Dark Matter Detection, Standard Model Parameters, and Intermediate Scale Supersymmetry

The vanishing of the Higgs quartic coupling at a high energy scale may be explained by Intermediate Scale Supersymmetry, where supersymmetry breaks at $(10^9$-$10^{12})$ GeV. The possible range of supersymmetry breaking scales can be narrowed down by precise measurements of the top quark mass and the strong coupling constant. On the other hand, nuclear recoil experiments can probe Higgsino or sneutrino dark matter up to a mass of $10^{12}$ GeV. We derive the correlation between the dark matter mass and precision measurements of standard model parameters, including supersymmetric threshold corrections. The dark matter mass is bounded from above as a function of the top quark mass and the strong coupling constant. The top quark mass and the strong coupling constant are bounded from above and below respectively for a given dark matter mass. We also discuss how the observed dark matter abundance can be explained by freeze-out or freeze-in during a matter-dominated era after inflation, with the inflaton condensate being dissipated by thermal effects.


Introduction
In 1985, Goodman and Witten proposed that halo dark matter could be detected directly in terrestrial experiments by observing small energy depositions from elastic scattering of dark matter particles from nuclei [1]. Their first illustration was of a neutral particle, such as a heavy neutrino, scattering via t-channel Z exchange with a cross section per nucleon of order σv ∼ G 2 F µ 2 red /2π, where µ red is the reduced mass of the dark matter and nucleon. They computed a signal of order 10 2 − 10 4 events per Kg per day for dark matter masses in the GeV to TeV range, depending on nuclear target. In the intervening 35 years, a succession of ever larger and more sensitive detectors have excluded this example by many orders of magnitude, so that the focus has shifted to theories where there is no contribution to the scattering from tree-level weak interactions. In fact, as the number density of dark matter particles scales as the inverse of its mass, present data constrains the mass of dark matter with tree-level Z exchange to be larger than 3 × 10 9 GeV [2]. Proposed detectors [3][4][5] will probe the mass range M DM,Z-exchange = (3 × 10 9 − 2 × 10 12 ) GeV. (1.1) The discovery of the Higgs boson at the Large Hadron Collider (LHC) completes the Standard Model (SM). Electroweak symmetry breaking arises from the potential via the ground state value of the Higgs field H = v 174 GeV. The Higgs boson mass is m 2 h = 4λv 2 . No other new particles have been discovered at the LHC so far, and in this paper we assume that the SM is valid to very high energies. All the SM couplings can be computed at high energies to high precision, including the Higgs quartic coupling [6]. As shown in Fig. 1, this running indicates that the Higgs quartic coupling vanishes at the scale µ λ = 10 9−12 GeV, (1.3) which we call the Higgs quartic scale. Indeed, within the context of the SM as an effective field theory to very high energies, a key result of the LHC is the discovery of this new mass scale.
In this paper we assume that physics beyond the SM first appears at µ λ , and the form of the new physics explains why the Higgs quartic is so small at this scale. It is interesting to note that, if dark matter couples to the weak interaction, the recent direct detection experiments have started to explore dark matter masses in the vicinity of the Higgs quartic scale. The mass range to be explored by the next generation of experiments, (1.1), will probe the entire range of (1.3).
Since the discovery of a Higgs with mass of 125 GeV, several proposals have been made for physics at µ λ that explains the small quartic coupling, including supersymmetry [7][8][9], extra dimension [10], Peccei-Quinn symmetry [11], and Higgs Parity symmetry [12][13][14][15][16]. In this paper we pursue the case of Intermediate Scale Supersymmetry (ISS), where the superpartner mass scalem is of order the Higgs quartic scale. The identification of µ λ withm is natural [7,8] since supersymmetry predicts a very small SM Higgs quartic at the scalem for a wide range of supersymmetry breaking parameters. Unlike in [7,8], we study the case of Higgsino or sneutrino Lightest Supersymmetric Particle (LSP) dark matter with mass of orderm, since this gives a direct detection signal that is correlated with the Higgs quartic scale.
