A realistic intersecting D6-brane model after the first LHC run

With the Higgs boson mass around 125 GeV and the LHC supersymmetry search constraints, we revisit a three-family Pati-Salam model from intersecting D6-branes in Type IIA string theory on the T6/(ℤ2 × ℤ2) orientifold which has a realistic phenomenology. We systematically scan the parameter space for μ < 0 and μ > 0, and find that the gravitino mass is generically heavier than about 2 TeV for both cases due to the Higgs mass low bound 123 GeV. In particular, we identify a region of parameter space with the electroweak fine-tuning as small as ΔEW ~ 24-32 (3-4%). In the viable parameter space which is consistent with all the current constraints, the mass ranges for gluino, the first two-generation squarks and sleptons are respectively [3, 18] TeV, [3, 16] TeV, and [2, 7] TeV. For the third-generation sfermions, the light stop satisfying 5σ WMAP bounds via neutralino-stop coannihilation has mass from 0.5 to 1.2 TeV, and the light stau can be as light as 800 GeV. We also show various coannihilation and resonance scenarios through which the observed dark matter relic density is achieved. Interestingly, the certain portions of parameter space has excellent t-b-τ and b-τ Yukawa coupling unification. Three regions of parameter space are highlighted as well where the dominant component of the lightest neutralino is a bino, wino or higgsino. We discuss various scenarios in which such solutions may avoid recent astrophysical bounds in case if they satisfy or above observed relic density bounds. Prospects of finding higgsino-like neutralino in direct and indirect searches are also studied. And we display six tables of benchmark points depicting various interesting features of our model. Note that the lightest neutralino can be heavy up to 2.8 TeV, and there exists a natural region of parameter space from low-energy fine-tuning definition with heavy gluino and first two-generation squarks/sleptons, we point out that the 33 TeV and 100 TeV proton-proton colliders are indeed needed to probe our D-brane model.

For the intersecting D-brane model building, the realistic SM fermion Yukawa couplings can be realized only in the Pati-Salam models. The three-family Pati-Salam models have been constructed systematically in Type IIA string theory on the T 6 /(Z 2 × Z 2 ) orientifold with intersecting D6-branes [23], and two of us (TL and DVN) with Chen and Mayes found that one model has a realistic phenomenology: the tree-level gauge coupling unification is realized naturally at the string scale, the Pati-Salam gauge symmetry can be broken to the SM close to the string scale, the small number of extra chiral exotic states may be decoupled JHEP08(2014)128 via the Higgs mechanism and strong dynamics, the SM fermion masses and mixings can be explained, the low-energy supersymmetric particle spectra might potentially be tested at the LHC, and the observed dark matter relic density may be generated for the lightest neutralino as the lightest supersymmetric particle (LSP), etc [44,45]. As far as we know, this is indeed one of the best globally consistent string models.
On the other hand, for the first run of the LHC, the big success is obviously the discovery of a SM-like Higgs boson with mass m h around 125 GeV in July 2012 [46,47], which is a little bit too large for the Minimal SSM (MSSM). Such large Higgs boson mass in the MSSM requires the multi-TeV top squarks with small mixing or TeV-scale top squarks with large mixing. In addition, the LHC supersymmetry (SUSY) searches have given strong constraints on the pre-LHC viable parameter space. For instance, the gluino mass mg should be heavier than about 1.7 TeV if the first two-generation squark mass mq is around the gluino mass mq ∼ mg, and heavier than about 1.3 TeV for mq mg [48,49]. Therefore, we should update the phenomenological study of this intersecting D-brane model. For this purpose, we have systematically scan the viable parameter space by considering µ < 0 and µ > 0 scenarios where µ is the bilinear Higgs mass term. We show that there indeed exists such viable parameter space which satisfies the collider and astrophysical bounds including the Higgs boson mass in the range [123,127] GeV. In particular, the absolute value of µ can be as small as 300 GeV in a region of parameter space, where the electroweak fine-tuning (EWFT) is small around ∆ EW ∼ 24-32 (3-4%). We identify another region of parameter space with |µ| 500 GeV and ∆ EW 300, where gluino masses are from 3 to 7 TeV, and the first two-generation squarks and sleptons are in the mass ranges of [4,7] TeV and [2,4] TeV, respectively. Because such parameter