Muon g-2 in Anomaly Mediated SUSY Breaking

Motivated by two experimental facts, the muon g-2 anomaly and the observed Higgs boson mass around 125 GeV, we propose a simple model of anomaly mediation, which can be seen as a generalization of mixed modulus-anomaly mediation. In our model, the discrepancy of the muon g-2 and the Higgs boson mass around 125 GeV are easily accommodated. The required mass splitting between the strongly and weakly interacting SUSY particles are naturally achieved by the contribution from anomaly mediation. This model is easily consistent with SU(5) or SO(10) grand unified theory.

In the minimal supersymmetic standard model (MSSM), the discrepancy of the muon g − 2 is explained if the smuons, chargino and neutralino are as light as O(100) GeV with tan β = O(10) [5,6]. Also, supersymmetry (SUSY) provides us with attractive features in addition to the explanation for the muon g − 2 anomaly: a solution to the hierarchy problem and a framework for the grand unified theory (GUT). Therefore, to consider SUSY models explaining ∆a µ is one of the important directions for physics beyond the SM.
However, there is an obstacle in this direction. The squarks and gluino have not yet been observed at the Large Hadron Collider (LHC), resulting in the lower bound on their mass at 1.4-1.8 TeV [7]. Moreover, the observed Higgs boson mass m h around 125 GeV [8] can be explained, only if there is a sizably large radiative correction from the heavy stop(s) [9], unless the large trilinear coupling of the stops exists. In fact, including higher order corrections beyond the 3-loop level, it is suggested that the stop is as heavy as 3-5 TeV [10] in the absence of the large trilinear coupling of the stops. Since squarks and sleptons belong to a same representation of SU (5) GUT gauge group and the gaugino masses unify at the high energy scale in a simple setup, it is rather nontrivial to obtain the heavy stop and light sleptons simultaneously. As a consequence, to construct a convincing SUSY scenario for the muon g − 2 is a rather difficult task.
Recently there has been a resurgence of interest in explaining both the muon g − 2 anomaly and the observed Higgs boson mass within a unified framework. It has been shown that the discrepancy of the muon g − 2 and the Higgs boson mass around 125 GeV can be explained simultaneously by introducing GUT breaking effects, 1 in the gauge mediation [14], gaugino 1 In Refs. [11,12], ∆a µ and the Higgs boson mass around 125 GeV are successfully explained without introducing a GUT breaking effect on the soft SUSY breaking masses. The models shown in Refs are based on the "Split-Family SUSY", where the third generation sfermions are much heavier than the first and second generation sfermions. Also, extensions of the MSSM allow us to explain ∆a µ without introducing a GUT breaking effect (see e.g. Ref. [13]). mediation [15,16], 2 and gravity mediation [19]. In most of these cases, the violation of the GUT relation among gaugino masses is at least required.
In this paper, we show that the required mass splitting among the strongly and weakly interacting SUSY particles, i.e. the GUT breaking effect on the soft SUSY breaking masses, is naturally induced from anomaly mediation [20,21] 3 : both the Higgs boson mass around 125 GeV and ∆a µ can be easily explained in our simple framework, which is consistent with SO(10) or SU (5) GUT.
The rest of the paper is organized as follows: in section 2 we propose the phenomenological AMSB (pAMSB) model used in our analysis. In section 3 we discuss the SUSY contribution to the muon g − 2 in our setup and show numerical results. A more fundamental realization of the pAMSB model is shown in section 4. Finally, section 5 is devoted to the conclusion and discussion.

