Revisiting the SUSY mu problem and its solutions in the LHC era

TL;DR

This paper reviews twenty solutions to the SUSY mu problem, analyzing their compatibility with the Little Hierarchy in light of LHC constraints, and discusses the theoretical and experimental implications of these solutions.

Contribution

It provides a comprehensive re-evaluation of existing mu problem solutions considering current LHC data and the naturalness of the Little Hierarchy, highlighting the role of symmetries and mechanisms involved.

Findings

Most solutions can accommodate the Little Hierarchy

Discrete R-symmetries are favored for mu solutions

Global symmetries face gravity-safety issues

Abstract

The supersymmetry preserving mu 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 mu problem. Its solution involves two steps: 1. first forbid mu, 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 mu~ m_{W,Z,h}~ 100 GeV so that a Little Hierarchy (LH) appears with mu << m_{soft}. We review twenty previously devised solutions to the SUSY mu 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 mu term while relating the mu 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 mu problem. Almost all solutions seem able to accommodate the LH.

Tables

Table 1. Table 1: ℤ 4 R superscript subscript ℤ 4 𝑅 \mathbb{Z}_{4}^{R} charge assignments for various superfields of the LRRRSSV model [ 40 ] .

multiplet	$H_u$	$H_d$	$Q_i$	$L_i$	$U_i^c$	$D_i^c$	$E_i^c$	$N_i^c$
${\mathbb{Z}}_4^R$ charge	0	0	1	1	1	1	1	1

Table 2. Table 2: Charge assignments for various superfields of nMSSM with a ℤ 5 R superscript subscript ℤ 5 𝑅 \mathbb{Z}_{5}^{R} discrete R 𝑅 R -symmetry.

multiplet	$H_u$	$H_d$	$Q_i$	$U_i^c$	$D_i^c$	$L_i$	$E_i^c$	$N^c$	$S$
${\mathbb{Z}}_5^R$	2	2	4	6	6	4	6	6	3

Table 3. Table 3: Charge assignments for various superfields of a U ​ ( 1 ) ′ 𝑈 superscript 1 ′ U(1)^{\prime} model  [ 80 , 79 ] .

multiplet	$H_u$	$H_d$	$Q_i$	$U_i^c$	$D_i^c$	$L_i$	$E_i^c$	$S$
$\left(2\sqrt{10}\right)Q^{\prime }$	-2	-3	1	1	2	2	1	5

Table 4. Table 4: Charge assignments for various superfields of the HPT U ​ ( 1 ) ′ 𝑈 superscript 1 ′ U(1)^{\prime} supergravity model  [ 78 ] .

multiplet	$H_u$	$H_d$	$Q_i$	$U_i^c$	$D_i^c$	$L_i$	$E_i^c$	$S$
$Q^{\prime }$	25	-31	0	-25	31	2	29	3

Table 5. Table 5: PQ charge assignments for various superfields of the KN model with n = 1 𝑛 1 n=1 . One may add multiples of weak hypercharge or B − L 𝐵 𝐿 B-L to these so their values are not unique.

multiplet	$H_u$	$H_d$	$Q_i$	$L_i$	$U_i^c$	$D_i^c$	$E_i^c$	X	Y	Z
PQ charge	$-1$	$-1$	$+1$	$+1$	0	0	0	+1	-1	0

Table 6. Table 6: PQ charge assignments for various superfields of the CKN model.

multiplet	$H_u$	$H_d$	$Q$	$Q^c$	$Q_i$	$U_i^c$	$D_i^c$
PQ charge	$-1$	$-1$	$1$	$1$	0	1	1

Table 7. Table 7: PQ charge assignments for various superfields of the EWK model.

multiplet	$H_u$	$H_d$	$\varphi$	$N$
PQ charge	$-1$	$-1$	$+2$	$-1$

Table 8. Table 8: Derived MSSM field R 𝑅 R charge assignments for various anomaly-free discrete ℤ N R superscript subscript ℤ 𝑁 𝑅 \mathbb{Z}_{N}^{R} symmetries which are consistent with S ​ U ​ ( 5 ) 𝑆 𝑈 5 SU(5) or S ​ O ​ ( 10 ) 𝑆 𝑂 10 SO(10) unification (from Lee et al. Ref.  [ 41 ] ).

multiplet	${\mathbb{Z}}_4^R$	${\mathbb{Z}}_6^R$	${\mathbb{Z}}_8^R$	${\mathbb{Z}}_{12}^R$	${\mathbb{Z}}_{24}^R$
$H_u$	0	4	0	4	16
$H_d$	0	0	4	0	12
$Q$	1	5	1	5	5
$U^c$	1	5	1	5	5
$E^c$	1	5	1	5	5
$L$	1	3	5	9	9
$D^c$	1	3	5	9	9
$N^c$	1	1	5	1	1

Table 9. Table 9: PQ charge assignments for various superfields of the CCK, MSY and SPM models of radiative PQ breaking.

multiplet	MSY	CCK	SPM
$H_u$	$-1$	$-1$	$-1$
$H_d$	$-1$	$-1$	$-1$
$Q$	$+1/2$	$3/2$	$+1/2$
$L$	$+1/2$	$3/2$	$+5/6$
$U^c$	$+1/2$	$-1/2$	$+1/2$
$D^c$	$+1/2$	$-1/2$	$+1/2$
$E^c$	$+1/2$	$-1/2$	$+1/6$
$N^c$	$+1/2$	$-1/2$	$+1/6$
$X$	$-1$	$+1$	$-1/3$
$Y$	$+3$	$-3$	$+1$

Table 10. Table 10: U ​ ( 1 ) R 𝑈 superscript 1 𝑅 U(1)^{R} and PQ charge assignments for various superfields of the CCL model for n = 2 𝑛 2 n=2 .

multiplet	$H_u$	$H_d$	$Q_i$	$L_i$	$U_i^c$	$D_i^c$	$E_i^c$	$N_i^c$	X	Y
$U{\left(1\right)}_R$ charge	$4$	$4$	$-\frac{4}{3}$	$-\frac{4}{3}$	$-\frac{2}{3}$	$-\frac{2}{3}$	$-\frac{2}{3}$	$-\frac{2}{3}$	$\frac{5}{3}$	$-3$
PQ charge	$3$	$3$	$-3$	$-2$	$0$	$0$	$-1$	$-1$	$1$	$-3$

Table 11. Table 11: ℤ 22 subscript ℤ 22 \mathbb{Z}_{22} and PQ charge assignments for various superfields of the MBGW model.

multiplet	$H_u$	$H_d$	$Q_i$	$L_i$	$U_i^c$	$D_i^c$	$E_i^c$	$N_i^c$	$X$	$Y$
${\mathbb{Z}}_{22}$ charge	$22$	$18$	$3$	$11$	$19$	$1$	$15$	$11$	$13$	$20$
PQ charge	$-1$	$-1$	$+1$	$+1$	$0$	$0$	$0$	$0$	$+1$	$-1$

Table 12. Table 12: ℤ 24 R superscript subscript ℤ 24 𝑅 \mathbb{Z}_{24}^{R} and PQ charge assignments for various superfields of the hyCCK model.

multiplet	$H_u$	$H_d$	$Q_i$	$L_i$	$U_i^c$	$D_i^c$	$E_i^c$	$N_i^c$	X	Y
${\mathbb{Z}}_{24}^R$ charge	16	12	5	9	5	9	5	1	-1	5
PQ charge	-1	-1	1	1	0	0	0	0	1	-3

Table 13. Table 13: ℤ 24 R superscript subscript ℤ 24 𝑅 \mathbb{Z}_{24}^{R} and PQ charge assignments for various superfields of the hySPM model.

multiplet	$H_u$	$H_d$	$Q_i$	$L_i$	$U_i^c$	$D_i^c$	$E_i^c$	$N_i^c$	X	Y
${\mathbb{Z}}_{24}^R$ charge	16	12	5	9	5	9	5	1	5	-13
PQ charge	-1	-1	1	1	0	0	0	0	-1/3	1

Table 14. Table 14: Summary of twenty solutions to the SUSY μ 𝜇 \mu problem and how they 1. admit a Little Hierarchy (LH), 2. solve the strong CP problem ( √ square-root \surd ) or not ( × \times ), 3. are expected gravity-safe, 4. Standard neutrino see-saw (SNSS) or other and 5. some experimental consequences.

