# MSSM Dark Matter in Light of Higgs and LUX Results

Authors: W. Abdallah; S. Khalil
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

The constraints imposed on the Minimal Supersymmetric Standard Model (MSSM) parameter space by the Large Hadron Collider (LHC) Higgs mass limit and gluino mass lower bound are revisited. We also analyze the thermal relic abundance of lightest neutralino, which is the Lightest Supersymmetric Particle (LSP). We show that the combined LHC and relic abundance constraints rule out most of the MSSM parameter space except a very narrow region with very largetan⁡β  (~50). Within this region, we emphasize that the spin-independent scattering cross section of the LSP with a proton is less than the latest Large Underground Xenon (LUX) limit by at least two orders of magnitude. Finally, we argue that nonthermal Dark Matter (DM) scenario may relax the constraints imposed on the MSSM parameter space. Namely, the following regions are obtained: m0≃O(4) TeV and m1/2≃600 GeV for low tan⁡β  (~10); m0~m1/2≃O(1) TeV or m0≃O(4) TeV and m1/2≃700 GeV for large tan⁡β  (~50).


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## Body

## 1. Introduction

The most recent observations by the Planck satellite confirmed that 26.8% of the universe content is in the form of DM and the usual visible matter only accounts for 5% [1]. The LSP remains one of the best candidates for the DM [2, 3]. It is a Weakly Interacting Massive Particle (WIMP) that can naturally account for the observed relic density of DM.Despite the absence of direct experimental verification, Supersymmetry (SUSY) is still the most promising candidate for a unified theory beyond the Standard Model (SM). SUSY is a generalization of the space-time symmetries of the quantum field theory that links the matter particles (quarks and leptons) with the force-carrying particles and implies that there are additional “superparticles” necessary to complete the symmetry. In this regard, SUSY solves the problem of the quadratic divergence in the Higgs sector of the SM in a very elegant natural way. The most simple supersymmetric extension of the SM, which is the most widely studied, is known as the MSSM [4–11]. In this model, certain universality of soft SUSY breaking terms is assumed at grand unification scale. Therefore, the SUSY spectrum is determined by the following four parameters: universal scalar mass m0, universal gaugino mass m1/2, universal trilinear coupling A0, and the ratio of the vacuum expectation values of Higgs bosons tan⁡β. In addition, due to R-parity conservation, SUSY particles are produced or destroyed only in pairs and therefore the LSP is absolutely stable, implying that it might constitute a possible candidate for DM, as first suggested by Goldberg in 1983 [12]. So although the original motivation of SUSY has nothing to do with the DM problem, it turns out that it provides a stable neutral particle and, hence, a candidate for solving the DM problem.The landmark discovery of the SM-like Higgs boson at the LHC, with mass ~125 GeV [13, 14], might be an indication for the presence of SUSY. Indeed, the MSSM predicts that there is an upper bound of 130 GeV on the Higgs mass. However, this mass of lightest Higgs boson implies that the SUSY particles are quite heavy. This may justify the negative searches for SUSY at the LHC run-I [15–18]. However, it is clearly generating a new “little hierarchy problem.”Moreover, the relic density data [1] and upper limits on the DM scattering cross sections on nuclei (LUX [19] and other direct detection experiments [20, 21]) impose stringent constraints on the parameter space of the MSSM [22–25]. In fact, combining the collider, astrophysics, and rare decay constraints [26–36] almost rules out the MSSM. It is tempting therefore to explore well motivated extensions of the MSSM, such as NMSSM [37, 38] and BLSSM [39, 40], which may alleviate the little hierarchy problem of the MSSM through additional contributions to Higgs mass [37, 38, 41] and also provide new DM candidates [42–45] that may account for the relic density with no conflict with other phenomenological constraints.In this paper, we analyze the constraints imposed by the Higgs mass limit and the gluino lower bound, which are the most stringent collider constraints, on the constrained MSSM (minimal SUGRA model, hereafter referred to as MSSM) parameter space. In particular, these constraints imply that the gaugino mass,m1/2, resides within the mass range: 620GeV≲m1/2≲2000 GeV, while the other parameters are much less constrained. We study the effect of the measured DM relic density on the MSSM allowed parameter space. We emphasized that in this case all parameter space is ruled out except for few points around tan⁡β~50, m0~1 TeV, and m1/2~1.5 TeV. We also investigate the direct detection rate of the LSP at these allowed points in light of the latest LUX result. Finally, we show that if one assumes nonstandard scenario of cosmology with low reheating temperature, where the LSP may reach equilibrium before the reheating time, then the relic abundance constraints on (m0,m1/2) can be significantly relaxed.The paper is organized as follows. In Section2, we briefly introduce the MSSM and study the constraints on (m0,m1/2) plane from Higgs and gluino mass experimental limits. In Section 3, we study the thermal relic abundance of the LSP in the allowed region of parameter space. We show that the combined LHC and relic abundance constraints rule out most of the parameter space except the case of very large tan⁡β. We also provide the expected rate of direct LSP detection at these points with large tan⁡β and TeV masses. Section 4 is devoted to nonthermal scenario of DM and how it can relax the constraints imposed on MSSM parameter space. Finally, we give our conclusions in Section 5.


## 2. MSSM after the LHC Run-I

The particle content of the MSSM is three generations of (chiral) quark and lepton superfields; the (vector) superfields are necessary to gaugeSU(3)C×SU(2)L×U(1)Y gauge of the SM, and two (chiral) SU(2) doublet Higgs superfields. The introduction of a second Higgs doublet is necessary in order to cancel the anomalies produced by the fermionic members of the first Higgs superfield and also to give masses to both up and down type quarks. The interactions between Higgs and matter superfields are described by the superpotential(1)W=hUQLULcH2+hDQLDLcH1+hLLLELcH1+μH1H2.Here, QL contains SU(2) (s)quark doublets and ULc, DLc are the corresponding singlets, (s)lepton doublets and singlets reside in LL and ELc, respectively.H1 and H2 denote Higgs superfields with hypercharge Y=∓1/2. Further, due to the fact that Higgs and lepton doublet superfields have the same SU(3)C×SU(2)L×U(1)Y quantum numbers, we have additional terms that can be written as(2)W′=λijkLiLjEkc+λijk′LiQjDkc+λijk′′DicDjcUkc+μiLiH2.These terms violate baryon and lepton number explicitly and lead to proton decay at unacceptable rates. To forbid these terms, a new symmetry, called R-parity, is introduced, which is defined as RP=(-1)3B+L+2S, where B and L are baryon and lepton number and S is the spin. There are two remarkable phenomenological implications of the presence of R-parity: (i) SUSY particles are produced or destroyed only in pair; (ii) the LSP is absolutely stable and, hence, it might constitute a possible candidate for DM.In the MSSM, a certain universality of soft SUSY breaking terms at grand unification scaleMX=3×1016 GeV is assumed. These terms are defined as m0, the universal scalar soft mass, m1/2, the universal gaugino mass, A0, the universal trilinear coupling, B, and the bilinear coupling (the soft mixing between the Higgs scalars). In order to discuss the physical implication of soft SUSY breaking at low energy, we need to renormalize these parameters from MX down to electroweak scale, which has been performed using SARAH [46], and the spectrum has been calculated using SPheno [47, 48]. In addition, the MSSM contains another two free SUSY parameters: μ and tan⁡β=〈H2〉/〈H1〉. Two of these free parameters, μ and B, can be determined by the electroweak breaking conditions: (3)μ2=mH12-mH22tan2⁡βtan2⁡β-1-MZ22,(4)sin⁡2β=-2m32m12+m22.Thus, the MSSM has only four independent free parameters, m0,m1/2,A0,tan⁡β, besides the sign of μ, which determine the whole spectrum.In the MSSM, the mass of the lightest Higgs state can be approximated, at the one-loop level, as [49–52] (5)mh2≤MZ2+3g216π2MW2mt4sin2βlog⁡mt~12mt~22mt4.Therefore, if one assumes that the stop masses are of order TeV, then the one-loop effect leads to a correction of order O(100) GeV, which implies that (6)mhMSSM≲90GeV2+100GeV2≃135GeV.The two-loop corrections reduce this upper bound by few GeVs [53–55]. Hence, the MSSM predicts the following upper bound for the Higgs mass: mh≲130 GeV, which was consistent with the measured value of Higgs mass (of order 125 GeV) at the LHC [13, 14].In Figure1, we display the contour plot of the SM-like Higgs boson: mh∈[124,126] GeV in (m0,m1/2) plane for different values of A0 and tan⁡β. It is remarkable that the smaller the value of A0 is, the smaller the value of m1/2 is needed to satisfy this value of Higgs mass. It is also clear that the scalar mass m0 remains essentially unconstrained by Higgs mass limit. It can vary from few hundred GeVs to few TeVs. Such large values of m1/2 seem to imply a quite heavy SUSY spectrum, much heavier than the lower bound imposed by direct searches at the LHC experiments in centre of mass energies s=7,8 TeV and total integrated luminosity of order 20fb-1. Furthermore, the LHC lower limit on the gluino mass, mg~≳1.4 TeV [56, 57], excluded the values of m1/2<620 GeV which was allowed by Higgs mass constraints for m0>4 TeV. Furthermore, this region is shown with dashed lines in Figure 1.Figure 1
MSSM parameter space fortan⁡β=10 (a) and 50 (b) with A0=0 and 2 TeV. The green region indicates 124≲mh≲126 GeV. The blue region is excluded because the lightest neutralino is not the LSP. The pink region is excluded due to the absence of radiative electroweak symmetry breaking (μ2 becomes negative). The gray shadow lines denote the excluded area because of mg~<1.4 TeV.
(a)
(b)


## 3. Dark Matter Constraints on MSSM Parameter Space

### 3.1. The LSP as Dark Matter Candidate

The neutralinosχi (i=1,2,3,4) are the physical (mass) superpositions of two fermionic partners of the two neutral gauge bosons, called gaugino B~0 (bino) and W~30 (wino), and of the two neutral Higgs bosons, called Higgsinos H~10 and H~20. The neutralino mass matrix is given by [58–61](7)MN=M10-MZcos⁡βsin⁡θWMZsin⁡βsin⁡θW0M2MZcos⁡βcos⁡θW-MZsin⁡βcos⁡θW-MZcos⁡βsin⁡θWMZcos⁡βcos⁡θW0-μMZsin⁡βsin⁡θW-MZsin⁡βcos⁡θW-μ0,where M1 and M2 are related due to the universality of the gaugino masses at the grand unification scale, M1=3g12/5g22M2, where g1, g2 are the gauge couplings of U(1)Y and SU(2)L, respectively. This Hermitian matrix is diagonalized by a unitary transformation of the neutralino fields, MNdiag=N†MNN. The lightest eigenvalue of this matrix and the corresponding eigenstate, say χ, has good chance of being the LSP. The lightest neutralino will be a linear combination of the original fields:(8)χ=N11B~0+N12W~0+N13H~10+N14H~20.The phenomenology and cosmology of the neutralino are governed primarily by its mass and composition. A useful parameter for describing the neutralino composition is the gaugino “purity” function fg=|N11|2+|N12|2 [58–61]. If fg>0.5, then the neutralino is primarily gaugino and if fg<0.5, then the neutralino is primarily Higgsino. Actually, if |μ|>|M2|≥MZ, the two lightest neutralino states will be determined by the gaugino components; similarly, the light chargino will be mostly a charged wino, while if |μ|<|M2|, the two lighter neutralinos and the lighter chargino are all mostly Higgsinos, with mass close to |μ|. Finally, if |μ|≃|M2|, the states will be strongly mixed.Here, two remarks are in order. (i) The abovementioned constraints inm1/2 from Higgs mass limit and gluino mass lower bound imply that mχ≳240 GeV, which is larger than the limits obtained from direct searches at the LHC. Moreover, an upper bound of order one TeV is also obtained (from Higgs mass constraint). (ii) In this region of allowed parameter space, the LSP is essentially pure bino, as shown in Figure 2. This can be easily understood from the fact that μ-parameter, determined by the radiative electroweak breaking condition, (3), is typically of order m0 and hence it is much heavier than the gaugino mass M1.Figure 2
The mass of lightest neutralino versus the purity function in the region of parameter space allowed by gluino and Higgs mass limits.


### 3.2. Relic Density

As advocated in the previous section, the LSP in MSSM, the lightest neutralinoχ, is a perfect candidate for DM. Here, we assume that χ was in thermal equilibrium with the SM particles in the early universe and decoupled when it was nonrelativistic. Once χ annihilation rate Γχ=〈σχannv〉nχ dropped below the expansion rate of the universe, Γχ≤H, the LSP particles stop to annihilate and fall out of equilibrium and their relic density remains intact till now. The above 〈σχannv〉 refers to thermally averaged total cross section for annihilation of χχ into lighter particles times the relative velocity, v.The relic density is then determined by the Boltzmann equation for the LSP number density(nχ) and the law of entropy conservation: (9)dnχdt=-3Hnχ-σχannvnχ2-nχeq2,dsdt=-3Hs,where nχeq is the LSP equilibrium number density which, as a function of temperature T, is given by nχeq=gχ(mχT/2π)3/2e-mχ/T. Here, mχ and gχ are the mass and the number of degrees of freedom of the LSP, respectively. Finally, s is the entropy density. In the standard cosmology, the Hubble parameter H is given by H(T)=2ππg∗/45T2/MPl, where MPl=1.22×1019 GeV and g∗ is the number of relativistic degrees of freedom, for MSSM g∗≃228.75. Let us introduce the variable x=mχ/T and define Y=nχ/s with Yeq=nχeq/s. In this case, the Boltzmann equation is given by (10)dYdx=13HdsdxσχannvY2-Yeq2.In radiation domination era, the entropy, as a function of the temperature, is given by (11)sx=2π245g∗sxmχ3x-3,which is deduced from the fact that s=(ρ+p)/T and g∗s is the effective degrees of freedom for the entropy density. Therefore, one finds (12)dsdx=-3sx.Thus, with assuming g∗≃g∗s, the following expression for the Boltzmann equation for the LSP number density is obtained: (13)dYdx=-πg∗45MPlmχσχannvx2Y2-Yeq2.If one considers the s-wave and p-wave annihilation processes only, the thermal average〈σχannv〉 then shows as (14)σχannv≃aχ+6bχx,where aχ and bχ are the s-wave and p-wave contributions of annihilation processes, respectively. The relic density of the DM candidate is given by(15)Ωh2=mχs0Yχ∞ρc/h2,where s0=2282.15×10-41GeV3, ρc=8.0992h2×10-47GeV4, and by solving the Boltzmann equation, one can find Yχ(∞) as follows [62]: (16)Yχ∞=1λχaχxTf+3bχx2Tf-1,where Tf is the freeze-out temperature, λχ=s(mχ)/H(mχ), and x(Tf) is given by(17)xTf=ln⁡αχλχcc+2xTfaχ+6bχxTf,where αχ=45/2π4π/8gχ/g∗s(Tf); the value c=1/2 results in a typical accuracy of about 5–10% more than sufficient for our purposes here.The lightest neutralino may annihilate into fermion-antifermion (ff¯), W+W-, ZZ, W+H-, ZA, ZH, Zh, and H+H- and all other contributions of neutral Higgs. For a bino-like LSP, that is, N11≃1 and N1i≃0, i=2,3,4, one finds that the relevant annihilation channels are the fermion-antifermion ones, as shown in Figure 3, and all other channels are instead suppressed. Also, the annihilation process mediated by Z-gauge boson is suppressed due to the small Zχχ coupling ∝N132-N142, except at the resonance when mχ~MZ/2, which is no longer possible due to the abovementioned constraints. Furthermore, one finds that the t-channel annihilation (first Feynman diagram in Figure 3) is predominantly into leptons through the exchanges of the three slepton families (l~L,l~R), with l=e,μ,τ. The squarks exchanges are suppressed due to their large masses.Figure 3
Feynman diagrams contributing to early-universe neutralinoχ annihilation into fermions through sfermions, Z-gauge boson, and Higgs.In Figure4, we display the constraint from the observed limits of Ωh2 on the plane (m0-m1/2) for A0=0,2000 GeV, tan⁡β=10,50, and μ>0. Here, we used micrOMEGAs [63] to compute the complete relic abundance of the lightest neutralino, taking into account the possibility of having coannihilation with the next-to-lightest supersymmetric particle, which is typically the lightest stau. Note that this type of coannihilation is not included in the approximated expressions in (14)–(17). In this figure, the red regions correspond to a relic abundance within the measured limits [1]: (18)0.09<Ωh2<0.14.It is noticeable that, with low tan⁡β  (~10), this region corresponds to light m1/2 (<500 GeV), where significant coannihilation between the LSP and stau took place. However, this possibility is now excluded by the Higgs and gluino mass constraints [64]. At large tan⁡β, another region is allowed due to possible resonance due to s-channel annihilation of the DM pair into fermion-antifermion via the pseudoscalar Higgs boson A at MA≃2mχ [65]. For A0=0, a very small part of this region is allowed by the Higgs mass constraint, while for large A0(~2 TeV) slight enhancement of this part can be achieved. In Figure 5, we zoom in on this region to show the explicit dependence of the relic abundance on the LSP mass and large values of tan⁡β. As can be seen from this figure, there is no point that can satisfy the relic abundance stringent constraints with tan⁡β<30.Figure 4
LSP relic abundance constraints (red regions) on(m0-m1/2) plane for tan⁡β and A0 as in Figure 1. The LUX result is satisfied by the yellow region. The other color codes are as in Figure 1.Figure 5
The relic abundance versus the mass of the LSP for different values oftan⁡β. Red points indicate 40≤tan⁡β≤50 and blue points 30≤tan⁡β<40. All points satisfy the abovementioned constraints.


### 3.3. Direct Detection

Perhaps the most natural way of searching for the neutralino DM is provided by direct experiments, where the effects induced in appropriate detectors by neutralino-nucleus elastic scattering may be measured. The elastic-scattering cross section of the LSP with a given nucleus has two contributions: spin-dependent contribution arising fromZ and q~ exchange diagrams and spin-independent (scalar) contribution due to the Higgs and squark exchange diagrams, which is typically suppressed. The effective scalar interaction of neutralino with a quark is given by(19)Lscalar=fqχ¯χq¯q,where fq is the neutralino-quark effective coupling. The scalar cross section of the neutralino scattering with target nucleus at zero momentum transfer is given by [2] (20)σ0SI=4mr2πZfp+A-Zfn2,where Z and A-Z are the number of protons and neutrons, respectively, mr=mNmχ/(mN+mχ), where mN is the nucleus mass, and fp, fn are the neutralino coupling to protons and neutrons, respectively. The differential scalar cross section for nonzero momentum transfer q can now be written: (21)dσSIdq2=σ0SI4mr2v2F2q2,0<q2<4mr2v2,where v is the neutrino velocity and F(q2) is the form factor [2]. In Figure 6, we display the MSSM prediction for spin-independent scattering cross section of the LSP with a proton (σSIp=∫04mr2v2‍dσSI/dq2|fn=fpdq2) after imposing the LHC and relic abundance constraints. It is clear that our results for σSIp are less than the recent LUX bound (blue curve) by at least two orders of magnitude. This would explain the negative results of direct searches so far.Figure 6
Spin-independent scattering cross section of the LSP with a proton versus the mass of the LSP within the region allowed by all constraints (from the LHC and relic abundance).


## 3.1. The LSP as Dark Matter Candidate

The neutralinosχi (i=1,2,3,4) are the physical (mass) superpositions of two fermionic partners of the two neutral gauge bosons, called gaugino B~0 (bino) and W~30 (wino), and of the two neutral Higgs bosons, called Higgsinos H~10 and H~20. The neutralino mass matrix is given by [58–61](7)MN=M10-MZcos⁡βsin⁡θWMZsin⁡βsin⁡θW0M2MZcos⁡βcos⁡θW-MZsin⁡βcos⁡θW-MZcos⁡βsin⁡θWMZcos⁡βcos⁡θW0-μMZsin⁡βsin⁡θW-MZsin⁡βcos⁡θW-μ0,where M1 and M2 are related due to the universality of the gaugino masses at the grand unification scale, M1=3g12/5g22M2, where g1, g2 are the gauge couplings of U(1)Y and SU(2)L, respectively. This Hermitian matrix is diagonalized by a unitary transformation of the neutralino fields, MNdiag=N†MNN. The lightest eigenvalue of this matrix and the corresponding eigenstate, say χ, has good chance of being the LSP. The lightest neutralino will be a linear combination of the original fields:(8)χ=N11B~0+N12W~0+N13H~10+N14H~20.The phenomenology and cosmology of the neutralino are governed primarily by its mass and composition. A useful parameter for describing the neutralino composition is the gaugino “purity” function fg=|N11|2+|N12|2 [58–61]. If fg>0.5, then the neutralino is primarily gaugino and if fg<0.5, then the neutralino is primarily Higgsino. Actually, if |μ|>|M2|≥MZ, the two lightest neutralino states will be determined by the gaugino components; similarly, the light chargino will be mostly a charged wino, while if |μ|<|M2|, the two lighter neutralinos and the lighter chargino are all mostly Higgsinos, with mass close to |μ|. Finally, if |μ|≃|M2|, the states will be strongly mixed.Here, two remarks are in order. (i) The abovementioned constraints inm1/2 from Higgs mass limit and gluino mass lower bound imply that mχ≳240 GeV, which is larger than the limits obtained from direct searches at the LHC. Moreover, an upper bound of order one TeV is also obtained (from Higgs mass constraint). (ii) In this region of allowed parameter space, the LSP is essentially pure bino, as shown in Figure 2. This can be easily understood from the fact that μ-parameter, determined by the radiative electroweak breaking condition, (3), is typically of order m0 and hence it is much heavier than the gaugino mass M1.Figure 2
The mass of lightest neutralino versus the purity function in the region of parameter space allowed by gluino and Higgs mass limits.


## 3.2. Relic Density

As advocated in the previous section, the LSP in MSSM, the lightest neutralinoχ, is a perfect candidate for DM. Here, we assume that χ was in thermal equilibrium with the SM particles in the early universe and decoupled when it was nonrelativistic. Once χ annihilation rate Γχ=〈σχannv〉nχ dropped below the expansion rate of the universe, Γχ≤H, the LSP particles stop to annihilate and fall out of equilibrium and their relic density remains intact till now. The above 〈σχannv〉 refers to thermally averaged total cross section for annihilation of χχ into lighter particles times the relative velocity, v.The relic density is then determined by the Boltzmann equation for the LSP number density(nχ) and the law of entropy conservation: (9)dnχdt=-3Hnχ-σχannvnχ2-nχeq2,dsdt=-3Hs,where nχeq is the LSP equilibrium number density which, as a function of temperature T, is given by nχeq=gχ(mχT/2π)3/2e-mχ/T. Here, mχ and gχ are the mass and the number of degrees of freedom of the LSP, respectively. Finally, s is the entropy density. In the standard cosmology, the Hubble parameter H is given by H(T)=2ππg∗/45T2/MPl, where MPl=1.22×1019 GeV and g∗ is the number of relativistic degrees of freedom, for MSSM g∗≃228.75. Let us introduce the variable x=mχ/T and define Y=nχ/s with Yeq=nχeq/s. In this case, the Boltzmann equation is given by (10)dYdx=13HdsdxσχannvY2-Yeq2.In radiation domination era, the entropy, as a function of the temperature, is given by (11)sx=2π245g∗sxmχ3x-3,which is deduced from the fact that s=(ρ+p)/T and g∗s is the effective degrees of freedom for the entropy density. Therefore, one finds (12)dsdx=-3sx.Thus, with assuming g∗≃g∗s, the following expression for the Boltzmann equation for the LSP number density is obtained: (13)dYdx=-πg∗45MPlmχσχannvx2Y2-Yeq2.If one considers the s-wave and p-wave annihilation processes only, the thermal average〈σχannv〉 then shows as (14)σχannv≃aχ+6bχx,where aχ and bχ are the s-wave and p-wave contributions of annihilation processes, respectively. The relic density of the DM candidate is given by(15)Ωh2=mχs0Yχ∞ρc/h2,where s0=2282.15×10-41GeV3, ρc=8.0992h2×10-47GeV4, and by solving the Boltzmann equation, one can find Yχ(∞) as follows [62]: (16)Yχ∞=1λχaχxTf+3bχx2Tf-1,where Tf is the freeze-out temperature, λχ=s(mχ)/H(mχ), and x(Tf) is given by(17)xTf=ln⁡αχλχcc+2xTfaχ+6bχxTf,where αχ=45/2π4π/8gχ/g∗s(Tf); the value c=1/2 results in a typical accuracy of about 5–10% more than sufficient for our purposes here.The lightest neutralino may annihilate into fermion-antifermion (ff¯), W+W-, ZZ, W+H-, ZA, ZH, Zh, and H+H- and all other contributions of neutral Higgs. For a bino-like LSP, that is, N11≃1 and N1i≃0, i=2,3,4, one finds that the relevant annihilation channels are the fermion-antifermion ones, as shown in Figure 3, and all other channels are instead suppressed. Also, the annihilation process mediated by Z-gauge boson is suppressed due to the small Zχχ coupling ∝N132-N142, except at the resonance when mχ~MZ/2, which is no longer possible due to the abovementioned constraints. Furthermore, one finds that the t-channel annihilation (first Feynman diagram in Figure 3) is predominantly into leptons through the exchanges of the three slepton families (l~L,l~R), with l=e,μ,τ. The squarks exchanges are suppressed due to their large masses.Figure 3
Feynman diagrams contributing to early-universe neutralinoχ annihilation into fermions through sfermions, Z-gauge boson, and Higgs.In Figure4, we display the constraint from the observed limits of Ωh2 on the plane (m0-m1/2) for A0=0,2000 GeV, tan⁡β=10,50, and μ>0. Here, we used micrOMEGAs [63] to compute the complete relic abundance of the lightest neutralino, taking into account the possibility of having coannihilation with the next-to-lightest supersymmetric particle, which is typically the lightest stau. Note that this type of coannihilation is not included in the approximated expressions in (14)–(17). In this figure, the red regions correspond to a relic abundance within the measured limits [1]: (18)0.09<Ωh2<0.14.It is noticeable that, with low tan⁡β  (~10), this region corresponds to light m1/2 (<500 GeV), where significant coannihilation between the LSP and stau took place. However, this possibility is now excluded by the Higgs and gluino mass constraints [64]. At large tan⁡β, another region is allowed due to possible resonance due to s-channel annihilation of the DM pair into fermion-antifermion via the pseudoscalar Higgs boson A at MA≃2mχ [65]. For A0=0, a very small part of this region is allowed by the Higgs mass constraint, while for large A0(~2 TeV) slight enhancement of this part can be achieved. In Figure 5, we zoom in on this region to show the explicit dependence of the relic abundance on the LSP mass and large values of tan⁡β. As can be seen from this figure, there is no point that can satisfy the relic abundance stringent constraints with tan⁡β<30.Figure 4
LSP relic abundance constraints (red regions) on(m0-m1/2) plane for tan⁡β and A0 as in Figure 1. The LUX result is satisfied by the yellow region. The other color codes are as in Figure 1.Figure 5
The relic abundance versus the mass of the LSP for different values oftan⁡β. Red points indicate 40≤tan⁡β≤50 and blue points 30≤tan⁡β<40. All points satisfy the abovementioned constraints.


## 3.3. Direct Detection

Perhaps the most natural way of searching for the neutralino DM is provided by direct experiments, where the effects induced in appropriate detectors by neutralino-nucleus elastic scattering may be measured. The elastic-scattering cross section of the LSP with a given nucleus has two contributions: spin-dependent contribution arising fromZ and q~ exchange diagrams and spin-independent (scalar) contribution due to the Higgs and squark exchange diagrams, which is typically suppressed. The effective scalar interaction of neutralino with a quark is given by(19)Lscalar=fqχ¯χq¯q,where fq is the neutralino-quark effective coupling. The scalar cross section of the neutralino scattering with target nucleus at zero momentum transfer is given by [2] (20)σ0SI=4mr2πZfp+A-Zfn2,where Z and A-Z are the number of protons and neutrons, respectively, mr=mNmχ/(mN+mχ), where mN is the nucleus mass, and fp, fn are the neutralino coupling to protons and neutrons, respectively. The differential scalar cross section for nonzero momentum transfer q can now be written: (21)dσSIdq2=σ0SI4mr2v2F2q2,0<q2<4mr2v2,where v is the neutrino velocity and F(q2) is the form factor [2]. In Figure 6, we display the MSSM prediction for spin-independent scattering cross section of the LSP with a proton (σSIp=∫04mr2v2‍dσSI/dq2|fn=fpdq2) after imposing the LHC and relic abundance constraints. It is clear that our results for σSIp are less than the recent LUX bound (blue curve) by at least two orders of magnitude. This would explain the negative results of direct searches so far.Figure 6
Spin-independent scattering cross section of the LSP with a proton versus the mass of the LSP within the region allowed by all constraints (from the LHC and relic abundance).


## 4. Nonthermal Dark Matter and MSSM Parameter Space

In the previous section, we assumed standard cosmology scenario where the reheating temperatureTRH is very large; namely, TRH≫Tf≃10 GeV. However, the only constraint on the reheating temperature, which could be associated with decay of any scalar field, ϕ, not only the inflaton field, is TRH≳1 MeV in order not to spoil the successful predictions of big bang nucleosynthesis.A detailed analysis of the relic density with a low reheating temperature has been carried out in [66]. It was emphasized that, for a large annihilation cross section, 〈σannv〉≳10-14GeV-2 so that the neutralino reaches equilibrium before reheating, and if there are a large number of neutralinos produced by the scalar field ϕ decay, then the relic density is estimated as [67](22)Ωh2=3mχΓϕ22π2/45g∗TRHTRH3σχannvh2ρc/s0.Here, the reheating temperature is defined as [62] (23)TRH=90π2g∗TRH1/4ΓϕMPl1/2,where the decay width Γϕ is given by (24)Γϕ=12πmϕ3Λ2.The scale Λ is the effective suppression scale, which is of order the grand unification scale MX. Therefore, for scalar field with mass mϕ≃107 GeV, one finds Γϕ≃10-11 GeV, and in our calculations, we have used g∗=10.75 due to the consideration of a low reheating temperature scenario.In Figure7, we show the constraints imposed on the MSSM (m0-m1/2) plane in case of nonthermal relic abundance of the LSP for tan⁡β=10,50 and A0=0,2 TeV. In this plot, we also imposed the LHC constraints, namely, the Higgs mass limit and the gluino mass lower bound, similar to the case of thermal scenario. It is clear from this figure that the stringent constraints imposed on the MSSM parameter space by thermal relic abundance are now relaxed and now low tan⁡β(~10) is allowed but with very heavy m0(~O(4)TeV) and m1/2≃600 GeV. In addition, the following two regions are now allowed with large tan⁡β(~50): (i) m0~m1/2≃O(1) TeV; (ii) m0≃O(4)TeV and m1/2≃700 GeV. The SUSY spectrum associated with these regions of parameters space could be striking signature for nonthermal scenario at the LHC.Figure 7
LSP nonthermal relic abundance constraints (red regions) on(m0-m1/2) plane for tan⁡β and A0 as in Figure 1. The color codes are as in Figure 1.


## 5. Conclusion

We have studied the constraints imposed on the MSSM parameter space by the Higgs mass limit and the gluino lower bound, which are the most stringent collider constraints obtained from the LHC run-I at energy 8 TeV. We showed thatm1/2 resides within the mass range 620GeV≲m1/2≲2000 GeV, while the other parameters (m0,A0,tan⁡β) are much less constrained. We also studied the effect of the measured DM relic density on the MSSM allowed parameter space. It turns out that most of the MSSM parameter space is ruled out except for few points around tan⁡β~50, m0~1 TeV, and m1/2~1.5 TeV. We calculated the spin-independent scattering cross section of the LSP with a proton in this allowed region. We showed that our prediction for σSIp is less than the recent LUX bound by at least two orders of magnitude. We have also analyzed the nonthermal DM scenario for the LSP. We showed that the constraints imposed on the MSSM parameter space are relaxed and low tan⁡β is now allowed with m0≃O(4)TeV and m1/2≃600 GeV. Also two allowed regions are now associated with large tan⁡β(~50); namely, m0~m1/2≃O(1) TeV or m0≃O(4)TeV and m1/2≃700 GeV.



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*Source: 5687463-2016-01-10.xml*