Naturalness, dark matter, and the muon anomalous magnetic moment in supersymmetric extensions of the standard model with a pseudo-Dirac gluino

We study the naturalness, dark matter, and muon anomalous magnetic moment in the Supersymmetric Standard Models (SSMs) with a pseudo-Dirac gluino (PDGSSMs) from hybrid $F-$ and $D-$term supersymmetry (SUSY) breakings. To obtain the observed dark matter relic density and explain the muon anomalous magnetic moment, we find that the low energy fine-tuning measures are larger than about 30 due to strong constraints from the LUX and PANDAX experiments. Thus, to study the natural PDGSSMs, we consider multi-component dark matter and then the relic density of the lighest supersymmetric particle (LSP) neutralino is smaller than the correct value. We classify our models into six kinds: (i) Case A is a general case, which has small low energy fine-tuning measure and can explain the anomalous magnetic moment of the muon; (ii) Case B with the LSP neutralino and light stau coannihilation; (iii) Case C with Higgs funnel; (iv) Case D with Higgsino LSP; (v) Case E with light stau coannihilation and Higgsino LSP; (vi) Case F with Higgs funnel and Higgsino LSP. We study these Cases in details, and show that our models can be natural and consistent with the LUX and PANDAX experiments, as well as explain the muon anomalous magnetic moment. In particular, all these cases except the stau coannihilation can even have low energy fine-tuning measures around 10.


I. INTRODUCTION
Supersymmetry (SUSY) provides a natural solution to the gauge hierarchy problem in the Standard Model (SM). In the supersymmetric SMs (SSMs) with R-parity, gauge coupling unification can be achieved, the Lightest Supersymmetric Particle (LSP) serves as a viable dark matter (DM) candidate, and electroweak (EW) gauge symmetry can be broken radiatively because of the large top quark Yukawa coupling, etc. On the other hand, gauge coupling unification strongly suggests Grand Unified Theories (GUTs), which can be realized from superstring theory. Thus, supersymmetry is an important bridge between the most promising new physics beyond the SM and the high-energy fundamental physics.
It is well-known that a SM-like Higgs boson (h) with mass m h = 125.09 ± 0.24 GeV was discovered at the LHC [1][2][3]. However, to obtain such a SM-like Higgs boson mass in the Minimal SSM (MSSM), we need either the multi-TeV top squarks with small mixing or TeV-scale top squarks with large mixing [4,5]. Also, the LHC SUSY searches give stringent constraints on the viable parameter space of the SSMs [6]. For example, the latest SUSY search bounds show that the gluino (g) mass is heavier than about 1.6-2.0 TeV, whereas the light stop (t 1 ) mass is heavier than about 800-1000 GeV (Assuming these particles' masses are well separated from the LSP mass, if not, the bound may not be so stringent). Thus, the big challenge is how to construct the natural SSMs, which can realize the correct Higgs boson mass, solve the SUSY electroweak fine-tuning problem, and evade the LHC SUSY search constraints. On the other hand, the dark matter direct detection experiments such as XENON100 [7], LUX [8], and PANDAX [9] experiments have given strong constraints on the dark matter-nucleon spin-independent scattering cross section. Moreover, the anomalous magnetic moment of the muon a µ = (g µ − 2)/2 is still one of strong hints for new physics since it is deviated from the SM prediction more than 3σ level. The discrepancy compared to its SM theoretical value is [10][11][12] ∆a µ = (a µ ) exp − (a µ ) SM = (28.6 ± 8.0) × 10 −10 . (1) In the SSMs, the light smuon, muon-sneutrino, Bino, Winos, and Higgsinos would contribute to ∆a µ [13][14][15][16][17][18]. Their contributions to ∆a µ from the neutralino-smuon and charginosneutrino loops can approximately be expressed as Ref. [15], we obtain that the 2σ bound on ∆a µ can be achieved for tan β = 10 if four relevant sparticles are lighter than 600 − 700 GeV. While for smaller tan β (∼3), the lighter sparticles ( 500 GeV) are needed. Therefore, to explain the muon anomalous magnetic moment, we do need the light smuon, muon-sneutrino, Bino, Winos, and Higgsinos.
Note that all the sparticles except gluino in the SSMs can be within about 1 TeV as long as the gluino is heavier than 3 TeV, which is clearly an simple modification to the SSMs before the LHC, we have proposed the SSMs with a pseudo-Dirac gluino (PDGSSMs) from hybrid F − and D−term SUSY breakings, which can explain the dark matter and muon anomalous magnetic moment simultaneously [19]. Such kind of models solves the following problems in the SSMs with Dirac gauginos or say supersoft SUSY [20][21][22][23][24][25][26][27][28][29][30][31] due to the F −term gravity mediation: µ problem cannot be solved via the Giudice-Masiero (GM) mechanism [32], the D-term contribution to the Higgs quartic coupling vanishes, and the right-handed slepton may be the LSP, etc [20]. There is another problem in supersoft SUSY: the scalar components of the adjoint chiral superfields may be tachyonic and then break the SM gauge symmetry, which was solved in Ref. [29].  [59]. We have solved the relevant problems mentioned in [59], and concentrate on low energy phenomenology(especially dark matter) here.
In this paper, we will study the naturalness, dark matter, and muon anomalous magnetic moment in the PDGSSMs. To obtain the observed dark matter density and explain the muon anomalous magnetic moment, we show that the low energy fine-tuning measures are larger than about 30 due to strong constraints from the LUX and PANDAX experiments.
Thus, to realize the natural PDGSSMs, we consider multi-component dark matter and then the relic density of the LSP neutralino is smaller than the correct value. We classify the dark matter models into six kinds: (i) Case A is a general case, which has small low energy fine-tuning measure and can explain the anomalous magnetic moment of the muon; (ii) Case B with the LSP neutralino and light stau coannihilation; (iii) Case C with Higgs funnel; (iv) Case D with Higgsino LSP; (v) Case E with light stau coannihilation and Higgsino LSP; (vi) Case F with Higgs funnel and Higgsino LSP. We discuss these Cases in details, and find that our models can be natural and consistent with the LUX and PANDAX experiments, and explain the muon anomalous magnetic moment as well. In particular, all these cases except the stau coannihilation can even have low energy fine-tuning measures around 10.

II. BRIEF REVIEW OF THE PDGSSMS
To obtain the Dirac gluino mass, we introduce a chiral superfield Φ in the adjoint rep-  Tables I and II, respectively. Similar to our previous paper, we still consider the model with ∆b = 4, while the model with ∆b = 3 will be studied elsewhere (For Dirac gaugino case, see Ref. [27].). In the model with ∆b = 4, the SU (2) L × U (1) Y Dirac gaugino masses are forbidden, and the neutrino masses and mixings can be generated via Type II seesaw mechanism [37].
Particles Quantum Numbers Particles Quantum Numbers Besides the MSSM superpotential, the new superpotential terms with universal vectorlike particle mass are where λ and λ are Yukawa couplings, and H u and H d are the Higgs doublet fields which give masses to the up-type quarks and down-type quarks/charged leptons, respectively. The λ and λ terms give the positive and negative contributions to Higgs boson mass via the nondecoupling effects, respectively. The Higgs mass is lifted through the non-decoupling effect which is mediated by the large mass splitting between m T + and M V . The details can be found in Appendix. m T + will contribute to the one-loop beta function of m 2 Hu due to the λ term, and thus the fine-tuning becomes very large. In order to avoid such a dangerous scenario, λ must be forbidden by some global symmetry consideration. We are left with λ and m T + , which realize the DiracTMSSM model. The corresponding Landau pole problem for λ strongly depends on its value at SUSY scale. The advantage of DiracTMSSM demonstrates that it only needs mild value of λ to achieve the observed Higgs mass. Therefore, the Landau pole problem can be removed easily. Then the corresponding SUSY breaking soft terms are

Particles Quantum Numbers Particles Quantum Numbers
where B T,D are bilinear soft terms, T λ is a trilinear soft term, m 2 φ are soft scalar masses, and M D is the Dirac gluino mass.
To realize the hybrid F − and D−term SUSY breakings, we consider the anomalous U (1) X gauge symmetry inspired from string models [19], and assume that all the SM particles and vector-like particles are neutral under U (1) X . Unlike Ref. [38], we introduce two SM singlet fields S and S with U (1) X charges 0 and −1 [19]. Generically, there might exist the other exotic particles Q X i with U (1) X charges q X i from any real string compactification. Therefore, the U (1) X D-term potential is where for example in the heterotic string compactification [39], the Fayet-Iliopoulos term is with M Pl the reduced Planck scale.
In order to have gravity mediation, we consider the following superpotential due to the instanton effect which breaks U (1) X gauge symmetry [19] W Instanton = M I SS .
This is the key difference between our model [19] and that in Ref. [38] where the superpotential is U (1) X gauge invariant and thus one cannot obtain the traditional gravity mediation.
Minimizing the potential, we get where F S and F S are the corresponding F-terms, D is the corresponding D-term. Note that S is neutral under U (1) X , the traditional gravity mediation can be realized via the non-zero F S [19]. The Dirac mass for gluino/Φ and soft scalar masses for Φ and T +/− can be generated respectively via the following operators [29] where for simplicity the coefficients are neglected, W 3,α is the field-strength superfield of SM gauge group SU(3), with α being the spinor index. X and X can both be Φ as well as respectively be T +/− and T −/+ , and M * can be either the reduced Planck scale for gravity mediation or the effective messenger scale. Thus, the Dirac mass for gluino/Φ and soft scalar masses for Φ and T +/− can be about 3-5 TeV from D-term contributions [19]. By the way, the above operators ameliorate several drawbacks of the purely supersoft supersymmetry [29], for example, the µ problem, and the vanishing Higgs quartic term problem, etc. Also, the Majorana masses of the adjoint fermions might result in a lighter Bino from seesaw mechanism. In particular, the new µ-term can give unequal masses to the up and down type Higgs fields, and the Higgsinos can be much heavier than the Higgs boson without finetuning. However, the unequal Higgs and Higgsino masses remove some attractive features of supersoft supersymmetry as well.
To solve the Landau pole problem for gauge couplings below the GUT scale, we require M V ≥ 3 TeV and M D ≥ 3 TeV. As we know, there is no fine-tuning problem for M D as large as 5 TeV due to supersoft supersymmetry [23]. However, large M D will lead to instability in numerical codes such as SPheno. Therefore, we choose M D = 3 TeV which can not only escape the LHC supersymmetry search constraints but also not introduce any finetuning issue and instability. Because the vector-like particles are introduced to retain gauge coupling unification, M V must be around M D as well. The Higgs boson mass is increased via a non-decoupling effect [19] as in the Dirac NMSSM [40,41] where tan β ≡ H u / H d , and Unlike the Dirac NMSSM, such contribution does not vanish at large tan β limit, which is very important to explain the muon anomalous magentic moment [19].
In this paper, we only study the simple low energy phenomenology. Thus, we consider the low energy fine-tuning measure which is defined as follows [42] where Compared to Ref. [42] we have extra C δm 2 Hu from the triplet threshold corrections to m H 2 u .

III. PHENOMENOLOGY STUDY
In this section, we will study the naturalness, dark matter, and muon anomalous magnetic Before we perform the numerical analysis, let us explain our convention. We define the dimensionless parameter l up ≡ −λ ef f 2 . For all the input mass parameters such as , mL ,3 , mẼ ,3 , we choose GeV unit. While for B µ , its unit is GeV 2 . In addition, the particle masses for all the benchmark points in the following tables are in GeV unit as well.
As the preferred range for the LSP neutralino relic density, we consider the 2σ interval combined range from Planck+WP+highL+BAO [53] However, we find that in this case, the low energy fine-tuning measures are generically larger than about 30. The reason is that the parameter spaces, which have the correct dark matter relic density and smaller fine-tuning measures, are excluded by the LUX and PANDAX experiments [9,10]. In Tables III and IV,    Therefore, we consider the multi-component dark matter in the following, and only require Other dark matter components(or candidates) can be sterile neutrino [54][55][56], axion [57,58], etc.
In the following paper, when we mention dark matter-nucleon scattering cross section of our candidate, we mean relative scattering cross section: Then we use relative scattering cross section to compare with PandaX/LUX data.
In our numerical studies, we consider the following six Cases: • Case A with general scan for the phenomenological preferred parameter space. To explain the anomalous magnetic moment of the muon and have the small low energy fine-tuning measures, we consider the input parameters given in Table V, and present the spin-independent elastic dark matter-nucleon scattering cross section, finetuning measure, and muon anomalous magnetic moment versus the LSP neutralino mass in Fig. 1 Tables VI and VII are close to the central value, while those for the   benchmark points in Tables VIII and IX are within 2σ range. Moreover, when the LSP neutralino mass increases, the fine-tuning measure decreases and increases for m χ 0 1 < 150 GeV and m χ 0 1 > 150 GeV, respectively. So the fine-tuning measure has a minimum around m χ 0 1 = 150 GeV. The reason is that for m χ 0 1 < 150 GeV, ∆ EW is given by the fine-tuning measure of m H 2 u , while m χ 0 1 > 150 GeV, ∆ EW is given by the fine-tuning measure of µ. This conclusion is valid for the Cases with Higgsino LSP as well.
• Case B with the LSP neutralino and light stau coannihilation, i.e., mτ 1 ≈ mχ 1 . With the input parameters given in Table X, we present the spin-independent elastic dark matter-nucleon scattering cross section, fine-tuning measure, and muon anomalous magnetic moment versus the LSP neutralino mass in Fig. 2. Only small parameter space is excluded by the LUX and PANDAX experiments, the fine-tuning measure is around 38.5 since we choose µ = 400 GeV, and the muon anomalous magnetic moments for most of the parameter space is within 2σ range. To be concrete, we present a benchmark point in Table XI with ∆a µ close to central value.
• Case C with Higgs funnel, i.e., 2mχ 1 ≈ m A . With the input parameters given in Table XII, we present the spin-independent elastic dark matter-nucleon scattering cross section, fine-tuning measure, and muon anomalous magnetic moment versus the LSP neutralino mass in Fig. 3. Similar to the Case B, only small parameter space is excluded by the LUX and PANDAX experiments, the fine-tuning measure is around 38.5, and the muon anomalous magnetic moments for most of the parameter space is within 2σ range. Also, we present a benchmark point in Table XIII with ∆a µ close to central value.
• Case D with Higgsino LSP. With the input parameters given in Table XIV, we present the spin-independent elastic dark matter-nucleon scattering cross section, fine-tuning measure, and muon anomalous magnetic moment versus the LSP neutralino mass in Fig. 4. Because the LSP neutralino relic density is small, the LUX and PANDAX experimental constraints are satisfied after rescale. The low energy fine-tuning measure is similar to Case (A), and the muon anomalous magnetic moment can be explained.
Moreover, we present a benchmark point in Table XV with fine-tuning measure around 8.87, and ∆a µ around central value.
• Case E is a hybrid scenario with light stau coannihilation and Higgsino LSP. With the input parameters given in Table XVI, we present the spin-independent elastic dark matter-nucleon scattering cross section, fine-tuning measure, and muon anomalous magnetic moment versus the LSP neutralino mass in Fig. 5, which are similar to the Case D. We also present a benchmark point in Table XVII with fine-tuning measure around 9.05, and ∆a µ close to central value.
• Case F is another hybrid scenario with Higgs funnel and Higgsino LSP. With the input parameters given in Table XVIII, we present the spin-independent elastic dark matternucleon scattering cross section, fine-tuning measure, and muon anomalous magnetic moment versus the LSP neutralino mass in Fig. 6. This Case is similar to the Case D except that the LSP neutralino mass is larger than about 180 GeV. We present a benchmark point in Table XIX with fine-tuning measure around 11.7, and ∆a µ close to central value.

IV. CONCLUSIONS
We studied the naturalness, dark matter, and muon anomalous magnetic moment in the PDGSSMs. In order to obtain the correct dark matter density and explain the muon anomalous magnetic moment, we found that the low energy fine-tuning measures are larger than about 30 due to strong constraints from the LUX and PANDAX experiments. Thus, to explore the natural PDGSSMs, we considered multi-component dark matter and then the relic density of the LSP neutralino is smaller than the observed value. We classified the dark matter models into six kinds: (i) Case A is a general case, which has small low energy fine-tuning measure and can explain the anomalous magnetic moment of the muon; (ii) Case B with the LSP neutralino and light stau coannihilation; (iii) Case C with Higgs funnel; (iv) Case D with Higgsino LSP; (v) Case E with light stau coannihilation and Higgsino LSP; (vi) Case F with Higgs funnel and Higgsino LSP. We studied these Cases in details, and showed that our models can be natural and consistent with the LUX and PANDAX experiments, as well as explain the muon anomalous magnetic moment. Especially, all these cases except the stau coannihilation can even have low energy fine-tuning measures around 10.

V. APPENDIX
The non-decoupling effect is calculated in terms of Mathematica, which can be found in our previous paper [19]. The whole process is tedious. So, we just show some key steps: 1. Getting the scalar potential part of Lagrangian.
Here, the scalar potential is expressed as a function of H 0 u , H 0 d , T 0 + and T 0 − . In addition to the conventional terms such as µ, B µ and m 2 φ , their quartic term is uniquely determined by the gauge couplings g 1 and g 2 . The general form of scalar potential can be illustrated as follows where φ i stands for the scalar particle that we are interested in.
2. Integrating out the massive scalar triplet particles.
In supersymmetric models, the heavy degrees of freedom can be integrated out through the equations ∂W/∂Φ = 0 due to the F-flatness conditions. After solving these equations, the heavy superfield Φ can be re-expressed in terms of light superfields. Then substituting the solution into superpotential yields an effective theory with light superfields. In this procedure, supersymmetry is preserved since we integrate out a supermultiplet. However, such an integrating out procedure only affect Higgs mass mildly which means all the heavy superfields are decoupled from the new sector. This strongly motivates us to consider another method where we only integrate out the scalar component of supermultiplet. Thus, we resort to solving the equation ∂V /∂φ = 0 which is called semi-soft supersymmetry breaking. In our model, the solution after taking the limit m 2 T > M 2 V are given by 3. Substituting the solution into scalar potential and obtain a new quartic coupling; After solving the equation, we can substitute the solution in equation (21) into original scalar potential. It is easy to find that we can obtain additional quartic coupling So it is clear to us that even the scalar soft mass m 2 T + is set to be very large, the λ ef f is still non-zero which is called non-decoupling effect. The interesting point is that the large m 2 T + does not appear in the renormalization equation of m 2 Hu , and thus does not have any effect on naturalness.
4. Solving the tadpole equation, obtaining Higgs mass matrix, and getting Higgs mass eigenvalues.