Yukawa-unified natural supersymmetry

Previous work on t-b-\tau Yukawa-unified supersymmetry, as expected from SUSY GUT theories based on the gauge group SO(10), tended to have exceedingly large electroweak fine-tuning (EWFT). Here, we examine supersymmetric models where we simultaneously require low EWFT ("natural SUSY") and a high degree of Yukawa coupling unification, along with a light Higgs scalar with m_h\sim125 GeV. As Yukawa unification requires large tan\beta\sim50, while EWFT requires rather light third generation squarks and low \mu\sim100-250 GeV, B-physics constraints from BR(B\to X_s\gamma) and BR(B_s\to \mu+\mu-) can be severe. We are able to find models with EWFT \Delta\lesssim 50-100 (better than 1-2% EWFT) and with Yukawa unification as low as R_yuk\sim1.3 (30% unification) if B-physics constraints are imposed. This may be improved to R_yuk\sim1.2 if additional small flavor violating terms conspire to improve accord with B-constraints. We present several Yukawa-unified natural SUSY (YUNS) benchmark points. LHC searches will be able to access gluinos in the lower 1-2 TeV portion of their predicted mass range although much of YUNS parameter space may lie beyond LHC14 reach. If heavy Higgs bosons can be accessed at a high rate, then the rare H, A\to \mu+\mu- decay might allow a determination of tan\beta\sim50 as predicted by YUNS models. Finally, the predicted light higgsinos should be accessible to a linear e+e- collider with \sqrt{s}\sim0.5 TeV.


Introduction
A striking feature in nature is that all the fermions of each generation fill out a complete 16-dimensional spinor multiplet of the gauge group SO(10) [1]. While ordinary grand unified theories (GUTs) suffer from the notorious gauge hierarchy problem, supersymmetric (SUSY) GUTs not only tame this hierarchy problem [2], but they also receive support from the well-known unification of gauge couplings [3]. In the simplest SO(10) SUSY GUT theories, where both MSSM Higgs doublets H u and H d occupy the same 10-dimensional representation, one also expects unification of third generation Yukawa couplings f t , f b and f τ at M GUT 2 × 10 16 GeV [4,5]. The t − b − τ Yukawa coupling unification is highly sensitive to both 2-loop renormalization group running (RGEs) and to threshold corrections when transitioning between MSSM and SM effective theories at the SUSY particle mass scale. Thus, the entire SUSY mass spectrum enters into a precise computation of Yukawa coupling unification.
Many groups have explored t−b−τ Yukawa unification (YU) in SUSY theories [6,7,8,9,10,11,12,13]. It has been found that, for µ > 0, YU can occur at the few percent level in either the Higgs splitting (HS) model or the DR3 model (D-term splitting, right-hand neutrino effects and third generation splitting), provided that the GUT scale soft SUSY breaking (SSB) terms are related as with A 0 < 0. Moreover, the GUT scale Higgs splitting m 2 Hu < m 2 H d is needed to allow for an appropriate radiative breakdown of electroweak symmetry. In these models, first generation squarks and sleptons are required to be in the multi-TeV range [8,10] while third generation sfermions are driven to much lighter TeV-scale masses. A benefit of these models is that the light SUSY Higgs boson mass m h tends naturally to be in the 125 GeV range, 1 as required by the recent LHC discovery [16,17]. The gaugino masses from YU SUSY are expected to be quite light, with typically mg 500 GeV (now excluded by LHC searches for gluino pair production along with cascade decay into states containing b-quarks [18]), although solutions with heavier gluinos ∼ 1 − 2 TeV can also be found [19]. In all these cases, the superpotential µ parameter, which is extracted from the electroweak minimization conditions, occupies values in the 1-10 TeV regime, leading to severe electroweak fine-tuning (EWFT).