Phenomenological AMSB Model
In SUSY models, masses of squarks and sleptons are required to be highly split in order to explain the Higgs boson mass around 125 GeV and the muon g − 2 anomaly simultaneously. Moreover, the bino and wino masses should be (much) smaller than the gluino mass at the high energy scale, otherwise the radiative corrections lift up the slepton masses and it becomes difficult to accommodate the experimental result of the muon g − 2.
The anomaly mediated SUSY breaking (AMSB) contributes to the masses of the colored and non-colored SUSY particles very differently: the squark and gluino masses obtain large contributions, while the slepton, bino and wino get negative or small contributions. This feature of AMSB is welcome for the Higgs mass around 125 GeV and the muon g − 2. Based on this observation, we propose a phenomenological AMSB (pAMSB) model, which can be easily accommodated into SU (10) or SU (5) grand unified theory.
Within a supergravity framework, we construct the pAMSB model with the following Kähler potential: where f hid is a function of hidden sector superfields, and Q SM is a chiral superfield in the MSSM. The reduced Planck mass is denoted by M P (M P 2.4 · 10 18 GeV). The superpotential is also assumed to be separated as W = W vis + W hid , where W vis and W hid are superpotentials for the visible sector and hidden sector superfields, respectively. (A concrete example of f hid and W hid is shown in Appendix A.2.) Here, ∆f is an additional source of the sfermion masses, and is defined later. In the case ∆f = 0, the Kähler potential is so-called sequestered form and the 2 The models shown in Refs. [15] are attractive, since they are free from the SUSY and strong CP problem as well as the SUSY flavor problem. Non-universal gaugino masses are naturally obtained based on the product group unification model, which solves the notorious doublet-triplet splitting problem [17,18]. 3 While completing this manuscript, Ref. [22] appeared in arXiv, which has some similarity in the starting point.
scalar masses vanish at the tree level. Scalar masses (gaugino masses) are generated at the two-loop level (one-loop level) from anomaly mediation (see Appendix A.1). The squark and slepton masses are estimated as where Q i ,Ū i andD i denote a left-handed quark, right-handed up-type quark and right-handed down-type quark, and L i andĒ i are left-handed lepton and right-handed lepton, respectively. The index i represents a generation of a chiral multiplet. The common mass scale from anomaly mediation is denoted by M 0 = m 3/2 /(16π 2 ), where m 3/2 is the gravitino mass. We evaluate the above soft masses at 2 TeV for tan β = 20, m t (pole) = 173.34 GeV and α s (M Z ) = 0.1185. The first term (second term) in the bracket comes from the gauge (Yukawa) interactions. The corrections from 1st and 2nd generation Yukawa couplings are neglected. Using one-loop betafunctions of gauge couplings, the gaugino masses are 4 where M 1 , M 2 and M 3 are the masses of the bino, wino and gluino, respectively: M 1 : M 2 : M 3 7 : 2 : −15. We see that from Eqs.(4) and (5) the masses of strongly interacting SUSY particles (M 3 , m Q , m Ū ) and weakly interacting ones (M 2 , m L , m Ē ) are highly split and it may be useful for explaining the muon g − 2 anomaly and the Higgs boson masses simultaneously. However, the slepton masses m L and m Ē are tachyonic, since it interacts only non-asymptotically free gauge interactions.
The tachyonic sleptons can be avoided if there is an additional source of the scalar masses, contained in ∆f : where H u and H d are up-type and down-type Higgs, respectively. Here, x = X + X † , and X is a moduli field which has a non-zero F -term F X : F X / x = O(m 3/2 /100). The above type of ∆f with the suppressed F -term, F X , arises if X couples to the matter fields. Note that F X / x ∼ (m 3/2 /100) is obtained with a KKLT-type superpotential [23] (see also Appendix A.2). The moduli X in ∆f gives corrections to the soft SUSY breaking masses of the MSSM 4 The signs of A klm and M a have been flipped by the R-rotation: A klm → e 2iθ R A klm and M a → e 2iθ R M a .
The definition of the A-term is given by fields comparable to those from anomaly mediation. These corrections uplift the tachyonic slepton masses. The setup in Eq. (6) is similar to that of the mixed modulus-anomaly mediation scenario [41,42], but allowing non-universal contributions to the soft masses from the moduli X.
Moreover, unlike the mixed modulus-anomaly mediation, we can independently chose the soft masses squared and the trilinear coupling of the stops A t determined by d Hu : large contributions to soft masses squared from X do not always lead to large A-terms. This significantly enlarges the parameter space for explaining the muon g − 2 anomaly and the Higgs boson mass around 125 GeV simultaneously, especially in cases that the Higgsino mass term µ is small (see discussion in Sec. 4). Note that ∆f is consistent with SU (5) GUT, and it is also consistent with SO(10) GUT if c 5 = c 10 .
With ∆f = 0, the scalar masses are modified from Eq.(4). The scalar masses including ∆f are given by where m 2 5,10 = c5 All the parameters are defined at the GUT scale (∼ 10 16 GeV), that is, a mass from anomaly is evaluated using the gauge and Yukawa couplings at the GUT scale. For simplicity, we set d H d = 0 here and hereafter. The trilinear coupling of stops and the mixed mass terms are The gaugino masses can be also modified by introducing couplings between field strength superfields of vector multiplets and X. The gauge kinetic functions are Then the gaugino masses get an additional contribution as where δM 1/2 ∼ (m 3/2 /100) and β a is the beta-function of the gauge coupling g a : an additional contribution to the gaugino masses comparable to those from anomaly mediation can arise.
The scalar masses are modified from Eq. (7) as where Here, γ k is the anomalous dimension of the superfield k, γ k = (∂ ln Z k )/(∂ ln µ) and C a (k) is a quadratic Casimir invariant of the field k (C 1 (k) = (3/5)Q 2 Y k ). So far, the SUSY breaking masses at the GUT scale in pAMSB are summarized as follows: where (m 2 k ) mixed is a sum of the contributions from Eqs. (8) and (12), and β Yt , β Y b and β Yτ are the beta-functions of the Yukawa couplings, Y t , Y b and Y τ , respectively. The soft SUSY breaking masses are written in terms of the following set of the parameters, In the limit m 2 10 = m 2 5 = δm 2 Hu = δm 2 H d and δA t = δM 1/2 = 0, the mass spectrum of the SUSY particles corresponds to that of the minimal AMSB [24]. 5 3 Muon g − 2 in the pAMSB In this section, we check whether the muon g − 2 anomaly and the observed Higgs boson mass around 125 GeV can be explained in the pAMSB model. The SUSY contribution to the muon g − 2, (δa µ ) SUSY , is sufficiently large in the following three cases: (a) The wino, Higgsino and muon sneutrino are light.
(b) The bino and left-handed smuon as well as the right-handed smuon are light. 5 See Refs. [25] for phenomenological aspects of the minimal AMSB, where the SUSY contribution to the muon g − 2 is also discussed. Also, in Ref. [26], the phenomenological aspects of anomaly mediation models are considered without imposing the muon g − 2 constraint. In the first case (a), the wino-Higgsino-(muon sneutrino) loop dominates (δa µ ) SUSY . This contribution is estimated as [6] where mν is the mass of the muon sneutrino, and we take µ = (1/2)mν and M 2 = mν in the second line. The soft mass parameters as well as µ in the R.H.S. of Eq. (17) are defined at the soft mass scale. A leading two-loop correction from large QED-logarithms is denoted by δ 2L , which is given by [27,28] To explain ∆a µ = (26.1 ± 8.0) · 10 −10 by (δa µ )W −H−ν , the masses of the wino and the muon sneutrino should be smaller than around 500 GeV.
where we take mμ L = 3M 1 and mμ R = 2M 1 in the second line. One can see that a very light bino with a mass ∼ 100 GeV is required to explain the muon g − 2 anomaly. Note that we do not need to consider the case (c). This is because the light bino and wino can not be obtained simultaneously. The bino and wino mass at 2 TeV are M 1 (2 TeV) = 0.43 δM 1/2 + 1.43M 0 , at the one-loop level. In the case the bino mass is small, say, M 1 (2 TeV) 0.2M 0 , the additional contribution to the gaugino masses is δM 1/2 = −2.9M 0 ; however, the wino mass becomes M 2 (2 TeV) −2.0M 0 , and hence, it is impossible to obtain the light bino and wino simultaneously. Because of this reason, we have only two possibilities (a) and (b) to explain ∆a µ .

