CP-safe Gravity Mediation and Muon g-2

We propose a CP-safe gravity mediation model, where the phases of the Higgs B parameter, scalar trilinear couplings and gaugino mass parameters are all aligned. Since all dangerous CP violating phases are suppressed, we are now safe to consider low-energy SUSY scenarios. As an application, we consider a gravity mediation model explaining the observed muon $g-2$ anomaly. The CP-safe property originates in two simple assumptions: SUSY breaking in the K\"ahler potential and a shift symmetry of a SUSY breaking field $Z$. As a result of the shift symmetry, the imaginary part of $Z$ behaves as a QCD axion, leading to an intriguing possibility: the strong CP problem in QCD and the SUSY CP problem are solved simultaneously.


Introduction
Large CP violation is a generic problem in the supersymmetric (SUSY) standard model (SSM). In particular, this problem becomes extremely serious when sleptons, binos and winos are as light as O(100) GeV. In such cases, SUSY contributions to the electric dipole moment (EDM) of the electron usually exceed the experimental bound [1] by many orders of magnitude, due to CP violation in the SUSY breaking sector [2]. On the other hand, the observed muon g − 2 anomaly [3] suggests such light sleptons, bino and wino; the experimental value of the muon g − 2 can be explained by the contributions from the bino, wino and sleptons of masses O(100) GeV [4]. Moreover, models with such light SUSY particles are fascinating possibilities for the ILC. Therefore, it is important to construct a mechanism to suppress CP violating phases of the SSM. The SSM has many sources of the CP violating phases: soft SUSY breaking masses of sfermions, gaugino mass parameters, scalar trilinear couplings and the Higgs B-term. The first source, linked to the flavor structure, can be eliminated by assuming that the soft SUSY breaking masses are universal or zero at the high-energy scale. Thus, we restrict our discussion to the other sources.
In this paper, we propose a CP-safe framework of the gravity mediation in the SSM, where all of the relevant phases are aligned. Consequently, the dangerous CP violating phases are suppressed enough, and hence low-energy SUSY scenarios become attractive.
Encouraged by this theoretical proposal, we construct a model consistent with the observed anomaly of the muon g − 2. We also show how to test this model at the LHC and linear colliders. The Lagrangian of the relevant part is given by

CP problem in the SSM
where λ 1 , λ 2 and λ 3 denote bino, wino and gluino, respectively. CP violation arises from the gaugino mass parameters M i , the trilinear couplings A e,d,u , the Higgsino mass parameter µ and the Higgs B-term B µ . Physical CP violating phases are given by combinations of the phases of B µ , µ, M i , A e,d,u as 2 θ i = arg(M i (µB * µ )), θ A e,d,u = arg(A e,d,u (µB * µ )), where we neglect small CP violating phases of the Higgs vacuum expectation values (VEVs).
In the following, we focus on the electron EDM, which gives the most stringent constraint for scenarios with light sleptons. SUSY contributions to the electron EDM at the one-loop level are roughly proportional to θ 1 , θ 2 and θ Ae = θ (Ae) 11 . From the current experimental bound on the electron EDM [1], we find that the phases are bounded from above as, assuming no cancellation among the contributions, Here, we take the common mass scale m SUSY for |µ|, |A e |, |M 1 |, |M 2 | and the masses of the sleptons for simplicity. Since the constraints on θ 1 and θ 2 are especially stringent, the 1 Masses of non-colored SUSY particles are still allowed to be O(100) GeV. 2 This fact can be understood by taking the convention in which µ and B µ are real, where the parameters are rotated as phases of B µ µ * , M 1 and M 2 must be very accurately aligned at the electroweak scale, or very small (see Eq. (2)).
It should be noted that the phase of B µ µ * depends on renormalization scale and is changed by those of the trilinear couplings A u,d,e and the gluino mass M 3 . The relevant renormalization group equations for B µ are given by [5] ( where we neglect small Yukawa couplings. Here, g i are the gauge couplings constants for SU(3), SU(2) and U (1) gauge groups. As seen from Eq. (5) However, it is challenging in general to align all of the phases at the high-energy scale.
To see this, we consider the following simple Kähler potential and the superpotential, so-called the Polonyi model [6]: where Z is the SUSY breaking field and C is a complex constant. Gaugino masses are generated from couplings between Z and field strength superfields, where M P is the reduced Planck mass. To focus on CP violation in the SUSY breaking sector, we assume that the phases of k i are universal as k i = |k i |e iθ in this section. For instance, the SU(5) GUT model satisfies this condition. Then, by a rotation of Z, the coefficients k i can be taken to be real. 3 In this basis, the scalar potential is given by 4 and the gaugino mass term of Eq. (7) is now written as the gaugino masses are proportional to F Z : The SUSY breaking occurs by the nonzero F-term of Z: Here, z is the VEV of Z, which is found to be for the stable SUSY breaking vacuum with vanishing cosmological constant V = 0. As a result, the phase of F Z turns out to be arg( F Z ) = − arg(µ 2 Z ). The constant term is found to be The Higgs B-term and the A-terms are generated from where W vis is the superpotential of the visible sector, W vis = µH u H d + W Yukawa . Using Eqs. (11)-(13), we have 3 By using R-rotation, the constant C can be taken as real. However, to clarify the point, we leave C as a complex parameter. 4 The replacement of the coupling, 1/(4g 2 i ) + Re(k i z) → 1/(4g 2 i ), has been done. The gauge kinetic functions are taken to be canonical.
In this basis, B µ µ * and A ijk have the same phases as C * , while the phases of the gaugino masses are given by arg(M i ) = arg(F Z ) = − arg(µ 2 Z ). 5 As arg(C) = arg(µ 2 Z ) in general, the phases of B µ µ * (A ijk ) and M i are different at the high energy. Thus the resultant CP violating phases become too large at the electroweak scale, and the EDM exceeds the constraints of Eq. (4) unless the sleptons are heavy as O(10) TeV.