space is natural from the low-energy fine-tuning definition while the gluino and first two-generation squarks/sleptons are out of the reach of 14 TeV LHC, this will provide a strong motivation for the 33 TeV and 100 TeV proton-proton colliders. There is some visible preference to achieve the viable parameter space consistent with constraints for µ < 0 case, but this is just an artifact of lack of statistics for µ > 0. Moreover, in order to have the Higgs boson mass from 123 GeV to 127 GeV, and satisfy the LHC low bounds on sparticles and the Bphysics bounds, we require gravitino mass 2 TeV for both cases of µ < 0 and µ > 0. We also present graphs in neutralino-sparticle planes showing various coannihilation scenarios such as neutralino-stau, neutralino-stop, neutralino-gluino, and A-resonance solutions. The solutions, which are consistent with the observed relic density, have gluino masses from 3 to 18 TeV. We also note that in our present data consistent with all bounds, the first two generation squarks are in the mass range [3,16] TeV and the first two generation sleptons can be heavier than 2 TeV but less than 6 TeV. On the other hand for third family squarks, the NLSP light stop satisfying 5σ WMAP bounds is in the mass of 0.5-1.2 TeV, in case of third family slepton, the light stau can be as light as 800 GeV. We have checked status of t-b-τ and b-τ Yukawa unification (YU) scenarios with both signs of µ in our data. For µ < 0 we find solutions with 10% or better YU with typical heavy spectra. The best YU we have achieved in our data set is about 5% consistent with all the constraints including the observed dark matter relic density bound. On the other hand, we do not have better than 12% YU t-b-τ for µ > 0 case. Since we did not perform any dedicate searches to study YU JHEP08(2014)128 in this project otherwise we may have solutions with much better YU. Relaxing the t-b-τ YU constraint to b-τ YU, we have plenty of solutions with 100% YU. For the points with Ωh 2 1 where the lightest neutralino is almost a pure bino, we introduce a lighter state axinoã as the LSP. Thus, the lightest neutralino is the Next to the LSP (NLSP) and can decay to axino viaχ 0 1 → γã. We calculate the lifetime of the NLSP neutralino for various choices of the axion decay constant f a in our data. For f a > 10 14 GeV, the lifetime of the NLSP bino is more than 1 second and may be ruled out by Big Bang Nucleosynthesis (BBN) constraints. We also note that in our data, there are solutions where the lightest neutralino can be a bino, wino, or higgsino type. The lightest neutralino masses are more than 1 TeV for both cases (µ < 0 and µ > 0) in the wino-type solutions, while they are less than 1 TeV in the bino-type solutions and in the mass range of 150-600 GeV in the higgsino-type solutions. Recent studies showed that the scenario with pure wino as dark matter is under siege [50,51]. In our model, the relic density of the wino dominant lightest neutralino can be smaller than the correct relic density, and then the above constraint can be escaped. Otherwise, to solve this problem, we suggest that the wino dominant neutralino is the NLSP and may decay toãγ and hence fulfil the relic density bounds, or we may invoke R-parity violation. Similarly, the higgsino-type solutions suffer underabundance of relic density problem. In such a case we assume that the higgsino-type neutralino makes up only a fraction of the dark matter relic density and the remaining abundance is comprised of axions. We also display graphs for direct and indirect searches for dark matter for our higgsino-like solutions and show that these solutions will be observed or ruled out by the XENON1T experiment. Finally, we present six tables of benchmark points, three for each sign of µ. These points depict various interesting scenarios of our model, namely points with minimum EWFT, various coannihilation and resonance solutions, bino-type, wino-type and higgsino-type solutions. Furthermore, because the lightest neutralino can be heavier than 1 TeV and up to about 2.8 TeV, how to search for such scenario at the 14 TeV LHC is still a challenging question. In short, we do need the 33 TeV and 100 TeV proton-proton colliders to probe such D-brane model. This paper is organized as follows. In section 2 we outline details of the supersymmetry breaking (SSB) parameters, the range of values employed in our scan, the scanning procedure and the relevant experimental constraints that we have employed. In section 3 we briefly describe our definition of EWFT and High scale (GUT) fine-tuning. We discuss results of our scans in section 4. A summary and conclusions are given in section 5.