model	admit LH?	strong CP?	gravity safe?	see-saw?	exp. cons.
GM	small ${\lambda }_{\mu }$	$\times$	$--$	$SNSS$	MSSM
CM	small ${\lambda }_{\mu }$	$\times$	$--$	$SNSS$	MSSM
$R$-sym	${\left(v_i/m_P\right)}^{n_i}\ll 1$	$\times$	$?$	$SNSS$	MSSM
${\mathbb{Z}}_4^R$	small ${\lambda }_{\mu }$	$\times$	$--$	$SNSS$	MSSM
Instanton	small $e^{-S_{cl}}$	$\times$	$--$	$SNSS$	MSSM
$G_{2}MSSM$	$⟨S_i⟩/m_P\ll 1$	$\times$	$--$	$SNSS$	$G_{2}MSSM$
NMSSM	small ${\lambda }_{\mu }$	$\times$	$--$	$SNSS$	extra Higgs/neutralino
nMSSM	small ${\lambda }_{\mu }$	$\times$	$--$	$SNSS$	extra Higgs/neutralino
$\mu \nu$SSM	small ${\lambda }_{\mu }$	$\times$	$--$	$bRPV$	$bRPV$, mixings
$U{\left(1\right)}^{\prime }$ (CDEEL)	small ${\lambda }_{\mu }$	$\times$	$--$	$SNSS$	$Z^{\prime }$
sMSSM	small ${\lambda }_{\mu }$	$\times$	$--$	$SNSS$	extra Higgs/neutralino
$U{\left(1\right)}^{\prime }$ (HPT)	small ${\lambda }_{\mu }$	$\times$	$--$	$bRPV$	$bRPV$, stable heavy hadrons
KN		$\surd$	$?$	$SNSS$	DFSZ axion
CKN		$\surd$	$?$	$SNSS$	DFSZ axion
BK/EWK	${\lambda }_{\mu }\sim {10}^{-10}$	$\surd$	$?$	$SNSS$	DFSZ axion
$\mathrm{HFD}$		$\surd$	$?$	$SNSS$	MSSM
MSY/CCK/SPM		$\surd$	$\times$	$RadSS$	DFSZ axion
CCL	small ${\lambda }_{\mu }$	$\surd$	$?$	$several$	DFSZ axion, $\tilde{G}$ or $\tilde{\nu }$ LSP
BGW	small ${\lambda }_{\mu }$	$\surd$	${\mathbb{Z}}_{22}$	$SNSS$	DFSZ axion
Hybrid CCK/SPM	small ${\lambda }_{\mu }$	$\surd$	${\mathbb{Z}}_{24}^R$	$SNSS$	DFSZ axion

Equations

$$m_Z^2/2=\frac{m_{H_d}^2+{\Sigma }_d^d-(m_{H_u}^2+{\Sigma }_u^u){\tan }^2\beta }{{\tan }^2\beta -1}-{\mu }^2\simeq -m_{H_u}^2-{\Sigma }_u^u({\tilde{t}}_{1,2})-{\mu }^2$$

$${\Delta }_{EW}\equiv max|largesttermonRHSofEq.\text{eqref}eq:mzs|/(m_Z^2/2)$$

$$|\mu |\sim m_{weak}\sim 100-300\mathrm{GeV}\ll m_{soft}\sim \mathrm{multi}-\mathrm{TeV}\lesssim m_{3/2}.$$

$$K\ni H_u^{†}{H}_u+H_d^{†}{H}_d+\left(\frac{{\lambda }_{\mu }}{m_P}{H}_u{H}_d{h}^{†}+h.c.\right).$$

$${\mu }_{\mathrm{eff}}={\lambda }_{\mu }\frac{⟨F_h^{\ast }⟩}{m_P}\sim {\lambda }_{\mu }{m}_{hidden}^2/m_P\sim {\lambda }_{\mu }{m}_{3/2}\sim m_{soft}.$$

$$W=W_0+{\lambda }_{\mu }{W}_0{H}_u{H}_d/m_P^2$$

$${\mu }_{\mathrm{eff}}\sim {\lambda }_{\mu }{m}_{hidden}^2/m_P\sim {\lambda }_{\mu }{m}_{3/2}\sim m_{soft}.$$

$$W\ni P_{\mu }({\varphi }_i)H_u{H}_d$$

$$W\sim \exp (-S_{\mathrm{cl}})M_s{H}_u{H}_d$$

$$W_{NMSSM}=W_{MSSM}(\mu =0)+{\lambda }_{\mu }S{H}_u{H}_d+{\xi }_{F}S+\frac{1}{2}{\mu }_S{S}^2+\frac{1}{3}\kappa S^3$$

$${\mathcal{L}}_{soft}^{NMSSM}={\mathcal{L}}_{soft}^{MSSM}-(a_{\lambda }S{H}_u{H}_d+B\mu H_u{H}_d+\frac{1}{3}{a}_{\kappa }{S}^3+\frac{1}{2}{b}_S{S}^2+tS+c.c.)-m_S^2|S{|}^2.$$

$${\mu }_{\mathrm{eff}}={\lambda }_{\mu }{v}_s.$$

$$W_{nMSSM}\ni {\lambda }_{\mu }S{H}_u{H}_d+f_{u}Q{H}_u{U}^c+f_{d}Q{H}_d{D}^c+f_{\ell }L{H}_d{E}^c+f_{\nu }L{H}_u{N}^c+\frac{1}{2}{M}_N{N}^c{N}^c$$

$$W_{nMSSM}^{tad}\ni {\xi }_{F}S$$

$$W$$

$${\mu }_{\mathrm{eff}}={\lambda }_{\mu i}⟨{\tilde{\nu }}_{Ri}⟩$$

$$W\ni {\lambda }_{\mu }S{H}_u{H}_d$$

$$W_{sMSSM}\ni {\lambda }_{\mu }S{H}_u{H}_d+{\lambda }_s{S}_1{S}_2{S}_3$$

$$W\ni {\lambda }_{\mu }{S}^2{H}_u{H}_d/M_p.$$

$$W_{\mu }\ni \frac{{\lambda }_{\mu }}{m_P^n}{X}^{n+1}{H}_u{H}_d$$

$$W_{PQ}={\lambda }_{PQ}Z\left(XY-v_{PQ}^2\right).$$

$$\mu ={\lambda }_{\mu }⟨X{⟩}^2/m_P.$$

$$W_{CKN}\ni \frac{{\lambda }_{\mu }}{m_P}Q{Q}^c{H}_u{H}_d.$$

$${\mu }_{\mathrm{eff}}\sim {\lambda }_{\mu }{\Lambda }^2/m_P.$$

$$W_{EWK}\ni {\lambda }_{\mu }\varphi H_u{H}_d+\kappa \varphi N^2$$

$$V(\varphi )\ni \frac{g}{m_P^{2m+n-4}}|\varphi {|}^{2m}{\varphi }^n+h.c.+c$$

$$\left\{\begin{tabular}[]{cc}$m_{G}$/2&$m_{G}$/2\\ $m_{G}$/2&$m_{G}$/2\\ \end{tabular}\right\}.$$

$$W^{(nonrenormalizable)}=\sum_{i,j=1,2}\Bigg{(}\frac{X^{(i)}\bar{X}^{(j)}}{m_{P}}\Bigg{)}H_{u}^{(i)}H_{d}^{(j)}+\sum_{ij}\sum_{kl}\Bigg{(}\frac{X^{(i)}\bar{X}^{(j)}}{m_{P}}\Bigg{)}H_{u}^{(k)}H_{d}^{(l)}$$

$$W^{(nonrenormalizable)}=\frac{{\lambda }_{\mu }{v}_{PQ}^2}{2m_P}(H_u^{(0)}+H_u^{(M_G)})(H_d^{(0)}+H_d^{(M_G)})$$

$$W=\frac{{\lambda }_{\mu }{X}^{(0)}{\bar{X}}^{(0)}}{2m_P}{H}_u^{(0)}{H}_d^{(0)}.$$

Full text

1. 1 Introduction: reformulating the $\mu$ problem for the LHC era

2. 2 A review of some solutions to the SUSY $\mu$ problem

3. 2.1 Solutions in supergravity/string construction

4. 2.1.1 Giudice-Masiero (GM)

5. 2.1.2 Casas-Munoz (CM)

6. 2.1.3 $\mu$ and a big hierarchy from approximate $R$-symmetry

7. 2.1.4 Solution via the discrete $R$-symmetry $\mathbb{Z}_{4}^{R}$

8. 2.1.5 String instanton solution

9. 2.1.6 Mu solution in ${\rm G_{2}MSSM}$

10. 2.2 Extended MSSM-type solutions

11. 2.2.1 NMSSM: Added singlet with $\mathbb{Z}_{3}$ discrete symmetry

12. 2.2.2 nMSSM

13. 2.2.3 Mu-from-nu SSM ($\mu\nu$SSM)

14. 2.3 $\mu$ from an extra local $U(1)^{\prime}$

15. 2.3.1 CDEEL model

16. 2.3.2 sMSSM model

17. 2.3.3 HPT model

18. 2.4 Solutions related to Peccei-Quinn symmetry breaking

19. 2.4.1 Kim-Nilles solution

20. 2.4.2 Chun-Kim-Nilles

21. 2.4.3 BK/EWK solution linked to inflation and strong CP

22. 2.4.4 Global symmetries and gravity

23. 2.4.5 Gravity-safe symmetries : gauge symmetries or $R$-symmetries: continuous or discrete

24. 2.4.6 Natural Higgs-Flavor-Democracy (HFD) solution to $\mu$ problem

25. 2.4.7 Radiative PQ breaking from SUSY breaking

26. 2.4.8 CCL model from gauged $U(1)_{R}$ symmetry

27. 2.4.9 MBGW model of PQ breaking from SUSY breaking

28. 2.5 Hybrid models of PQ breaking from SUSY breaking

29. 2.5.1 Hybrid CCK Model

30. 2.5.2 Hybrid SPM Model

31. 2.5.3 Hybrid MSY model

32. 3 Are the various $\mu$ solutions experimentally distinguishable?