CP-safe gravity mediation
The Polonyi model has a large CP violation unless arg(µ 2 Z ) = arg(C). As one can see from Eqs. (10) and (14), the phases of the B-terms and A-terms are determined by those of F Z (∂K)/(∂Z) and W * , while those of the gaugino masses are determined by the phase of F Z . When the phases of µ 2 Z and C are aligned, Z becomes real (see Eq. (12)). Then, (∂K)/(∂Z) is real and arg( F Z ) = arg( W * ) is satisfied (see Eq. (11)). As a result, the phases of the B-term, A-terms and gaugino masses are all aligned. However, there is no reason why the phases of µ 2 Z and C are aligned. We now propose a generic gravity mediation model where the CP-safe conditions, (a) (∂K)/(∂Z) ∈ R and (b) arg( F Z ) = arg( W * ), are automatically satisfied. We adopt the gravitational SUSY breaking scenario [7] (see also Appendix B for details), where the SUSY is broken in the Kähler potential rather than in the superpotential. Furthermore, we assume a shift symmetry of Z, which guarantees (∂K)/(∂Z) to be real as shown below.
Consequently, the conditions (a) and (b) are automatically satisfied (see Eq. (11)). 6 To demonstrate the above CP-safe gravity mediation, we consider the following La- 5 Of course the superpotential can be more generic; arg(B µ µ * ) = arg(A ijk ) is not satisfied in the general case (see Appendix A). 6 In Ref. [8], the CP violation in the SUSY breaking is discussed with a Lagrangian motivated by the string theory. In their setup, parameters in a superpotential of SUSY breaking moduli can be taken to be real by approximate shift symmetries of moduli, resulting in no CP violation in the gaugino mass parameters and trilinear couplings. However, it is not clear if the CP violating phase of the Higgs B parameter is sufficiently aligned with the other phases in their approaches. 7 As shown in Appendix D, the sequestered form of the Kähler potential is also consistent with a CP-safe gravity mediation.
where s(x) is a real function of x, and x = (Z +Z * ). The Lagrangian has a shift symmetry where R is a real constant. Notice that the R-charge of (H u H d ) is 2, which forbids c. This is very important, since otherwise we may have a CP violating phase in the Higgs B term.
The scalar potential is then given by The SUSY breaking F-term of the hidden sector field Z is proportional to the constant Note that the shift symmetry of Z guarantees (∂s)/(∂x) ∈ R, and the phases of F Z is the same as that of W * = C * . Now, it is clear that the CP-safe conditions (a) and (b) are satisfied.
The condition for the vanishing cosmological constant, V = 0, is satisfied with appropriate choice of the Kähler potential rather than the tuning of the constant C; where the VEV x is determined by a stationary condition, A concrete example model which satisfies Eq. (20) and Eq. (21) with a stable minimum is shown in Appendix C. The constant C can be written using the gravitino mass m 3/2 as The Higgs B-term and A-term arises from the second term of Eq. (18) and they are proportional to the constant C * as 8 Notice that the phases of B µ µ * and A ijk are aligned with that of F Z . By using Eq. (19), they are rewritten as Scalar masses are the same as the gravitino mass: Finally, let us consider the gaugino masses, which arise from the couplings between Z and field strength superfields W i : However, the required ZW i α W α i terms violate the shift symmetry, and hence the terms must vanish, k i = 0. However, we consider the ZW i α W α i terms are generated by gauge anomalies of the shift symmetry, resulting in k i ∈ R. The constants k i depend on unknown high-energy physics, and hence, we take them as free parameters in this paper. Now, we see that all relevant phases are aligned, arg(B µ µ * ) = arg(M i ) = arg(A ijk ) and the SUSY contributions to the EDM are successfully suppressed. We call this a CPsafe gravity mediation. Note that this feature is not affected by the anomaly mediation effect [9]. This is because the contributions from the anomaly mediation to A ijk , B µ /µ and gaugino masses are also aligned with C * (= arg(F Z )) [10], and any additional CP violating phases are not introduced by the anomaly mediation.

