Goldilocks Supersymmetry: Simultaneous Solution to the Dark Matter and Flavor Problems of Supersymmetry

Neutralino dark matter is well motivated, but also suffers from two shortcomings: it requires gravity-mediated supersymmetry breaking, which generically violates flavor constraints, and its thermal relic density \Omega is typically too large. We propose a simple solution to both problems: neutralinos freezeout with \Omega ~10-100, but then decay to ~1 GeV gravitinos, which are simultaneously light enough to satisfy flavor constraints and heavy enough to be all of dark matter. This scenario is naturally realized in high-scale gauge-mediation models, ameliorates small scale structure problems, and implies that ``cosmologically excluded'' models may, in fact, be cosmologically preferred.

PACS numbers: 95.35.+d, 04.65.+e, 12.60.Jv Supersymmetric extensions of the standard model of particle physics are among the prime candidates for new microphysics. Among their many virtues, supersymmetric models naturally predict new particles that are candidates for dark matter. The most well studied of these are thermal relic neutralinos [1], superpartners of the Higgs and electroweak gauge bosons. The thermal relic density of neutralinos is dependent on unknown supersymmetry parameters. However, order-of-magnitude estimates yield relic densities that are consistent with [2] Ω DM h 2 = 0.1050 +0.0041 −0.0040 (1σ) , where Ω DM is the observed energy density of nonbaryonic dark matter in units of the critical density, and h ≃ 0.73 is the normalized Hubble parameter. This remarkable fact has not only motivated supersymmetry, but has also focused attention on "cosmologically preferred" models, in which the neutralino thermal relic density is exactly that required for dark matter. Such studies have implications for a large range of experiments, from direct and indirect dark matter searches to those at the Large Hadron Collider (LHC) at CERN. The neutralino dark matter scenario is not without its blemishes, however. First, for the neutralino to be stable, it must be the lightest supersymmetric particle (LSP). In particular, it must be lighter than the gravitino. This requires gravity-mediated supersymmetry breaking models, in which low energy bounds on flavor and CP violation are generically violated by several orders of magnitude. Gauge-mediated supersymmetry breaking (GMSB) models [3] elegantly avoid these constraints, but such models have gravitino LSPs and so are incompatible with neutralino dark matter.
Second, although general arguments imply that the neutralino thermal relic density is of the right order of magnitude, in concrete models, it is often too large: Neutralinos are Majorana fermions, and so annihilation to quarks and leptons is P -wave suppressed. In addition, gauge coupling unification and radiative electroweak symmetry breaking typically imply Bino-like neutralinos, which suppresses annihilation to gauge and Higgs bosons. These effects together enhance relic densities to values that may far exceed those given in Eq. (1).
These two shortcomings of neutralino dark matter are usually considered unrelated and addressed separately. One may, for example, consider gravity-mediated scenarios, such as minimal supergravity, where low energy constraints are satisfied by unification assumptions. One then further focuses on special regions of parameter space in which the neutralino relic density is reduced to acceptable levels through, for example, resonant annihilation [4], stau co-annihilation [5], or significant Bino-Higgsino mixing [6]. Alternatively, one may simply abandon the hope that the order-of-magnitude correctness of the neutralino thermal relic density is a significant lead in the hunt for dark matter and explore other mechanisms for dark matter production. For example, one may consider GMSB models with thermally produced gravitinos [7]. (Note, however, that recent Lyman-α constraints requiring mG ≥ 2 keV [8] imply that the gravitino thermal relic density Ω th G h 2 ≈ 1.2 (mG/keV) must be significantly diluted through late entropy production [9] for this possibility to be viable.) More recently, GMSB-like models with gravitino dark matter produced by late decaying gauge singlets have also been proposed [10].
In this work, we consider the possibility that the two shortcomings described above are not separate issues, but are in fact pointing to a single resolution. We propose that neutralinos do, in fact, freezeout with very large densities. However, they then decay to gravitinos, which are light enough to accommodate the GMSB solution to the flavor and CP problems, but heavy enough to be all of dark matter. In analogy to Goldilocks planets, which have temperatures that lie within the narrow window required to support life, these supersymmetric models have gravitino masses in the narrow window required to satisfy both particle physics and cosmological constrants, and so we call this "Goldilocks Supersymmetry." The essential features of this scenario may be illustrated by simple scaling arguments. Consider models in which there are two mass scales: the scale of the standard model superpartner massesm, and the gravitino mass mG. The freezeout density of neutralinos is inversely proportional to the neutralino annihilation cross section, and so by dimensional analysis, Ω χ h 2 ∼ σv −1 ∼m 2 . The gravitino relic density is therefore ΩGh 2 = (mG/m)Ω χ h 2 ∼ mGm. At the same time, a natural solution to the supersymmetric flavor and CP problems requires mG ≪m. We find, then, that we can always make ΩG large enough to explain dark matter by raising mG andm together with their ratio fixed. The essential question, then, is whether the scenario may be realized withm < ∼ TeV, as required for a natural solution to the gauge hierarchy problem, and whether it passes all other particle physics and astrophysical constraints.