Phenomenological constraints and scanning procedure
In our realistic intersecting D-brane model, if we do not consider CP violation, the supersymmetry breaking (SSB) soft terms from the non-zero F-terms F u i and F s can be parametrized by Θ 1 , Θ 2 , Θ 3 , Θ 4 ≡ Θ s , and gravitino mass m 3/2 where 4 i=1 Θ 2 i = 1 [45]. Thus, we can reparametrize Θ i with i = 1, 2, 3 in terms of γ 1 , γ 2 and Θ 4 as follows Thus, the supersymmetry breaking soft terms are [45] where M 1,2,3 are the gauginos masses respectively for U(1) Y , SU(2) L and SU(3) c gauge groups, A 0 is the trilinear scalar coupling, m L and m R are the soft mass terms respectively for the left-handed and right-handed squarks and sleptons, and m H u,d are the SSB Higgs soft mass terms. We employ the ISAJET 7.84 package [52] to perform random scans over the parameter space given below. In this package, the weak scale values of gauge and third generation Yukawa couplings are evolved to M GUT via the MSSM renormalization group equations (RGEs) in the DR regularization scheme. We do not strictly enforce the unification condition g 3 = g 1 = g 2 at M GUT , since a few percent deviation from unification can be assigned to unknown GUT-scale threshold corrections [53][54][55]. With the boundary conditions given at M GUT , all the SSB parameters, along with the gauge and Yukawa couplings, are evolved back to the weak scale M Z .
In evaluating Yukawa couplings the SUSY threshold corrections [56] are taken into account at the common scale M SUSY = √ mt L mt R . The entire parameter set is iteratively run between M Z and M GUT using the full 2-loop RGEs until a stable solution is obtained. To better account for leading-log corrections, one-loop step-beta functions are adopted for gauge and Yukawa couplings, and the SSB parameters m i are extracted from RGEs at appropriate scales m i = m i (m i ). The RGE-improved 1-loop effective potential is minimized at an optimized scale M SUSY , which effectively accounts for the leading 2-loop corrections. Full 1-loop radiative corrections are incorporated for all sparticle masses.
The requirement of radiative electroweak symmetry breaking (REWSB) [57][58][59][60][61] puts an important theoretical constraint on the parameter space. Another important constraint comes from limits on the cosmological abundance of stable charged particle [62]. This excludes regions in the parameter space where charged SUSY particles, such asτ 1 ort 1 , become the LSP. We accept only those solutions for which one of the neutralinos is the LSP and saturates the dark matter relic abundance bound observed by WMAP9.

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We have performed Markov-chain Monte Carlo (MCMC) scans for the following parameter range where tan β is the ratio of the vacuum expectation values (VEVs) of two Higgs fields. We use m t = 173.3 GeV [63], and m DR b (M Z ) = 2.83 GeV which is hard-coded into ISAJET. We have done our scans with both µ < 0 and µ > 0, and find that our results are not too sensitive to one or two sigma variation in the value of m t [64].
In scanning the parameter space, we employ the Metropolis-Hastings algorithm as described in [65,66]. The collected data points all satisfy the requirement of REWSB, with the neutralino in each case being the LSP. After collecting the data, we require the following bounds (inspired by the LEP2 experiment) on particle masses: We also use IsaTools package [67][68][69][70][71] and ref. [72] to implement the following B-physics constraints: As far as the muon anomalous magnetic moment a µ is concerned, we require that the benchmark points are at least as consistent with the data as the Standard Model.

Fine-tuning
We use the latest (7.84) version of ISAJET [52] to calculate the fine-tuning (FT) conditions at the electroweak scale (EW) M EW and at the high scale (M HS ). Brief description of these parameters is given in this section.

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The Z boson mass M Z , after including the one-loop effective potential contributions to the tree level MSSM Higgs potential, is given by the following relation: where Σ u u and Σ d d are the contributions coming from the one-loop effective potential defined in [80]) and tan β ≡ H u / H d . All parameters in eq. (3.1) are defined at the M EW .

Electroweak scale fine-tuning
We follow [80] in order to measure the EW scale fine-tuning condition, the following definitions are used: with each C Σ u,d u,d (k) less than some characteristic value of order M 2 Z . Here, k labels the SM and SUSY particles that contribute to the one-loop Higgs potential. For the fine-tuning condition we have It is important to note that ∆ EW depends only on the weak scale parameters of the theory, therefore fixed by the particle spectrum. Hence, it is independent of how SUSY particle masses arise. Lower values of ∆ EW correspond to less fine tuning, for example, ∆ EW = 10 implies ∆ −1 EW = 10% fine tuning. Moreover, this condition of EW scale fine-tuning is different from the fine-tuning definition in refs. [81,82] beyond the tree level (for more details see [83]).