An application: SUSY solving muon g − 2 anomaly
We have seen that the model we proposed, the CP-safe gravity mediation, provides SUSY breaking without CP violation. This feature is very helpful for SUSY scenarios with light, i.e., O(100) GeV, SUSY particles. In this section, focusing on this advantage, we will consider an application of the CP-safe gravity mediation model.
The anomalous magnetic moment of muons, or the muon g − 2, has a 3σ-level discrepancy between the experimental value measured in the Brookheaven E821 experiment [3] and theoretical predictions based on the Standard Model [11,12]. SUSY is capable to  [14], while blue stars are those which we have checked is not excluded in any collider searches. Collider status of the other points (gray dots) are not determined in this study. The parameter space with α 1 = α 2 = 0 is described with the red-dashed line as a reference.
solve the anomaly with contributions of loop diagrams in which smuons, sneutrinos, neutralinos and charginos are involved [4]. The SUSY contribution, which we call ∆a SUSY µ , can be large enough to solve the discrepancy if the masses of bino and/or winos are 1 TeV and those of smuons and/or muon sneutrinos are at the same order (see, e.g., Ref. [13] for recent study). However, as we have discussed, this scenario generally confronts too large CP violation because of the small SUSY particle masses. This is the main reason why we apply the CP-safe gravity mediation model to explain the anomaly of the muon g − 2.
In this section, we consider a model with slightly extended Kähler potential, where α 1 and α 2 are real constants and x = Z +Z * . A normalization factor r = 1+α 1 x + α 2 x 2 is introduced for canonical kinetic terms. Note that α 1 = α 2 = 0 corresponds to the model discussed in the previous section (Eq. (16)), and that this extension introduces no additional CP phases. 9 The Higgs B-term and A-terms are given by and the universal scalar mass is by where α 1 ≡ (α 1 + 2α 2 x )/r and α 2 ≡ 2α 2 /r. is restricted to the one-loop level. 10 We also utilizes SDECAY 1.3 [21] 9 The original model with α 1 = α 2 = 0 has a strict constraint that the universal scalar mass parameter at the high-energy scale is related to the Higgs B term as (B µ /µ) 2 = 4m 2 0 (cf. Eqs. (24) and (25)). 10 See Ref. [20] for detailed discussion on two-loop level contributions in scenarios with hierarchical SUSY mass spectrum. to obtain decay rates and branching ratios of the SUSY particles. For the analysis of the vacuum stability, we utilized the fitting function in Ref. [16].
When we fix M 2 and increase M 3 , ∆a SUSY µ becomes larger and the discrepancy of the muon g − 2 is relaxed, but with too large M 3 we face vacuum instability or the stau LSP.
This feature can be understood with the following discussion on the renormalization group evolution from a high-energy scale of ∼ 10 16 GeV to the SUSY scale. Firstly, the large M 3 increases squark masses during the evolution. Then, the large scalar-top mass, which on the one hand raises the Higgs boson mass, affects the soft mass of the up-type Higgs.
For successful EWSB, the µ parameter is forced to be large, as shown in Fig. 1. It results in a large mixing betweenμ L andμ R , which enhances the bino-smuon contribution (a loop diagram ofB-μ L -μ R ). Note that the other contributions, e.g., sneutrino-chargino contribution, are insignificant because Higgsinos are decoupled. However, too large mixing is not allowed it makes the lighter stau too light or induces vacuum instability [15,16], and this is why M 3 is bounded from above. 11 In this section we use the term "lepton" ("l") for electrons and muons, not taus.  Refs. [26,14,27]. Further information on the electroweakino masses can be found on whole of the 1σ region in Fig. 1 will be examined at the 14 TeV LHC.
Another promising production channel is wino pair production: pp →χ 0 2χ + 1 , which has the largest production cross section. Future prospects of this channel are widely discussed [29, 30, 31, 32] in W Z + / E T signature, i.e., with assuming that produced winos exclusively decay asχ 0 2χ + 1 → (Zχ 0 1 )(W +χ0 1 ). Reportedly, winos of mass 700-800 GeV are searched for at 14 TeV LHC with the integrated luminosity of 300 fb under the assumption. 12 However, the branching ratio is negligible in our benchmark points; the winos mostly decays via sleptons, sneutrinos or staus, and produces multi-(e, µ, τ ) plus / E T signature. As capability of searches for these signature seriously dependent on mass gaps among SUSY particles as well as resultant lepton species [28], we will here just comment that multi-l + / E T signature generally provides tighter constraint than W Z + / E T while multi-τ + / E T gives looser bound [25,26].
For experiments at linear colliders, e.g., ILC or CLIC, the stau is a particularly inter- 12 In Ref. [33] future prospects of this channel is discussed in W H + / E T signature.
esting target for its small mass and large mixing. After staus are discovered, stau mixing angle as well as stau mass should be measured, because it will be a test of this scenario.
Furthermore, if all the sleptons are within the reach of linear colliders, we can estimate the size of ∆a SUSY µ through measurements of the stau mixing angle, slepton production cross section and masses [34].