To analyze this question concretely, we consider the example of minimal GMSB models [11]. Such models are specified by the 4+1 parameters M m , Λ, N m , tan β, and sign(µ), where M m is the messenger mass, Λ = F/M m , where F is the supersymmetry breaking scale in the messenger sector, N m is the number of 5+ 5 messenger pairs, tan β = H 0 u / H 0 d , and µ is the supersymmetric Higgsino mass. In terms of these parameters, the gaugemediated contributions to squark and slepton masses are where , and C f i = 0 for gauge singlets, 3 4 for SU(2) L doublets, and 4 3 for SU(3) C triplets. The gaugino masses are where i = 1, 2, 3 for the U(1) Y , SU(2) L , and SU(3) C groups, c 1 = 5 3 , and c 2 = c 3 = 1. As indicated, these masses are generated at the energy scale M m . We determine physical masses through renormalization group evolution to the weak scale and radiative electroweak symmetry breaking with SoftSUSY 2.0 [12].
In addition to the gauge-mediated masses, there are gravity-mediated contributions. These generate the gravitino mass mG = F0 √ 3M * , where F 0 is the total supersymmetry breaking scale and M * ≃ 2.4 × 10 18 GeV is the reduced Planck mass. Because F 0 receives contributions from all supersymmetry breaking F -terms, F 0 ≥ F . For direct gauge mediation, F 0 ∼ F , but this is modeldependent. Here, we assume F 0 = F , and so Our results are not changed significantly for F 0 > F . Gravity-mediation also generates flavor-and CPviolating squark and slepton mass parameters (m f ij ) AB , where i, j = 1, 2, 3 label generation, A, B = L, R label chirality, and f = l, u, d. The chirality-preserving parameters are naturally ∼ mG; for concreteness, we assume The chirality-violating masses require the breaking of electroweak gauge symmetry (and possibly horizontal symmetries); we assume where the λ f ij are Yukawa couplings. Finally, we assume O(1) CP-violating phases for both the gravity-and gauge-mediated masses, as detailed below.
Given these assumptions, the most stringent constraints are the flavor-changing observables ∆m K and ǫ K , and the CP-violating, but flavor-preserving, electron and neutron electric dipole moments (EDMs) [13,14,15]: In the mass insertion approximation, these constrain (δ f ij ) AB ≡ (m f ij ) AB /mf , wheremf is an averagef mass.
The leading constraints are from ∆m K on Re (δ d 12 ) LL (δ d 12 ) RR , from ǫ K on Im (δ d 12 ) LL (δ d 12 ) RR , and from the EDMs on the gauge-mediated masses.
The supersymmetric contributions to the kaon observables are ∆m SUSY K = Re(M ) and ǫ SUSY K = Im(M )/( √ 8 ∆m exp K ), with M as given in Ref. [16]. For concreteness, we choose the δ phases to maximize the supersymmetric contribution for each kaon observable. The constraints from ∆m K and ǫ K are therefore not simultaneously applicable, but the most stringent constraint smoothly interpolates between these as the phase varies. For the EDMs, we first use micrOMEGAs 1.3.7 [17] to determine the supersymmetric contribution to a µ , the anomalous magnetic moment of the muon. The EDMs are, then, d e = me 2m 2 µ a µ tan θ CP and d n = 1 3 (4d d + d u ), where d d and d u are determined from d e with α → α s , M 1 → M 3 , ml → md ,ũ , and the introduction of appropriate color factors [16]. We set tan θ CP = 1 in the EDMs. Note that the EDMs may be suppressed, depending, for example, on the origin of the µ and B parameters.
The resulting constraints are given in Fig. 1. The observables ∆m K and ǫ K require mG < ∼ 30 GeV (500 GeV) for neutralino mass m χ ∼ 100 GeV (1 TeV). In contrast, the EDMs are insensitive to mG, since they do not rely on gravity-mediated contributions. They are found to require m χ > ∼ 1 TeV, in agreement with earlier work [18]. These results are, of course, subject to the assumptions we have made. However, they imply that in any model in which gravity-mediated contributions are at their natural scale and all mass parameters have O(1) phases, the standard model superpartners must be heavy, and the LSP is the gravitino, not the neutralino.