Small µ case
First, we consider the small µ case with δM 1/2 = 0. In this case, the gaugino masses are same as those in anomaly mediation. As shown in Eq.(5), the wino is the lightest gaugino, and it is expected that (δa µ )W −H−ν is enhanced if µ is small. On the other hand, it is difficult to enhance (δa µ )B −μ L −μ R because of the large bino mass. Therefore we concentrate on the wino-Higgsino-(muon sneutrino) contribution.
In our numerical calculation, the SUSY mass spectrum is calculated using Suspect 2.43 [29] with a modification suitable for our purpose. The Higgs boson mass (m h ) as well as the SUSY contribution to the muon g − 2 ((δa µ ) SUSY ) is evaluated using FeynHiggs 2.10.4 [30]. In the region where both Higgsino and wino are light, the branching ratio of Br(b → sγ) is enhanced due to the SUSY contribution. We demand that the SUSY contribution do not exceed 2σ bound: where Here, we use the SM prediction in Ref. [31] and the experimental value in Ref.
[32]. We use SuperIso package [33] to calculate ∆Br(b → sγ). Note that the constraint from Br(B s → µ + µ − ) [34] is not stringent in the parameter space of our interest, since the CP-odd Higgs boson mass m A is rather large. In Fig. 1, we plot the contours of m h and the region consistent with ∆a µ . We take m 10 = m5, which is consistent with SO (10)  In the orange (yellow) region, the discrepancy of the muon g − 2 from the SM prediction is reduced to 1σ (2σ) level. The gray region is excluded due to the stop LSP (left-bottom) or stau LSP (right). In the green region, ∆Br(b → sγ) exceeds the 2σ bound in Eq. (21). The constraint from Br(b → sγ) is rather severe and the region with large δA t is excluded. Note that one can not cancel between the chargino contribution and the charged Higgs contribution to Br(b → sγ) by taking smaller m A , since the both contributions are constructive to the SM value for A t , µ > 0 at the soft mass scale. Still, as one can see the discrepancy of the muon g − 2 can be reduced to 1σ level. The calculated Higgs boson mass m h is consistent with the observed value around 125 GeV.
Combined CMS and ATLAS measurement of Higgs mass allow a range from 124.6 to 125.6 GeV at 2σ [8]. On top of it the experimental uncertainty in the top mass measurement [35] and theoretical uncertainty estimated by FeynHiggs 2.10.4 allow for at least ±3 GeV uncertainty in the Higgs boson mass value. Thus in all the plots we show the Higgs boson mass in the range 122-126 GeV.
Next, we relax the condition m 10 = m5. In this case, the muon g − 2 anomaly and the Higgs boson mass around 125 GeV are more easily explained. We show the contours of m h and the region consistent with ∆a µ in Fig. 2 for m 10 = √ 3m5. Because the heavier stops are allowed ((Ū 3 , Q 3 )∈ 10 in SU(5) GUT gauge group), the constraint from ∆Br(b → sγ) becomes less sever than the previous case with m 10 = m5. Moreover, the right-handed stau can be heavier and the region with tachyonic stau is reduced. As a result, the region which can explain the muon g − 2 anomaly and the observed Higgs boson mass simultaneously becomes wider.
Also, we show sample mass spectra of different model points in Table 1. P1 (P2) is consistent with SO(10) (SU (5)) GUT, where m 10 /m5 = 1.0 ( √ 2) is taken. In both of the model points, the calculated Higgs boson mass m h is consistent with the observed value, and the discrepancy of the muon g − 2 from the SM prediction is reduced to 1σ level. The squark  . The lightest neutralino is Higgsino-like mixed with the wino, therefore the relic abundance of this neutralino is too small to explain the observed dark matter abundance: we need another dark matter candidate, e.g. axion in the small µ cases. 6 Note that the existence of the small δM 1/2 is also helpful in the small µ case: it enlarges the  parameter space which can explain the muon g − 2 anomaly. This is because the small mass of the wino can always be obtained by choosing δM 1/2 , regardless of the gravitino mass (see Eq. (20)).
If one takes the bino mass to be small with δM 1/2 = 0, the wino mass becomes large (see Eq. (20)). Then, (δa µ )W −H−ν is suppressed. In this case, we need (δa µ )B −μ L −μ R 1.8 · 10 −9 to explain the muon g − 2 anomaly. Since (δa µ )B −μ L −μ R is proportional to µ tan β and large tan β easily leads to tachynic staus via radiative corrections, we consider the case with large µ and moderate tan β for this purpose.