High scale fine-tuning
From eq. (3.1) it is evident that ∆ EW does not give any informations about the possible high scale origin of the parameters in the equation. In order to address fully the fine-tuning condition we need to write down weak-scale parameter m 2 H u,d in eq. (3.1) and with their explicit dependence on the (HS) as: Here m 2 H u,d (M HS ) and µ 2 (M HS ) are the corresponding parameters renormalized at the high scale, and δm 2 H u,d , and δµ 2 measure how the given parameter is changed due to Renormalization Group Equation (RGE) evolution. Eq. (3.1) can be re-expressed in the form As we did before, we follow ref. [80] and introduce the following parameters (3.6) and the high scale fine-tuning measure ∆ HS is defined to be In short, ∆ EW includes information about the minimal amount of fine-tuning present in the low scale model for a given SUSY spectrum, while ∆ HS better represents the finetuning that is present in high scale model.

Numerical results
In figure 1, we present graphs for various parameter given in eq. (2.3). The left and the right panels show solutions for µ < 0 and µ > 0 scenarios, respectively. Color coding is given as, grey points satisfy REWSB and neutralino as an LSP conditions. Aqua points satisfy the mass bounds and B-physics bounds. Magenta points are subset of aqua points and also represent 123 GeV m h 127 GeV. Red points are subset of magenta points and also satisfy WMAP9 5σ bounds.
We see that in our scans, in Θ 1 − Θ 2 plane for both cases, the range of red points for Θ 1 is −0.6 Θ 1 0.6, but most of the points are concentrated in the range -0.4 to 0.4, while for Θ 2 most of the points are in the range of large values 0.4-0.8. But we also have some red points -0.6 to -0.4. On the other hand, magenta points can be more or less anywhere in the plot. We see that for Θ 1 , we have solutions for its entire range in contrast to Θ 2 where points mostly have relatively large absolute values. In Θ 1 − Θ 3 plane we see that red points favor positive values of Θ 1 and Θ 3 as we have also seen in Θ 1 − Θ 2 plane. We also see some red points for small negative values of Θ 1 and but large negative values of Θ 3 . Magenta points are every where but in contrast to Θ 1 − Θ 2 plane, here large density of points are around the centre of the plot. In the last panel we have plot in Θ 3 − Θ 2 plane. Here too, we see that the red points lie mostly in large positive ranges of Θ 2 and Θ 3 . In case of magenta points, as compared to other panels, here we have some kind of polarisation and we do not have magenta points in the center.
We calculate (SSB) parameters using eqs.         maximum value of m L ≈ 10 TeV. In the tan β − m H u,d plane, right panel seems to be more populated in red points as compared to the left panel. We can see that in the right panel red points are 10 tan β 60 with 0 m H u,d 7 TeV. This apparent difference is due to lack of data in the case of µ < 0. By generating more data, we can reduce the apparent differences. In tan β − A 0 plane too, we see the same situation. But one thing is clear from both panels which is that data favours A 0 < 0.
Plots in µ − ∆ EW and ∆ HS − ∆ EW planes are shown in figure 4. Color coding is same as in figure 1. The top left and right panels depict plots with large ranges of parameters as compared to the bottom left and right panels. Moreover, the left and right panels represent µ < 0 and µ > 0 scenario, respectively. With large parameter ranges, the top two panels almost look like the mirror images of each other. But from the left panel we see that it is relatively easy to have WMAP9 compatible red points with µ < 0 as compared right panel with µ > 0 where the minimal value of ∆ EW for red points is about 2800. In order to investigate further we redraw the same plot with small ranges of parameters. We immediately note that there are some red points below ∆ EW 200. We also note that the minimal values of ∆ EW with and without WMAP9 bounds are 56(1.78% FT) and 24(4.1% JHEP08(2014)128 FT) respectively. On the other hand in the right panel we see that the minimum value of ∆ EW for magenta points is 31(3.2% FT). We have also checked that in the right panel, points with relatively small values of ∆ EW have relic density of about 1. This shows that if we try more harder we can get some solutions with small ∆ EW and compatible with the WMAP9 bounds. In the bottom left and right panels we show plots in ∆ HS − ∆ EW plane. Here we see that for the entire data ∆ HS ∆ EW . We note that for µ < 0 case the minimal value of ∆ HS is 1125 (0.08% FT) with ∆ EW value of 297(0.33% FT), while we have 963(0.1% FT) and 285(0.35% FT) for ∆ HS and ∆ EW respectively for µ > 0. It was shown in dedicated studies of