For N m = 1, the lightest standard model superpartner is the lightest neutralino χ throughout parameter space. In Fig. 1 we also show the freezeout density Ω χ h 2 , that is, the relic density if neutralinos were stable, determined using micrOMEGAs [17]. These results illustrate the difficulties for neutralino dark matter. At the weak scale, typically µ, M 2 > M 1 , and χ is Bino-like. Its annihilation is therefore suppressed for the reasons noted above. For m χ = 100 GeV, Ω χ h 2 ∼ 1 is already far too large, and for the heavier superpartner masses favored by the EDM constraints, it grows to values of ∼ 10 − 100.
In the scenario proposed here, however, neutralinos are not stable, but decay to gravitinos. The resulting gravitino relic density is given in Fig. 2. In the dark green shaded region, ΩGh 2 is in the range required to account for all of non-baryonic dark matter. We see that parts of this shaded region are consistent with low energy flavor and CP constraints. In this scenario, very large neutralino freezeout densities are a virtue, not a problem, as they allow light gravitinos to have the required relic density, despite the significant dilution factor mG/m χ . In this simple example of minimal GMSB, the Goldilocks window, in which both relic density and low energy constraints are satisfied, has mG ∼ 1 − 10 GeV.
So far, we have considered constraints from particle physics and Ω DM . We now turn to astrophysical constraints. In the preferred band, the gravitino is light and dominantly couples through its Goldstino components. The neutralino decay widths are Γ(χ → γG) = (cos 2 θ W /48π)(m 5 χ /m 2 G M 2 * ) and Γ(χ → ZG) = (sin 2 θ W /48π)(m 5 χ /m 2 G M 2 * ) 1 − m 2 Z /m 2 χ 4 . As shown in Fig. 3, these imply lifetimes τ > ∼ 0.01 s in the preferred band. Such late decays are constrained by entropy production, µ distortions of the cosmic microwave background, Big Bang nucleosynthesis (BBN) [19], and small and lifetime τ (χ →G) (dotted) in the (mG, Λ) plane, for Nm = 1, tan β = 10, µ > 0, and top quark mass mt = 175 GeV. In the light yellow (medium blue) shaded region, hadronic (electromagnetic) showers from χ decays produce discrepancies with BBN observations. The band with the correct ΩGh 2 is as in Fig. 2, and the neutralino LSP region and fixed GMSB parameters are as in Fig. 1. scale structure [20]. We find that the last two are most stringent, and so focus on them here.
Standard BBN agrees reasonably well with observations. This agreement constrains electromagnetic (EM) and hadronic energy release in late decays, which may be parameterized by ξ i ≡ ǫ i B i Y χ , where i = EM, had, ǫ i is the EM/hadronic energy released in each neutralino decay, B i is the branching fraction into EM/hadronic components, and Y χ ≡ n χ /n BG γ , where n BG γ = 2ζ(3)T 3 /π 2 . We have determined the ξ i following the prescription of Refs. [21] and compared them to the constraints given in Ref. [22]. The BBN constraints are shown in Fig. 3 and are stringent -in this scenario, neutralinos are longlived and greatly overproduced, resulting in large energy release. In the region of parameter space with 0.097 < ΩGh 2 < 0.113, the EM (hadronic) constraint requires lifetimes τ < ∼ 10 5 s (0.1 s) and m χ > ∼ 200 GeV (1 TeV). Dark matter produced in late decays also may suppress structure on small scales [20]. The free-streaming scale λ FS = in the present context, where u τ ≡ | pG|/mG at decay time τ , and we have neglected the effect of m Z on kinematics and other small effects. Values of λ FS are given in Fig. 3; they are essentially independent of mG. Current constraints [8] require λ FS < ∼ 0.2 Mpc, but values near this bound may be preferred by observations. Remarkably, constraints from small scale structure are satisfied in the region of parameter space allowed by BBN, flavor and CP bounds, but just barely -Goldilocks supersymmetry therefore predicts "warm" dark matter and may explain the suppression of power on scales ∼ 0.1 Mpc.
In summary, we have proposed a simple model in which the flavor and overdensity problems of neutralino dark matter are simultaneously solved. In the specific framework considered here, the preferred model is high-scale GMSB, with mG ∼ 1 GeV, √ F ∼ 10 9 GeV, Ω χ ∼ 100, and m χ ∼ 2 TeV. This last mass scale is unnaturally high, but is dictated by EDM constraints, irrespective of cosmology. More generally, this scenario de-emphasizes "cosmologically preferred" models with Ω χ ∼ 0.1, and implies that models typically considered excluded by neutralino overclosure may, in fact, be viable and preferred.
Acknowledgments -We thank Eva Silverstein for stimulating conversations in early stages of this work. JLF is supported in part by NSF grants PHY-0239817 and PHY-0653656, NASA grant NNG05GG44G, and the Alfred P. Sloan Foundation. BTS is supported in part by NSF grant PHY-0239817. FT is supported in part by NSF grant PHY-0355005.