In this paper, we examine the correlation in ISS between the dark matter detection signal via Z exchange and the precision measurement of the top quark mass, m t , the strong coupling constant, α s (m Z ), (and to a lesser extent, of the Higgs boson mass, m h ). A dark matter signal will determine the mass of the LSP and precision measurements will greatly reduce the uncertainties in the Higgs quartic scale. In particular, we find that the discovery of a direct detection signal implies an upper bound on the top quark mass and a lower bound on the strong coupling constant. The effects on the running of the Higgs quartic in reducing the uncertainties in m t , α s (m Z ) and m h are shown by the colored bands in Fig. 1. Future uncertainties in m t (0.01GeV), α s (m Z ) (0.0001), and m h (0.01 GeV) from measurements at future lepton colliders [17][18][19][20][21], improved lattice calculations [22], and the high-luminosity LHC [23], will substantially reduce the uncertainty in µ λ to within a few tens of percents, as shown by the solid black strip in Fig. 1 which is centered at the current central values of m t , α s (m Z ), and m h .
In 1977, Lee and Weinberg showed dark matter, if coupled to the weak interaction, could be produced in the early universe by freezing-out, losing thermal equilibrium while nonrelativistic [24]. Indeed, they discovered that a heavy neutrino, with a GeV-scale mass, could yield the observed dark matter abundance. Many other electroweak dark matter candidates arising from freeze-out were studied, with masses up to several TeV. Apparently our proposal of Higgsino or sneutrino dark matter with a mass of 10 9 − 10 12 GeV leads to a huge overproduction of dark matter. However, we find that the observed abundance can result from freeze-out or freeze-in during a matter-dominated era after inflation. The inflaton mass must be below the dark matter mass, otherwise the O(1) branching fraction of the inflaton into sparticles leads to an overproduction of dark matter. Then during freeze-out or freeze-in, the inflaton is dissipated by scattering reactions rather than by decays. If the products of the scattering reactions are thermalized at a high enough temperature, freeze-out occurs; otherwise, the abundance is set by freeze-in from non-thermal radiation. Either way, determining the dark matter mass from direct detection will provide a correlation between the reheat temperature after inflation and the inflaton mass.
In section 2, building on [7,8], we show that if the UV completion of the SM EFT is provided by ISS, there is a large region of parameter space where the SM quartic coupling is predicted to be very small atm, and hencem ∼ µ λ . In section 3 we compute the present limits on Higgsino and sneutrino dark matter, and compute the reaches expected for XENONnT, LZ, and DARWIN. We then study the correlation between the dark matter signal and future precision measurements of m t , α s (m Z ), and m h . In section 4 we study how this correlation is affected by supersymmetric threshold corrections to the Higgs quartic coupling in the Minimal Supersymmetric Standard Model (MSSM). We find that these threshold corrections can be significant and derive an upper bound on the Higgsino or sneutrino LSP mass as a function of the top quark mass and the strong coupling constant. An observable direct detection signal is predicted for top masses above a critical value. In section 5 we compute the supersymmetric threshold corrections in a scheme where the supersymmetry breaking parameters are constrained to a universal form at unified scales. In section 6 we compute the relic dark matter abundance from freeze-out or freeze-in during a matter dominated era where the inflaton condensate is dissipated by scattering reactions. Finally, we draw conclusions in section 7.

The Tree-Level Boundary Condition on the SM Quartic Coupling
We take the SM to be the effective theory below the scale of supersymmetry breaking,m. In this section, we review the tree-level prediction for the SM Higgs quartic coupling, λ tree . At scalem, we assume that there is no gauge symmetry breaking and the theory contains a single pair of Higgs doublets, (H u , H d ), and no weak singlets or triplets which have a zero hypercharge and couple to the Higgs doublets. For a wide range of parameters of this Higgs sector, we find λ(m) 0.01; remarkably there are large regions with λ(m) 0.001, and the supersymmetry breaking scalem may be identified with the Higgs quartic scale µ λ .
The Higgs potential is where µ is the supersymmetric Higgs mass parameter, while m 2 Hu , m 2 H d , and Bµ are supersymmetryviolating mass parameters. These parameters are all taken real, without loss of generality, and have sizes determined by the scale of supersymmetry breaking,m. The constants g and g are the SU (2) and U (1) gauge couplings. Requiring electroweak symmetry to be unbroken atm and one combination of the Higgs doublets to be much lighter thanm requires that where tan 2 β = (µ 2 + m 2 H d )/(µ 2 + m 2 Hu ), and we take β in the first quadrant. Matching the two theories atm gives the tree-level value for λ(m) 06 and hence at tree levelm µ λ . Furthermore, over a wide range of values for m 2 Hu , m 2 H d , and µ the cos 2 2β factor gives a significant further suppression of λ(m) tree , as shown in Fig. 2. Indeed, cos 2β → 0 in the limit that either µ 2 |m 2 H u,d | or m 2 Hu → m 2 H d ; in these limitsm is identified with µ λ . The gray-shaded region is excluded since µ 2 + m 2 Hu < 0 or µ 2 + m 2 H d < 0 and there is no stable vacuum with a large hierarchy between the weak scale and the supersymmetry breaking scale. In the blue-shaded region, where λ(m) tree > 0.01,m is predicted to be below a few 10 9 GeV. As we will see in the next section, the Higgsino or sneutrino LSP then gives too large a direct detection rate. However, there is a remarkably large region of parameter space in Fig. 2 with λ(m) tree < 0.003.

Direct Detection of Dark Matter
In this section, we discuss direct detection of the Higgsino or sneutrino LSP dark matter in nuclear recoil experiments and show that detection rates are correlated with SM parameters through the connection betweenm and µ λ . An observable direct detection signal is predicted for top masses below a critical value.

Higgsino dark matter
The neutral components and the charged component of the Higgsino are degenerate in mass in the electroweak symmetric limit. With elecroweak symmetry breaking, the charged component becomes heavier than the neutral components by O(100) MeV via one-loop quantum corrections [25]. The neutral components slightly mix with the bino and the wino and obtain a small mass splitting (3.1) The two mass eigenstates are Majorana fermions. For a soft mass scale above ∼ 10 9 GeV, however, the splitting is smaller than the typical nucleon recoil energy of O(10 − 100) keV, and the Majorana nature does not affect the rate of dark matter signals. Specifically, Z boson exchange leads to the up-scattering of the ligher state into the heavier state, which almost behaves as scattering of a Dirac fermion.

Sneutrino dark matter
The sneutrino is lighter than its charged SU (2) partner because of electroweak symmetry breaking and quantum corrections. The two components of the sneutrino obtain a small mass splitting from the A term of the Majorana neutrino mass term, which is negligibly small. Sneutrino dark matter interacts with nucleon via Z boson exchange as a complex scalar field. If the slepton and squark masses are universal at the unification scale, the sneutrino cannot be the LSP because renormalization running makes the right-handed stau the lightest among them. Non-universality is required for the sneutrino LSP. We note that the sneutrino LSP is consistent with SU (5) unification, since the sneutrinos and the right-handed sleptons are not unified, and the right-handed down type squarks become heavier than the sneutrinos by renormalization running.

Direct detection rate and standard model parameters
Both Higgsino and sneutrino dark matter scatter with nuclei, with an effective dark matternucleon scattering cross section given by where G F is the Fermi constant, m n is the nucleon mass, A is the mass number, Z is the atomic number, and θ W is the Weinberg angle. The current constraint by XENON1T [2] and the future sensitivities of LZ with an exposure of 15 ton·year, XENONnT with an exposure of 20 ton·year, and DARWIN with an exposure of 1000 ton·year [3][4][5] are given by Once dark matter signals are found in recoil experiments, within the framework of Higgsino or sneutrino dark matter in ISS, the dark matter mass is fixed from the observed signal rates. Since λ(m) tree is positive and m DM = m LSP <m, we obtain a bound on SM parameters including an upper bound on the top quark mass. Conversely, for given SM parameters, m DM is bounded from above. The prediction for the top quark mass for givenm and λ(m) tree is shown in Fig. 3. The right vertical axis shows cos2β corresponding to λ(m) tree . For a given m DM , the prediction on m t for λ(m) tree = 0 andm = 0 can be understood as an upper bound on m t . For a given m t ,m such that λ(m) tree = 0 in an upper bound on m DM . To obtain those bounds precisely, we include threshold corrections to λ(m) in the next section.

Including Threshold Corrections to the Higgs Quartic
The full prediction for λ(m) in ISS is where λ tree is the the tree-level result, (2.3), and δλ the quantum corrections that arise on integrating out heavy sparticles. The largest contributions arise from sparticles with the largest couplings to the light Higgs; hence the most important mass parameters are the masses of the third generation doublet squark mq, the third generation up-type squark mũ, the bino M 1 , the wino M 2 , the heavy Higgs m A , and the A term of the top quark yukawa A t . We choose the matching scale to be the lighter of mq and mũ, which we denote as m − . Since quantum corrections are greater than λ tree only for tan β 1, we neglect corrections which vanish in this limit. Using the results in [26], the corrections are given by Here, X t ≡ (A t − µ) 2 /mũmq, and the functions F, G, f 1 , f 2 are given by .  They are normalized so that they are unity when the arguments are unity. For a degenerate mass spectrum and negligible X t , δλ(m − ) −0.002. In Fig. 4, we evaluate Eq. (4.2) and show how δλ varies as a function of sparticle masses. The left and right panels correspond to A t positive and negative, respectively. Each curve corresponds to varying one of (A t , µ, m + , m A , M 1 , M 2 ), while keeping all the others fixed at m − . With all these parameters near m − , the correction is δλ(m − ) −0.002 for A t > 0 or +0.002 for A t < 0. For |X t | 10m − , the electroweak vacuum is unstable, as shown by the sudden discontinuation of the A t and µ curves. The bound on X t from the instability is derived in Appendix A. The Higgsino can be the LSP on the solid curves, but is not the LSP on the dashed part of the curves for µ, M 1 and M 2 . The slepton mass parameter ml may be taken small enough to give sneutrino LSP anywhere on the lines.
We show contours of the prediction for m t in the (m − , λ(m − )) plane in Fig. 5, with the strong coupling constant varied within ±1σ uncertainty from its central value in the top and bottom panels. The right axis shows cos2β corresponding to λ(m − ) when δλ λ tree . The lower bound on the dark matter mass from XENON1T is shown in green, and the lower bound on threshold corrections to λ(m − ) is shown in red. Together, these bounds require m t 174.2 GeV. The reach of the DARWIN experiment, shown by the dashed green line, will strongly limit the top quark mass to m t 172.4 GeV, if no signals are found. For the central values of SM parameters, the dark matter mass is required to be below 7 × 10 10 GeV, and LZ and XENONnT can cover most of the parameter space.
The bounds on the dark matter and top quark masses may be relaxed by hierarchical  sparticle masses. As shown in Fig. 4, large wino or bino masses give negative threshold corrections to the quartic coupling, thereby relaxing the upper bounds on the top quark mass and the dark matter mass. In Fig. 6, we show the upper bound on the dark matter mass as a function of the top quark mass or, equivalently, the upper bound on the top quark mass as a function of the dark matter mass. The blue curve is without threshold corrections, the orange curve has threshold corrections for a degenerate mass spectrum with A t µ, and on the green curve, the degeneracy is lifted by taking M 1,2 = √ 10m − . With this hierarchy, the upper bound on the dark matter mass is relaxed by a factor of 2, and that on the top quark mass is relaxed by 100 MeV. (Assuming a high mediation scale of supersymmetry breaking, a larger hierarchy is destabilized by quantum corrections from the gauginos to the soft scalar masses.) In Fig. 7, the upper bound on the dark matter mass or, equivalently, the upper bound on the top quark mass or the lower bound on the strong coupling constant, is shown. Here we impose δλ(m − ) > −0.002. The current 2σ uncertainty of m t and α s (m Z ) are shown by wide bands. The uncertainty of α s (m Z ) can be reduced by a factor of 10 by measurements at future lepton colliders [21] or improved lattice calculations [22]. The uncertainty of m t can be reduced down to few 10 MeV by future lepton colliders [17][18][19][20]. At this stage, the theoretical computation of the running of the Higgs quartic coupling should be improved; the most recent computation [6] has a theoretical uncertainty equivalent to the shift of the top quark mass by 100 MeV.

Supersymmetry Breaking Constrained by Unification
In this section, we discuss the quartic coupling at the supersymmetry breaking scale,m, starting from boundary conditions at the unification scale ∼ 10 16 GeV. We show that the tree-level quartic coupling is typically 0.001 − 0.01.
As we have seen, the quartic coupling atm is small when m 2 Hu ∼ m 2 H d . A relation m 2 Hu = m 2 H d can be naturally realized at a high energy scale by a symmetry relating H u with H d , such as a discrete symmetry or SO(10) gauge symmetry, or a universality of scalar masses. The relation is necessarily destabilized by quantum correction from the top quark Yukawa coupling, where the ellipsis denotes terms independent of the top Yukawa. We compute the renormalization group running of the MSSM from a scale 10 16 GeV down tom with a UV boundary condition motivated from SU (5) unification, In Fig. 8, we show the tree-level quartic coupling as a function of m 1/2 /m H for several representative boundary conditions; the left (right) panels have m H = 10 10 GeV (10 12 GeV). We fix the renormalization scale to be the matching scale used in the previous section, m − , the lighter of mq and mũ. The boundary condition for m 2 5 is not specified as it does not affect the renormalization group running of m Hu . Note that the bino,b, is the lightest gaugino and the right-handed slepton,ẽ, is the lightest scalar in the matter tenplet. We define m (b,ẽ) to be the smaller of mb and mẽ. On the five lines, µ is fixed to be As µ is increased, the tree-level quartic coupling decreases rapidly, as expected from (2.3), (2.4) and Fig. 2. For large values of m 2 5 the Higgsino is the LSP above the green dot-dashed line, and the region below the line is excluded because at low (high) m 1/2 the LSP is the bino (a charged right-handed slepton). For small values of m 2 5 the tau sneutrino can be the LSP throughout the plane, although at low µ the Higgsino LSP is also possible. In the blue shaded region, the top quark mass must be below 171.86 GeV, more than 3σ away from the central value, in order for λ(m − ) to be consistent with the running of the Higgs quartic coupling. To derive a conservative bound, we take α s (m Z ) = 0.1189, 1σ above the central value, and δλ = −0.002, the smallest realistic threshold correction.
We see that smaller values of λ tree result for larger m H , which gives less running, larger values of µ/m H and smaller values of m 10 /m H and A t,G /m H . For m H = 10 12 GeV, λ tree < 0.003 over much of the parameter space. Including threshold corrections, Fig. 5 shows that this is ideal for consistency with the observed Higgs mass, and requires a low value of the top quark mass. For m H = 10 10 GeV, λ tree < 0.01 over much of the parameter space, except at low values of µ, which from Fig. 5 again shows excellent consistency with the observed Higgs mass, and leads to the expectation that Higgsino/sneutrino dark matter will be discovered at planned experiments.

Cosmological Abundance of LSP with Intermediate Scale Mass
In this section, we discuss how the heavy LSP dark matter can be populated in the early universe. Most of the discussion in this section is applicable to generic heavy dark matter with electroweak interactions. Standard freeze-out during the radiation dominated era overproduces the LSP because of its large mass. To avoid this, the reheating temperature of the universe must be smaller than the LSP mass, and the LSP must be produced during the reheating process. We discuss reheating by the inflaton φ, but, if the LSPs produced during inflaton reheating are subdominant, the following discussion also applies to the case where some other particle or condensate dominates the energy density of the universe.

Direct decay of the inflaton
The inflaton can directly decay into sparticles if its mass is more than double the LSP mass. The energy density of the LSP normalized by the entropy density is where N LSP is the number of LSPs produced per inflaton decay. Because of supersymmetry, N LSP is at the smallest O(1). When m φ m DM and the inflaton dominantly decays into SM charged particles, showering leads to N LSP 1 [27,28]. Giving the lower bound T RH > 4 MeV [29][30][31], it is difficult to produce the correct LSP abundance in this way. Hence, the inflaton mass must be below the sparticle mass scale. (In this case, production of the LSP via scattering among the inflaton decay products and the thermal bath [32][33][34] is absent.)

Production during the inflaton dominated era
We first derive the evolution of the temperature of the universe. We consider the case where the dissipation of the inflaton occurs by perturbative processes, with dissipation rates given by For T < m φ , dissipation is governed by the zero-temperature decay rate Γ 0 , while for m φ < T , thermal effects should be taken into account. n = 1 arises when dissipation is caused by a dimensionless coupling, while n = −1 arises when dissipation is caused by a dimension-3 coupling, such as φhh † . The dependence of the temperature on the Hubble scale is given by : m φ < T, , (6.4) where H RH = π 2 g * /90 T 2 RH /M Pl is the Hubble scale at the completion of reheating. We implicitly assumed that the radiation is thermalized, which is not satisfied for small T RH and/or large T . We discuss thermalization while discussing the production of the LSP below.
Case 1: T FO < m φ < 2m DM During freeze-out, when T FO = m φ /x FO < m φ , radiation is created from the zero-temperature decay of the inflaton and the temperature of the universe is given by the first line of Eq. (6.3). Such a case is studied in the literature assuming efficient thermalization [35,36].
After freeze-out, the LSP number density, normalized by the inflaton energy density, is (6.5) Using ρ φ /s 3T RH /4 at the completion of reheating, we obtain Here, we assume that radiation thermalizes around the freeze-out temperature. This assumption is valid if 4πα 2 T FO > H FO , requiring If this condition is violated, the radiation produced from the inflaton does not reach thermal equilibrium by the would-be freeze-out. We expect that the distribution of radiation in this case is close to that after preheating [37,38]. Since scattering is efficient at lower energies, the lower energy modes are populated. The typical energy of the radiation is below the would-be temperature and the radiation is in an over-occupied state. The energy distribution has a cutoff, above which the scattering is inefficient and the distribution is exponentially suppressed.
For large m DM , the reheating temperature to reproduce the observed abundance from Eq. (6.6) is in fact smaller than T RH,th . Then the LSP abundance is exponentially suppressed and LSPs are under-produced. As T RH approaches T RH,th , the LSP production is not suppressed, and the freeze-out picture is applicable. Since T RH ∼ T RH,th is larger than that to produce an appropriate amount of LSPs according to Eq. (6.6), LSPs are over-produced. Thus, the observed dark matter abundance can be reproduced for T RH slightly below T RH,th . We call this scenario non-thermal freeze-in.
The required reheating temperature to produce the observed dark matter abundance by LSP production during reheating is shown in Fig. 9. Above the black dashed line, T FO < m φ < 2m DM and the analysis shown above is applicable. To the left of the black dot-dashed line, the LSP abundance is determined by freeze-out, while to the right, the abundance is determined by the exponentially suppressed production just before thermalization.
For the inflaton mass between T RH and T FO , the temperature of the universe during freezeout is given by the second line of Eq. (6.3). By a computation similar to that which leads to Figure 9. Contours of the reheating temperature T RH required to produce the observed dark matter abundance by LSP production during reheating. In the blue region, direct decay of the inflaton into sparticles overproduces the LSP. To the right of the dot-dashed line, radiation is not thermalized by the would-be freeze-out, and the LSP production occurs just before the completion of thermalization.
Eq. (6.6), we obtain The reheating temperature required to produce the observed dark matter abundance is shown in Fig. 9. The above analysis is applicable between the dashed and dotted lines.
For the inflaton mass below the reheating temperature, the temperature during freeze-out is given by Eq. (6.4). The LSP density is given by The reheating temperature required to produce the observed dark matter abundance is shown in Fig. 9. This analysis is applicable below the dotted line.

Other possibilities
Is is possible that the maximal temperature of the universe is the reheating temperature. This occurs when reheating is instantaneous, the dissipation rate of the inflaton increases towards the end of inflation [39], or a kinematically available decay channel opens suddenly [40]. In this case, the correct LSP abundance is obtained if the reheating temperature is about m DM /10, so that the LSP production is exponentially suppressed. The evolution of the early universe may include an era of domination by primordial black holes (PBHs). If the initial Hawking temperature of the PBHs is below m DM , the PBHs emit LSPs only after they lose most of their mass by Hawking radiation into light particles, and the LSP abundance is suppressed. As a result the correct LSP abundance can be obtained for sufficiently large initial PBH masses [41,42].

Conclusions
In recent decades, the theoretical and experimental investigations of supersymmetry were focused on weak scale supersymmetry. The discovery of the Higgs with a mass of 125 GeV has revealed a new scale of the SM, the Higgs quartic scale µ λ = 10 9−12 GeV, at which the SM Higgs quartic coupling vanishes. In this paper, we focused on Intermediate Scale Supersymmetry where supersymmetry is broken near the Higgs quartic scale. In this framework, including threshold corrections we found a small SM Higgs quartic coupling for a wide range of supersymmetry breaking parameters. The LSP is a dark matter candidate, and we studied the cases of Higgsino and sneutrino LSP, which scatter with nuclei via tree-level Z boson exchange. Direct detection experiments have already excluded the LSP mass below 3 × 10 9 GeV, and will probe it up to 10 12 GeV.
The Higgs quartic scale is sensitive to SM parameters. Currently, the uncertainty of the scale is dominated by the top quark mass and the strong coupling constant. We derived an upper bound on the LSP mass as a function of the top quark mass and the strong coupling constant shown in Fig. 7. Around the central value of SM parameters, dark matter signals should be discovered by near future experiments. Conversely, the figure shows an upper bound on the top quark mass and a lower bound on the strong coupling constant as a function of the LSP mass.
We also discussed how this LSP dark matter may be populated in the early universe. Because of the large LSP mass, the standard freeze-out mechanism overproduces the LSP. We avoid this by taking the reheating temperature after inflation below the LSP mass. We find that the observed dark matter abundance can be obtained during the reheating era, and in most of the parameter space, the inflaton condensate is dissipated by thermal effects during LSP production. We determined the required reheating temperature as a function of the inflaton mass and the LSP mass. Once the LSP mass is fixed by the signal rate at direct detection experiments, the reheating temperature is predicted from the inflaton mass. form = (10 10 − 10 12 ) GeV. The upper bound excludes large values of A − µ that would give a negative threshold correction to λ. For A + µ = 0, the bound becomes stronger. Larger m A slightly relaxes the bound, but not enough to enable a negative threshold correction to λ from the trilinear coupling.