Analysis of the TeV-scale mirage mediation with heavy superparticles

We discuss effective models derived from a supersymmetric model whose mediation mechanism of supersymmetry (SUSY) breaking is namely mirage mediation. In this model, light higgsino mass, that is required by the natural realization of the electroweak scale, is achieved by the unification of the soft SUSY breaking parameters at the low scale. Besides, we find that extra Higgs fields are also possibly light in some cases. Then, the effective model is a two Higgs doublet model (2HDM) with higgsinos, and it is distinguishable with namely type-II 2HDM which is widely discussed. In this paper, we study the mass spectrum of SUSY particles and the extra Higgs fields, and summarize the phenomenology in the effective model. We survey the current experimental bounds from the LHC and the dark matter experiments as well as the flavor physics. Then, we point out the expected mass scale of the SUSY particles and reveal the future prospects for the direct and indirect searches. We also discuss the difference between our effective model and the 2HDM in the bottom-up approach.


Introduction
One elegant explanation for the origin of the electroweak (EW) scale is given by the supersymmetric extension of the Standard Model (SM) 1 . In the Minimal Supersymmetric Standard Model (MSSM), the superpartners of the SM particles are introduced and quadratic divergence is canceled out in the Higgs mass squared. The natural realization of the EW scale predicts the superpartners to be the EW-scale, so that a lot of efforts have been devoted to discover the new particles. The latest LHC results on the supersymmetry (SUSY) search, however, show that SUSY particles do not exist below a few TeV in a simple scenario. In addition, the Higgs discovery around the 125 GeV mass [3] indirectly constraints the SUSY scale, since the MSSM predicts that the mass of a neutral Higgs particle is much lower than 125 GeV without large radiative corrections. Then, we might conclude that the SUSY scale is much higher than the EW scale.
On the other hand, it is true that the relation between the SUSY scale and the realization of the EW scale is not so simple in the MSSM. In fact, we can find some explicit SUSY models that are able to be consistent with the experimental results and explain the origin of the EW scale naturally. One simple way is to consider the MSSM with non-universal gaugino masses at the unification scale (∼ 10 16 GeV) [4,5]. A specific mass ratio of wino to gluino can realize the EW scale naturally, even if gluino is heavy. Then, we can evade the strong bounds from the direct SUSY search and explain the 125 GeV Higgs mass [6,7]. The mass ratio is non-trivial, but it is known that such a unique mass spectrum is predicted by a mediation mechanism: namely mirage mediation [8,9]. In this mechanism, the moduli mediation [10][11][12] and the anomaly mediation [13,14] are compatible, and the renormalization-group (RG) correction of the moduli-mediation contribution is canceled by the anomaly-mediation. Phenomenology of the mirage mediation have been studied before the Higgs boson discovery [15][16][17] and after the discovery [18][19][20][21][22]. The mirage mediation in the Next-to-MSSM 2 has been also studied in Refs. [18,19,21].
In the last few years, the LHC run-II excludes light SUSY particles (sparticles); especially, colored sparticles, namely squarks and gluinos, have to be heavier than 2 TeV. Therefore it is worth to study the scenarios that heavy colored sparticles can be compatible with the natural explanation of the EW scale.
In this paper, we reconsider the mirage mediation and discuss the phenomenology based on the latest experimental results. Even if the sparticles, except for the superpartner of the Higgs field (higgsino), are much heavier than the LHC reach, the extra Higgs bosons and the higgsino become lighter than TeV scale in our scenario. The light particles can be tested by the direct LHC search, flavor experiments, precision measurements of the Higgs boson couplings and dark matter (DM) searches. Based on the integrated research, we point out the expected mass scale of the SUSY particles and reveal the future prospects for the direct and indirect searches. We also discuss the difference between our effective model and the 2HDM in the bottom-up approach.
This paper is organized as follows. The mirage mediation is briefly reviewed in Section 2, This alignment is predicted by the TeV-scale mirage mediation, as discussed in previous section.
We have the following parameters: Note that the CP-odd Higgs boson mass m A and the µ-parameter are treated as input parameters instead of soft parameters m 2 Hu and m 2 H d . The size of modulus mediation M 0 is fixed to explain the SM-like Higgs boson mass.