natural supersymmetry [84,85]      We know that the LHC is a color particle producing machine. Among the color particles, gluinos are the smoking guns for the SUSY signals. Recent analysis have put limits of gluino mass mg 1.7 TeV (for mg ∼ mq) and mg 1.3 TeV (for mg mq) [48,49]. In figure 6 we present plots in mg − ∆ EW and mg − µ planes. Color coding is same as in figure 1 except we do not apply gluino mass bounds mentioned in section 2. The top left and right panels depict plots with large ranges of parameters as compared to the bottom left and right panels. Moreover, the left and right panels represent µ < 0 and µ > 0 scenario receptively. Here we show that in both scenarios we have heavy gluinos as M 3 is a free parameter in our model. Such solutions can easily evade the above mentioned LHC bounds on gluino and squarks. In top left frame, we see that we have mg 3 TeV for small values of ∆ EW in case of red points. Interestingly, there exists a region of parameter space with |µ| 500 GeV and ∆ EW 300, where gluino masses are from 3 to 7 TeV, and the first two-generation squarks and sleptons are respectively in the mass ranges [4,7] TeV and [2,4] TeV. Because such parameter space is natural from low-energy fine-tuning definition while the gluino and first two-generation squarks/sleptons can not be probed at the 14 TeV LHC, this will JHEP08(2014)128 provide a strong motivation for 33 TeV and 100 TeV proton-proton colliders. In the top right frame, we have red points around mg ∼ 5 TeV with ∆ EW ∼ 2000. Even if we consider magenta points, we see that we lose very tiny amount of data because of LHC bounds on gluino mass and most of our data remains intact. We also note that in our model ∆ EW can be small over the gluino mass range of 2 to 10 TeV (magenta points). It is shown in [86] that the squarks/gluinos of 2.5 TeV, 3 TeV and 6 TeV may be probed by the LHC14, High Luminosity (HL)LHC14 and High Energy (HE) LHC33, respectively. This clearly shows that our models have testable predictions. Moreover, in future if we have collider facility with even higher energy, we will be able to probe over even larger values of sparticle masses.
We present results with neutralino mass versesτ 1 , A andχ ± 1 masses in figure 7. Color coding is same as in figure 1 and same panel description. Solid black lines are just to guide the eyes, where we can expect to have coannihilation and resonance solutions. In the top left and right panels we have plots in mχ0 1 − mτ 1 plane. We here note that in the left panel, if do not care about the stringent WMAP9 5σ bounds we haveτ 1 nearly degenerate with χ 0 1 from 0.3 TeV to 2.8 TeV. But the solutions satisfy WMAP9 5σ bounds have stau mass in the range of 1.2 TeV to 2.8 TeV. In the right panel, stau mass range is 0.8 TeV to 2.6 TeV while we see the solutions without WMAP9 bonds have the have same stau mass range as we have in the left panel. Here we also note that the next to NLSP (NNLSP) mχ± 1 is close to NLSP mτ in mass. Their masses also lie within the 20% of LSPχ 0 1 mass. In the bottom left and right panels of the figure we present plots in mχ0 1 − m A plane. We see that, in both JHEP08(2014)128 panels we have A-resonance solutions for more than 1 TeV m A without WMAP9 bounds. But if WMAP9 5σ consistent points have m A 2 TeV. Graphs in mχ0 1 − mt 1 and mχ0 1 − mg planes are shown in figure 8 with the same color coding and panel description given in figure 1, except in middle and bottom panels we do not apply gluino bounds mentioned in section 2. From top left panel we see that we have two red points compatible with the WMAP9 bounds and representing neutralino-stop coannihilation scenario with mass around 570 GeV and 1.2 TeV respectively. On the other hand in the right panel we do not have red points along the line but we know that it is just because of lack of statistics (for studies on color and charge breaking vacua in the MSSM, see [87][88][89]). In the middle left and right panels we show graphs in mχ0 1 − mg. In both cases we see that there are no WMAP9 compatible red points. But we do note that we have some magenta solutions where gluino and neutralino masses are almost degenerate and Ωh 2 < 1. In the right panel we see only one magenta point near the black line but we can always generate more data around this point. Graphs in the bottom panels show that in our model, we can accommodate gluinos as heavy as 18 TeV consistent with WMAP9 5σ bounds. Such a scenario suggest that there should be very high energy collider in order to probe such model points.
We quantify t-b-τ and b-τ the Yukawa coupling unification (YU) via the R-parameter where y t , y b and y τ are Yukawa couplings at the scale of the Grand Unified Theory (GUT).