Large µ case
Here, we consider the model with non-zero M 1/2 . In this case, there is a region where (δa µ )B −μ L −μ R dominates (δa µ ) SUSY . To obtain (δa µ )B −μ L −μ R 1.8 · 10 −9 , it is required that µ is as large as ∼ 3 TeV and the smuons and bino are as light as 100 -300 GeV.
In large µ case, the Higgs soft masses are not required to be tuned for realizing successful electroweak symmetry breaking; therefore, we set δm 2 Hu = δm 2 H d = 0, for simplicity. In Fig. 3, we show the contours of the Higgs boson mass and the region explaining ∆a µ . Here, tan β = 8. We take M 1 (M GUT ) as an input parameter instead of δM 1/2 . Also, mĒ(M GUT ) and mL(M GUT ) are input parameters, which corresponds to choosing m 2 5 and m 2 10 . The sign of µ is chosen such that (δa µ ) SUSY is positive (same sign of the bino mass). One can see that there is a region where the discrepancy of the muon g − 2 from the SM prediction is reduced to 1σ level (orange) for m 2 L (M GUT ) < 0. The negative soft mass squared at the GUT scale is required, since the wino mass is rather large and it gives large positive radiative correction to the left-handed Table 1: The mass spectra for small µ cases. We take δM 1/2 = 0, α s (M Z ) = 0.1185 and m t (pole) = 173.34 GeV. 3.9 · 10 −5 slepton masses: to make the left-handed sleptons light, the fine-tuning of m 2 L (M GUT ) is needed. Consequently, in the case M 1 < 0 (right panel), the region which can explain ∆a µ is smaller due to the larger wino mass, compared to the case M 1 > 0 (left panel). The gray region is excluded since the stau becomes LSP. On the edge of the gray region, the relic abundance of the lightest neutralino explains the observed value of the dark matter, Ω DM h 2 0.12 [39], via the coannihilation with the stau [40].
Also, we show sample mass spectra of two model points in Table 2. The squark and gluino are heavier than the previous case, δM 1