In this setup, all of sparticles, except for higgsinos, reside far above the LHC reach, while the Higgs bosons are expected to be around TeV-scale. One promising way is the direct search for extra Higgs bosons and the higgsino at collider experiments. Besides, the extra Higgs bosons lead deviations from the SM predictions in some processes such as, B s → µ + µ − , B → X s γ and the decay of the 125-GeV Higgs boson. The higgsino becomes the lightest supersymmetric particle (LSP) in this setup. Hence the neutral component of the higgsino is a good DM candidate and the direct and indirect DM searches give significant bounds on our model.
The mass spectrum and the values of observables at several benchmark points are summarized in Table 1. We calculated the mass spectrum of sparticles and their decays by using SuSpect-v2.41 [30] and SDECAY [31] interfaced by SUSY-HIT-v1.5a [32]. The Higgs boson masses, decays and couplings to the SM fermions are calculated by using FeynHiggs-v2.12.2 [33][34][35][36][37][38]. M 0 should be larger than about 6.8 TeV in order to explain the 125 GeV Higgs boson mass, so that superpartners except higgsino are much higher than the LHC reach.

Electroweak symmetry breaking
First of all, we analyze the condition for the EW symmetry breaking. In the MSSM, the Z-boson mass m Z is related to the parameters of the MSSM as where ∆V is loop corrections to the effective potential.
According to the assignment of the modular weights in Eq. (10), the m 2 Hu at a scale Q can be expressed as [26] where the anomalous dimension for the up-type Higgs boson is given by Note that the modulus contribution to the Higgs bosons is absent at Q = M mir because of c Hu = c H d = 0. The RG effects through the U (1) Y gauge coupling are also vanishing in our setup of the modulus mediation. Let us discuss the sensitivity of the Z-boson mass with respect to the parameters in the mirage mediation [39]: Here we do not consider tuning of δm H d and the b-term because they are suppressed by tan 2 β and expected to be small [8]. The degrees of tuning of the µ-parameter and δm 2 Hu are given by where we treat δm 2 Hu as a fundamental parameter that is independent of other parameters. We also assume that the µ-parameter is a fundamental parameter since it is an unique supersymmetry preserving parameter. Then, ∆ µ increases as µ-parameter increases quadratically. We see that ∆ µ ∼ ∆ δm 2 Hu since the minimization condition Eq. (15) requires |µ| 2 δm 2 Hu . We assume that some suitable mechanism fixes α to satisfy M mir = M SUSY . If we consider the KKLT-like setups behind the mirage mediation, the value of α may be fixed by rational parameters and it is irrelevant to this tuning argument. Thus the size of the modulus mediation M 0 is important for the argument.
In Eq. (17), M mir implicitly depends on M 0 through Eq. (8). When we assume that the Kähler potential of the moduli is and only the modulus T gets non-zero F-term vacuum expectation value, the size of the anomaly mediation F C /C relates to M 0 and α as where K T ≡ ∂K/∂T . Using above relations, we obtain Note that c ∼ 1 because m 3/2 M 0 and α ∼ 2 when M mir ∼ M 0 . Therefore the sensitivity to M 0 is We see that the sensitivity is suppressed by the loop factor. In our numerical analysis, we evaluate the correction from the 1-loop Coleman-Weinberg potential [40,41] induced by the third-generation quarks and squarks. Figure 1 shows the degree of tuning of M 0 and the µ-parameter. We see that ∆ M 0 100, 250 and 1000 when M 0 4.6, 6.8 and 14 TeV, respectively. ∆ µ ≤ 100 when |µ| 650 GeV. In the case µ 1.1 TeV where the thermal relic density of the higgsino saturates the observed value of DM, ∆ µ 290 and the degree of tuning is about 0.34%. Since the 125 GeV Higgs boson mass requires M 0 6.0 TeV as we will see later, the tuning of M 0 becomes severer than that of the µ-parameter as long as |µ| 1.1 TeV. In these regions, the electroweak symmetry breaking occurs with the parameter tuning about 0.3%-level. This is much better than other scenarios with sparticles heavier than about 5 TeV.