In figure 9 we present graphs in tan β−R tbτ and tan β−R bτ planes. Color coding is same as in figure 1. The left panels represent graphs in tan β −R tbτ and tan β −R bτ planes in case of µ < 0 and the right panels represent plots in tan β − R bτ and tan β − R tbτ for µ > 0. The horizontal black dashed line represents 10% or better t-b-τ (b-τ ) YU. We see in the top left panel that in our scans we have 10% or better YU solutions for tan β ∼ 50−60. The minimal value for R tbτ we have is 1.05 (5% YU). We note that red points below the dashed line have m h ∼ 125 GeV, gluino in the mass range of 8 TeV to 10 TeV, the first two generation squarks and sleptons are in the mass ranges of 8 TeV to 9.5 TeV and 3 TeV 3.5 TeV respectively. The third generation squarks and sleptons lie in the mass ranges of 5 TeV to 7.5 TeV and 1.3 TeV to 3 TeV. They also have large values for |µ| (∼ -5 TeV to -4 TeV) and ∆ EW ∼ 4000 to 8500. More or less magenta points also have the similar mass spectrum. It was shown in more exhausted studies (see e.g [90][91][92][93] and references there in) with non-universal gaugino masses that one can have 100% YU with the LHC testable predictions.
In the bottom left panel we have b-τ YU solutions. Since this is a less constraint situation, we have 10% or better YU solutions for a wider range of tan β, i.e., 30 tan β 60. Here, the minimal value of R bτ is about 1.04 (4% YU). Moreover, the particle mass spectra also have slightly wider ranges as compared to t-b-τ YU case. We also note that those magenta points, which do not satisfy WMAP9 bounds, have more or less the same mass ranges as given above.

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In the top right panel, we see that we do not have even magenta solution with 10% or better t-b-τ YU with µ > 0. It was noticed that in a SUSY SO(10) GUT with non-universal SSB gaugino masses at M GUT and µ > 0, t-b-τ Yukawa unification [94][95][96] can lead one to predict the lightest CP even Higgs boson mass to be 125 GeV [97][98][99]. Even if we consider gaugino-universality 10% or better t-b-τ YU can be achieved consistent with the LHC bounds [100] but very hard to satisfy relic density bounds. In our scans, we have solutions with 12% t-b-τ YU, if we do dedicated searches for better YU solutions, we can get them. Since here, we are not so keen to have 100% YU solutions but to give a flavor that our model can admit such solutions. In the bottom right panel we see that we have only three red points. On the other hand we have plenty of magenta points with 10% or better b-τ YU. In fact, we have R bτ =1, i.e., 100% b-τ YU solutions for tan β ∼ 30-55. In order to save such solutions we have to add some extra physics to the MSSM. In such scenario we can consider SUSY models augmented with Peccei-Quinn(PQ) solution to the strong CP problem [101][102][103][104] (PQMSSM). In SUSY context the axino field is just one element of an axion supermultiplet. The axion supermutiplet contains a complex scalar field, whose real part is the R-parity even saxion field s(x) and whose imaginary part is the axion field a(x). The supermutliplet also contains an R-parity odd spin half Majorana field, the axinoã(x) [105][106][107]. 6 In case where Ωh 2 1, one way to have relic density within the observed range if we assume theχ 0 1 may not be the LSP, but instead decays to much lighter state, such asχ 0 1 → γã, whereã is axino. In such a scenario we have mixed axion/axino (aã) dark matter [109,110]. In this way the neutralino abundance is converted into an axino abundance with [111,112] It is important to know the life time (τ ) of decaying neutralino. If it is more than 1 second, it can disturb Big Bang Nucleosynthesis (BBN) (see [113] and references there in). We first calculate mã for a given mχ0 1 and its relic density Ω 2 χ 0 1 by assuming relic density of axino Ωãh 2 =0.11 by using eq. (4.2). We then follow [114] to calculate the lifetime for the decaying NLSP neutralino. We use C aY Y = 8/3 in the DFSZ model [115,116], N =6 (the color anomaly model dependent factor). We present our calculations in figure 10, where we display the NLSP bino-like neutralino mass (mχ0 1 ) versus its lifetime (τ ). Panel description is same as in figure 1. Here points in various colors correspond to various choices of the axion decay constant f a values. The light-green, light-blue, orange, black, dark-green and blue represent f a = 10 10 − 10 15 GeV, respectively. The black solid line show τ =1 second. In the left panel, the plot appears somewhat flat as compare to the right plot in right panel because of small mass range. From both the frames, it is clearly visible that for f a = 10 15 GeV, life time of NLSP bino is more than 1 second. For f a = 10 14 GeV, in the right panel, points with bino mass greater than 1 TeV are allowed.
