Dark Matter Detection, Standard Model Parameters, and Intermediate Scale Supersymmetry
PPrepared for submission to JHEP
Dark Matter Detection, Standard Model Parameters,and Intermediate Scale Supersymmetry
David Dunsky , Lawrence J. Hall , Keisuke Harigaya Department of Physics, University of California, Berkeley, California 94720, USA Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720,USA School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA
Abstract:
The vanishing of the Higgs quartic coupling at a high energy scale may beexplained by Intermediate Scale Supersymmetry, where supersymmetry breaks at (10 -10 )GeV. The possible range of supersymmetry breaking scales can be narrowed down by precisemeasurements 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 GeV. We derive the correlation between the dark matter mass and precision measurementsof standard model parameters, including supersymmetric threshold corrections. The darkmatter mass is bounded from above as a function of the top quark mass and the strongcoupling constant. The top quark mass and the strong coupling constant are bounded fromabove and below respectively for a given dark matter mass. We also discuss how the observeddark matter abundance can be explained by freeze-out or freeze-in during a matter-dominatedera after inflation, with the inflaton condensate being dissipated by thermal effects. a r X i v : . [ h e p - ph ] N ov ontents – 1 – Introduction
In 1985, Goodman and Witten proposed that halo dark matter could be detected directly interrestrial experiments by observing small energy depositions from elastic scattering of darkmatter particles from nuclei [1]. Their first illustration was of a neutral particle, such as aheavy neutrino, scattering via t -channel Z exchange with a cross section per nucleon of order σv ∼ G F µ / π , where µ red is the reduced mass of the dark matter and nucleon. Theycomputed a signal of order 10 − events per Kg per day for dark matter masses in theGeV to TeV range, depending on nuclear target. In the intervening 35 years, a successionof ever larger and more sensitive detectors have excluded this example by many orders ofmagnitude, so that the focus has shifted to theories where there is no contribution to thescattering from tree-level weak interactions. In fact, as the number density of dark matterparticles scales as the inverse of its mass, present data constrains the mass of dark matterwith tree-level Z exchange to be larger than 3 × GeV [2]. Proposed detectors [3–5] willprobe the mass range M DM ,Z -exchange = (3 × − × ) GeV . (1.1)The discovery of the Higgs boson at the Large Hadron Collider (LHC) completes theStandard Model (SM). Electroweak symmetry breaking arises from the potential V SM ( H ) = − m | H | + λ | H | , (1.2)via the ground state value of the Higgs field (cid:104) H (cid:105) = v (cid:39)
174 GeV. The Higgs boson massis m h = 4 λv . No other new particles have been discovered at the LHC so far, and in thispaper we assume that the SM is valid to very high energies. All the SM couplings can becomputed at high energies to high precision, including the Higgs quartic coupling [6]. Asshown in Fig. 1, this running indicates that the Higgs quartic coupling vanishes at the scale µ λ = 10 − GeV , (1.3)which we call the Higgs quartic scale . Indeed, within the context of the SM as an effective fieldtheory 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 thenew physics explains why the Higgs quartic is so small at this scale. It is interesting to notethat, if dark matter couples to the weak interaction, the recent direct detection experimentshave started to explore dark matter masses in the vicinity of the Higgs quartic scale. Themass range to be explored by the next generation of experiments, (1.1), will probe the entirerange of (1.3).Since the discovery of a Higgs with mass of 125 GeV, several proposals have been madefor physics at µ λ that explains the small quartic coupling, including supersymmetry [7–9],extra dimension [10], Peccei-Quinn symmetry [11], and Higgs Parity symmetry [12–16]. In thispaper we pursue the case of Intermediate Scale Supersymmetry (ISS), where the superpartner– 2 – igure 1 . Running of the SM quartic coupling with current and future uncertainties in m t , α s ( m Z ),and m h . Their central values are m t = 172 .
76 GeV, α s ( m Z ) = 0 . m h = 125 .
10 GeV. mass scale ˜ m is of order the Higgs quartic scale. The identification of µ λ with ˜ m is natural[7, 8] since supersymmetry predicts a very small SM Higgs quartic at the scale ˜ m for a widerange of supersymmetry breaking parameters. Unlike in [7, 8], we study the case of Higgsinoor sneutrino Lightest Supersymmetric Particle (LSP) dark matter with mass of order ˜ m , sincethis 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 detectionsignal via Z exchange and the precision measurement of the top quark mass, m t , the strongcoupling constant, α s ( m Z ), (and to a lesser extent, of the Higgs boson mass, m h ). A darkmatter signal will determine the mass of the LSP and precision measurements will greatlyreduce the uncertainties in the Higgs quartic scale. In particular, we find that the discoveryof a direct detection signal implies an upper bound on the top quark mass and a lower boundon the strong coupling constant. The effects on the running of the Higgs quartic in reducingthe uncertainties in m t , α s ( m Z ) and m h are shown by the colored bands in Fig. 1. Futureuncertainties in m t (0 . α s ( m Z ) (0 . m h (0 .