A realization of the pAMSB
We consider a more fundamental realization of the pAMSB, motivated by the mixed modulusanomaly mediation scenario [41,42]. Here, we consider the following Kähler potential and superpotential: where we have taken the unit of M P = 1 and the MSSM matter superfields couple to a moduli field X in the Kähler potential. The superpotential for a SUSY breaking field Z, w(Z), contains a constant term, which is around the gravitino mass m 3/2 . The moduli X has a F -term of F X /(2 Re X ) ∼ m 3/2 /100: corrections to the soft SUSY breaking masses are comparable with those from anomaly mediation. The SUSY breaking de Sitter vacuum is obtained thanks to a coupling between X and Z, f (X +X † ) s+1 |Z| 2 [42]. The detailed explanations are shown in Appendix A.2. It is also assumed that X couples to the field strength superfield of the vector multiplets, giving tree level gaugino masses. Then, together with contributions from anomaly where (m 2 i ) AMSB is a contribution from anomaly mediation and (m 2 i ) mixed is a mixed contribution from the moduli and anomaly mediation. Here, x = X + X † . The detailed mass formulae are shown in Eq. (43) in Appendix A.2. In this model, we can write the soft SUSY breaking masses using the following parameters: n 10 , n 5 , n u , n d , δM 1/2 , m 3/2 , F X x .
With these parameters, we can easily reproduce the results of the large µ case.

Conclusion and discussion
We have proposed a simple anomaly mediation model, namely the phenomenological anomaly mediated SUSY breaking (pAMSB) model, in order to explain the Higgs boson mass around 125 GeV and the muon g − 2 anomaly. The pAMSB can be regarded as a generalization of mixed modulus-anomaly mediation. We have shown that the muon g − 2 anomaly and the observed Higgs boson mass are easily explained. Moreover, our model can be accommodated into SU (5) or SO(10) GUT without difficulty, since required GUT breaking effects to obtain the mass splitting among the strongly and weakly interacting SUSY particles are induced by anomaly mediation. We have also presented a possible realization of the pAMSB. When the muon g −2 anomaly is explained by the wino-Higgsino-(muon sneutrino) diagram, the gluino and squark masses can be as small as 2 -3 TeV; therefore our scenario is expected to be tested at the LHC with √ s = 14 TeV. Even in the other case, where theB −μ L −μ R diagram dominates the SUSY contribution, the sleptons masses are around 300 GeV, and hence, the existence of the these light sleptons can be checked easily.
Finally let us briefly comment on the cosmological aspects of the pAMSB. Since the gravitino is as heavy as ∼ 100 TeV, the cosmological gravitino problem is relaxed. In our model, there exists the moduli field X, which lifts up the slepton masses via its F -term. The decay of the moduli into the gravitinos with a large branching fraction may spoil the success of the standard cosmology and may be problematic [45]; however, it can be solved if the moduli strongly couples to the inflaton [46,47].

