Muon $g-2$ in Gauge Mediation without SUSY CP Problem

We discuss gauge mediated supersymmetry breaking models which explain the observed muon anomalous magnetic moment and the Higgs boson mass simultaneously. The successful explanation requires the messenger sector which violates the relation motivated by the grand unification theory (GUT). The naive violation of the GUT relation, however, ends up with the CP problem. We propose a model in which the phases of the gaugino masses are aligned despite the violation of the GUT relation. We also consider a model which generates the $\mu$-term and the additional Higgs soft masses squared without causing CP violation. As a result, we find a successful model which explains the muon anomalous magnetic moment and the Higgs boson mass. The model is also free from the CP, flavor-changing neutral current and the lepton flavor violation problems caused by the subdominant gravity mediation effects.


Introduction
The Muon g − 2 experiment at Fermilab reported the first results on the measurement of the muon anomalous magnetic moment, a µ = (g − 2) µ /2. The reported value of the combined result of Muon g − 2 experiment at Fermilab and Brookhaven National Laboratory is a exp µ − a SM µ = (25.1 ± 5.9) × 10 −10 , (1.1) from the CP problem. As a result, we find that the extended GMSB can explain a µ and the observed Higgs boson mass. We also discuss the LHC constraints on the SUSY spectra which explain a µ and the Higgs boson mass. The organization of the paper is as follows. In Sec. 2, we discuss the necessity of the violation of the GUT relation in the messenger sector. In Sec. 3, we discuss a model which evades the relative phases of the gaugino masses. In Sec. 4, we discuss the origin of the µ-term. We also discuss the SUSY CP and LFV problems due to the subdominant gravity mediation. The LHC signatures are discussed in Sec. 5. The final section is devoted to our conclusions.

GMSB with GUT relation
Let us first review the SUSY contribution to a µ in the GMSB. In the minimal setup, the messenger chiral multiplets (Φ,Φ) are in the 5 +5 representation of the minimal GUT gauge group, SU (5). The messenger multiplets couple to the SUSY breaking field Z via the superpotential, where k D,L are coupling constants. We decompose the messenger multiplets into Φ = (ΦD, Φ L ) and Φ = (Φ D ,ΦL) in accordance with the MSSM gauge charges. To maintain the successful coupling unification, we require k D ∼ k L ∼ k at the messenger scale. For a while, we treat the SUSY breaking field as a spurious chiral supermultiplet which breaks supersymmetry with the vacuum expectation value (VEV), (2.2) By using the phase rotation of Z and superspace coordinate, θ, i.e., R-symmetry rotation, we take A Z and F Z real positive. We also take k D,L real positive by the phase rotation of the messenger multiplets. Thus, in the minimal setup, there is no source of the CP violation.
In the minimal setup, the gaugino masses at the messenger mass scale, M mess = kA Z , are given by while the soft SUSY breaking masses squared of the MSSM scalar fields (φ) are given by (2.4) [43,48]. Here, we assume that there are N 5 pairs of the messenger multiplets. The upper limit on N 5 is about N 5 + 1 150/ log(M GUT /M mess ), which is imposed from the perturbativity of the t a n β = 5 t a n β = 1 0 t a n β = 2 0 t a n β = 4 0 (a) N 5 = 1 t a n β = 5 t a n β = 1 0 t a n β = 2 0 t a n β = 4 0 are the quadratic Casimir invariants of the representations r φ a , and Q φ Y is the U(1) Y charges of the scalar field φ. We have assumed F Z /kA 2 Z 1. The SUSY breaking trilinear A-terms vanish at the messenger scale. The mediated SUSY breaking masses are independent of the coupling constant k at the leading order. 1 We call Eqs. (2.3) and (2.4) the GUT relation.
In Fig. 1, we show the predicted Higgs boson mass and a µ in the minimal GMSB for N 5 = 1 (left) and N 5 = 5 (right). In the figure, we vary F Z /A Z and M mess ∈ [10 4 , 10 16 ] GeV for a given tan β. In our analysis, we have used the programs SOFTSUSY 4.1.10 [49] to estimate the SUSY mass spectrum, FeynHiggs 2.18.0 [50] for the Higgs mass calculation, and GM2Calc 1.7.5 [51] for the a µ estimation. In our analysis, we adopt the PDG average of the top mass measurement m t = 172.76 ± 0.30 GeV [52]. The Higgs mass is measured as m h = 124.97 ± 0.24 GeV by the ATLAS collaboration [53] and m h = 125.38 ± 0.16 GeV by the CMS collaboration [54]. In addition to the experimental error of the Higgs mass measurement, there are theoretical uncertainties of the Higgs mass estimation originated from missing higher-order corrections and the experimental and the theoretical errors of the top mass (see e.g., Ref. [55]). In this analysis, we assume the theoretical uncertainty of the Higgs mass boson is 1 GeV.
As the figures show, a µ | SUSY is below 10 −10 when the Higgs boson mass is m h 125 GeV. Thus, we find that the minimal GMSB fails to explain a µ and the Higgs boson mass simultaneously. Note that the ratio between the slepton masses and the squark masses deviates from the GUT relation 1 The k dependence of the soft masses appears in higher order terms, O F 2 Z /k 2 A 4 Z , at the messenger scale. The soft masses also has a logarithmic dependence on k through the messenger mass, M mess = kA Z .
for F Z /kA 2 Z → 1. We have checked that the minimal GMSB model cannot explain a µ and the Higgs boson mass simultaneously even in such a parameter region.
Before closing this subsection, let us comment on the GMSB models with a messenger-Higgs mixing. In the presence of the messenger-Higgs mixing, a rather large trilinear A-term can be generated [56][57][58]. With a large A-term, the observed Higgs boson mass can be obtained for relatively light gluino/squarks [58][59][60]. In fact, a µ and the observed Higgs boson mass can be explained simultaneously without violating the GUT relation [59]. The parameter region discussed in Ref. [59] predicts a very light sparticles, which are severely constrained by the LHC searches. In this paper, we focus on the case without a messenger-Higgs mixing, and hence, the trilinear A-terms vanish at the messenger scale as in the minimal setup of GMSB.