01 GeV) from measurements atfuture lepton colliders [17–21], improved lattice calculations [22], and the high-luminosityLHC [23], will substantially reduce the uncertainty in µ λ to within a few tens of percents, asshown 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, couldbe produced in the early universe by freezing-out, losing thermal equilibrium while non-relativistic [24]. Indeed, they discovered that a heavy neutrino, with a GeV-scale mass, couldyield the observed dark matter abundance. Many other electroweak dark matter candidatesarising from freeze-out were studied, with masses up to several TeV. Apparently our proposalof Higgsino or sneutrino dark matter with a mass of 10 − GeV leads to a huge over-production of dark matter. However, we find that the observed abundance can result fromfreeze-out or freeze-in during a matter-dominated era after inflation. The inflaton mass mustbe below the dark matter mass, otherwise the O (1) branching fraction of the inflaton intosparticles leads to an overproduction of dark matter. Then during freeze-out or freeze-in, the– 3 –nflaton is dissipated by scattering reactions rather than by decays. If the products of thescattering reactions are thermalized at a high enough temperature, freeze-out occurs; other-wise, the abundance is set by freeze-in from non-thermal radiation. Either way, determiningthe dark matter mass from direct detection will provide a correlation between the reheattemperature after inflation and the inflaton mass.In section 2, building on [7, 8], we show that if the UV completion of the SM EFT isprovided by ISS, there is a large region of parameter space where the SM quartic coupling ispredicted to be very small at ˜ m , and hence ˜ m ∼ µ λ . In section 3 we compute the present limitson 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 futureprecision measurements of m t , α s ( m Z ), and m h . In section 4 we study how this correlationis affected by supersymmetric threshold corrections to the Higgs quartic coupling in theMinimal Supersymmetric Standard Model (MSSM). We find that these threshold correctionscan be significant and derive an upper bound on the Higgsino or sneutrino LSP mass asa function of the top quark mass and the strong coupling constant. An observable directdetection signal is predicted for top masses above a critical value. In section 5 we computethe supersymmetric threshold corrections in a scheme where the supersymmetry breakingparameters are constrained to a universal form at unified scales. In section 6 we compute therelic dark matter abundance from freeze-out or freeze-in during a matter dominated era wherethe inflaton condensate is dissipated by scattering reactions. Finally, we draw conclusions insection 7. 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 scale ˜ m , we assume that there is no gauge symmetry breaking and the theory containsa single pair of Higgs doublets, ( H u , H d ), and no weak singlets or triplets which have a zerohypercharge and couple to the Higgs doublets. For a wide range of parameters of this Higgssector, we find λ ( ˜ m ) (cid:28) .
01; remarkably there are large regions with λ ( ˜ m ) (cid:46) . m may be identified with the Higgs quartic scale µ λ .The Higgs potential is V ( H u , H d ) = ( µ + m H u ) H † u H u + ( µ + m H d ) H † d H d + ( Bµ H u H d + h . c . )+ g H † u (cid:126)σH u + H † d (cid:126)σH d ) + g (cid:48) H † u H u − H † d H d ) , (2.1)where µ is the supersymmetric Higgs mass parameter, while m H u , m H d , and Bµ are supersymmetry-violating 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 (cid:48) are the SU (2) and U (1) gauge couplings. Requiring electroweak symmetry to be unbrokenat ˜ m and one combination of the Higgs doublets to be much lighter than ˜ m requires that– 4 – igure 2 . Regions of parameter space showing the smallness of the ISS tree-level prediction for theHiggs quartic coupling at the scale ˜ m . λ ( ˜ m ) tree is less than 10 − if µ is much greater than m H u and m H d , or if m H u and m H d are nearly degenerate. The tree-level prediction is zero when m H d = m H u ,as indicated by the black horizontal line. In the gray region, one of the Higgs doublets has a negativemass squared. With Higgsino or sneutrino LSP, the blue region is excluded by XENON1T. µ + m H u,d are both positive. The fine tune for a light doublet requires that Bµ is taken tobe the geometric mean of µ + m H u,d . The light SM Higgs doublet is H = sin β H u + cos β H † d , (2.2)where tan β = ( µ + m H d ) / ( µ + m H u ), and we take β in the first quadrant.Matching the two theories at ˜ m gives the tree-level value for λ ( ˜ m ) λ ( ˜ m ) tree = g ( ˜ m ) + g (cid:48) ( ˜ m ) β (2.3)with cos β = (cid:32) m H u − m H d m H u + m H d + 2 µ (cid:33) . (2.4)ISS gives 0 ≤ λ ( ˜ m ) tree ≤ ( g ( ˜ m ) + g (cid:48) ( ˜ m ) ) / (cid:39) .