In another approach to reduce relic density is to assume the additional late decaying scalar fields are present in the model. These fields may get produced at large rates via coherent oscillations. If they temporarily dominate the energy density of the Universe, and then decay to mainly SM particles, they may inject considerable entropy into the cosmic JHEP08(2014)128 soup, thus diluting all relics which are present at the time of decay. Entropy injection can occur at large rates for instance from saxion production in the PQMSSM [117][118][119][120][121][122][123][124][125], or from moduli production and decay, as is expected in string theory [126][127][128][129][130][131][132]. However, it was shown in [133] that the efforts to dilute the relic density of neutralino below the observed dark matter relic density through entropy injection from saxion decays such as saxion decays to gluon violate the CMB bound on ∆N eff , where ∆N eff is the apparent number of additional effective neutrinos.
On the other hand, the solutions with good YU may also have small relic density Ωh 2 ∼ 10 −5 − 10 −2 . In such cases the neutralino abundance can be augmented in the PQMSSM case where mã > mχ0 1 and additional neutralinos are produced via thermal axino production and decay mã → mχ0 1 γ [121][122][123][124][125]. In these cases, the CDM tends to be neutralino dominated with a small component of axions.
In figure 11 we show graphs in mχ0 1 − mχ± 1 plane with the same panel description as in figure 1. The top left and right frames have same color coding as in figure 1. From these frames, it is apparent that we have solutions from 0.1 TeV to 2.8 TeV. In bottom frames we further analyse these points on the basis of neutralino composition. Here orange, green and brown points represent neutralino with more than 90% wino, more than 80% bino and more than 50% higgsino composition, respectively. It is to be noted that orange and the green points satisfy all constraints given section 2 but brown point do not satisfy relic density bounds. Here, we want to show that in our scans where the neutralino and chargino masses are almost degenerate, and neutralino LSP can be of bino, wino and higgsino like. We immediately see that in both cases (µ < 0 and µ > 0), wino-type neutralino have masses more than 1 TeV. On the other hand bino-like solutions have masses less than 1 TeV while higgsino-type solutions have mass range of 150 to 600 GeV. It is shown in [50,51] that for NFW and Einasto distribution, the entire mass range of thermal wino dark matter from 0.1 to 3 TeV may be excluded. In a recent study [134], wino as dark matter candidate is excluded in the mass range bellow 800 GeV from antiproton and between 1.8 TeV to 3.5 TeV from the absence of a γ-ray line feature toward the galactic center. Since our bino-like points have some admixture of higgsinos and that is why they have large nucleon-neutralino scattering cross section. Such solutions are also under stress because of the current upper bound set by XENON100 [135]. Here, we argue that such wino-like (bino-like) neutralino solutions may avoid the above mentioned bounds. For example, the wino-like neutralino density is smaller than the observed density. Otherwise, instead of treating them as the LSPs we assume that they are the NLSP and may decay to axino and γ as we have discussed above. Similarly, we can also assume the mechanism of late decaying fields via coherent oscillations or production of moduli and their decay as we argued previously. In addition to it, we can also invoke R-parity violation scenario, where the bino LSP and similarly winolike neutralino can decay to the SM fermions via sfermion exchange. In order to address the issue of underabundance of higgsino-like solutions we argue that mainly higgsino-like neutralino by itself does not make a good cold dark matter candidate and we need additional dark matter candidates to match the observed dark matter relic density. For this purpose we assume that the higgsino could make up only a fraction of the relic dark matter and the remaining abundance is comprised of axions produces through the vacuum misalignment JHEP08(2014)128 mechanism [137][138][139][140][141]. This is why we could expect the higgsino relic density somewhat suppressed between 1 − 15 in the present universe. This not only provides us with the opportunity to look for higgsinos, despite the fact that they would only constitute a fraction of the measured relic dark matter abundance but also the possibility to detect axions. We would also like to mention that our higgsino-like solutions especially for ∆ EW 50 more or less look like the solutions form radiative natural SUSY [142]. Since such solutions tend to have large direct and indirect neutralino detection rates, let us check the status of our higgsino-like solutions. We will follow [143]. In the left panel of figure 12 we plot rescaled higgsino-like neutralino spin-independent cross section ξσ SI (Z 1 p) versus m(higgsino) (in this figure for both panels we have combined solutions with µ < 0 and µ > 0). The orange solid line represents the current upper bound set by the CDMS experiment and black solid line depicts upper bound set by XENON100 [135], while the orange (black) dashed line represents future reach of SuperCDMS [144] (XENON1T [145]). We rescale our result by a factor ξ = ΩZ 1 h 2 /0.11 in order to account for the fact that the local relic density might be much less than the usually assumed value ρ local 0.3 GeV/cm 3 as pointed out in [146]. Here, we see that all the points lie below the current upper bounds set by CDMS XENON100 experiments. It is very clear that the future experiments like XENON1T will be able to probe almost all of our model points. This shows our results are in agreement with [143] where it was shown that all higgsino points could be tested by the XENON1T and one could discover neutralino (WIMPs) or exclude the concept of electroweak naturalness in R-parity conserving natural SUSY models. In right panel of figure 12, we have a plot of (non-rescaled) higgsino-like neutralino spin-dependent cross section σ SD (Z 1 p) versus m(higgsino). The IceCube DeepCore and future IceCube DeepCore bounds are shown in black solid line and black dashed line [147]. Color coding is same as in left panel. Here we do not rescale our results because the IceCube detection depends on whether the Sun has equilibrated its core abundance between capture rate and annihilate rate [148]. It was shown in [149] that for the Sun, equilibrium is reached for almost all of SUSY parameter space. In this plot we see that the future IceCube DeepCore searches will be able to probe our entire set of solutions in our present scans.
In tables 1-3, we list benchmark points for µ < 0 case. All of these points satisfy the sparticle mass, B-physics and Higgs mass constraints described in section 2. In table 1, point 1(2) represents the minimal value of ∆ EW not consistent and consistent with WMAP9 5σ bounds, while points 3-5 respectively correspond to the minimal value of ∆ HS , best point with t-b-τ and b-τ YU, an example of heavy gluino solution. Points 3 and 4 also satisfy WMAP9 5σ bounds. In table 2, points 1, 2, 3 and 4 display neutralino-stau, neutralinostop, m A -resonance and neutralino-gluino coannihilation, respectively. Point 4 is the case where relic density is below WMAP9 5σ bounds. In table 3 In tables 4-6, we display benchmark points for µ > 0 case consistent with the sparticle mass, B-physics and Higgs mass constraints described in section 2. In table 4, points 1-4 respectively correspond to the minimal value of ∆ EW , minimal value of ∆ HS , best point with b-τ YU, an example of heavy gluino solution. Points 3 and 4 also satisfy WMAP9 5σ bounds. Table 5 and table 6 have similar description as table 2 and table 3.

Discussions and conclusion
The three-family Pati-Salam models have been constructed systematically in Type IIA string theory on the T 6 /(Z 2 × Z 2 ) orientifold with intersecting D6-branes [23]. It was found that one model has a realistic phenomenology [44,45]. Considering the Higgs boson mass around 125 GeV and the LHC supersymmetry search constraints, we have revisited this three-family Pati-Salam model in details. We systematically scanned the viable parameter space for µ < 0 and µ > 0, and found that in general the gravitino mass is heavier than about 2 TeV for both cases because of the Higgs boson mass low bound 123 GeV. In particular, we identified a natural region of parameter space where the electroweak finetuning can be as small as ∆ EW ∼ 24-32 (3-4%). Also, we found another interesting region of parameter space with |µ| 500 GeV and ∆ EW 300, where the mass ranges for the gluino, and first two-generation squarks and sleptons are [3,7] TeV, [4,7] TeV, and [2,4] TeV, respectively. This will provide a strong motivation for 33 TeV and 100 TeV proton-proton colliders since it is natural from low-energy fine-tuning definition while the gluino and first two generation squarks/sleptons are heavy. In the whole viable parameter space which is consistent with all the current experimental constraints including the dark matter relic density bounds, the gluino mass range is [3,18] TeV, the first two-family squarks have masses from 3 to 16 TeV, and the first two-family sleptons have masses from 2 to 7 TeV. Thus, the viable parameter space with heavy gluino and squarks is even out of reach of the 100 TeV proton collider [150]. On the other hand, for the third-family sfermions, the NLSP light stop satisfying 5σ WMAP bounds is in the mass range [0.5, 1.2] TeV, and the light stau can be as light as 800 GeV. We also showed various coannihilation and resonance scenarios through which the observed dark matter relic density can be achieved. Interestingly, the certain portions of parameter space have excellent t-b-τ and b-τ Yukawa coupling unification. Also, we highlighted the regions of parameter space where the LSP neutralino can be a bino, wino, or higgsino. We discussed various scenarios in which such solutions may avoid recent astrophysical bounds in case if they satisfy or above the correct dark matter relic density bounds. Prospects of finding higgsino-like neutralino in direct and indirect searches were shown and discussed as well. To be concrete, we displayed six benchmark tables depicting various interesting features of our model. Furthermore, because the LSP neutralino can be heavier than 1 TeV and up to about 2.8 TeV, how to test such scenario at the 14 TeV LHC is still a big question. Therefore, the 33 TeV and 100 TeV proton-proton colliders are indeed needed to probe our D-brane model.

JHEP08(2014)128
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