Direct LHC search
In this setup, the extra Higgs bosons are expected to be lighter than the TeV scale and they could be discovered by the LHC experiments. The neutral Higgs bosons H/A dominantly decay to a pair of bottom quarks with a branching fraction ∼ 80% and sub-dominantly decay to a pair of tau leptons with a branching fraction ∼ 10%. When tan β is small and the bottom and the tau Yukawa couplings are suppressed, the Higgs bosons also decay to a pair of top quarks. The searches for H, A → τ τ however give the most stringent bound on their masses, since there are large amounts of backgrounds for the Higgs bosons decaying to quarks. The experimental analyses constrain where the Higgs bosons are produced by the gluon fusion process or b-associated process [42]. In our analysis, we calculate the production cross section σ(pp → H/A) associated with bottom quark by using MadGraph-v2.5.4 with 5-flavor scheme [43]. We calculate the experimental limits based on the result of Ref.
[42] obtained by 36.1 fb −1 data with the center of energy √ s =13 TeV. The higgsino can be also detected by the collider experiments. There are two neutral and two charged components in the higgsino. Their masses tend to be mostly degenerate and mass differences are smaller than sub-GeV in our scenario. The mass differences come from the mixing with gauginos, and decrease as gaugino masses increase. The mass difference ∆m + ≡ mχ± 1 −mχ0 1 is smaller than about O (0.1GeV) for M 0 6.0 TeV at the tree-level. ∆m + get smaller for larger M 0 , but there are radiative corrections from SM gauge boson loops [44,45]. The radiative correction for the mass difference is given by where Thus the size of the correction is ∆m rad α 2 m Z s 2 W /2 ∼ 350 MeV and would dominate the mass difference. The values of ∆m + including the loop effect in Eq.(26) are shown in Table 1. The LEP experiment gives the most stringent bound on such degenerate higgsinos [46]: the higgsino has to be heavier than about 90 GeV.
On the other hand, the expected signals are so weak that these are buried under the backgrounds in hadron collider experiments and the mass limits could not be severer than the one at the LEP. Recently, the higgsino search exploiting disappearing tracks are proposed in Refs. [47,48]. When the mass difference ∆m + ≡ mχ± 1 − mχ0 1 is smaller than about 0.6 GeV, the 8 chargino dominantly decays to eν e , µν µ and π + and the lightest neutralino. The partial decay widths are given by [45,49] where x l,π = m l,π /∆m + , G F is the Fermi constant, f π ∼ 91.9 MeV is the pion decay constant, θ C is the Cabbibo angle and The decay length of the chargino could be longer than O(0.1cm) in our setup, so that the future hadron collider experiments would give stronger bounds than the LEP experiment. The decay lengths calculated from the above three decay modes are shown in Table 1. The expected exclusion limits at the HL-LHC and the 33-TeV hadron collider are shown in the next section, referring the result shown in Ref. [48]. We refer the most optimistic cases in the future collider experiments, where there is no background events and the location of the second layer of a pixel detector is at a radius 3 cm in the 33-TeV collider.

Precision Higgs coupling measurement
The light extra Higgs bosons change the couplings between the 125-GeV Higgs boson and SM particles; especially, couplings with bottom quark and tau lepton can be largely deviated from the SM prediction [50][51][52][53][54]. There are recent studies about the precision measurements of the Higgs couplings [55][56][57]. Now we define a ratio of a coupling in the MSSM to the one in the SM, where g I hf f (I = MSSM, SM) is the coupling between the 125-GeV Higgs (h) and the SM fermions (f ): κ f are written as where α h is a mixing angle between two CP-even Higgs bosons. The first factor comes from purely Higgs boson mixing, and then it also exists in the type-II two Higgs doublet model. The factor, ∆ f , comes from radiative corrections induced by sparticles. ∆ b , ∆ τ are approximately given by where I(a, b, c) is defined as Note that these corrections are enhanced by tan β. We calculate κ f , using FeynHiggs 2.12.2 which includes the re-summation of the ∆ b corrections [58] and the two-loop SUSY QCD corrections [59]. We have checked that the other couplings of the SM-like Higgs boson are very close to the SM value. The current accuracy of the Higgs coupling measurements at the LHC is O(10%) [60,61]. The future sensitivity may reach a few % in the HL-LHC and the measurement may be more accurate than 1% in the future lepton collider experiments such as ILC and TLEP [62,63]. The HL-LHC will be able to measure κ τ more precisely, while κ b can be determined more precisely at the lepton collider experiments.

Flavor physics
In our model, the flavor violating couplings involving sparticles are only given by the CKM matrix. Then, we can evade the strong bounds from flavor physics. It is known that the stringent bound on this kind of model comes from the rare B meson decays: B → X s γ [64][65][66] and B s → µ + µ − [67,68]. If there is only one extra Higgs doublet that couples to the righthanded down-type quarks as in the type-II 2HDM, there is an one-loop correction involving charged Higgs which does not depend on tan β. In our supersymmetric model, there is also a superpartner of Higgs field, namely higgsino, below 1 TeV. There may be a cancellation between the charged Higgs loop and the higgsino loop [69].

DM physics
The neutral component of the higgsino is the LSP and a good candidate for the dark matter in our scenario 3 . There are several ways to produce the higgsino in the early universe.
If the mirage mediation is realized by the KKLT-like setup and the moduli and the gravitino masses are below PeV-scale, late-time decays of these particles produce the higgsino. However, it is known that the higgsino dark matter produced in such a way tends to overclose the universe [77]. Hence, the production of the higgsino should be suppressed, or the produced higgsino should be diluted by e.g. thermal inflation [78,79]. If the SUSY breaking scale is enough heavy M 0 O(10) TeV and the higgsino mass is enough light m χ O(100) GeV that the higgsinos produced by the late-time decays annihilate enough efficiently, the overclose problem could be circumvented and the relic density could be explained by the higgsino produced by the non-thermal way [80]. Otherwise, the model should be extended to have another light supersymmetric dark matter, such as axino [81,82]. In this case, the dark matter is no longer the higgsino, while the dark matter could be the higgsino dark matter in the other cases.
It would be possible that the relic density of the higgsino is explained by the usual thermal freeze-out mechanism, when the higgsino mass is about 1.1 TeV [44,83]. This occurs if the gravitino and moduli fields decay earlier than the higgsino freezing-out, although the decay widths of the gravitino and the moduli should be larger than the ones naively expected from the KKLT-like setups. Another possibility is that the mass spectrum of the mirage mediation is realized by some other mechanisms than the KKLT-like setup. This scenario looks interesting from the bottom-up point of view.
In Fig. 2, we consider the case that the higgsino dark matter is only thermally produced and saturates the observed relic density : Ω DM h 2 = 0.1188 ± 0.001 [84]. In other case, we study the constraints from the direct and indirect detections of the dark matter, assuming that the observed relic density is fully occupied by the higgsino dark matter. We calculate the thermal relic density of the higgsino, spin-independent cross section and the annihilation cross section by using micrOmegas-4.3.2 [70][71][72].
Note that the discussion about the DM in this Subsection would become irrelevant, if the observed DM relic density is explained by some particle(s) other than the higgsino. If the higgsino does not dominate the dark matter density, the limits from the DM observations are relaxed by the rescaling factor, ξ ≡ Ω χ /Ω DM , where Ω χ is the higgsino density and Ω DM is the total DM density. The cross section for the direct detection should be rescaled by the factor ξ and the cross section for the indirect detection, which observes the cosmic rays originated from the annihilation of the higgsinos, should be rescaled by ξ 2 , when our predictions are compared with the experimental results.
There are studies about the DM in the mirage mediation and other similar setups [15,85,86]. In the previous works, the size of modulus mediation is below sub-TeV, so that the 125-GeV Higgs boson mass can not be realized. Furthermore, the constraints from the direct detections now become very strong, so that the gaugino masses should be heavier than sub-TeV as far as the higgsino dominates the DM relic density. This situation is similar to the one in Ref. [87], where the phenomenology of the higgsino DM is studied in the Non-Universal Gaugino Mass scenario. The most important difference with the previous work in Ref. [87] is that the sparticle masses are much heavier in our scenario, because the size of A-term is fixed by the mediation mechanism at relatively small values and the heavy top squark is required to explain the Higgs boson mass around 125 GeV. Moreover, there are relatively light exotic Higgs bosons.
The direct-detection experiments constrain a cross section of DM-nucleon scattering; espe-cially, the spin-independent cross section gives stringent bounds on the parameter space of the MSSM. The spin-independent cross section for the higgsino DM is approximately given by, where m N is the nucleon mass, m N f N Tq = N |m qq q|N and f T G = 1 − q=u,d,s f Tq . t q = − cot β for q = u, c, t and t q = tan β for q = d, s, b. ± corresponds to the relative sign of the gaugino masses and the µ-parameter. λ hχχ , λ Hχχ are the higgsino-higgsino-Higgs boson couplings, where c W , t W , s 2β and c 2β are short for cos θ W , tan θ W , sin 2β and cos 2β, respectively. We replaced α h to β by taking the decoupling limit m A m h and drop all contributions from the sparticles.
The XENON1T experiment [88,89] and the PandaX-II experiment [90] give the most severe bounds on the spin-independent cross section. The current limit is about 10 −10 pb for m χ ∼ 100 GeV and 10 −9 pb for m χ ∼ 1100 GeV. There are future experiments such as LZ experiment [91], XENON-nT experiment and so on. These experiments will probe wide parameter space as long as the spin-independent cross section times the rescaling factor ξσ SI is on the so-called neutrino floor [92]. If ξσ SI is below the neutrino floor, the signals of the dark matter are buried under the neutrino background.
If the dark matter is dominated by the higgsino, we will observe cosmic rays originated from the higgsino annihilation. In most parameter region, a higgsino pair annihilates to a pair of W-bosons and of Z-bosons through the t-channel higgsino exchange except for the case with m A 2m χ . An important fact is that these processes are independent of other sparticles masses, because these are mediated by the higgsino itself. Thus the indirect detections constrain the higgsino mass itself. If m A 2m χ is satisfied, a higgsino pair annihilate into a pair of bottom quarks (sub-dominantly tau leptons) through the s-channel CP-odd Higgs boson exchange. Although this contribution potentially becomes sizable, the higgsino-higgsino-Higgs boson coupling is suppressed by the gaugino masses as in the spin-independent cross section, so that these processes can not dominate the total annihilation process unless the Higgs boson mass is at the resonance region precisely. We could not observe such enhancement of the annihilation cross section in our numerical analysis in next section.
The current limit on the annihilation cross section comes from the AMS-02 experiment [93] which detects anti-protons. Analyses in Refs. [94,95] show the upper limits on the cross section, and the higgsino mass lighter than about 500 GeV has been already excluded [87] if the higgsino saturates the dark matter density. The future experiments, such as the CTA experiment [96], would reach the cross section, ∼ 1.0 × 10 −26 cm 3 /s, although there is a large uncertainty due to the unknown profiles of the dark matter. Figure 2 shows the observables in the case that the thermal relic density of the higgsino saturates the observed value: Ω χ h 2 = 0.1188 ± 0.001. In this case, the higgsino mass is about 1.1 TeV. In The yellow region shows that the spin-independent cross section is below the neutrino floor. That means that the yellow region is very difficult to be probed by the direct detections even if the dark matter is saturated by the higgsino. The yellow lines show the spin-independent cross section in the unit of ×10 −11 pb. We see that the contribution from the heavy Higgs boson exchange reduces (enhances) the cross section when sgn(µM 1,2 ) = −1 (+1). Figure 3 shows the results on µ-m A planes with the fixed tan β. The meanings of the red, brown, green and yellow regions and lines are the same as in Fig.2. The gray region is excluded by the chargino search at the LEP experiment. The purple region is excluded by the AMS-02 experiment if the higgsino saturates the dark matter relic density. The purple dashed lines show the annihilation cross section in the zero velocity limit in the unit of ×10 −25 cm 3 /s.

Numerical Analysis
The expected upper bounds on µ from the long-lived higgsino search at the HL-LHC (the 33-TeV hadron collider) are described by thick (dashed) red lines in the panel with tan β = 10. Most of the region with ∆ µ ≤ 100 could be covered by the long-lived particle search at the 33 TeV hadron collider. This region can be covered by the indirect detections for the dark matter only if the higgsino dominates the dark matter density. Thus the long-lived search is crucial to test the region where ∆ µ ≤ 100 in the TeV-scale mirage mediation. Unfortunately, the long-lived particle search could not probe our scenario for larger tan β because of the lighter When M 0 10 TeV is satisfied, m A , µ are expected to be below sub-TeV due to the vanishing modulus mediation for the Higgs soft masses. The current limit on the parameter space comes from the direct search for the extra Higgs bosons except large µ-parameter region with tan β = 50, where the bound from Br(B s → µ + µ − ) becomes severe. Those limits are highly dependent on tan β. Note that the 125-GeV Higgs boson couplings to the bottom quarks and the tau leptons deviate from the SM value at 1%-level when m A 1.5 TeV.
The yellow lines show the spin-independent cross section. The cross section is under the neutrino floor at tan β = 10, while whole region in the figures will be covered by the future experiment if the higgsino is the dark matter. The cross section depends on the heavier Higgs boson mass significantly and it increases (decreases) as m A increases, when µ is same (opposite) sign as the gaugino masses as can be read from Eq. (37).
The annihilation cross section at zero-velocity limit is above 1.0 × 10 −26 cm 3 /s at |µ| 1.0 TeV, so that the future indirect detections could cover the region where the degree of tuning the µ-parameter is O(0.1)%-level. The event rate of the annihilation reduces significantly as the abundance of the higgsino decreases because the event rate is suppressed by a square of the rescaled factor ξ = Ω χ /Ω DM .

Conclusion
In this paper, we have studied the TeV-scale mirage mediation scenario that can explain the EW scale naturally without conflict with the current experimental results.
A specific feature of the mirage mediation is that the RG effects for the soft parameters given by the modulus mediation are compensated by the anomaly mediated contributions below the unification scale, and both contributions cancel out at the mirage scale. This cancellation also happens in the soft SUSY scalar masses as well as the gaugino masses, if the modulus mediation respects the condition Eq. (7), namely the mirage condition. Exploiting this unification feature, m 2 Hu and the µ-parameter, which are relevant to the EW symmetry breaking, can be smaller than the other soft parameters when the modulus mediated contributions to the Higgs soft parameters are vanishing at the gauge-coupling unification scale (M U ) 4 and the mirage scale is near the TeV scale.
The mirage condition leads small m 2 Hu , but it also predicts small top squark mixing. Then the top squark mass should be heavier than about 5 TeV to realize the 125-GeV Higgs. This fact indicates that all superpartners except the higgsino are also heavier than 5 TeV in our scenario and consistent with the stringent bounds on masses of superpartners with color at the LHC.
The assignment Eq. (10) leads light extra Higgs bosons, and then the effective theory below the SUSY scale is like the type-II two-Higgs double model (2HDM) accompanied with the higgsinos. The apparent difference from the 2HDM is the existence of the higgsinos. The higgsino could be the dark matter and can be detected by the dark matter experiments. The higgsino in our scenario will be searched by not only the lepton collider but also the hadron collider exploiting disappearing tracks if tan β 10. The light higgsino also influences physical observables such as Br(B s → µ + µ − ), Br(b → sγ) and the decay of the 125-GeV Higgs boson, so that these would be different from the predictions in the 2HDM. The differences are getting significant as tan β increases.
We emphasize that the TeV-scale mirage mediation discussed in this paper still holds motivations of the low-scale supersymmetry. The EW scale can be explained by tuning the parameters at 0.3%-level. The mirage unification features allows us to control the soft parameters at low energy and the parameters relevant to the EW symmetry breaking can remain below the TeV-scale. In our scenario, the higgsino is a candidate for the dark matter. The extra Higgs bosons and the higgsinos are expected to be below TeV-scale, so that they can be explored by the current and upcoming experiments in both the direct and indirect ways.