ m 2 hidden /m P where m hidden is an intermediate mass scale associated with the supergravity hidden sector. Thus, the aforementioned Little Hierarchy µ m sof t ensues for f a < m hidden [40]. Also, since now the PQ scale f a is comparable to the hidden sector SUSY breaking scale, the expected axion dark matter density is generated in the cosmologically allowed range. In addition, a Majorana neutrino scale is generated as  Unfortunately, these very appealing radiative PQ breaking scenarios are beset by the issue of the postulated global U (1) P Q symmetry suffering from the previously mentioned gravity spoliation problem. One way to deal with the gravity spoliation is to assume instead a gravitysafe discrete gauge symmetry Z M of order M . The Z M discrete gauge symmetry can forbid the offending terms of Eq. 3 while allowing the necessary terms [41]. Babu, Gogoladze and Wang [39] have found a closely related model (written previously by Martin [36] thus labelled MBGW) with which is invariant under a Z 22 discrete gauge symmetry. These Z 22 charge assignments have been shown to be anomaly-free under the presence of a Green-Schwarz (GS) term [42] in the anomaly cancellation calculation. The PQ symmetry then arises as an accidental approximate global symmetry as a consequence of the more fundamental discrete gauge symmetry. The PQ charges of the MBGW model are also listed in Table 1. The discrete gauge symmetry Z M might arise if a charge M e field condenses and is integrated out of the low energy theory while charge e fields survive (see Krauss and Wilczek, Ref. [7]). 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 [43] of theories not consistent with a UV completion in string theory. In addition, the charge assignments [39] are not consistent with SU (5) or SO(10) grand unification which may be expected at some level in a more ultimate theory. Beside the terms in Eq. 8, the lowest order PQV term in the superpotential is (Y ) 11 m 8 P : thus this model is gravity safe. An alternative very compelling approach is to implement a discrete R symmetry Z R N of order N . 3 In fact, in Lee et al. Ref. [47], it was found that the requirement of an anomaly-free discrete symmetry that forbids the µ term and all dimenion four and five baryon and lepton number violating terms in Eq. 3 while allowing the Weinberg operator LH u LH u and that commutes with SO(10) (as is suggested by the unification of each family into the 16 of SO(10)) has a unique solution: a Z R 4 R-symmetry. If the requirement of commutation with SO(10) is weakened to commutation with SU (5), then further discrete Z R N symmetries with N being an integral divisor of 24 are allowed [48]: N = 4, 6, 8, 12 and 24. Even earlier [49], the Z R 4 was found to be the simplest discrete R-symmetry to realize R-parity conservation whilst forbidding the µ term. In that reference, the µ term was regenerated using Giudice-Masiero [50] which would generate µ ∼ m sof t (too large).
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 2 θ) = −2. For the Lagrangian L d 2 θW to be invariant under R-symmetry, then the superpotential W must carry Q R (W ) = 2. Discrete R symmetries should be gravity-safe since they are expected to emerge as remnants of 10-d Lorentz symmetry under compactification of extra dimensions in superstring theory. The Z R N symmetry gives rise to a universal gauge anomaly ρ mod η where the remaining contribution ρ is cancelled by the Green-Schwartz (GS) axio-dilaton shift and η = N (N/2) for N odd (even). The anomaly free R charges of various MSSM fields are listed in Table 2 for N values consistent with grand unification.
We have examined whether or not the three radiative PQ breaking models of Table 1 (CCK, MSY and SPM) can be derived from any of the more fundamental Z R N symmetries in Table 2. In almost all cases, the hXN c N c operator is disallowed: then there is no large Yukawa coupling present to drive the PQ soft term m 2 X negative so that PQ symmetry is broken. And since the PQ symmetry does not allow for a Majorana mass term M N N c N c , then no see-saw scale can be developed. One exception is the MSY model under Z R 4 symmetry with charge assignments Q R (X) = 0 and Q R (Y ) = 2: then a Y H u H 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  [48]).
X and Y , we can see that the R-charges for X and Y should be such that the term XY H u H d is allowed and since the R-charges of H u and H d are 0, then a term M XY would always be allowed: this term breaks PQ at high order and is not gravity safe. A second exception is SPM under the Z R 6 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 2 Y which break PQ are always allowed but are not gravity safe.
We have also examined the MBGW model of Table 1 which does allow for the M N c N c seesaw term but where PQ and Z R N symmetry breaking is triggered by large negative soft terms instead of radiative breaking. To check gravity safety, we note that additional superpotential terms of the form λ 3 X p Y q may be allowed for given Z R N charge assignments and powers p and q. Such terms will typically break the PQ symmetry and render the model not gravity safe if scalar potential terms V (φ) develop which are not suppressed by at least eight powers of 1/m P [8]. The largest dangerous scalar potential terms develop from interference between λ 2 (XY ) 2 /m P and λ 3 X p Y q /m p+q−3 P when constructing the scalar potential V F = φ |∂W/∂φ|φ →φ (here, thê φ label chiral superfields with φ being their leading components). We find the MBGW model to be not gravity safe under any of the Z R N discrete R-symmetries of Table 2. Next, we will adopt a hybrid approach between the radiative breaking models and the MBGW model by writing a superpotential: along with PQ charge assignments given under the GSPQ (gravity-safe PQ model) heading of Table 1. For this model, we have checked that there is gravity spoliation for N = 4, 6, 8 and 12. But for Z R 24 and under R-charge assignments Q R (X) = −1 and Q R (Y ) = 5, then the lowest order PQ violating superpotential operators allowed are X 8 Y 2 /m 7 P , Y 10 /m 7 P and X 4 Y 6 /m 7 P . These operators 4 lead to PQ breaking terms in the scalar potential suppressed by powers of 4 The X 8 Y 2 /m 7 P term was noted previously in Ref. [48]. (1/m P ) 8 . For instance, the term X 8 Y 2 /m 7 P leads to V P Q 24f λ * 3 X 2 Y X * 7 Y * 2 /m 8 P + h.c. which is sufficiently suppressed by enough powers of m P so as to be gravity safe [8]. We have also checked that hybrid model using the MSY XY H u H d /m P term is not gravity-safe under any of the discrete R-symmetries of Table 2 but the hybrid SPM model with Y 2 H u H d /m P and charges Q R (X) = 5 and Q R (Y ) = −13 is gravity-safe under only Z R 24 . We augment the hybrid CCK model scalar potential V F = |3f φ 2 X φ Y /m P | 2 + |f φ 3 X /m P | 2 of the GSPQ model by the following soft breaking terms and then minimize the resultant scalar potential. The minimization conditions are already written down in Ref. [40] Eq's 17-18. In the case of the GSPQ model, the PQ symmetry isn't broken radiatively, but instead can be broken by adopting a sufficiently large negative value of A f (assuming real positive couplings for simplicity). The scalar potential admits a non-zero minimum in the fields φ X and φ Y for A f < 0 as shown in Fig. 1 which is plotted for the case of m X = m Y ≡ m 3/2 = 10 TeV, f = 1 and A f = −35.5 TeV. For these values, we find v X = 10 11 GeV, v Y = 5.8 × 10 10 GeV, v P Q = 1.15 × 10 11 GeV and f a = v 2 X + 9v 2 Y = 2 × 10 11 GeV. These sorts of numerical values lie within the mixed axion/higgsino dark matter sweet spot of cosmologically allowed values and typically give dominant DFSZ axion CDM with ∼ 5 − 10% WIMP dark matter [51,52,53]. Under these conditions, the model develops a µ parameter µ = λ µ v 2 X /m P and for a value λ µ = 0.036 then we obtain a natural value of the µ parameter at 150 GeV.
The allowed range of GSPQ model parameter space is shown in Fig. 2 where we show Such high values of m 3/2 also allow for a resolution of the early universe gravitino problem [54] (at higher masses gravitinos may decay before the onset of BBN) and such high soft masses serve to ameliorate the SUSY flavor and CP problems as well [55,56]. They are also expected in several well-known string phenomenology constructions including compactification of Mtheory on a manifold of G 2 holonomy [57], the minilandscape of heterotic strings compactified on orbifolds [58] and the statistical analysis of the landscape of IIB intersecting D-brane models [31].
As far as phenomenological consequences go, we may expect rather heavy sparticle masses which may require high-luminosity or high-energy LHC for verification of natural SUSY [33]. The required light higgsinos with mass ∼ µ may be observable at LHC [59] or at a √ s > ∼ 500 GeV e + e − collider [60]. The under-dense higgsino-like WIMP dark matter should eventually be detectable at ton-scale noble liquid WIMP detectors [61]. Regarding axion searches, the axion anomaly contribution E/N to the aγγ coupling [62] is found (by including higgsino contributions to the anomaly diagram) to be E/N = 6/3: this value is slightly larger than the chiral contribution to the aγγ coupling: together, the two contributions nearly cancel leaving the axion visibility at microwave cavity experiments to be a very difficult prospect [63].
Summary: We have found that the several DFSZ axionic extensions of the MSSM which generate PQ breaking radiatively as a consequence of SUSY breaking and a large PQ-neutrino Yukawa coupling cannot be realized as a consequence of gravity-safe Z R N symmetries which are consistent with GUTs and forbid the µ term. The MBGW model, where PQ breaking results from a large quartic soft term, also does not turn out to be gravity safe under Z R N symmetries which are consistent with GUTs. However, the MBGW model does turn out to be gravity safe under Z 22 discrete gauge symmetry but this fundamental symmetry is inconsistent with GUTs. We have found (two) gravity-safe hybrid type models GSPQ with PQ superpotential as in the radiative models, but with an explicit see-saw neutrino sector which is unrelated to SUSY or PQ breaking. Instead, the PQ breaking results as a consequence of a large quartic (Plancksuppressed) soft term so that it generates an axionic solution to the strong CP problem along with a natural value of the MSSM µ term. As emphasized in Ref. [48], the gravity-safe Z R 24 symmetry (which may emerge as a remnant of 10-d Lorentz symmetry which is compactified to four spacetime dimensions) yields an accidental approximate global PQ symmetry as implemented in the GSPQ model of PQ symmetry breaking as a consequence of SUSY breaking. The Z R 24 (PQ) symmmetry breaking leads to µ m sof t as required by electroweak naturalness and to PQ energy scales f a ∼ 10 11 GeV as required by mixed axion-higgsino dark matter. The Z R 24 symmetry also forbids the dangerous dimension-four R-parity violating terms. Dimension-five proton decay operators are suppressed to levels well below experimental constraints [48]. Overall, our results show the axionic solution to the strong CP problem is enhanced by the presence of both supersymmetry and extra spacetime dimensions which give rise to the gravity-safe Z R 24 symmetry from which the required global PQ symmetry accidentally emerges.