GMSB without GUT relation
As we have discussed, GMSB with the GUT relation does not explain the observed a µ and the Higgs boson mass simultaneously. In this section, we discuss the models with violation of the GUT relation. The simplest model which violates the GUT relation is given by, where the two types of the messenger multiplets have the independent mass parameters. We also changed the spurious SUSY breaking field to the one which only has the F -term VEV, In this model, the GUT relation is violated and the resultant GMSB soft masses at the messenger scale are modified to, and (2.10) Here, we have defined, Here, we have assumed |k D,L F Z /A 2 Z D,L | 1. With the violation of the GUT relation, it is possible to explain a µ and the observed Higgs boson mass simultaneously by taking |Λ D GMSB | |Λ L GMSB | (see e.g., Ref. [61]). 2 The GUT violating messenger interactions, however, introduce new sources of CP violation. Unlike the messenger coupling in Eq (2.1), we can not eliminate all of the complex phases of the parameters in Eq. (2.5) by field redefinitions. As a result, there is a relative phase of O(1) between Λ D GMSB and Λ L GMSB , which propagates to the gaugino masses and B through the renormalization group (RG) equations. Once the gaugino masses and B have relative phases of O(1), the resultant soft parameters can induce the non-vanishing electric dipole moments (EDMs). In particular, the electron EDM, d e , is roughly correlated with a µ where e is the QED coupling constant and m e is the electron mass. 3 By comparing this equation with the current upper bound on the electron EDM given by ACME [62], we see that an accidental tuning is required.

CP-Safe GMSB without GUT Relation
In the above discussion, we have found that: • GUT violating messenger coupling is required to explain the a µ and the Higgs mass simultaneously • Naive GUT violation of the messenger coupling ends up with a too large electron EDM.
In this section, we propose a model of the GUT violating messenger sector which avoids the CP violating phases.