Acknowledgments
We thank Luca Silvestrini for useful discussion and careful reading of the manuscript. The research leading to these results has received funding from the European Research Council under the European Unions Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement N o 279972 "NPFlavour".

A Soft mass parameters
In this appendix, we list the formulae for the soft mass parameters. We use the unit where the reduced Planck mass is set to unity in the following discussions.

A.1 AMSB
The soft SUSY breaking parameters with a sequestered Kähler potential are listed. Here, we consider the case that there is no tree level gaugino mass term. The scalar masses from anomaly mediation are [21] where b i are the coefficients of the one-loop beta-functions for gauge couplings: b i = (33/5, 1, −3). For third generation sfermions, there are terms proportional to the Yukawa couplings and their beta-function. Here, we have neglected first and second generation Yukawa couplings. The gaugino masses are given by at the one-loop level. Trilinear couplings are given by A.2 A model with KKLT type potential Following Ref. [42], we consider the following Kähler potential and superpotential: where X is a moduli field and Z is a SUSY breaking field. The superpotential for Z is denoted by w(Z), which contains the constant term: w(Z = 0) = C. The parameter A and constant term C are taken to be real positive by the shift of X and U (1) R transformation without loss of generality. Provided Z 1, 10 the relevant part of the Kähler potential is written as where x = X + X † . Then, the scalar potential is given by V = Abe −bx 3x 2 Abx + 6A − 6Ce bx/2 cos(b Im(X)) +

∂w ∂Z
The imaginary part of X is stabilized at Im(X) = 0, and the scalar potential for x is where s = s+3 and D = |∂w(Z)/∂Z| 2 . Using the minimization condition (∂V /∂x) = 0 and the condition for the vanishing cosmological constant V = 0, the minimum is found for b x ∼ 70 with the equation: 3Ce y/2 (4 − 2s + y) + A[−12 − 7y − y 2 + s (6 + y)] = 0, where y = bx. Here, we consider the case of C ∼ 10 −13 and A ∼ 1. We see that F X / x is suppressed by a factor y ∼ 70 compared to the gravitino mass.
Note that further suppression is possible if one consider more general Kähler potential and super potential for X [48]. Now, we couple X to the matter fields such that the soft SUSY breaking masses which are comparable to those from anomaly mediation are obtained. The couplings are given by ∆f = (X + X † ) n 10 (Q † Q +Ū †Ū +Ē †Ē ) + (X + X † ) n 5 (L † L +D †D ) The Kähler potential is replaced as K = −3 ln[−(f + ∆f )/3]. The canonically normalized Q k is obtained by Q c k = [ x n k −1 ] 1/2 Q k . Then, scalar masses at the tree level are The trilinear couplings are given by A u = (n 10 + n 10 + n u ) F X x , A d = A e = (n 10 + n 5 + n d ) The gaugino masses are generated from the gauge kinetic functions: and Here, Re X l = 1/g 2 . Including the contributions from AMSB, we obtain m 2 k = n k F X x 2 + (m 2 k ) AMSB + (m 2 k ) mixed , M a = l 2 F X X + β a g a m 3/2 = δM 1/2 + β a g a m 3/2 A t = −(n 10 + n 10 + n u ) where (m 2 k ) AMSB is the contribution coming purely from AMSB shown in Eq. (27), and (m 2 k ) mixed is (m 2 k ) mixed = 1 2 m 3/2 16π 2 c k a g 2 a −δM 1/2 + h.c.
Here, we have flipped the signs of A-terms and M i by the U (1) R rotation. The coefficients c k a and d k can be read from the anomalous dimension of the field k: