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Unified Models of the QCD Axion and Supersymmetry Breaking

Similarities between the gauge meditation of supersymmetry breaking and the QCD axion model suggest that they originate from the same dynamics. We present a class of models where supersymmetry and the Peccei-Quinn symmetry are simultaneously broken. The messengers that mediate the effects of these symmetry breakings to the Standard Model are identical. Since the axion resides in the supersymmetry breaking sector, the saxion and the axino are heavy. We show constraints on the axion decay constant and the gravitino mass.

Second, if the PQ symmetry breaking field resides in the SUSY breaking sector, the super partners of the axion, namely the saxion and the axino, may obtain large masses [13][14][15][16]. Such a model is free from the cosmological problems associated with light saxions and axinos (see [17] and references therein).
Finally, one realization of the PQ mechanism, the KSVZ model [18,19], has the following superpotential, where Z is a PQ charged field with a non-zero vacuum expectation value (VEV), and Q andQ are PQ and standard model gauge (especially SU (3) c ) charged fields. If the chiral field Z also obtains a non-zero F term VEV, the SUSY breaking is mediated to super partners of standard model particles via the gauge interaction. This is nothing but the gauge mediation of SUSY breaking [20][21][22][23][24] with messenger fields Q andQ.
Motivated by these hints, we propose a model where SUSY and the PQ symmetry are simultaneously broken, and the messenger fields that mediate SUSY breaking and the anomaly of the PQ symmetry are in fact the same. The model provides a unification for the physics of SUSY breaking and the PQ mechanism.

II. UNIFICATION OF SUSY AND PQ SYMMETRY BREAKING
A. Simultaneous SUSY and PQ symmetry breaking in a single sector We introduce chiral fields M + and M − , whose U (1) PQ charges are +1 and −1, respectively. The PQ symmetry is broken by introducing a chiral field X and a superpotential coupling, where κ and v are constants. SUSY is broken by lifting the flat direction M + M − = v 2 . To achieve this, we introduce chiral fields Z + and Z − , and couple them to M ± via mass terms. The superpotential of this minimal model is then given by where λ and r are constants. By phase rotations of chiral fields, we take all constants in Eq. (3) to be real. The simultaneous breaking of the PQ symmetry and SUSY via the superpotential in Eq. (3) is discussed in [13,14]. As is shown in section III, this model is the low energy effective theory of a dynamical SUSY breaking model with a deformed moduli constraint (the IYIT model) [25,26], and is studied by [16] in the context of the heavy scalar scenario [27][28][29][30][31]. A direct coupling between the SUSY and the PQ breaking sector is also analysed in [15] using an effective field theory.
For λ < κ, the VEVs of the fields are given by up to a U (1) PQ rotation. The PQ symmetry is broken by the VEVs of the charged fields M ± and Z ± , where z is undetermined at tree level. If λ > κ, the VEVs of M ± and Z ± vanish, and the PQ symmetry is not broken.

arXiv:1702.00401v1 [hep-ph] 1 Feb 2017
Thus we will adopt the above hierarchy and also assume that λ κ for simplicity. SUSY is predominantly broken by the F terms of Z ± , B. Mass spectrum The chiral field X and a linear combination of M ± obtain a large mass κv. We may integrate them out and parametrize M ± as where A is a chiral field. The effective superpotential of Z ± and A is then given by where f ≡ v (r 2 + 1/r 2 )/2 and λ ≡ 2λ /(r 2 + 1/r 2 ). We note that most of the following discussion relies only on this effective superpotential, and not on the UV completion in Eq. (3). Let us first calculate the masses of scalar components of Z ± and A. We decompose scalar components as Expanding the scalar potential, we obtain the mass terms, The mass eigenstates and eigenvalues are given by Scalar fields ρ L and θ L are massless at tree level but obtain masses through quantum corrections, as we will see later. The remaining massless field, a, is the axion. Next we consider the masses of the fermionic components of Z ± and A. The quadratic terms of δZ ± ≡ Z ± −z and A in the superpotential in Eq. (3) are The mass eigenstates ψ ± and eigenvalues are where ψ A and ψ Z H are the fermionic components of A and Z H ≡ (Z + − Z − )/ √ 2, respectively. The fermionic component of Z L ≡ (Z + + Z − )/ √ 2 is the goldstino and is eaten by the gravitino via the super Higgs mechanism.
The expressions for the mass eigenstates are simplified in the limit 1 or 1. In the limit 1, where the PQ symmetry is dominantly broken by the VEVs of M ± , we have In the limit 1, where the PQ symmetry is dominantly broken by the VEVs Z ± , we obtain where the masses of σ − = ρ H and ψ − = ψ Z H are suppressed by the large Majorana masses λz of σ + = s and ψ + = ψ A .
C. Sgoldstino potential in the minimal model As we have seen, the directions ρ L and θ L , which correspond to the sgoldstino components, are massless at tree level. Accordingly, z is undetermined at tree level. Here we discuss the stabilization of the sgoldstino in the mimimal model given by Eq. (3).
Quantum corrections generate a potential for the scalar where m 3/2 is the gravitino mass. We take m 3/2 to be real by a U (1) R rotation. The gravitino mass is related with the magnitude of the SUSY breaking by the (almost) vanishing cosmological constant condition The tadpole term induces the VEV of Z L and the messenger scale [32]. Assuming | Z L | f , we obtain For small λ, the formula (24) yields | Z L | > f . In such a parameter region, the potential of Z L given by the quantum correction is logarithmic, and cannot stabilize Z L against the tadpole term. Instead Z L is stabilized around Z L ∼ M Pl by the supergravity effect. Later, we couple Z ± to the messenger field. If Z L is as large as M Pl , the gauge mediated soft masses of supersymmetric standard model (SSM) particles are smaller than the gravitino mass. Thus, in the following, we concentrate on the parameter region where Z L M Pl . Then in the minimal model, Z L is at the most O(f ).

D. Simultaneous mediation of SUSY breaking and the anomaly of the PQ symmetry
The simplest possibility of the mediation is to introduce a pair of standard SU (3) c charged chiral fields Q andQ with the coupling, To avoid tachyonic masses for the messenger fields, we require that On the other hand, the quantum correction from the messenger loop generates a potential term for the SUSY breaking field, where N Q is the multiplicity of the messenger field. By requiring that this potential does not destabilize the SUSY breaking vacuum, we obtain The bounds on y in Eqs. (27) and (29) are compatible if In Fig. 1, we show the upper bound on N Q as a function of Z L . Here we have evaluated m Z using ∆V ± in Eq. (21). It is evident that the upper bound is too severe and is inconsistent with N Q > ∼ 3, which leads us to extend the model to stabilize the sgoldstino.

E. Stabilization of the sgoldstino in extended models: model-independent analysis
By coupling the sgoldstino to other chiral multiplets, quantum corrections from these multiplets give addi-tional contributions to the mass of the sgoldstino. Here we assume that a positive squared mass m 2 Z is generated from a quantum correction. (For setups which generate a negative squared mass, see [33][34][35][36][37][38].) Even in this generic situation, we show that there is a lower bound on the axion decay constant and the gravitino mass.
The VEV of Z L is given by and the gauge mediated gluino mass is given by For given λ, f , and mg, m 2 Z is fixed, There are two bounds that must be considered. One is Eq. (30), Another is Otherwise we need fine-tuning between ∆V ± and additional contributions to obtain a required value of m 2 Z . In Fig. 2, we show the constraints on (λ, f ) as well as the contours of the axion decay constant f a , and the gravitino mass m 3/2 . Here we assume that the messenger is in the 5 representation of the SU (5) GUT group, so N Q = 5. For the most part, the axion decay constant is dominated by the VEVs of M ± in the left half of the parameter space and the VEVs of Z ± in the right half. The blue shaded region is excluded as the messenger field becomes tachyonic. The region below a black dashed line calls for fine-tuning. We obtain lower bounds from them, MeV.

F. Cosmology
We now address several cosmological topics that may affect the parameter space of our model. Our model contains a SUSY preserving vacuum where the messengers obtain nonzero VEVs, so we must ensure that the SUSY breaking vacuum is selected during cosmological evolution. Following the discussion in [39], in the early Universe we assume that the SSM particles are in thermal equilibrium and therefore the sgoldstino field potential obtains finite temperature corrections from the messenger fields. We also take the sgoldstino field to be stabilized at the origin initially due to a positive Hubbleinduced mass. The messenger potential becomes unstable about Z L = 0 as the universe cools, which causes the messengers become tachyonic and develop VEVs. To reach the SUSY-breaking vacuum, the sgoldstino field must leave the origin before this occurs, which requires that [39] y Combining this with Eq. (35) and Eq. (27), we obtain f a 2.6 × 10 10 mg 3TeV MeV.
Hence vacuum selection raises the lower bounds by a factor O(10).
Another potential concern is that the sgoldstino, which may be produced in the early universe by thermal or nonthermal processes, might affect Big Bang Nucleosynthesis (BBN). The relevant decay modes of the sgoldstino are Z L → aa and Z L → gg with decay rates respectively. Looking to the parameter space in Fig. 2, the former decay dominates in most of the area where Z ± controls the axion decay constant, while the latter decay dominates for a majority of the remaining allowed parameter space. Sgoldstino decay into gravitinos dominates in the upper right portion of the parameter space but the gravitino is heavy in this region and so it is not favored. In both of the relevant regions, the decay time is short enough that the sgoldstino does not affect BBN.
It should also be noted that the super partners of the axion obtain large masses. This is a merit of the setup described above [13][14][15][16]. In general, the super partners of the axion obtain only small masses, typically smaller than the masses of SSM particles. Since they couple to SSM particles very weakly while being light, they cause various cosmological problems (see [17] and references therein). These problems are particularly serious in gauge mediation, where the SUSY breaking scale is small and the super partners of the axion are light. In our setup, since the axion multiplet resides in the SUSY breaking sector, the super partners of the axion can be much heavier than SSM particles and do not cause cosmological problems.

G. Alignment of CP phases
An interesting feature of our model is that the phases of the gravitino mass and the gaugino masses are aligned with each other. This is because the phase of the VEV of Z ± , which generates the messenger scale, is aligned with the gravitino mass in a phase convention where the F term of the SUSY breaking field Z L is real. Thus, the CP phase of the Bµ term (in a convention where the µ term is real) due to the supergravity effect [51] is absent in our model. This feature would be advantageous if one requires that SUSY particles are light (e.g. to explain the experimental anomaly of the muon anomalous magnetic moment [52][53][54] by SUSY particles [55][56][57]) while the gravitino mass is large (e.g. to be consistent with a large reheating temperature.)