Discussion and conclusions
We have discussed the CP-safe gravity mediation models, which are free from the SUSY CP problem. The SUSY CP problem is serious obstacle for the low-energy SUSY. The CP-safe gravity mediation models assume that the SUSY is broken in the Kähler potential with a non-vanishing constant term in the superpotential. Together with a shift symmetry of the SUSY breaking field, the Higgs B-term, trilinear couplings as well as gaugino masses have a common phase determined by the constant term in the superpotential, and hence, their phases are all aligned.
Note that the CP-safe gravity mediation is also consistent with the sequestered form of the Kähler potential. (A concrete model is shown in Appendix D.) In this case, the Higgs B-term, A-terms and scalar masses vanish at the tree level; then, the gaugino mediation with a large gravitino mass is possible [35], 13 and the gravitino problem may be relaxed. It is very important that the stable electroweak symmetry breaking vacuum is easily realized, since the Higgs B-term (B µ /µ) becomes order of gaugino masses rather than the gravitino mass.
In the CP-safe gravity mediation, the Lagrangian has the shift symmetry of the SUSY breaking field Z: Z → Z + iR (R is a real constant). Therefore, the imaginary part of Z is massless at the perturbative level. This imaginary part gets a mass via the QCD non-perturbative effect, since it couples to the gluon field strength superfield as shown in Eq. (26); 14 the imaginary part of Z behaves as a QCD axion [36] with the decay constant, It is very intriguing that the strong CP problem and the SUSY CP problem are solved simultaneously in the present framework. If the inflation scale is high as in the chaotic inflation, we have an axion iso-curvature problem. This problem may be solved in more complicated frameworks [37], which is, however, beyond the scope of the present paper.

Acknowledgment
We thank K. Harigaya for helpful discussion. This work was supported by JSPS KAK-

A General superpotential
The scalar potential is given by where We take the unit of M P = 1. The gravitino mass is m 2 Assuming the minimal Kähler potential, K = Z * Z + (Q SM ) * i (Q SM ) i , the scalar potential is written as 14 One might introduce a shift symmetry breaking term, e.g., K 2 |Z| 2 (K 4 |Z| 4 ). With this shift breaking term, the axionic part of Z can get a larger mass of ∼ 2 m 3/2 (∼ | 4 |m 3/2 ) without inducing a CP violating phase. However, such explicit breaking terms must be extremely small, otherwise the Peccei-Quin mechanism does not work for solving the strong CP problem in QCD.
where z = Z and W vis = µH u H d + W Yukawa . The Higgs B-term and the A parameter is given by where F Z = −e |z| 2 /2 (∂W * hid )/(∂Z * ) + (∂K)/(∂Z * )(W * hid + C * ) . The gaugino mass arises from the coupling between Z and the gauge kinetic function.
The resultant gaugino masses are M i = 2k i g 2 i F Z .

B The Gravitational SUSY breaking
Here, we consider the model where the SUSY is broken with a constant superpotential.
The SUGRA Lagrangian is written as where ϕ is the chiral compensator and ϕ = 1 + F θ 2 . Here, W vis is a function of the SSM fields. Then, the scalar potential is given by where Q i contains both the SUSY breaking field Z and SSM fields. The equations of motions are Solving equation of motions, F i is written as where thef −1 is the inverse of the matrix, The scalar potential is The scalar potential in the Einstein frame is obtained by the rescaling, The vanishing cosmological constant is obtained by requiring V = 0, resulting in Then, the Kähler potential K = −3 log(−f /3) must satisfy the condition: To satisfy the condition for the vanishing cosmological constant, the SUSY must be broken when C = 0 [7]. This is because F i (∂f )/(∂Q i ) can not be zero for F = 0 (see Eq. (38)).
Since F = 0, the Higgs B-term and the A-terms arise from V −(∂W vis )/(∂Q i )F i .

C A model of CP-safe gravity mediation
We consider the following Kähler potential of the hidden sector; where, f is a function of x = Z + Z † and is invariant under the shift, Z → Z + iR. The superpotential is taken as W = C; the SUSY is broken with a constant superpotential.
By using the equations of motions, the scalar potential is written as V = −3CF , where The vanishing cosmological constant is given by F = 0, which leads to The minimum of x is determined by the equation (∂V )/(∂x) = 0; The F-term of Z is Now, let us consider the following f ; f = c 1 x+c 2 x 2 +c 3 x 3 +c 4 x 4 . Then, the conditions ∂f ∂x = 0, ∂f 2 ∂x 2 = 0, ∂ 3 f ∂x 3 = 0 give The SUSY breaking vacuum is stable for 768c 3 4 + 64c 1 c 3 c 2 4 − 3c 4 3 > 0.

D A model with a sequestered Kähler potential
Here, we consider the sequestered form of the Kähler potential and superpotential; where, f hid is a function of x = Z + Z † ; f is invariant under the shift, Z → Z + iR. The superpotential in the visible sector is W vis = (µ ij /2)Q i Q j + (y ijk /6)Q i Q j Q k .
If one choose f vis = (Q SM ) * i (Q SM ) i , the Higgs B-term, A-terms, and scalar masses vanish. This can be seen clearly in the base that the SM fields are rescaled as (Q SM ) i → (Q SM ) i φ. Then, the scalar potential is written as where f hid = 3 − f hid . The equations of motions are given by Using equations of motions, we have where F i = −(∂W * vis )/(∂Q * i ). Since the vanishing cosmological constant is satisfied with F = 0 [7], B-terms and A-terms as well as scalar masses vanish at the tree level. Note that, in this setup, the stable electroweak symmetry breaking vacuum easily realized even if the gravitino is large as O(10) TeV. This is because B vanishes at the tree level. This may be very useful for gaugino mediation models [35].