In this work, we examine to what extent it is possible to reconcile YU with low EWFT. Minimization of the SUSY scalar potential allows one to relate the Z mass scale to the superpartner mass scale via the well-known relation The radiative corrections Σ u and Σ d are given in the 1-loop approximation of the Higgs effective potential by: where ∆V is the one-loop correction to the tree-level potential, and the derivative is evaluated in the physical vacuum: i.e. the fields are set to their vacuum expectation values after evaluating the derivative. At the one-loop level and in the limit of setting first/second generation Yukawa couplings to zero, Σ u contains 18 and Σ d contains 19 separate contributions from various particles/sparticles [20]. We include contributions from W ± , Z,t 1,2 ,b 1,2 ,τ 1,2 , W 1,2 , Z 1,2,3,4 , t, b and τ , h, H and H ± . We adopt a scale choice Q 2 = mt 1 mt 2 to minimize the largest of the logarithms. The dominant contribution to the terms Σ u,d arise from superpotential Yukawa interactions of third generation squarks involving the top quark Yukawa coupling. For instance, the dominant contribution to Σ u is given by . This expression thus grows quadratically with the stop mass. We adopt the fine-tuning measure from [20], which requires that each of the 40 terms on the right-hand-side (RHS) of Eq. (2) should be of order ∼ m 2 Z /2. Labeling each term as GeV, depending on how much EWFT one is willing to tolerate. This measure of fine-tuning is similar to (but not exactly the same as) Kitano-Nomura [21] but different from Barbieri-Giudice [22] beyond the tree-level. In the following, we will use the fine-tuning parameter where lower values of ∆ correspond to less fine-tuning, and e.g. ∆ = 20 would correspond to ∆ −1 = 5% fine-tuning. Our goal in this paper is to search for parameter choices which 1. Maximize the degree of Yukawa coupling unification, i.e. minimize with each Yukawa coupling evaluated at the GUT scale. Thus, a value of R yuk = 1 would give perfect Yukawa coupling unification.
2. Have as low EWFT as possible (in practice, we will require ∆ 100, or better than 1% EWFT).
3. Have m h ≈ 125 GeV in accord with the recent LHC discovery of a Higgs-like resonance. In practice, we will require 122 GeV < m h < 128 GeV to allow for a roughly 2-3 GeV error in the RG-improved one-loop effective potential calculation of the Higgs mass m h .
We recognize that in addition to EWFT, there also exists a fine-tuning associated with generating particular weak scale SUSY spectra from distinct GUT scale parameters [22], (see also [23,24] for related discussions). Here we adopt the less restrictive weak scale fine-tuning condition, which nonetheless turns out to be indeed very restrictive. In this vein, we regard particular GUT scale parameters as merely a parametrization of our ignorance of the mechanism of SUSY breaking and soft term generation. 2 For instance, in this paper we will require rather low values of superpotential µ parameter to avoid excessive EWFT in Eq. (2). In generic SUSY models, the value of µ is expected to be of order M Planck since it is a dimensionful SUSY-preserving parameter. Excessively large µ can be avoided ala Giudice-Masiero [26] where the µ superpotential term is forbidden by some high scale symmetry, but then is regenerated as a soft SUSY breaking term via a Higgs-Higgs coupling to the hidden sector.
For the remainder of this paper, in Section 2 we present details of our scan over SUSY parameter space, and which constraints are invoked in our analysis. In Section 3, we present the results of our parameter space scans. We will find that requiring ∆ 100 and m h ∼ 125 GeV only allows for R yuk as low as ∼ 1.2 − 1.3. While this degree of Yukawa unification is not optimal, we feel it is still useful in that it might guide model builders towards models including additional GUT scale threshold corrections or extra matter or above-GUT-scale running which may ameliorate the situation. In Section 5, we discuss observable consequences of YUNS for LHC, ILC and dark matter searches. We pay some attention to methods which might allow one to distinguish YUNS from generic NS models at lower tan β values. In Section 6 we present a summary and conclusions.

Parameter space and Yukawa unification
For our calculations, we adopt the ISAJET 7.83 [27] SUSY spectrum generator ISAS-UGRA [28]. ISASUGRA begins the calculation of the sparticle mass spectrum with input DR gauge couplings and f b , f τ Yukawa couplings at the scale Q = M Z (f t running begins at Q = m t ) and evolves the 6 couplings up in energy to scale Q = M GUT (defined as the value Q where g 1 = g 2 ) using two-loop RGEs. At Q = M GUT , we input the soft SUSY breaking parameters as boundary conditions, and evolve the set of 26 coupled MSSM RGEs [29] back down in scale to Q = M Z . Full two-loop MSSM RGEs are used for soft term evolution, while the gauge and Yukawa coupling evolution includes threshold effects in the one-loop beta-functions, so the gauge and Yukawa couplings transition smoothly from the MSSM to SM effective theories as different mass thresholds are passed. In ISASUGRA, the values of SSB terms of sparticles which mix are frozen out at the scale Q ≡ M SU SY = √ mt L mt R , while non-mixing SSB terms are frozen out at their own mass scale [28]. The scalar potential is minimized using the RG-improved one-loop MSSM effective potential evaluated at an optimized scale Q = M SU SY which accounts for leading two-loop effects [30]. Once the tree-level sparticle mass spectrum is computed, full one-loop radiative corrections are calculated for all sparticle and Higgs boson masses, including complete one-loop weak scale threshold corrections for the top, bottom and tau masses at scale Q = M SU SY [31]. Since the GUT scale Yukawa couplings are modified by the threshold corrections, the ISAJET RGE solution must be imposed iteratively with successive up-down running until a convergent sparticle mass solution is found. Since ISASUGRA uses a "tower of effective theories" approach to RG evolution, we expect a more accurate evaluation of the sparticle mass spectrum for models with split spectra (this procedure sums the logarithms of potentially large ratios of sparticle masses) than with programs which make an all-at-once transition from the MSSM to SM effective theories. The fine-tuning measure ∆ described in Sec. 1 has been implemented in ISAJET 7.83 [27]. In models of "natural SUSY" (NS) [32,21,33,34,35,36,37,20], the first requirement to gain a low EWFT is that the µ parameter be of the order of ∼ M Z , while in Yukawaunified SUSY, it is necessary to invoke some manner of Higgs soft term splitting at the GUT scale in order to obtain raditive EWSB. For these reasons, the model parameter space chosen in this study is the two-parameter non-universal Higgs model (NUHM2) where the weak scale values of µ and m A are input in lieu of the GUT scale values of m 2 Hu and m 2 H d . In addition, to allow for (sub)TeV-scale third generation masses (as required by EWFT from Eq. (2) along with at least a partial decoupling solution to the SUSY flavor and CP problems, we allow for split first/second and third generations at the GUT scale. Thus, the parameter space we choose is given by Here, m 16 (1, 2) and m 16 (3) are the first/second and third generation sfermion soft masses, respectively; m 1/2 ≡ M 1 = M 2 = M 3 is the universal gaugino mass parameter; and A 0 ≡ A t = A b = A τ is the universal trilinear coupling. These parameters are defined at M GUT , while tan β, µ, and m A are defined at the weak scale. The top quark mass is set to m t = 173.2 GeV. We search for mass spectra with low EWFT ∆ and low R yuk by performing a vast random scan over the following parameter ranges (masses in GeV units): 50 < tan β < 60, 100 GeV < µ < 250 GeV, 500 GeV < m A < 5000 GeV.
The lower limit on m 1/2 comes from the approximate LHC bound of mg 900 GeV (for mq mg) [38], while the lower bound on m A comes from LHC searches for A, H → τ + τ − which require m A > 500 GeV at tan β ∼ 50 [39]. We require of our solutions that 1. electroweak symmetry be radiatively broken (REWSB), 2. the neutralino Z 1 is the lightest MSSM particle, 3. the light chargino mass obeys the rather model independent LEP2 limit that m W 1 > 103.5 GeV [40], and 4. the light Higgs mass falls within the window m h = 122-128 GeV, where we adopt ±3 GeV as theoretical error on the Higgs mass calculation. 3) The light blue triangles obey the BR(B s → µ + µ − ) constraint, but deviate from the measured BR(B → X s γ) by more than 3σ. Finally, the pink squares satisfy both the BR(B s → µ + µ − ) and BR(B → X s γ) constraints. Only points with ∆ < 1000 and R yuk < 1.5 are shown.
Regarding B-physics constraints, we consider BR(B → X s γ) = (3.55 ± 0.34) × 10 −4 , where experimental and theoretical uncertainties have been added in squares, and BR(B s → µ + µ − ) < 4.2 × 10 −9 at 95% CL. In the following we will impose the BR(B → X s γ) constraint only at the 3σ level (the reason for this is that calculating BR(B → X s γ) with both IsaTools and SuperISO, we observe deviations of the order of 5% in the relevant region of parameter space). For the BR(B s → µ + µ − ) constraint, we assume a theoretical uncertainty of 20%. This leads to the the following limits which we will use throughout the numerical analysis.
Regarding neutralino relic density, we remark here that models of natural SUSY contain a higgsino-like lightest neutralino with thermal abundance of typically Ω TP . This thermal under-abundance can be regarded as a positive feature of NS models in the sense that if one invokes the axion solution to the strong CP problem, then one expects mixed axion-higgsino dark matter, where the higgsino portion is typically enhanced by thermal axino production and decay to higgsinos in the early universe. Thus, a thermal under-abundance leaves room for additional non-thermal higgsino production plus an axion component to the dark matter [41].

Scan results
As our first result, we show in Fig. 1 points from our parameter space scan in the R yuk vs. ∆ plane. All points have m h = 122-128 GeV and obey the current LEP and LHC SUSY  mass limits. The dark blue triangles however violate the B-physics constraints of Eq. (9). These points may still be valid if additional small flavor violating terms are allowed in the theory. The light blue triangles obey the BR(B s → µ + µ − ) constraint, but deviate from the measured BR(B → X s γ) by more than 3σ. Finally, the pink squares satisfy both the BR(B s → µ + µ − ) and BR(B → X s γ) constraints. This color scheme is used throughout the remainder of the paper. We see already from this plot that flavor physics constraints significantly affect the parameter space of Yukawa-unified natural SUSY. This is to be expected, since naturalness requires lighter third generation squarks while Yukawa unification requires tan β ∼ 50: both these effects bolster SUSY contributions to Bphysics observables. (Analogous observations were made in [10,42] in the context of generic YU models.) Without flavor constrains, we can obtain R yuk as low as ∼ 1.2. The BR(B s → µ + µ − ) constraint pushes this up to R yuk 1.27, and the BR(B → X s γ) constraint to R yuk 1.3. Aside from the B-physics constraints, it is intriguing that points with lowest ∆ also have lowest values of R yuk . This is because low ∆ requires light third generation squark masses, while at the same time Yukawa unification requires large SUSY threshold corrections which also require lighter third generation squarks.
In Fig. 2, we show the value of R yuk versus various input parameters for ∆ < 100 (less than 1% fine-tuning). Here, the behavior deviates considerably from t − b − τ unified models with large µ > 0 and a spectrum derived from the radiatively driven inverted  scalar mass hierarchy [7,8,10]. From frame a), we see that YUNS models actually prefer large m 1/2 whereas generic Yukawa unified SUSY models (with R yuk 1.1 but arbitrary EWFT), YUS for short, prefer low m 1/2 . The preference for large m 1/2 helps to avoid LHC constraints on the gluino mass. If we impose the B-physics constraints, then the distribution flattens out with some preference for lower m 1/2 values. In frame b), where R yuk is plotted vs. m 16 (3), we see that lowest R yuk values prefer m 16 (3) ∼ 2 TeV, which leads to rather light third generation squarks and typically violation of B-physics constraints. If we respect B-constraints, then larger values of m 16 (3) ∼ 3 − 7 TeV are preferred, at the cost of larger values of R yuk . In frame c), we plot versus A 0 /m 16 (3). For YUS models, there is a strong preference for A 0 ∼ −2m 16 [7], while for YUNS, the lowest R yuk values are obtained for smaller |A 0 |. Imposing B-constraints, the preference moves to A 0 ∼ (−2 to 0) × m 16 (3). Frame d) shows R yuk vs. tan β. As expected, tan β ∼ 50 is preferred.
The dependence of ∆ on the input parameters is shown in Fig. 3 for points with R yuk < 1.4. In frame a), we see a mild preference by ∆ for low m 1/2 , i.e. the gluino mass can't be too heavy, lest it pushes the stop masses too high, leading to large Σ u .     (3) is again preferred. The large negative A 0 values lead to larger m h values and also can lower the EWFT [24]. In frame d), we see that there is only mild preference for tan β ∼ 48 − 58 values unless B-constraints are respected, in which case tan β 52 is preferred.
In Fig. 4 we show R yuk versus various sparticle and Higgs masses. In frame a), the distribution versus m h is shown, and we see that low R yuk prefers the lower range of m h . This is understandable since low EWFT prefers lower top and bottom squark masses, which may not feed a sufficient radiative correction into the m h computation. These low third generation squark mass solutions also tend to give large contributions to B-constraint. If we impose B-constraints, then m h can live in the 122 − 128 GeV range, at the cost of larger R yuk . In frame b), we show the distribution versus mg. While all solutions-especially low R yuk ones-favor the heavier range of mg (mg ∼ 2 − 5 TeV, likely beyond LHC reach), the solutions obeying B-constraints tend to slightly favor lower mg, possibly within range of LHC14 searches. In frame c), the distribution versus mt 1 is shown. Here, we see a clear demarcation: B-constraints favor mt 1 1.5 TeV to suppress SUSY loop contributions to BR(B → X s γ). This constraint forces the minimum R yuk to move form ∼ 1.18 to about 1.3. Additional small flavor-violating contributions to the  MSSM Lagrangian could alter the predicted BR(B → X s γ) and/or BR(B s → µ + µ − ) rates and thus allow the lower R yuk solutions [47]. In frame d), we show the distribution versus pseudoscalar. Higgs mass m A . We see low R yuk favors the lower range of m A , although values up to and beyond 5 TeV are also possible (at the cost of R yuk ∼ 1.4, however). Since tan β ∼ 50, LHC searches for A, H → τ + τ − will access a significant range of m A in this case. It is also possible for some range of m A 1 TeV for LHC to access bA → bµ + µ − production [48].
In Fig. 5, we show scatter plots of YUNS points with R yuk < 1.4 and ∆ < 100 in a) mt 1 vs. mb 1 and b) mt 1 vs. mb 1 space. From frame a), we see as expected that mt 1 and mb 1 are correlated, with some solutions reaching well below mt 1 ∼ 500 GeV, which should be accessible to LHC searches. However, in this case these points all violate B-constraints, so that requiring B-constraints within measured range requires instead mt 1 , mb 1 1.5 TeV, likely beyond the 14 TeV LHC reach. In frame b), we see also that mb 2 and mt 1 are correlated. This is different from usual NS, where mb 2 can be far above mt 1,2 and mb 1 . The reason here is that f b is large and so there is also a non-negligible contribution to Σ u fromb 1,2 . Imposing B-constraints, we find mt 2 and mb 2 both 2 TeV (the former aids in lifting m h into its measured range).

Benchmark points
In this section we present several YUNS benchmark points, see Table 1, and compare to one YUS benchmark point (HSb, from the "just-so" HS model) from Ref. [43].
For HSb, R yuk ∼ 1.02, which is nearly perfect Yukawa coupling unification. The Higgs mass m h 127.8 GeV is also sufficiently heavy. Unfortunately, the point is now excluded by LHC7 searches for multi-jet+E miss T plus one b-tag searches, since mg is only 351 GeV. We also list in Table 1 Table 1: Parameters and masses in GeV units for HSb [43] and three Yukawa-unified natural SUSY (YUNS) benchmark points. We also show B-decay constraints and dark matter relic density and (in)direct detection cross sections. relaxing R yuk to 1.37. For YUNS1, mg 1.3 TeV, with first/secnd generation squarks at ∼ 19 TeV. The point is likely beyond LHC8 reach, but should be accessible to LHC14 with 10-100 fb −1 . In column 4, we list YUNS2 with R yuk = 1.3, as low as allowed by B-constraints, but with ∆ ∼ 100. This point has mg ∼ 4.3 TeV and mq ∼ 17.5 TeV, so it is likely beyond LHC reach, including a high-luminosity upgrade. While the higgsinolike chargino is only 112.5 GeV, it decays via 3-body mode into a higgsino-like Z 1 with m Z 1 = 106.2 GeV, so that visible decay products are very soft, and likely impossible to observe above SM backgrounds at the LHC. Point YUNS3, listed in column 5, features R yuk as low as 1.19, with ∆ = 48. This point has BR(B → X s γ) = 2.4 × 10 −4 , and BR(B s → µ + µ − ) = 2.4×10 −8 , so it falls out of the B-physics allowed range. While gluinos and first/second generation squarks are beyond LHC reach, the rather light top and bottom squarks may be accessible to LHC searches. All these points have Ω Z 1 h 2 0.11, leaving room for non-thermal higgsino production and axions. This contrasts point HSb, which has a much too thermal abundance and so would need an extremely light axino or huge late-time entropy production to tame this over-abundance [41].
In Fig. 6, we show the Yukawa coupling evolution of f t , f b and f τ versus renormalization group scale Q, from m weak to M GUT for benchmark points YUNS1 and YUNS3. These can be compared to similar plots for YUS, as in e.g. Fig. 6 of Ref. [44]. The SUSY threshold corrections implemented at the scale Q = √ mt 1 mt 2 show up as jumps in the curves. In the case of YUS models, the m b threshold correction is positive due to a largẽ t i W j loop contribution which goes like δ b ∼ (f 2 t /32π 2 )(µA t /m 2 t ) tan β, where both µ and A t are extremely large. For YUNS models, with rather low µ, these loops are suppressed and in the case of YUNS1, thegb i loops actually dominate, and are of opposite sign to thet i W j loops, leading to the slight downward jump of f b and thus bad Yukawa coupling unification. For YUNS3, thet i W j loops are larger, and the jump goes upwards, thus providing better Yukawa coupling unification.

Yukawa-unified natural SUSY: LHC, ILC and DM searches
In this Section, we discuss the observable consequences of YUNS for LHC, ILC and direct and indirect WIMP and also axion searches.

YUNS at LHC
In natural SUSY models, it is favorable to have multi-TeV first/second generation squarks and sleptons because 1. they are safely beyond current LHC searches, 2. they provide at least a partial solution to the SUSY flavor and CP problems and 3. they provide additional suppression of third generation scalar masses via large 2-loop RGE effects. However, this means they are likely beyond any conceivable LHC reach. Third generation squarks may be much lighter, and in generic NS models are naively expected to be below the TeV scale (but see Ref. [24] where 1-4 TeV third generation squarks work just fine, and lift the value of m h into its measured range). In the case of YUNS, with tan β ∼ 50, the combined light squarks and large tan β usually imply violation of B-constraints, and if these are imposed, then top and bottom squarks are beyond 1.5 TeV, and likely inaccessible to LHC searches. If additional sources of flavor violation are invoked, then the B-constraints may be invalid, and then the solutions with much lower R yuk ∼ 1.2 are accessible, along with much lighter top and bottom squarks, potentially accessible to LHC searches. For YUNS, the gluino mass may lie anywhere in the 1-5 TeV range. It has been estimated in Ref. [45] that LHC14 with 100 fb −1 should be able to access gluino pair production in the case of heavy squarks for mg up to 1.8 TeV. In the case of YUNS with heavier top and bottom squarks, theg is expected to dominantly decay via 3-body modes into tt Z i and tb W i final states. The gluino pair events will thus contain multi-jets plus missing energy plus isolated leptons plus several identifiable b-jets [46]. While the higgsino-like chargino and neutralino production cross sections can be large, their decays to soft visible particles, arising from the small energy release in their 3-body decays, will be difficult to detect at LHC above SM backgrounds [35].
In the case where m A 1 TeV, then it may be possible to detect A, H → τ + τ − , especially if these are produced in association with b-jets, e.g. pp → bA, bH production. It may also be possible to detect bA, bH production with A, H → µ + µ − [48], since the production and decay are enhanced at large tan β. In this case, the A, H mass and width may be determined by reconstructing m(µ + µ − ). At large tan β, this width is typically in the tends of GeV range and is very sensitive to tan β. This reaction offers a method to distinguish YUNS from NS, in that the former is expected to occur at tan β ∼ 50.

YUNS at ILC
A linear e + e − collider operating at √ s ∼ 250 − 500 GeV would in many ways be an optimal discovery machine for YUNS. The reason is that by construction µ 250 GeV, so chargino and neutralino pair production should always be available. While the small energy release in W 1 and Z 2 decay is problematic at LHC, it should be much more easily observable in the clean environment of an e + e − collider. In this sense, an e + e − collider operating at √ s ∼ 250 − 500 GeV would be a higgsino in addition to a Higgs factory. It is  Figure 7: Thermal neutralino relic density Ωh 2 versus Z 1 mass (left) and versus Z 1 higgsino fraction (right) for scan points with R yuk < 1.4 and ∆ < 100. Same color code as in Fig. 1.
also possible that some lighter third generation squarks are accessible to ILC with √ s ∼ 1 TeV or CLIC with √ s = 3 TeV, depending if one avoids B-constraints and accepts the low mass, low R yuk solutions.

Higgsino-like WIMPs
A generic prediction of both NS and YUNS models is that the LSP is a higgsino-like WIMP with a typical under-abundance of thermally produced (TP) neutralinos Ω TP Z 1 h 2 ∼ 0.002 − 0.01. However, in cases where m 1/2 is as low as ∼ 300 GeV and µ is as large as 200 − 250 GeV, then there can be substantial bino-higgsino mixing, boosting Ω TP Z 1 h 2 up to 0.11 or even beyond. The situation is illustrated in Fig. 7, where we plot Ω TP Z 1 h 2 versus m Z 1 and versus the Z 1 higgsino fraction from YUNS models satisfying R yuk < 1.4 and ∆ < 100.
The typical under-abundance is an appealing feature if one invokes the Peccei-Quinn solution to the strong CP problem, in which case one must introduce an axion superfield a which contains a pseudoscalar axion a as well as a spin-1/2 axinoã and a spin-0 saxion s. In such models, one expects both saxion and axino masses at or around the SUSY breaking scale, so that dark matter is comprised of an axion-WIMP admixture. In this case, thermal production of axinos and subsequent decay to states such agg bolster the WIMP abundance beyond its TP-value. In addition, axions are produced as usual via coherent oscillations. It is also possible to suppress the WIMP abundance in the mixed a Z 1 cosmology via late-time entropy production from saxion production and decay, although this case tends to be highly constrained by maintaining successful Big Bang nucleosynthesis (BBN). The upshot is that in the mixed a Z 1 dark matter scenario, it may be possible to detect both a WIMP and an axion.
In the case of the YUNS model, the higgsino-like neutralinos have a substantial spinindependent (SI) direct detection cross section, as illustrated in Table 1, where σ SI ( Z 1 p) ∼ 10 −8 pb. While this level of direct detection cross section is now highly constrained by recent XENON100 [49] results, one must bear in mind that the higgsino-like WIMPs would constitute only a portion of the dark matter, so their local abundance might be up to a factor of 30 lower than is commonly assumed. In Table 1, we also list σ SD ( Z 1 p) (relevant for WIMP detection at IceCube) and σv | v→0 , relevant for detection of dark matter annihilation into gamma rays or anti-matter throughout the cosmos. While these cross sections are also at potentially observable levels, again one must take into account that the overall WIMP abundance may be up to a factor of about 30 below what is commonly assumed. Plots of σ SI ( Z 1 p), σ SD ( Z 1 p) and σv | v→0 have been presented in the case of higgsino-like WIMPs in Ref's [35] and [37] and so similar plots will not be reproduced here.

Summary and conclusions
Previous analyses of t − b − τ Yukawa-unified models suffer from two problems: 1. they tend to predict a light gluino mg 500 GeV (alhough solutions are possible for much heavier gluinos) which is now excluded by LHC searches, and 2. they suffer from extreme fine-tuning in the electroweak sector. In this paper, we examined how well the Yukawa couplings could unify in the natural SUSY context, where µ ∼ 100 − 250 GeV, while at the same time requiring the light Higgs mass m h ∼ 122 − 128 GeV. The small value of µ suppresses the larget i W j loop contributions to the b-quark Yukawa coupling which seem to be needed for precision Yukawa coupling unification. Nonetheless, by scanning over NUHM2 parameters with split third generation, we are able to find solutions with R yuk as low as 1.18. These solutions, with very light third generations squarks and tan β ∼ 50 tend to violate B-physics constraints. If B-physics constraints are imposed, then only R ∼ 1.3 can be achieved. However, the B-physics calculations can be modified if additional small flavor-violating terms are allowed in the MSSM Lagrangian, so it is not clear how seriously R 1.3 should be taken.
The Yukawa-unified natural SUSY spectra have important differences from previous YU spectra. The gluino mass can easily be in the 1-4 TeV range, thus avoiding LHC constraints from SUSY searches. The light higgsino-like charginos and neutralinos decay to soft particles, also avoiding LHC searches. However, light higgsinos should be easily accessible to an ILC with √ s ∼ 0.25 − 05 TeV, as is typical of all NS models. In addition, as in all NS models, the lightest neutralino is higgsino-like with a typical thermal underabundance of WIMP dark matter. We regard this as a positive feature in that the WIMP abundance is typically increased in non-standard (but more attractive) cosmologies such as those conatining mixed axion-neutralino cold dark matter.
The question arises as to how to distinguish YUNS from ordinary NS. The YUNS model requires tan β ∼ 50, which leads to large production cross sections for heavy Higgs bosons A and H at LHC, and large widths for these particles. If the rare decays A, H → µ + µ − can be identified with suficiently high statistics (perhaps at a luminosity upgraded LHC), then the widths may be measured with precision, allowing one to highly constrain tan β, and perhaps verify that it is consistent with YUNS models. hospitality of the Aspen Center for Physics which is supported by the National Science Foundation Grant No. PHY-1066293.