Alignment of CP phases
To avoid the unwanted CP phases in the GUT violating messenger coupling, let us introduce two independent SUSY breaking fields, Z D and Z L . As we will see shortly, they obtain the VEVs of We will also see that it is possible to align all the phases of A Z D,L and F Z D,L . The Down-type messengers and the Lepton-type messengers couple to Z D,L via By appropriate phase redefinitions ofΦ D ΦD andΦ L ΦL, we can always take k D,L real-positive valued. Thus, if we can prepare the SUSY breaking fields in Eq. (3.1) with all of A Z D,L and F Z D,L real-positive, we achieve the GUT violating messenger coupling without CP violation. Now, let us discuss how we can prepare the SUSY breaking sector in which the phases of the VEVs of Z D and Z L are aligned. We assume that there are two independent SUSY breaking sectors where each SUSY breaking is caused by the F -component VEVs of Z D and Z L , respectively. We also assume that the mass scales of the two sectors are not very different. Then, the effective theory of the pseudo-flat directions Z D,L is described by the Kähler potential and the superpotential, Here, m 3/2 is the gravitino mass and M Pl is the reduced Planck scale. The mass parameters w 2 D,L and Λ 2 D,L ∈ R encapsulate the ultraviolet (UV) completion of the two SUSY breaking sectors. We neglected the dimension 6 or higher order terms. Each SUSY breaking sector can be the low energy effective theory of, for example, the O'Raifeartaigh-type SUSY breaking model [63] (see also [44][45][46][47] and the Appendix A).
We assume symmetries under the phase rotations of Z D,L , U(1) D,L , which are explicitly broken only by the mass parameters, w 2 D,L , respectively. By giving U(1) D,L charges toΦ D ΦD andΦLΦ L , we forbid the mixings such as Z DΦL Φ L . We also assume the R-symmetry which is broken only by m 3/2 . By U(1) D,L and the R-symmetry rotation, we can take without loss of generality. A Z D,L are determined by the minimum of the scalar potential of Z D,L , where we neglected the dimension 8 or higher order terms. We also neglected the terms of O(m 2 3/2 ). which is the generic feature of the O'Raifeartaigh models [64][65][66]. With this scalar potential, we obtain the VEVs of the SUSY breaking field as [44][45][46][47], and their F components 4 As we have aligned the phases of the parameters as in Eq. (3.6), this setup provides appropriate SUSY breaking fields in Eq. (3.1) with all the CP phases aligned, that is, With the condition for the vanishing cosmological constant, we parametrize as (3.14) Accordingly, the combinations relevant for the GMSB spectrum in Eqs. (2.7)-(2.10) are written as, For successful GMSB, we require Λ D,L GMSB = 10 5-6 GeV, and hence, To explain a µ and the Higgs boson mass simultaneously, we take Λ D GMSB /Λ L GMSB 5-10, which is achieved for, for example, κ D 1 and Finally, let us discuss the fermion components of Z D,L . As Z D,L break global SUSY independently, the fermion components are massless goldstini,G D,L . One linear combination of them becomes the gravitino with a mass, m 3/2 , and the other obtains a mass, 2m 3/2 , through the super-Higgs mechanism [67].

Vacuum Stability
Since the SUSY breaking fields couple to the messenger fields in superpotential Eq. (3.3), there is a supersymmetric vacuum at with vanishing Z D,L . Hence, the SUSY breaking vacuum in Eq. (3.9) is at best meta-stable [45]. The squared masses of the messenger scalars around the meta-stable vacuum is given by, Hence, the meta-stability condition, det M 2 D,L > 0, leads to The couplings to the messenger fields also induce the Coleman-Weinberg potentials to the pseudo flat directions, where d D = 3 and d L = 2. These terms contribute to the mass matrix of As a result, another meta-stability condition, det M 2 Z D,L > 0, leads to In summary, the meta-stable vacuum conditions restrict the range of the messenger scale, Hence, the messenger scale lies in the range, (3.24) Finally, let us discuss the tunneling rate of the meta-stable vacuum into the supersymmetric vacuum in Eq. (3.17). The VEVs in Eq. (3.9) are much larger than the VEVs of the messenger fields in Eq. (3.17). The displacement between the meta-stable vacuum and the unwanted color-breaking supersymmetric vacuum is of order of ∆A D,L ∼ Λ 2 D,L /M Pl . The tunneling rate per unit volume, Γ/V ∝ e −S E , is estimated in Ref. [68] where Therefore, the meta-stable vacuum is stable enough, for example, for Λ D,L Λ D,L GMSB so that S E 500. 5

GUT violating messenger multiplets
In Eq. (3.3), we assume the Down-type and the Lepton-type messenger multiplets which couple to Z D and Z L , respectively. The simplest realization of such GUT violating messenger multiplets is to consider the product group unification models [69][70][71][72][73]. The product group unification is motivated to solve the infamous doublet-triplet splitting problem of the Higgs multiplets in the conventional GUT.
As a concrete example, let us consider the product group GUT model based on the GUT gauge group, SU(5)×U(2) H [69,74]. 6 In this model, SU(5)×U(2) H is spontaneously broken down to the MSSM gauge groups by the VEV of the chiral multiplets of the vector-like bi-fundamental representation, (Q,Q). Their VEVs are, where v denotes the VEV of the order of the GUT scale. In this model, the nominal coupling unification of the MSSM gauge group at the GUT scale is explained in the strong coupling limit of U(2) H gauge interaction. Now, let us introduce a messenger multiplet (Φ 5 ,Φ 5 ) in the (anti-)fundamental representation of SU(5), a vector-like multiplet (Φ 2 ,Φ 2 ) in the (anti-)fundamental representation of SU(2) H . The messenger fields couple to the SUSY breaking field Z D through, where λ andλ are coupling constants. Here, we can take all the parameters real-positive by field redefinitions. Due to the second and third terms, the doublet components in (Φ 5 ,Φ 5 ) obtain masses of the GUT scale, and decouple. Thus, W D provides the Down-type messenger in Eq. (3.3). The Lepton-type messengers can be also obtained by introducing the SU(2) H doublet (Φ L ,ΦL) with U(1) H changes, ∓1/2. By assuming that (Φ L ,ΦL) couple to Z L ,

Sweet Spot Supersymmetry
In the previous section, we show how to achieve the gaugino masses whose CP phases are aligned while the GUT relation is violated. To discuss the SUSY CP problem, however, we also need to specify the origin of the µ-term as well as the B-term. Also note that we are interested in the model with light sleptons and heavy squarks to explain a µ and the Higgs boson mass simultaneously. In this case, the large squark masses induce the large Higgs soft masses squared, m 2 Hu and m 2 H d , at the TeV scale through the RG running. With large m 2 H u,d , the required size of the µ-term is also large to achieve the correct electroweak symmetry breaking vacuum. With a large |µ| term, the stau tends to be light and causes the stability problem [77].
To avoid a too large µ-term, we introduce additional contributions to m 2 H u,d in addition to the GMSB contributions [44][45][46][47] (see also Refs. [61,78,79]). The additional contributions offset the RG contributions. In summary, for successful explanation of a µ and the Higgs boson mass, we consider models with: • The mechanism which generates µ-term without causing new CP phase • The additional source of m 2 H u,d other than GMSB to achieve a small µ-term.

Higgs mass parameters
As developed in Refs. [44][45][46][47], we can generate the µ-term and the additional Higgs soft masses squared simultaneously by coupling the Higgs doublets with a SUSY breaking sector. Here, we assume that the Higgs doublets couple to the SUSY breaking sector of Z D . Then, the direct coupling induces the effective Kähler potential, The cutoff parameters Λ 2 u,d are real valued by definition, while we can take Λ µ real-positive by the Peccei-Quinn (PQ) rotation of H u H d . We discuss the details of the origin of the Kähler potential in the Appendix. A, where we find Λ 2 u,d > 0. The resultant µ-term and the additional Higgs soft masses squared are given by, 7 where µ is real-positive. For successful explanation of a µ , we require that µ is within a TeV range, and hence, Similarly, the requirement that δm 2 H u,d is in a few TeV range leads to With the closeness of Λ L,D,µ,u,d for m 3/2 = O(1) GeV, we call this scenario the (extended) Sweet Spot Supersymmetry which is originally proposed in Ref. [47]. In Eq. (4.1), we assumed that the Higgs doublets do not couple to Z L . Such a model is possible by combining the Z D phase rotation symmetry in Eq. (3.4) with the PQ phase rotation (see also the Appendix A). The same symmetry also forbids the terms such as Z As a result, the B-term from the Kähler potential in Eq.
Note that these contributions do not bring CP violating phases. Similarly, the induced A-terms are also suppressed and do not have CP violating phases. Since these A and B-terms are harmless and overwhelmed by the RG contributions at the electroweak scale, we neglect them in the following analysis.
In the Appendix A, we discuss a perturbative UV completion of the effective Kähler potentials in Eqs.
Here, the coupling constants, λ, h andh defined in the Appendix A are taken real-positive without loss of generality. The coefficient functionsf andg are given in Eqs. (A.4) and (A.6) (see also Fig. 7). Figure 2: The parameter region satisfying Eq. (4.8) (cyan) and Eq. (4.9) (pink) for given range of M 2 and µ. We take h =h = 1 to avoid too large hierarchy between the µ-parameter and δm H u,d . We also fix κ D = 1 and κ L = 0.1, which is motivated to explain a µ and the Higgs boson mass simultaneously. In the gray shaded region, the VEV of A Z D becomes too large (see Eq. (A.9)).
Note that the ratio,g 1/2 /|f |, is of O(1) for a wide range of parameters. The mass parameter M * denotes the scale at which the higher dimensional operators in Eqs.

Renormalization Group Analysis
In the present model, the additional Higgs soft-mass squared and the µ-term are generated at the scale M * , which is independent from M mess . From Eqs. Since we are mostly interested in the case m 3/2 = O(100) MeV to O(1) GeV, we take 10 11 GeV M * 10 14 GeV in the following analysis. The gaugino masses and the sfermion masses are, on the other hand, generated at the messenger scale, M mess , which is assumed to be common for the Down-type and the Lepton-type messengers for simplicity. The two step mediation at M * and M mess predicts a peculiar spectrum [47].
In Fig. 3a, we show the RG running of the soft parameters. As the figure shows, δm 2 H u,d generated at M * offset the negative RG contributions to m 2 H u,d at the TeV scale. This feature makes a small µ-parameter compatible with successful electroweak symmetry breaking. It also shows that δm 2 H u,d give negative contributions to the soft sfermion masses of the third generation. We also show an example of the spectrum which explains the a µ and the observed Higgs boson mass simultaneously. 10 As we have seen, Bµ is dominated by RG contributions from the gaugino mass through the RG evolution, and hence, Bµ is not a free parameter. Besides, there is no large contributions to Bµ   In both cases, the predicted tan β is typically ∼ 30 − 40.
from subdominant gravity mediation effects. Thus, we take Bµ to be vanishing above the messenger scale. Accordingly, tan β is not a free parameter but is a prediction. In Fig. 4, we present the result of the parameter scan of the present model. We adopt log-flat priors for M mess , M * , and linear-flat priors for Λ L,D GMSB , ∆m 2 Hu , ∆m 2 H d , imposing M * > 10 11 GeV and M mess > 10 6 GeV. We show the MSSM parameters which is consistent with a µ , the Higgs mass and the vacuum stability bound on the stau direction [77]. The model predicts light Higgsinos and the Winos. The impact on the electroweak precision measurements are found to be minor [80].

Effects of gravity mediated SUSY breaking
As the Sweet Spot Supersymmetry assumes a rather large gravitino mass of O(1) GeV, the gravity mediated SUSY breaking effects could cause FCNC/LFV and CP problems even if they are subdominant. First, let us consider the gravity mediated contributions to the B-term [81]. In supergravity, the µ-term from the Kähler potential in Eq. (4.1) is shifted to with µ 0 = F D /Λ µ in Eq. (4.2). The associated B-term induced by the gravity mediation effects is where we have neglected the small contribution to Bµ in Eq. (4.6). Note that B = m 3/2 due to the coupling between the holomorphic term, A Z D H u H d /Λ µ , in the Kähler potential and W = w 2 D Z D + w 2 L Z L (see e.g., Ref. [82]). 11 Those effects do not induce CP violating phases, since we have set m 3/2 > 0.
A CP violating phase appears from the additional origin of the µ-term, with c µ being a complex valued coupling constant of O(1). This term is consistent with the PQ symmetry which is identified with the U(1) D symmetry. This term shifts the µ-term and B-term by, where ∆Bµ/∆µ = m 3/2 . Thus, we find that the phases of the total µ-term and the Bµ-term are no more aligned due to c µ = 0, which induces a CP violating phase. The CP violating phase is estimated as follows. In the phase convention in the previous section, the phase of the total µ-term induced by c µ is Thus, in the phase convention where the total µ-term, µ tot > 0, the total Bµ-parameter at the scale M * becomes where the first term is real-positive. At the TeV scale, the Bµ-term is dominated by the gaugino mass contributions through the RG running, which is real valued in the present model. As a result, the Bµ-term at the TeV scale is given by, where m A is the mass of the CP-odd Higgs. As a result, the effective CP-violating phase appearing in the Bµ-term is of Thus, we find that the CP violating phase due to the gravity mediated effects on the Bµ-term is suppressed by O(10 −9 ) × (m 3/2 /GeV) 3 . Thus, the expected electron EDM from Eq. (2.12) is much lower than the current limit for m 3/2 10 GeV. 11 We assume that the effective Kähler potentials appear in the Einstein frame. Even if they appear in the conformal frame, Bµ/µ = m 3/2 , and hence, we obtain the similar effective CP phase in Eq. (4.21).
Next, let us consider the effects of the subdominant gravity mediated soft masses squared of the sfermions, where φ's are MSSM matter chiral fields and c ij the O(1) coefficients. In general, they are not flavor diagonal and have CP violating phases. The CP violating sleptons squared mass matrix contributes to the electron EDM, which roughly correlates with the Bino contributions to a µ | SUSY (see e.g., Ref [38]) as, Here, m τ is the tau lepton mass, δ LL 13 ∼ m 2 3/2 /m 2 and δ LL 13 ∼ m 2 3/2 /m 2 e R with m 2 and m 2 e R being the left-handed and the right-handed slepton squared masses. The left-right mixing parameters are suppressed since neither GMSB nor gravity mediation generates large A-terms. As a result, the electron EDM is roughly given by, where we used a µ | Bino a µ | SUSY . Thus, the gravity mediated contribution to the electron EDM through the slepton mass is consistent with the current upper limit on the electron EDM for m 3/2 = O(1) GeV.
Finally, let us comment on the subdominant gravity mediation contribution to the gaugino masses and the trilinear A-term. If the SUSY breaking fields are completely singlets under any symmetries, we expect the gravity mediated effects on those parameters of O m 3/2 . In our setup, however, the SUSY breaking fields are charged under U(1) D,L and R-symmetry. Accordingly, the gravity mediated effects are of O m 3 3/2 /M 2 Pl , which are negligible. Besides, the anomaly mediated contributions [93,94] are aligned with GMSB, since m 3/2 in the superpotential is taken to be real positive. Therefore, there are no SUSY CP problems from the gravity/anomaly mediated contributions to the gaugino masses and the A-terms.

LHC Signatures
Here we discuss the LHC constraints on the present model. To achieve large a µ | SUSY , the masses of the relevant SUSY particles are rather small, which suffer from the LHC constraints [95][96][97][98][99]. In the present GMSB models, the lightest SUSY particle (LSP) is the gravitino and, all the MSSM particles can decay into the goldstini. For example, partial decay rate of the Wino into the goldstino/gravitino is given by (5.1) Unless the gravitino is much lighter than O(1) MeV, the MSSM particles cannot decay inside the LHC detector. 14

Higgsino and Wino system
In the present model, so-called the GUT relation among the gaugino masses are violated. In the typical parameter region of interest, the next-to-lightest SUSY particle (NLSP) is the Wino or Higgsino. 15 The other particles are heavier than these particles and play less important roles at 13 C R denotes the conversion rate in a nucleus divided by the muon capture rate by a nucleus. 14 The relic abundance of the NLSP is severely constrained by the Big-Bang Nucleosynthesis (BBN) when its lifetime is longer than O 10 2 sec [100]. In the present model, the NLSP abundance depends on the cosmological evolution of the pseudo-flat directions [46], which will be discussed in future work. 15 Strictly speaking, the NLSP is the massive goldstino in our model. In the following, however, we call the lightest SUSY particle in the MSSM sector the NLSP as in the conventional context of the GMSB phenomenology. the LHC, compared to the Wino and Higgsinos. Therefore, a simplified setup of the Higgsino-Wino system is useful to see the collider constraints on the present model.
The collider signature significantly depends on the nature of the NLSP.

Wino NLSP
After the electroweak symmetry breaking, the Wino particles are decomposed into a neutralinoχ 0 1 and a charginoχ ± 1 . The tree-level mass difference between the charged Wino and neutral Wino is approximately given by . For a sizable a µ | SUSY , tan β should be large and accordingly, sin(2β) 2/ tan β is suppressed. Therefore the tree-level mass splitting is severely suppressed. In addition to the tree-level mass splitting, the electroweak loop correction provides the mass difference around 165 MeV [102]. In the present model, if the Higgsino mass is greater than 300 GeV, the Wino mass splitting is less than 1 GeV and the charged Wino can be long-lived and the decay length can be O(1) cm. This charged tracks are detected as a disappearing charged track at colliders. This signature is intensively studied in the anomaly mediation model [103][104][105][106][107][108][109][110][111]. The latest ATLAS search of the disappearing charged tracks with 139 fb −1 data excludes the Wino lighter than 660 GeV for a large Higgsino mass [112].

Higgsino NLSP
By the electroweak symmetry breaking, the Higgsinos are decomposed into two neutralinosχ 0 1,2 and a charginoχ ± 1 . As in the case of the Wino, there are mass splittings among these particles. The mass difference between the charged Higgsino and lightest neutral Higgsino is approximately given as [109,113,114], The mass difference between the two neutralinos is approximately twice of this mass splitting. If the Wino mass is less than 1 TeV, the chargino-neutralino mass splitting is greater than 5 GeV. Therefore we do not expect the disappearing charged tracks like the Wino NLSP case. There is, however, another important signature comes from the decay of theχ 0 2 →χ 0 1 + − . This soft lepton signature can give the Higgsino mass constraint up to 200 GeV [115].

Next-to-NLSP decay
In addition to the constraint on the direct production of the NLSP, the decay of the next-to-NLSP into the NLSP also provides the clue for the collider searches. If the Higgsino mass is much greater than the Wino mass, the Higgsinos decay into the Wino with Z, W ± and h 0 bosons with almost equal branching fractions. Such bosons decay into leptons, photons and b-jets, which are characteristic signatures.
In the present analysis, we study the following analysis.

Disappearing charged track
We directly apply the ATLAS result of the Wino LSP searches with 139 fb −1 data [112].

Soft di-lepton
We directly adopt the Higgsino LSP search with 139 fb −1 data at the ATLAS [115].
Tri-lepton mode from W and Z(h 0 ) decays We study the search of leptonic decays of W ( * ) and Z ( * ) with 139 fb −1 data at the ATLAS [116]. We use model-independent inclusive event selections (12 on-shell channels and 17 offshell channels).
One lepton + two b-jets from W and h 0 decays We study the search of a leptonic W decay and b-jets from the 125 GeV Higgs boson with 139 fb −1 data at the ATLAS [117]. 16 We adopt the signal regions, SR-LM, SR-MM and SR-HM. In our estimation, we found this search cannot give a constraint on the present Higgsino-Wino system.

Di-lepton from two W (˜ ) bosons decay
We study the search of two leptonic W decays or two slepton decays into a lepton and a dark matter with 139 fb −1 data at the ATLAS [119]. We adopt eight signal regions with di-lepton. In our estimation, this search cannot give a constraint on the present Higgsino-Wino system.

Mono-jet
The ATLAS search of mono-jet events [120] aims the direct production of the dark matter with high energy initial state radiations. The constraint of this search on the Wino is comparable to the LEP chargino searches [121], if we adopt the leading order cross section.
In our analysis, we have used the programs MadGraph5 MC@NLO [122,123], PYTHIA8 [124] and DELPHES 3 [125] (with FastJet [126] incorporated). We adopt the cross sections provided by the LHC SUSY Cross Section Working Group [127], which are based on works [128][129][130][131][132][133][134][135]. If constraints on the simplified model provided by the LHC is directly applicable to the present model, we recast the constraints. In addition to the direct production of the SUSY particles, Wino and Higgsinos affect the SM processes. Precision measurement of the Drell-Yan process can give indirect signature of the SUSY particles at the LHC [136][137][138][139]. Although such searches can also provide the constraint on the low mass Wino, we need to know the detailed information on the systematic uncertainty of the measurement and the SM signature estimation at the LHC. At present, we cannot get a reliable constraint from the precision measurement at the LHC.
In Fig. 5, we show the current LHC and LEP chargino constraints on the plane of the Wino and Higgsino mass parameters M 2 and µ with tan β = 40. We show the excluded regions by the disappearing charged track in green, soft di-lepton in blue and tri-lepton in red.
In the present model, the dominant contributions to a µ | SUSY come from the left-handed slepton-Wino-Higgsino loops. Thus, for given M 2 , µ and tan β, we can predict the left-handed slepton mass to explain a µ . In Fig. 5, we also show the rough upper limit on the left-handed slepton mass to explain the observed a µ by the Higgsino-Wino contribution at the one-loop level. The figure shows that the LHC constraint, m˜ L > 660 GeV, favors the Higgsino-Wino mass within 100 GeV-600 GeV.

Scalar lepton constraint
To explain a µ , the scalar leptons should also be light. In the present model, the left-handed slepton is typically lighter than the right-handed slepton since Λ L GMSB Λ D GMSB . Therefore the constraint on the left-handed sleptons is relevant for the present model, where the left-handed sleptons dominantly decay into the Wino-like chargino and neutralino.
If the Wino is the NLSP, we can directly apply the constraint on simplified model of˜ L → χ 0 1 ( = e, µ) provided by the ATLAS [119]. In this model, the ATLAS searches for the di-leptons from the process pp →˜ +˜ − → + −χ0 1χ 0 1 are relevant. If the mass ofχ 0 1 is less than around 300 GeV, the current upper bound on the cross section of the slepton pair production in the simplified model is around 0.3 fb.

Heavy Higgs constraint
In the present model, the B term is zero at the UV scale and radiatively generated through the RG effects (Fig. 3a). Therefore the heavier Higgs mass tends to be light (Fig. 4a). Moreover to enhance the muon g − 2, the value of the tan β is large. In this case, the production of the heavier CP-odd Higgs A 0 is significantly enhanced and can be constrained by the LHC experiments. The CMS and ATLAS provide constraints on the process pp → A 0 → τ + τ − [140,141] and these constraints have a significant impact on the present model. In this model, there are SUSY particles lighter than the CP-odd Higgs mass, and hence, A 0 can also decay into such SUSY particles. For the large tan β, however, the branching fractions into the SUSY particles are small and we directly apply the constraint on the m A 0 − tan β provided by the ATLAS [141]. In Fig. 4b, we show the posterior distribution of the SUSY parameters after imposing the collider constraints. In our estimation so far, the light Wino and Higgsino of masses 100-300 GeV are consistent with the LHC constraints. Compared to the constraints on simplified models studied by the CMS and ATLAS, our constraints look rather conservative. One reason of such weak constraints will be the mass degeneracy of the Higgsino and Wino with which the SUSY events at the LHC are less energetic. Moreover, in the present model, the Higgsino and Wino have various decay channels. Therefore, the current LHC searches optimized for simplified models are not so effective. For the typical masses of the Higgsinos and Wino in the present model, the production cross section is rather large. Thus, if we can optimize the LHC searches for the present model, we have a large chance of the discovery of the extended Sweet Spot Supersymmetry for muon g − 2. Study of such prospects are beyond the focus of this paper. We will discuss this point in the future work.
Let us comment on the case ofτ NLSP case. In this case, the stau is long-lived, and the LHC signatures are massive charged tracks. The direct constraint on the stau mass is 430 GeV [142]. In this stau NLSP case, a portion of the stau is stopped in the LHC detectors. By measuring the latetime decay of the stopped stau into goldstini [143,144], we can obtain the fundamental information on the SUSY breaking sectors [145].
In the present model, the Wino and Higgsino are light ∼ 300 GeV. Although the direct production of such particles will be out of reach of the ILC250, such light particles have significant impact on the SM processes through quantum corrections. Therefore, the precision measurement of the di-fermion process e + e − → ff can probe the most of the parameter space consistent with the muon g − 2 [146]. The slepton mass is, on the other hand, predicted to be relatively high and it will be difficult to probe the slepton directly at the ILC500 [147].

Conclusions
In this paper, we discussed the GMSB models which explain a µ and the Higgs boson mass simultaneously. There have been two known major types of gauge-mediated models that explain the observed mass of the Higgs boson and the a µ of the muon. The first type is a model that produces a large A-term by mixing the Higgs fields with messenger fields. This class of the models predict the existence of relatively light squarks and gluinos. Therefore, in those models, it is expected that particles with color charges are produced at the LHC, which puts severe collider constraints. In the second type of model, the mass of the Higgs boson is realized by heavy squarks, which evades severe LHC constraints. As we have discussed, however, the GUT relation of the GMSB should be violated since the explanation of a µ requires light sleptons. The naive violation of the GUT relation ends up with the SUSY CP problem. To avoid such problems, we proposed a model in which the phases of the gaugino masses are aligned despite the violation of the GUT relation.
The successful explanation of a µ and the Higgs boson mass requires a rather light µ-parameter and heavy squarks. To achieve such a SUSY spectrum, we need additional sources of the Higgs soft masses squared other than the GMSB contributions. The model also requires the origin of the µ-term which is free from the CP violation. For these purposes, we utilized the (extended) Sweet Spot Supersymmetry [47]. As we have shown, the model can explain a µ and the Higgs boson mass in the GMSB model without causing the SUSY CP problem. The model evades the LHC constraints so far. We also found that the SUSY CP, FCNC, LFV processes caused by the subdominant gravity mediation are also suppressed.
Several comments are in order. In our set up, we only considered the Down-type and the Lepton-type messengers. By utilizing the product GUT models, it is also possible to have GUT violating messenger fields of more various representations, such as the adjoint representation (see e.g., Refs. [78,79]). The extended Sweet Spot Supersymmetry allows those complicated messenger sector without causing unwanted CP violation.
Since we assume Λ D,L 10 15-16 GeV, the masses of the pseudo-flat directions in Eq. (3.8) are in the hundreds GeV to a few TeV region for m 3/2 = O(1) GeV. The cosmological evolution of a light pseudo-flat direction has been discussed in Ref. [46]. Note that there are two pseudo-flat directions in the present model. Besides, there is a goldstino with a mass 2m 3/2 in addition to the gravitino. Since the massive goldstino has a cosmological lifetime [67], cosmic ray signatures of the very late time decay of the goldstino could give us a smoking gun of the present model. We will discuss details of cosmology of the model including the dynamics of the pseudo-flat directions, the constraints for the gravitino/goldstino dark matter in future work.

A Higgs Coupling to SUSY Breaking
In Sec. 4, we consider the effective Kähler potential which generates the µ-term as well as the additional soft masses squared of the Higgs doublets. In this appendix, we discuss an example of the UV completion [44][45][46][47]. The simplest example is based on the O'Raifeartaigh model with the superpotential, W = w 2 Z + 1 2 λZX 2 + M XY XY + hH uq X +hH d qX + M q qq + m 3/2 M 2 Pl . (A.1) Here, Z, X, Y are gauge singlet fields, H u,d are the Higgs doublets, and (q,q) are the vector-like SU(2) L doublet fields. All the phases of the coupling constants, λ, h, andh, as well as those of the mass parameters M XY and M q can be rotated away without loss of generality. We assume the PQ symmetry with the charges, P Q(H u,d ) = 1, P Q(X) = −1, P Q(Y ) = 1 and P Q(Z) = 2, which is explicitly broken by w 2 . In the Sweet Spot Supersymmetry in Sec. 4, we identified Z in Eq. (A.1) with Z D .
The coefficient of the |Z| 4 term in the effective Kähler potential is given by, Here, E is the Euclidean loop momentum. The coefficient of Z † H u H d term is given by, The coefficient of Z † ZH u H u term is given by, The coefficient of Z † ZH † d H d is given by replacing h withh. In Fig. 7, we show the functionf and g. As is clear from the integrands of Eqs.
where the ratio |f |/g 1/2 is of O(1) in a wide range of M 2 XY /M q . Thus, to provide the appropriate Λ µ and Λ u,d in Eqs. (4.4) and (4.5), we find that h andh are of O(1). With this choice, the generated µ-parameter is parametrically smaller than the additional Higgs soft term by an order of magnitude.
Note also that Z obtains non-vanishing A-term VEV due to the supergravity effect,