III. EXAMPLE OF AN EXTENDED MODEL: LOW ENERGY THEORY OF THE IYIT MODEL
A. Effective theory of the IYIT model Let us consider a vector-like SUSY breaking sector based on SU (2) hidden strong gauge dynamics [25,26]. We introduce four chiral fields which are in the fundamental representation of SU (2), q i (i = 1-4), and six singlet chiral fields, Z + , Z − , Z 0,a (a = 1-4). We assume U (1) PQ charges shown in Table I, and consider the following superpotential, + Z 0,a λ 13 a q 1 q 3 + λ 14 a q 1 q 4 + λ 23 a q 2 q 3 + λ 24 a q 2 q 4 , where the λ's are constants, and summation over a is assumed. The genericity of the superpotential can be guaranteed by symmetries. One concrete example of U (1) R and Z 4 charges is shown in Table I.
Below the dynamical scale of the hidden SU (2), Λ, the theory is described by meson fields M ij q i q j /ηΛ with the deformed moduli constraints, PfM ij = Λ 2 /η 2 [58].
Here and hereafter, we assume the naive dimensional analysis to count factors of η ∼ 4π [59,60]. The deformed moduli constraint may be expressed by introducing a Lagrange multiplier field X, The tree-level superpotential in Eq. (44) becomes We define Then the effective superpotential in Eq. (45) is given by By SU (4) rotations of M 0,a and Z 0,a , the total superpotential can be simplified as with c ab as a unitary matrix. We will work with only one pair of neutral fields (Z 0 , M 0 ), which corresponds to the generic case that there exists a mild hierarchy in the neutral coupling constants so that the effect of only one neutral field dominates. Therefore, after a redefinition of constants, we have the effective superpotential The coupling constant κ originates from strong dynamics and is expected to be large. The absolute value of the coupling constant c is at maximum unity. To maximize the quantum correction, we assume |c| = 1 in the following. We also assume that λ 0 v 2 > λf 2 , since otherwise M 0 obtains a VEV instead of M ± . The vacuum is then given by B. Stabilization of the sgoldstino by neutral fields in the IYIT model To estimate the quantum correction from Z 0 and M 0 , we use the parametrization [61] Here we have neglected the dependence on A, which is irrelevant for the quantum correction from Z 0 and M 0 to Z L . The effective superpotential of Z L and Z 0 , M 0 is given by The quantum correction to the potential of Z L from Z 0 and M 0 is given by In this model, m 2 Z is given by C. Parameter window of the IYIT model Let us now discuss constraints on the parameter space. The constraint from the stability of the vacuum, λ 0 v 2 > λf 2 , is Constants λ r, λ /r and λ 0 are dimensionless coupling constants in the IYIT model, and are at the most O(1). This gives upper bounds on λ 0 and R, Finally, the potential of Z L becomes logarithmic for λR 2 Z L > λ 0 f , and cannot stabilize the sgoldstino against the tadpole term, so By combining the bounds in Eqs. (56)(57)(58)(59), we obtain upper bounds on R 4 /λ 0 , Eqs. (57), (59) λ −10/3 h 10/3 Eqs. (58), (59) These give upper bounds on m 2 Z . In Fig. 3, we show the constraints on (λ, f ). The meaning of the blue shaded region and the black dashed line are the same as in Fig. 2. In the green shaded region, the bound on m 2 Z from Eqs. (55) and (60) is inconsistent with the required value of m 2 Z shown in Eq. (33). We obtain the bounds on the axion decay constant f a and the gravitino mass m 3/2 10 9 GeV f a 10 12 GeV, for a gluino mass O(TeV). It is interesting that the allowed range of f a is consistent with the axion dark matter scenario [62][63][64][65].

IV. SUMMARY
In this letter, we have presented a model that tackles several outstanding issues in the Standard Model and its supersymmetric extension.
We have examined a minimal hidden sector that consists of a superpotential with a U (1) symmetry, which we identify with the PQ symmetry, and messenger quarks that carry SU (3) c charges. Supersymmetry and this PQ symmetry are spontaneously broken while lowest order supergravity effects create the messenger scale. Quantum effects generate a potential for the sgoldstino and force constraints on model parameters to ensure the stability of the SUSY-breaking vacuum. These constraints proved to be too stringent and required that we supplement the minimal model with extra matter fields. We have shown that classes of models that share features with ours, such as a quantum mechanically induced sgoldstino mass and a minimal messenger sector, automatically obtain lower bounds on the axion decay constant and gravitino mass. This fact encouraged us to supplement our minimal model in the hopes that such attractive features could be preserved and expanded upon in a stable extended model.
An IYIT model with SU (2) gauge dynamics is a natural candidate for such an extended model since the minimal model is easily embedded in the U (1) charged subsector of this larger model. Combining the inequalities from vacuum stability and IYIT coupling constants, upper bounds for the sgoldstino mass were derived. The resulting window in parameter space was found to restrict the gravitino mass to lie between 0.1 MeV m 3/2 10 MeV and the axion decay constant to 10 9 GeV f a 10 12 GeV, which is the suitable range for invisible axion dark matter.