33. 4 Conclusions

1 Introduction: reformulating the $\mu$ 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[1, 2]. 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) [3],
2. the measured value of $m_{t}$ falls in the range required for a radiatively-driven breakdown of electroweak symmetry [4],
3. the measured value of the Higgs boson mass falls squarely within the narrow allowed range required by the MSSM [5, 6] and
4. the measured values of $m_{W}$ and $m_{t}$ favor the MSSM with heavy superpartners [7]. In spite of these successes, so far no direct signal for SUSY has emerged at LHC leading to mass limits $m_{\tilde{g}}\gtrsim 2$ TeV and $m_{\tilde{t}_{1}}\gtrsim 1$ TeV while the rather large value of $m_{h}\simeq 125$ GeV also seemingly requires multi-TeV highly mixed top squarks [6]. The new LHC Higgs mass measurement and sparticle mass limits seem to have exacerbated the so-called Little Hierarchy problem (LHP) [8]: why doesn’t the Higgs mass blow up to the soft SUSY breaking scale $m_{soft}\gtrsim$several TeV, or what stabilizes the apparent hierarchy $m_{h}\ll m_{soft}$? The LHP opens up the naturalness question: how can it be that the weak scale $m_{weak}\sim m_{W,Z,h}\sim 100$ 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 [2]. A straightforward calculation[2] reveals that

[TABLE]

where $\tan\beta\equiv v_{u}/v_{d}$ is the ratio of Higgs field vevs, $\mu$ 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 $\Sigma_{u}^{u}$ and $\Sigma_{d}^{d}$ terms contain a large assortment of loop corrections (see the Appendix of Ref. [9] for expressions) the largest of which are usually the $\Sigma_{u}^{u}(\tilde{t}_{1,2})$ from the top-squark sector.

We can see immediately from the right-hand-side of Eq. (1) 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 $\Delta_{EW}$ has been introduced [9, 10]–

[TABLE]

– to quantify the weak-scale fine-tuning required to maintain $m_{Z}$ at its measured value. While a low value of $\Delta_{\rm EW}$ seems to be a necessary condition for naturalness within the MSSM, the question is: is it also sufficient? It is argued in Ref’s [11, 12, 13, 14] 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 $\Delta_{\rm HS}\simeq\delta m_{h}^{2}/m_{h}^{2}$ and $\Delta_{\rm BG}\equiv max_{i}|\frac{\partial\log m_{Z}^{2}}{\partial\log p_{i}}|$ (where the $p_{i}$ are fundamental model parameters) collapse to $\Delta_{\rm EW}$ so that $\Delta_{\rm EW}$ 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 $\Delta_{\rm EW}$ as a measure of naturalness in fundamental theories with the MSSM as the weak scale effective theory. In Ref. [15], it is shown that the fine-tuning already turns on for values of $\Delta_{\rm EW}\sim 20-30$. We will adopt a value of $\Delta_{\rm EW}<30$ as a conservative choice for natural models of SUSY.

For a natural theory– where $m_{W,Z,h}\sim 100$ GeV because the RHS contributions to Eq. (1) are comparable to or less than the measured value of $m_{Z}^{2}/2$– then evidently

• •

$m_{H_{u}}^{2}(weak)\sim-(100-300)^{2}$ GeV2 and

• •

$|\mu|\sim 100-300$ GeV [16, 17],

• •

the largest of the radiative corrections (usually $\Sigma_{u}^{u}(\tilde{t}_{1,2})$) 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 $m_{H_{u}}^{2}(\Lambda=m_{GUT})$ 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 [18, 19]. 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[20]. 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 $\Sigma_{u}^{u}(\tilde{t}_{1,2})\sim 100-300$ 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 $\Sigma_{u}^{u}(\tilde{t}_{1,2})$ values also increase the Higgs mass to its measured value $m_{h}\sim 125$ GeV [10, 9].

The second condition– that the superpotential $\mu$ parameter is of order the weak scale– brings up the famous SUSY $\mu$ problem [21]: since $W_{\rm MSSM}\ni\mu H_{u}H_{d}$ is SUSY preserving, naively one expects the dimensionful parameter $\mu$ to be of order $m_{P}\simeq 2.4\times 10^{18}$ GeV while phenomenology requires $\mu\sim m_{weak}$. In this paper, we focus attention on the SUSY $\mu$ problem as occurs in gravity-mediation. The SUSY $\mu$ problem in gauge-mediated supersymmetry breaking (GMSB) is summarized in Ref. [22]. 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 $m_{h}\simeq 125$ GeV [23, 24].111 We also do not consider SUSY models with non-holonomic soft terms[25] or multiple $\mu$ terms; it is not clear whether such models have viable UV completions[26, 27].

There are two parts to solving the SUSY $\mu$ problem:

• •

First, one must forbid the appearance of $\mu$, usually via some symmetry such as Peccei-Quinn (PQ) or better a continuous or discrete gauge or $R$-symmetry, and then

• •

re-generate $\mu$ at the much lower weak scale $|\mu|\sim 100-300$ GeV (the lower the more natural) via some mechanism such as symmetry breaking.

Many solutions to the SUSY $\mu$ problem have been proposed, and indeed in Sec. 2 we will review twenty of these. In most of these solutions, the goal (for gravity-mediation) was to re-generate $\mu\sim m_{3/2}$ 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[28]. When many of these $\mu$ solutions were proposed– well before the LHC era– it was commonly accepted that $m_{3/2}\sim m_{weak}$ which would also solve the SUSY naturalness problem. However, in light of the above discussion, the SUSY $\mu$ problem needs a reformulation for the LHC era: any solution to the SUSY $\mu$ problem should first forbid the appearance of $\mu$, but then re-generate it at the weak scale, which is now hierarchically smaller than the soft breaking scale:

[TABLE]

Our goal in this paper is to review various proposed solutions to the SUSY $\mu$ problem and confront them with the Little Hierarchy as established by LHC data and as embodied by Eq. 3. While many solutions can be tuned to maintain the Little Hierarchy, others may offer compatibility with or even a mechanism to generate Eq. 3. Thus, present LHC data may be pointing to favored solutions to the SUSY $\mu$ problem which may be reflective of the way nature actually works.

With this end in mind, in Sec. 2 we will review a variety of mechanisms which have been offered as solutions to the SUSY $\mu$ problem. We organize the twenty solutions according to:

• •

solutions from supergravity/superstring constructions,

• •

extended MSSM solutions,

• •

solutions from an extra local $U(1)^{\prime}$ and

• •

solutions involving Peccei-Quinn (PQ) symmetry and axions.

Many of these solutions tend to relate the $\mu$ parameter to the scale of soft SUSY breaking which would place the $\mu$ parameter well above the weak scale and thus require significant EW fine-tuning. One such example is the original Kim-Nilles (KN) [29] model (Subsec. 2.4.1) which generates a $\mu$ parameter $\mu\sim v_{PQ}^{2}/m_{P}$ and relates $v_{PQ}\sim m_{hidden}$ (where $m_{hidden}$ is a mass scale associated with hidden sector SUGRA breaking) and thus obtains $\mu\sim v_{PQ}^{2}/m_{P}\sim m_{hidden}^{2}/m_{P}\sim m_{3/2}$. However, the LHP can also be accomodated by allowing for $v_{PQ}\ll m_{hidden}$ so that $\mu\ll m_{3/2}$. While KN allows this possibility to be implemented “by hand”, the later MSY [30], CCK [31] and SPM [32] models (Subsec. 2.4.7) implement radiative PQ breaking as a consequence of SUSY breaking with the result that $v_{PQ}\ll m_{hidden}$ and hence $\mu\ll m_{soft}$ [33].

A prominent criticism of the $\mu$ solutions based on the existence of a global PQ or discrete symmetry is that such symmetries are incompatible with gravity at high scales [34, 35, 36, 37, 38], i.e. that including the presence of gravity could spoil any global or discrete symmetries which may be postulated. In Subsec. 2.4.4, we discuss possible ways around the gravity spoliation of global or discrete symmetries. The MBGW model [39] (Subsec. 2.4.9) adopts a gravity-safe PQ symmetry thanks to a more fundamental discrete gauge symmetry $\mathbb{Z}_{22}$ 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 $\mu$ parameter and other dangerous proton decay operators, and are consistent with an underlying grand unification structure[40, 41]. Such discrete $R$-symmetries are expected to arise from compactification of extra dimensions in string theory. The $\mathbb{Z}_{4}^{R}$ 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 $\mu$ problem, then two hybrid models based on $\mathbb{Z}_{24}^{R}$ are examined (Subsec. 2.5). 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. 3 we discuss the issue of experimental testability and distinguishability of various solutions to the $\mu$ problem. In Sec. 4, we present a convenient Table 14 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 [2].

2 A review of some solutions to the SUSY $\mu$ problem

In this Section, we review some solutions to the SUSY $\mu$ problem. In the solutions reviewed here, the $\mu$-term is typically generated by breaking the symmetry which originally prohibits the $\mu$-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 $U(1)^{\prime}$ and 4. those which include also a solution to the strong CP problem with Peccei-Quinn symmetry breaking.

2.1 Solutions in supergravity/string construction

2.1.1 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. [42]) to forbid the usual MSSM $\mu$ term, then one may regenerate it via the Higgs fields coupling to hidden sector fields $h_{m}$ via non-renormalizable terms in K [42]:

[TABLE]

If we arrange for SUSY breaking in the hidden sector, then the auxilliary component of $h$ develops a vev $\langle F_{h}\rangle\sim m_{hidden}^{2}$ so that the gravitino gets a mass $m_{3/2}\sim m_{hidden}^{2}/m_{P}$. A $\mu$ term is generated of order

[TABLE]

Thus, in the GM case, the $\mu$ parameter arises which is typically of order the soft breaking scale unless the coupling $\lambda_{\mu}$ is suppressed at the $\sim 0.01-0.1$ level.

2.1.2 Casas-Munoz (CM)

Casas and Munoz [43] propose a string theory inspired solution to the SUSY $\mu$ problem. In string theory, dimensionful couplings such as $\mu$ 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

[TABLE]

where $W_{0}$ is the usual superpotential of the MSSM (but without the $\mu$ 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 $\mu$ term is forbidden in $W_{0}^{vis}$, the non-renormalizable contribution in Eq. (6) 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 $\langle W_{0}^{hid}\rangle\sim m_{hidden}^{2}m_{P}$ (as is easy to see in the simplest Polonyi model for SUSY breaking with $W_{Polonyi}=m_{hidden}^{2}(h+\beta m_{P})$ where $\beta$ is a dimensionless constant). Under these conditions, then a $\mu$ term develops with

[TABLE]

Ref. [43] 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 $\Lambda_{h}$ (where then we would associate $m_{hidden}^{2}\simeq\Lambda_{h}^{3}/m_{P}$).

A benefit of the CM solution is that it should be consistent with any stringy UV completion [44] as it avoids the presence of some global (PQ) symmetry. A possible drawback to CM is that the $\mu$ term is naturally expected to be of order $m_{soft}$ instead of $m_{weak}$ unless $\lambda_{\mu}$ 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. (6) since it would violate the PQ symmetry.

2.1.3 $\mu$ 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 $\mu$ term is forbidden since the gauge-invariant bilinear term of Higgs pair $H_{u}H_{d}$ carries zero $R$-charge while the superpotential must have $R_{W}=+2$. However, $H_{u}H_{d}$ may couple to various other superfields $\phi_{i}$ which carry non-trivial $R$-charges so that

[TABLE]

where $P_{\mu}(\phi_{i})$ is a sum over monomials in the fields $\phi_{i}^{n}$. Unbroken $R$-symmetry requires a vanishing $\langle P_{\mu}(\phi_{i})\rangle$ but if the $R$-symmetry is approximate then non-vanishing $P_{\mu}(\phi_{i})$ contributions will develop at higher orders in powers of the field vevs $\langle(\phi_{i}/m_{P})\rangle\lesssim 1$. Thus, a mild hierarchy in the field vevs $\langle\phi_{i}/m_{P}\rangle\lesssim 1$, when raised to higher powers $\langle(\phi_{i}/m_{P})^{n_{i}}\rangle\ll 1$, can generate a much larger hierarchy of scales [45]. In this solution to the $\mu$ problem, which is essentially a UV completion of the CM solution, then $\mu\sim m_{3/2}\sim\langle W\rangle$ is expected to arise.

2.1.4 Solution via the discrete $R$-symmetry $\mathbb{Z}_{4}^{R}$

A particularly attractive way to solve the $\mu$ problem in some string constructions is via a discrete Abelian $R$-symmetry $\mathbb{Z}_{4}^{R}$ [46, 47, 48]. 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. [49], the $\mathbb{Z}_{4}^{R}$ symmetry was invoked to forbid the $\mu$ term as well as dimension-4 baryon- and lepton-number violating operators while dangerous dimension-5 operators leading to proton decay are highly suppressed [40, 41]. The desirable Weinberg neutrino mass operator is allowed. The $\mathbb{Z}_{4}^{R}$ 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 $\mathbb{Z}_{4}^{R}$ are shown in the second row of Table 1.

The charge assignments are consistent with embedding the matter superfields into a single ${\bf 16}$ of $SO(10)$ while the split Higgs multiplets would arise from Wilson-line breaking of gauge symmetry. The $\mathbb{Z}_{4}^{R}$ 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 $m_{soft}\sim m_{3/2}$. A $\mu$ term may arise via GM (Sec. 2.1.1) and/or CM (Sec. 2.1.2) so that $\mu\sim\langle W\rangle/m_{P}^{2}\sim m_{3/2}\sim m_{soft}$. Although the discrete $\mathbb{Z}_{4}^{R}$ $R$-symmetry is broken, the discrete matter/$R$-parity remains unbroken so that the LSP remains absolutely stable. This sort of solution to the $\mu$ problem is expected to be common in heterotic string models compactified on an orbifold [41]. Other possibilities for $\mathbb{Z}_{N}^{R}$ with $N>4$ also occur[41] and in fact any $N$ value is possible under anomaly cancellations provided one includes additional exotic matter into the visible sector [50].

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 [51, 52, 53]. For the case of gravity mediation, the domain walls would be expected to form around the SUSY breaking scale $T\sim 10^{12}$ 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 $T\sim 10^{6}$ GeV [54].

2.1.5 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 $\mu$ term in the superpotential [55, 56]. In some D-brane models which include the MSSM at low energy, then the superpotential $\mu$ 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 $\mu$ term of the form

[TABLE]

can be induced where then $\mu\simeq\exp(-S_{\rm cl})M_{s}$ and $M_{s}$ is the string mass scale. The exponential suppression leads to the possibility of a $\mu$ term far below the string scale. Of course, in this case one might expect the $\mu$ term to arise at any arbitrary mass scale below the string scale rather than fortuitously at the weak scale. If the $\mu$ 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 $\mu$ (the naturalness condition) to ensure a universe with anthropically required electroweak symmetry breaking [19].

2.1.6 Mu solution in ${\rm G_{2}MSSM}$

In Ref. [57] (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 $\mu$ problem is required. Witten suggested the existence of an additional discrete symmetry which forbids the $\mu$ term from appearing but which allows the Higgs triplets to gain large enough masses so as to evade proton decay constraints [58]. In Ref. [59], it is shown that a $\mathbb{Z}_{4}$ symmetry is sufficient to forbid the $\mu$ term and other dangerous RPV operators while allowing massive Higgs triplets. The $\mathbb{Z}_{4}$ discrete symmetry is assumed to be broken via moduli stabilization so that a small $\mu$ term develops.

In the $G_{2}MSSM$, the gravitino gains mass from non-perturbative effects (such as gaugino condensation) in the hidden sector so that $m_{3/2}\sim\Lambda_{h}^{3}/m_{P}^{2}\sim 10-200$ TeV. Matter scalar soft masses are expected at $m_{\phi}\sim m_{3/2}$ so should be very heavy (likely unnatural in the context of Eq. (1)). 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: $m_{\lambda}\sim$TeV scale and in fact these may have dominant AMSB contributions [60] (with comparable moduli-mediated SUSY breaking contributions) so that the wino may be the lightest of the gauginos. The dominant contribution to the $\mu$ parameter arises from Kähler contributions ala Giudice-Masiero and these are expected to be $\mu\sim c\frac{\langle S_{i}\rangle}{m_{p}}m_{3/2}\sim 0.1m_{3/2}$ (where $c$ is some constant $\sim 1$) and thus is suppressed compared to scalar soft masses, but perhaps comparable to gaugino masses.

2.2 Extended MSSM-type solutions

2.2.1 NMSSM: Added singlet with $\mathbb{Z}_{3}$ 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 [61]. Some motivation for the NMSSM can originate in string theory models such as heterotic orbifolds where the $\mu$-term arises as an effective term from couplings of the Higgs pair to a singlet field [44]. Without imposing any symmetry to forbid singlet couplings, we can write a generic NMSSM superpotential as follows:

[TABLE]

and corresponding soft terms

[TABLE]

Here $W_{MSSM}(\mu=0)$ denotes the superpotential for the MSSM but without the $\mu$-term. The tadpole $t$ in Eq. (11) may have destabilizing quadratic divergences and must be suppressed [62]. A $\mathbb{Z}_{3}$ discrete symmetry is usually imposed wherein chiral superfields transform as $\phi\rightarrow e^{2\pi i/3}\phi$ which sends the dimensionful couplings $\xi_{F}$, $\mu$, $\mu_{S}$, $B\mu$, $b_{S}$ 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 [63]. (Some means of avoidance of domain walls are proposed in Ref’s [64].) 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 $\mu$ term emerges with

[TABLE]

An attractive alternative choice for $\mu$-forbidding symmetry than the (perhaps ad-hoc) $\mathbb{Z}_{3}$ would be one of the anomaly-free discrete $R$-symmetries $\mathbb{Z}_{4}^{R}$ or $\mathbb{Z}_{8}^{R}$ [41]. Like the $\mathbb{Z}_{3}$ discrete symmetry, the $\mathbb{Z}_{8}^{R}$ symmetry also forbids the dangerous divergent tadpole term. The $\mathbb{Z}_{4}^{R}$ 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 [41].

Thus, all the advantages of the $\mathbb{Z}_{3}$ discrete symmetry can be obtained by imposing instead either a $\mathbb{Z}_{4}^{R}$ or $\mathbb{Z}_{8}^{R}$ symmetry: this then avoids the disadvantages–ad-hocness and introduction of domain walls into the early universe after electroweak phase transition– inherent in the $\mathbb{Z}_{3}$ 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 $v_{s}\sim v_{u,d}$. So far, LHC Higgs coupling measurements favor a SM-like Higgs so one might expect $v_{s}\gg v_{u,d}$ which may lead one to an unnatural value of $\mu_{\rm eff}$. The superfield $S$ also contains a spin-$1\over 2$ singlino $\tilde{s}$ which may mix with the usual neutralinos and might even be the LSP [65]. 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 $m_{h}\simeq 125$ GeV[66].

2.2.2 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) [67, 68]. The nMSSM, like the NMSSM, solves the $\mu$ problem via an added singlet superfield $S$. But in the nMSSM, the model is founded on a discrete $R$-symmetry either $\mathbb{Z}_{5}^{R}$ or $\mathbb{Z}_{7}^{R}$. Discrete $R$-charge assignments for $\mathbb{Z}_{5}^{R}$ are shown in Table 2. The tree level superpotential is given by

[TABLE]

so that unlike the NMSSM with $\mathbb{Z}_{3}$ symmetry, the $\kappa S^{3}$ 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

[TABLE]

is induced at six-loop or higher level where $\xi_{F}\sim m_{3/2}^{2}$ (along with a corresponding soft SUSY breaking term). Due to lack of the discrete global $\mathbb{Z}_{3}$ 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 [69, 70, 71, 72, 73]. 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 $\chi_{2}^{0}$ or a singlino $\chi_{1}^{0}$, 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[71, 72].

2.2.3 Mu-from-nu SSM ($\mu\nu$SSM)

The $\mu$-from-$\nu$SSM ($\mu\nu$SSM) [74] 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 $\mu$ term. The $\mu\nu$SSM first requires a $\mathbb{Z}_{3}$ symmmetry to forbid the usual $\mu$ term (and also a usual Majorana neutrino mass term $M_{i}N^{c}N^{c}$). The superpotential is given by

[TABLE]

If the scalar component of one of the RHN superfields $\tilde{\nu}_{Ri}$ of $N_{i}^{c}$ gains a weak scale vev, then an effective $\mu$ term develops:

[TABLE]

along with a weak scale Majorana neutrino mass term $M_{Njk}\sim\kappa_{ijk}\langle\tilde{\nu}_{Ri}\rangle$. By taking small enough neutrino Yukawa couplings, then a weak scale see-saw develops which can accommodate the measured neutrino masses and mixings.

The $\mu\nu$SSM develops bilinear $R$-party violating terms via the superpotential $f_{\nu}LH_{u}N^{c}$ term so that the lightest $\mu\nu$SSM particle is not stable and doesn’t comprise dark matter: $\tilde{\chi}_{1}^{0}\rightarrow W^{(*)}\ell$ and other modes. As an alternative, a gravitino LSP is suggested with age longer than the age of the universe: it could decay as $\tilde{G}\rightarrow\nu\gamma$ and possibly yield gamma ray signals from the sky [75]. The phenomenology of the $\mu\nu$SSM 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 [76].

While the $\mu\nu$SSM may be considered the most minimal model to solve the $\mu$ problem, it suffers the same $\mathbb{Z}_{3}$ domain wall problem as the NMSSM (and perhaps the same routes to avoidance [64]). 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.

2.3 $\mu$ from an extra local $U(1)^{\prime}$

In this class of models [77, 78, 79, 80, 81], a SM singlet superfield $S$ is introduced which is charged under a new $U(1)^{\prime}$ gauge interaction, so terms with mass dimensions in Eq. (10) are forbidden. Due to the $U(1)^{\prime}$ gauge charges of $S$, the cubic coupling $S^{3}$ is also absent. We will see below three representative realizations of this class of model.

2.3.1 CDEEL model

Cvetic-Demir-Espinosa-Everett-Langacker [77] (CDEEL) propose a $U(1)^{\prime}$ extended gauge symmetry model as emblematic of fermionic orbifold string compactifications. While the usual $\mu$ term is forbidden by the extended gauge symmetry, the superpotential term

[TABLE]

is allowed and under $U(1)^{\prime}$ breaking then $S$ develops a vev $\langle S\rangle\sim m_{weak}$ such that a $\mu$ term is generated $\mu_{\rm eff}=\lambda_{\mu}\langle S\rangle$ along with an additional weak scale $Z^{\prime}$ gauge boson. Forbidding the $\mu$ term via a gauge symmetry avoids the gravity spoliation/global symmetry problem. In addition, the $\mu$ term is linked to EW symmetry breaking and this would be expected to occur at $m_{weak}$ rather than $m_{soft}$. The $U(1)^{\prime}$ 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 $Z^{\prime}$ gauge bosons with exotic decays to light higgsinos [80].

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 $U(1)^{\prime}$ gauge charges are listed in Table 3.

2.3.2 sMSSM model

An alternative $U(1)^{\prime}$-extended MSSM (abbreviated as sMSSM)[82, 83] also solves the $\mu$ problem by invoking multiple SM singlet superfields charged under $U(1)^{\prime}$ 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 $Z^{\prime}$ gauge boson by introducing as well a secluded sector containing three additional singlets $S_{1},\ S_{2},\ S_{3}$ charged under $U(1)^{\prime}$. The superpotential is given by

[TABLE]

so that the secluded sector has a nearly $F$- and $D$-flat scalar potential. The $U(1)^{\prime}$ 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 $\lambda_{s}$ is small, then the additional $Z^{\prime}$ 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. 2.2.2) which has very different manifestations from what is expected from the CDEEL $U(1)^{\prime}$ model.

2.3.3 HPT model

The Hundi-Pakvasa-Tata (HPT) model [78] also solves the SUSY $\mu$ problem by positing an additional $U(1)^{\prime}$ gauge symmetry in a supergravity context. The $U(1)^{\prime}$ charges of the multiplets in the HPT scheme are shown in Table 4. With these $U(1)^{\prime}$ charge assignments, the $\mu$ term is forbidden in the superpotential but (unlike the CDEEL model) a dim-4 term as $\mu$ solution à la Kim-Nilles is allowed:

[TABLE]

The $U(1)^{\prime}$ gauge symmetry also forbids trilinear RPV couplings and dangerous $p$-decay operators. When the $U(1)^{\prime}$ breaks (at an intermediate scale $Q\sim 10^{11}$ GeV), the $S$ field acquires a vev to yield an effective $\mu$ 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 [84]. 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 $\tilde{\chi}_{1}^{0}\rightarrow\ell W^{(*)}$ or $\nu Z^{(*)}$ and may lead to displaced vertices in collider events. Dark matter must be comprised of some other particles (e.g. axions). Also, the $U(1)^{\prime}$ is broken at the intermediate scale $Q\sim 10^{11}$ GeV so that the additional $Z^{\prime}$ has a mass far beyond any collider reach.

Since solving the $\mu$ problem as well as generating the neutrino mass scale of suitable order requires introduction of a new gauge group $U(1)^{\prime}$, care must be taken so that associated anomalies are cancelled. Anomaly cancellation requires introducing various additional exotic fields including color triplets $K_{i}$ and $K_{i}^{\prime}$ 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.

2.4 Solutions related to Peccei-Quinn symmetry breaking

In this Subsection, we examine natural $\mu$-term solutions related to the PQ symmetry used to solve the strong CP problem. In this class of models, the $\mu$-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. 2.4.1, 2.4.2, and 2.4.3, we review $\mu$-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. 2.4.4, we discuss a criterion for protecting the PQ solution to the strong CP problem, and in Subsec. 2.4.5 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. 2.4.6.

Finally, we review $\mu$-term generation by breaking of PQ symmetry from SUSY breaking: radiative breaking of PQ symmetry (Subsec. 2.4.7), breaking of an accidental approximate PQ symmetry from a gauged $U(1)_{R}$ symmetry (Subsec. 2.4.8) and a $\mathbb{Z}_{22}$ discrete gauge symmetry (Subsec. 2.4.9) by a large negative trilinear term.

2.4.1 Kim-Nilles solution

Kim and Nilles (KN) [29] presented the first formulation of the SUSY $\mu$ 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 [85] axion model222In the DFSZ axion model [85], 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 $\mu$ 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 $Q_{PQ}(X)=+2/(n+1)$:

[TABLE]

for $n\geq 1$.

It is arranged to spontaneously break PQ by giving the $X$ field a vev $\langle X\rangle$ which also generates a (nearly) massless axion $a$ which solves the strong CP problem. To obtain cosmologically viable axions– with $\langle X\rangle\sim 10^{11}$ GeV and with $m_{p}\simeq 2.4\times 10^{18}$ GeV, we can obtain the $\mu$ 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 5.

The intermediate PQ breaking scale can be gained from a PQ superpotential of the form:

[TABLE]

The scalar components of $X$ and $Y$ develop vevs $\langle X\rangle=\langle Y\rangle=v_{PQ}$ such that a $\mu$ term is generated:

[TABLE]

This value of the $\mu$ term $\mu\sim\lambda_{\mu}v_{PQ}^{2}/m_{P}$ is to be compared to the soft breaking scale in models of gravity-mediation: $m_{soft}\sim m_{3/2}\sim m_{hidden}^{2}/m_{P}$. Here, $v_{PQ}$ is identified as $v_{PQ}\sim m_{hidden}$ and thus $\mu$ is obtained as $\mu\sim m_{3/2}$. But, a value $\mu\sim m_{weak}\ll m_{soft}\sim m_{3/2}$ 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.333 In models with SUSY breaking arising from e.g. gaugino condensation at an intermediate scale $\Lambda_{h}$, then $m_{3/2}\sim\Lambda_{h}^{3}/m_{P}^{2}$ in which case we would define $m_{hidden}^{2}\sim\Lambda_{h}^{3}/m_{p}$. 444 The model [86] 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 $\mu\sim m_{weak}$ when SUSY is broken in the hidden brane through the shining mechanism [87]. A virtue of the KN solution is that it combines a solution to the strong CP problem with a solution to the SUSY $\mu$ problem which also allows for a Little Hierarchy. A further benefit is that it provides an additional dark matter particle– namely the DFSZ [85] 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 [88, 89]. 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 [90]. For larger $v_{PQ}$ values, then non-thermal processes such as saxion and axino [91] 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 $N_{eff}=3.13\pm 0.32$ from the recent Particle Data Group tabulation [92]). In the context of the DFSZ model embedded within the MSSM, then the presence of higgsinos in the $a\gamma\gamma$ 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 [93].

2.4.2 Chun-Kim-Nilles

In the CKN model [94], it is assumed that SUSY is broken in the hidden sector due to gaugino condensation $\langle\lambda\lambda\rangle\sim\Lambda_{h}^{3}\sim(10^{13}$ 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 6:

This allows for the presence of a superpotential term

[TABLE]

Along with gauginos condensing at a scale $\Lambda_{h}$ to break SUGRA with $m_{3/2}\sim\Lambda_{h}^{3}/m_{P}^{2}$, the hidden sector scalar squarks condense at a scale $\Lambda<\Lambda_{h}$ to break the PQ symmetry and to generate a $\mu$ term

[TABLE]

Thus, this model provides a framework for $\mu<m_{soft}$. 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) [95]. The CC arises from the very low mass MI axion field slowly settling to the minimum of its potential.

2.4.3 BK/EWK solution linked to inflation and strong CP

In Ref’s [96, 97], a model is proposed with superpotential

[TABLE]

where the $\phi$ field plays the role of inflaton and the $N$ field is a waterfall field leading to hybrid inflation in the early universe [98]. Although the model appears similar to the NMSSM, it is based on a PQ rather than $\mathbb{Z}_{3}$ symmetry with charges as in Table 7. Thus, it avoids the NMSSM domain wall problems which arise from a postulated global $\mathbb{Z}_{3}$ symmetry. Augmenting the scalar potential with soft breaking terms, then the $\phi$ and $N$ fields gain vevs of order some intermediate scale $Q\sim 10^{12}$ GeV so that Yukawa couplings $\lambda_{\mu}$ and $\kappa$ are of order $10^{-10}$. Such tiny Yukawa couplings might arise from type-I string theory constructs [99]. To fulfill the inflationary slow-roll conditions, then the field $\phi$ 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.

2.4.4 Global symmetries and gravity

It is well known that gravitational effects violate global symmetries, as has been considered via black hole “no hair” theorems [34] and wormhole effects [35]. 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 [36, 38, 37]. Indeed, Kamionkowski and March-Russell [38] (KMR) considered the effect of gravitational operators such as

[TABLE]

involving PQ charged fields $\phi$ 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 $G_{\mu\nu A}\tilde{G}_{A}^{\mu\nu}$ settles to a non-zero minimum thus destroying the PQ solution to the strong CP problem. To maintain $\bar{\theta}\lesssim 10^{-10}$, 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 [102]. 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. 1 the Kim diagram is shown [100, 101]. 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.

2.4.5 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 $\mu$ term. Some models based on an extra local $U(1)^{\prime}$ were examined in Subsec. 2.3. 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 [102, 103]. Such $\mathbb{Z}_{M}$ symmetries may emerge from a local $U(1)^{\prime}$ when a charge $M$ object (charged under the new $U(1)^{\prime}$) condenses at very high energy leaving a discrete $\mathbb{Z}_{M}$ gauge symmetry in the low energy effective theory. Since the $\mathbb{Z}_{M}$ emerges from a local gauge theory, it remains gravity-safe. In Subsec. 2.4.9, the MBGW model [39] which is based on a $\mathbb{Z}_{22}$ discrete gauge symmetry is examined. The model under $\mathbb{Z}_{22}$ is found to be anomaly-free and is used to not only forbid the $\mu$ term but to generate a PQ symmetry needed to solve the strong CP problem. The lowest order PQ violating term allowed by the $\mathbb{Z}_{22}$ is sufficiently suppressed so that PQ arises as an accidental approximate global symmetry thereby rendering the model to be gravity-safe. The $\mathbb{Z}_{22}$ 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. 2.1.3.555See also Ref. [104]. In the interest of minimality, it is noted that continuous $R$ symmetries are not consistent with the particle content of just the MSSM [105]. Then it is also of interest to examine the possibility of discrete remnant $R$-symmetries $\mathbb{Z}_{N}^{R}$ 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 $\theta$ carry non-trivial $R$-charge: in the simplest case, $Q_{R}(\theta)=+1$ so that $Q_{R}(d^{2}\theta)=-2$. For the Lagrangian ${\cal L}\ni\int d^{2}\theta W$ to be invariant under $\mathbb{Z}_{N}^{R}$-symmetry, the superpotential $W$ must carry $Q_{R}(W)=2+$integer multiples of $N$.

These remnant discrete $R$-symmetries $\mathbb{Z}_{N}^{R}$– 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 8 (as derived in Ref’s [40, 41]) 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 $\mu$ 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 $\mathbb{Z}_{N}^{R}$ symmetries of Table 8 have been shown to be the only anomaly-free symmetries which allow for fermion masses and suppress the $\mu$ 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 [106].

2.4.6 Natural Higgs-Flavor-Democracy (HFD) solution to $\mu$ problem

In Ref. [107], the $\mu$ 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 $S_{2}(H_{u})\times S_{2}(H_{d})$. This is Higgs flavor democracy (HFD). Besides solving the $\mu$ 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 $S_{2}(H_{u})\times S_{2}(H_{d})$ . Then the Higgsino mass matrix has a democratic form given by:

[TABLE]

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 $\mu$. In order to fulfill this criteria, the HFD must be broken and this mechanism results in $\mu$ $\approx$ $\mathcal{O}$(TeV).

Generation of $\mu$: The minimal Kahler potential is considered as $K=\Phi_{i}\Phi_{i}^{\dagger}$ where $\Phi_{i}$ ( i =1, 2) is a doublet under the gauge group such as the Higgs superfield and $X_{i}$ and $\bar{X_{i}}$ (i=1,2) are singlets under the gauge group. Both $\Phi_{i}$ and $X_{i}$ and the corresponding barred fields obey the $S_{2}\times S_{2}$ symmetry. $X^{(0)}$ and $\bar{X}^{(0)}$ are SM singlet fields containing a very light QCD axion for $10^{9}$ GeV $\leq$ $v_{PQ}$ $\leq$ $10^{12}$ GeV. With this construct, the $S_{2}(L)\times S_{2}(R)$ symmetric nonrenormalizable term is:

[TABLE]

With the HFD breaking minimum at $\langle X_{1}\rangle$ = $\langle\bar{X_{1}}\rangle$ = $v_{PQ}$ and $\langle X_{2}\rangle$ = $\langle\bar{X_{2}}\rangle$ = 0, Eq. (26) becomes

[TABLE]

This choice of HFD breaking minimum is spontaneous. Thus we obtain $\mu$ = $\frac{\lambda_{\mu}v_{PQ}^{2}}{2m_{P}}$. With $10^{10}$ GeV $\leq$ $v_{PQ}$ $\leq$ $10^{12}$ GeV and $\lambda_{\mu}$ $\approx$ $\mathcal{O}$(1), we obtain $\mu$ $\approx$ $\mathcal{O}(0.1-10^{3}\text{ TeV})$. The LH can be accomodated for the lower range of $v_{PQ}$ or if $\lambda_{\mu}<1$.

Light QCD Axion - Integrating out the heavy fields in Eq. (27), one obtains

[TABLE]

The PQ charges of Higgs multiplets are obtained from their interaction with the quarks and PQ charges of $X^{(0)}$ and $\bar{X}^{(0)}$ are defined by Eq. (28). Thus, a term $m_{3/2}\frac{\lambda^{2}}{4m_{P}^{2}}\frac{1}{M_{G}}H_{u}H_{d}(XX^{c})^{2}$ is obtained which violates PQ and hence adds a tiny correction to $\mu$. 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.

2.4.7 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 $\mu\ll m_{soft}$. A drawback to the KN model is that it inputs the PQ scale “by hand” via the superpotential Eq. (20). 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) [30].

In radiative PQ breaking models, the MSSM superpotential is

[TABLE]

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

[TABLE]

and where

[TABLE]

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

[TABLE]

For simplicity, we assume a diagonal coupling $h_{ij}=h_{i}\delta_{ij}$. The model may be defined as applicable at the reduced Planck scale $m_{P}\simeq 2.4\times 10^{18}$ GeV and the corresponding Renormalization Group Equations (RGEs) can be found in Ref. [30] at 1-loop and Ref. [33] 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 $V=V_{F}+V_{D}+V_{\rm soft}$. 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

[TABLE]

Augmenting this with $V_{\rm soft}$, we minimize $V$ at a scale $Q=v_{PQ}$ to find the vevs of $\phi_{X}$ and $\phi_{Y}$ ($v_{X}$ and $v_{Y}$):

[TABLE]

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 $\phi_{X}$ and $\phi_{Y}$ obtains vevs $v_{X}$ and $v_{Y}$ at the intermediate mass scale, taken here to be $v_{PQ}=\sqrt{v_{X}^{2}+9v_{Y}^{2}}$. The corresponding axion decay constant is given by $f_{a}=\sqrt{2}v_{PQ}$.666For axion interactions, the actual decay constant is $f_{A}\equiv f_{a}/N_{DW}$ 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 $\mu$ parameter, which is originally forbidden by PQ symmetry, is generated with a value

[TABLE]

and a Majorana neutrino mass, also initially forbidden by PQ symmetry, is generated at

[TABLE]

Since the $\mu$ term depends on an arbitrary coupling $g_{MSY}$, one may obtain any desired value of $\mu$ 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. $g_{MSY}\gg 1$ or $g_{MSY}\ll 1$, we might need to introduce an additional physical scale to explain the $\mu$ term. To generate a value of $\mu=150$ GeV, then values of $g_{MSY}$ as shown in Fig. 2 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 $\mu\ll m_{soft}$ for the case where $v_{PQ}<m_{hidden}$.

Detailed numerical calculations in the MSY model have been carried out in Ref. [33]. There, it is found that for generic $W_{\mu}^{MSY}$ couplings $g_{MSY}\sim 0.1-1$, then a $\mu$ parameter $\mu\sim 100-200$ GeV can easily be generated from TeV-scale soft breaking terms. Furthermore, since the $\mu$ 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 $m_{a}\simeq 0.48f_{\pi}m_{\pi}/f_{a}=6.25\times 10^{-3}\ {\rm GeV}/f_{a}$ is related to the Higgs mass $m_{h}$ and the higgsino masses $m_{\widetilde{W}_{1},\widetilde{Z}_{1,2}}\sim\mu$. The required PQ charges for the MSY model are listed in Table 9.

Other closely related models make different choices for which fields enter into $W_{\mu}$. We can also have:

[TABLE]

The above three possibilities for $W_{\mu}$ correspond to Ref’s [30] (MSY), [31] (CCK) and [32] (SPM). The corresponding PQ charges for the three radiative PQ breaking models are listed in Table 9.

We list in Fig’s 3 and 4 also the values of $g_{CCK}$ and $g_{SPM}$ which are needed to generate a value of $\mu\simeq 150$ 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 $\mu$ from multi-TeV values of $m_{3/2}$: i.e. $\mu\ll m_{3/2}$ 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 $\mathbb{Z}_{N}^{R}$ symmetries. We have examined whether or not the three radiative PQ breaking models of Table 9 (CCK, MSY and SPM) can be derived from any of the more fundamental $\mathbb{Z}_{N}^{R}$ symmetries in Table 8 [108]. 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 $m_{X}^{2}$ negative so that PQ symmetry is broken. And since the PQ symmetry does not allow for a Majorana mass term $\frac{1}{2}M_{N}N^{c}N^{c}$, then no see-saw scale can be developed. One exception is the MSY model under $\mathbb{Z}_{4}^{R}$ symmetry with charge assignments $Q_{R}(X)=0$ and $Q_{R}(Y)=2$: then a $YH_{u}H_{d}$ term is allowed which would generate a $\mu$ 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 $XYH_{u}H_{d}$ 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 $\mathbb{Z}_{6}^{R}$ 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 $M^{2}Y$ 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.

2.4.8 CCL model from gauged $U(1)_{R}$ symmetry

In the model of Choi, Chun and Lee [104] (CCL), the $\mu$ term is generated in a manner similar to the SPM model [32], 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

[TABLE]

with $U(1)_{R}$ and $PQ$ charges for the $n=2$ case given in Table 10.

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 $\mu\sim m_{soft}$. 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 $m_{1/2}\gg m_{scalars}>m_{3/2}$ ($m_{1/2}$: gaugino mass). Scalar soft masses are fixed in terms of $U(1)_{R}$ $D$-terms and typically lead to large negative $m_{H_{u}}^{2}$ at the weak scale which then requires a large, unnatural $\mu$ term which would violate the $\mu\ll m_{soft}$ 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.

2.4.9 MBGW model of PQ breaking from SUSY breaking

The Martin-Babu-Gogoladze-Wang (MBGW) model [32, 39] begins with a superpotential

[TABLE]

which is augmented by soft SUSY breaking terms

[TABLE]

so that the scalar potential is

[TABLE]

with

[TABLE]

The scalar potential admits non-zero minima in the fields $\phi_{X}$ and $\phi_{Y}$ for $C<0$. The scalar potential for the case of $m_{X}=m_{Y}\equiv m_{s}=10^{4}$ GeV and $C=-3.5\times 10^{4}$ GeV is shown in Fig. 5.

It is found in Ref. [39] that the model admits a remnant $\mathbb{Z}_{22}$ 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. (43), the lowest order PQ-violating term in the superpotential is $\frac{(Y)^{11}}{m_{P}^{8}}$: thus this model is gravity safe according to the KMR criterion. An approximate PQ symmetry emerges as an accidental consequence of the discrete $\mathbb{Z}_{22}$ gauge symmetry. The $\mathbb{Z}_{22}$ and PQ charges are listed in Table 11.

By taking $\langle\phi_{X}\rangle\equiv v_{x}$ and $\langle\phi_{Y}\rangle\equiv v_{Y}$, then the scalar potential minimization conditions read

[TABLE]

A simplifying assumption of $m_{X}^{2}=m_{Y}^{2}\equiv m_{s}^{2}$ and $v_{X}=v_{Y}\equiv v_{s}$ leads to

[TABLE]

so that the $\mu$ term is

[TABLE]

with $v_{s}^{2}\simeq\frac{|C|}{12\lambda_{2}}m_{P}$. Taking $m_{s}\simeq m_{3/2}=10^{4}$ GeV with $\mu=150$ GeV and $C=-3.5\times 10^{4}$ GeV leads to $v_{s}\simeq v_{PQ}\simeq 10^{11}$ GeV for $\lambda_{2}=0.7$ and $\lambda_{\mu}\simeq 0.036$. Thus, the MBGW model admits a Little Hierarchy $\mu\ll m_{3/2}$ whilst generating the PQ scale $v_{PQ}\sim 10^{11}$ GeV (which generates mainly axion dark matter with a smaller portion of higgsino-like WIMPs [89, 90, 93]). The allowed range of MBGW model parameter space is shown in Fig. 6 where we show contours of $\lambda_{\mu}$ values which lead to $\mu=150$ GeV.

As mentioned previously, the MBGW model appears gravity-safe under the $\mathbb{Z}_{22}$ discrete gauge symmetry, The discrete gauge symmetry $\mathbb{Z}_{M}$ 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. [108]). 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 [109] 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.

Alternatively, it is worth checking whether MBGW is gravity-safe under any of the discrete $R$-symmetries listed in Table 8. To check gravity safety, we note that additional superpotential terms of the form $\lambda_{3}X^{p}Y^{q}$ may be allowed for given $\mathbb{Z}_{N}^{R}$ 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 $V(\phi)$ includes terms which are not suppressed by at least eight powers of $1/m_{P}$ [38]. The largest dangerous scalar potential terms develop from interference between $\lambda_{2}(XY)^{2}/m_{P}$ and $\lambda_{3}X^{p}Y^{q}/m_{P}^{p+q-3}$ when constructing the scalar potential $V_{F}=\sum_{\hat{\phi}}|\partial W/\partial\hat{\phi}|_{\hat{\phi}\rightarrow\phi}^{2}$ (here, the $\hat{\phi}$ label chiral superfields with $\phi$ being their leading components). We find the MBGW model to be not gravity safe under any of the $\mathbb{Z}_{N}^{R}$ discrete $R$-symmetries of Table 8.

2.5 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 [108] 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 $MN^{c}N^{c}/2$ 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 :

2.5.1 Hybrid CCK Model

The superpotential for the hybrid CCK model (hyCCK) is given by [108]:

[TABLE]

Thus when the PQ symmetry breaks, the $\mu$ parameter is obtained as

[TABLE]

We have checked that the hyCCK model is not gravity-safe under the $\mathbb{Z}_{N}^{R}$ symmetries for $N=4,6,8$ or 12. However, it does turns out to be gravity-safe under $\mathbb{Z}_{24}^{R}$ symmetry with the $\mathbb{Z}_{24}^{R}$ charge and PQ charge assignments as shown in Table 12.

The scalar potential for hyCCK is found to be

[TABLE]

and is shown in Fig. 7 vs. scalar field values $\phi_{X}$ and $\phi_{Y}$. For large negative values of soft term $A_{f}$, then a $\mathbb{Z}_{24}^{R}$ and $PQ$ breaking minimum develops.

The lowest order PQ violating terms in the superpotential are $X^{8}Y^{2}/m_{P}^{7}$, $X^{4}Y^{6}/m_{P}^{7}$ 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. 8 where we show contours of $\lambda_{\mu}$ values which lead to $\mu=200$ 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 $\lambda_{\mu}\sim 0.015-0.2$ are generally sufficient for a natural $\mu$ term and are easily consistent with soft mass $m_{soft}\sim m_{3/2}\sim 2-30$ TeV as indicated by LHC searches. We also note that for $m_{3/2}\sim 5-20$ TeV, then $v_{PQ}\sim 10^{11}$ GeV which corresponds to the sweet spot for axion cold dark matter.

2.5.2 Hybrid SPM Model

The superpotential for the hybrid SPM model (hySPM) is given by [104, 108]

[TABLE]

In this case, when PQ symmetry breaks, the $\mu$ parameter is generated to be

[TABLE]

This model also turns out to be not gravity-safe under $\mathbb{Z}_{N}^{R}$ symmetries for $N=4,6,8$ and 12 but is gravity-safe for $\mathbb{Z}_{24}^{R}$ symmetry. The gravity-safe $\mathbb{Z}_{24}^{R}$ charge and PQ charge assignments are shown in Table 13.

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 $Y^{8}X^{2}/m_{P}^{7}$, $Y^{4}X^{6}/m_{P}^{7}$ 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. 9 where we show contours of $\lambda_{\mu}$ values which lead to $\mu=150$ GeV in the $m_{3/2}$ vs. $-A_{f}$ plane for $f=1$. We also show several representative contours of $v_{PQ}$ values.

2.5.3 Hybrid MSY model

The superpotential in the hybrid MSY model (hyMSY) is given as [108]:

[TABLE]

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 8.

3 Are the various $\mu$ solutions experimentally

distinguishable?

An important question arises: are the various solutions to the SUSY $\mu$ 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 $\mu$ 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. (1). Upper limits have been derived on sparticle masses within the context of unified SUSY models with no fine-tuning [9, 15, 110] (i.e. $\Delta_{EW}\lesssim 30$). These imply typically $m_{\tilde{g}}\lesssim 6$ TeV and $m_{\tilde{t}_{1}}\lesssim 3$ TeV and $|\mu|\lesssim 360$ GeV. To explore such high sparticle masses, then about 15 ab*-1* of $pp$ collisions at $\sqrt{s}\gtrsim 27$ TeV is required for a hadron collider [110] or $\sqrt{s}\gtrsim 720$ GeV is needed for an $e^{+}e^{-}$ collider [111]. 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 $\mu$ 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 [112].

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 [113] 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 $a\gamma\gamma$ triangle vertex– has a much smaller coupling [93]. It is not clear whether present technology has the capability to probe such tiny $a\gamma\gamma$ 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 $\mu$ 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 $\mu$ 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 [61, 65].

The $U(1)^{\prime}$ $\mu$ solutions also include distinctive new particle predictions. The CDEEL model [77] requires the presence of an additional weak scale $Z^{\prime}$ boson which could decay to higgsinos as well as SM particles [81, 80]. For the HPT model [78], the $Z^{\prime}$ 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.

4 Conclusions

In this paper, we have re-examined the SUSY $\mu$ problem with perspective gained from experimental results from LHC through Run 2 with 150 fb*-1* of data. The two parts to the SUSY $\mu$ solutions are

1. first forbid the $\mu$ 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 $\mu$ should be generated of order $m_{weak}\sim m_{W,Z,h}\sim 100-300$ GeV whilst the soft SUSY breaking terms likely inhabit the multi-TeV regime. Thus, a Little Hierarchy (LH) should now be included in SUSY $\mu$ solutions where $|\mu|\ll m_{soft}$. This is different from pre-LHC expectations where solutions sought to generate $|\mu|\simeq m_{soft}$.

To gain an updated perspective on the SUSY $\mu$ problem, we examined twenty solutions. These solutions are summarized in Table 14 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 $\lambda_{\mu}$), some actually generate $\mu\sim m_{weak}\ll m_{soft}$ with $\lambda_{\mu}\sim 1$ (such as the radiative PQ breaking models MSY, CCK and SPM).

Also, early attempts to solve the SUSY $\mu$ problem could appeal to an underlying global symmetry such as PQ to suppress the $\mu$ 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 ($U(1)^{\prime}$) 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 [40, 41], which can arise from compactification of extra dimensions in string theory, seems to provide the cleanest suppression symmetry for the $\mu$ term. A delineation of anomaly-free (including a GS term) $\mathbb{Z}_{N}^{R}$ 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 $\mu$ problem. For $N=4,6,8,12$ and 24, these symmetries forbid $\mu$ along with RPV trilinear terms and dimension-5 $p$-decay operators whilst allowing the required Yukawa couplings and neutrino mass operators. Of these, the $\mathbb{Z}_{4}^{R}$ 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 $\mathbb{Z}_{24}^{R}$ 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, $U(1)^{\prime}$, instanton, G2MSSM).

If the naturalness edict is followed– which requires $|\mu|$ not too far from $m_{weak}\sim 100$ 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 [114]. 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 $\mu$ problem (while also suppressing dangerous $p$-decay operators and allowing for see-saw neutrino masses), then the hybrid models based on $\mathbb{Z}_{24}^{R}$ discrete $R$-symmetry stand out as a rather complete answer.

Overall, the SUSY $\mu$ 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.