06 and hence at tree level ˜ m (cid:46) µ λ .Furthermore, over a wide range of values for m H u , m H d , and µ the cos β factor gives asignificant further suppression of λ ( ˜ m ) tree , as shown in Fig. 2. Indeed, cos 2 β → µ (cid:29) | m H u,d | or m H u → m H d ; in these limits ˜ m is identified with µ λ . Thegray-shaded region is excluded since µ + m H u < µ + m H d < λ ( ˜ m ) tree > .
01, ˜ m is predicted to be below a few 10 GeV.As we will see in the next section, the Higgsino or sneutrino LSP then gives too large a direct– 5 –etection rate. However, there is a remarkably large region of parameter space in Fig. 2 with λ ( ˜ m ) tree < . In this section, we discuss direct detection of the Higgsino or sneutrino LSP dark matter innuclear recoil experiments and show that detection rates are correlated with SM parametersthrough the connection between ˜ m and µ λ . An observable direct detection signal is predictedfor top masses below a critical value. The neutral components and the charged component of the Higgsino are degenerate in massin the electroweak symmetric limit. With elecroweak symmetry breaking, the charged com-ponent becomes heavier than the neutral components by O (100) MeV via one-loop quantumcorrections [25]. The neutral components slightly mix with the bino and the wino and obtaina small mass splitting ∆ m ∼ g v M ≈
10 keV (cid:18) M GeV (cid:19) − . (3.1)The two mass eigenstates are Majorana fermions. For a soft mass scale above ∼ GeV,however, the splitting is smaller than the typical nucleon recoil energy of O (10 − Z bosonexchange leads to the up-scattering of the ligher state into the heavier state, which almostbehaves as scattering of a Dirac fermion. The sneutrino is lighter than its charged SU (2) partner because of electroweak symmetrybreaking and quantum corrections. The two components of the sneutrino obtain a smallmass splitting from the A term of the Majorana neutrino mass term,∆ m ∼ Am ν m ˜ ν , (3.2)which is negligibly small. Sneutrino dark matter interacts with nucleon via Z boson exchangeas a complex scalar field.If the slepton and squark masses are universal at the unification scale, the sneutrinocannot be the LSP because renormalization running makes the right-handed stau the lightestamong them. Non-universality is required for the sneutrino LSP. We note that the sneutrinoLSP is consistent with SU (5) unification, since the sneutrinos and the right-handed sleptonsare not unified, and the right-handed down type squarks become heavier than the sneutrinosby renormalization running. – 6 – igure 3 . Prediction for the top quark mass as a function of the sparticle mass scale, ˜ m , and thetree-level Higgs quartic coupling at ˜ m . Contours of m top span 3 σ above and below the current centralvalue for m top , (172 . ± .
30) GeV. For Higgsino or sneutrino LSP dark matter, the green shadedregion is excluded by XENON1T and dotted green lines show the sensitivities of future experiments.Values of m t are experimentally disfavored in the dark blue region Both Higgsino and sneutrino dark matter scatter with nuclei, with an effective dark matter-nucleon scattering cross section given by σ n = G F m n π (cid:20) ( A − Z ) − (1 − θ W ) ZA (cid:21) , (3.3)where G F is the Fermi constant, m n is the nucleon mass, A is the mass number, Z is theatomic number, and θ W is the Weinberg angle. The current constraint by XENON1T [2] andthe future sensitivities of LZ with an exposure of 15 ton · year, XENONnT with an exposureof 20 ton · year, and DARWIN with an exposure of 1000 ton · year [3–5] are given by σ n < × − GeV − m DM GeV (XENON1T, current) . (3.4) σ n < × − GeV − m DM GeV (LZ, XENONnT, future) . (3.5) σ n < × − GeV − m DM GeV (DARWIN, future) , (3.6)which translates into the constraint on and the sensitivity to the Higgsino or sneutrino darkmatter mass of m DM > × GeV (XENON1T, current) , (3.7) m DM > × GeV (LZ, XENONnT future) , (3.8) m DM > × GeV (DARWIN, future) . (3.9)– 7 –nce dark matter signals are found in recoil experiments, within the framework of Hig-gsino or sneutrino dark matter in ISS, the dark matter mass is fixed from the observed signalrates. Since λ ( ˜ m ) tree is positive and m DM = m LSP < ˜ m , we obtain a bound on SM parametersincluding 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 given ˜ m and λ ( ˜ m ) tree isshown in Fig. 3. The right vertical axis shows cos2 β corresponding to λ ( ˜ m ) tree . For a given m DM , the prediction on m t for λ ( ˜ m ) tree = 0 and ˜ m = 0 can be understood as an upper boundon m t . For a given m t , ˜ m such that λ ( ˜ m ) tree = 0 in an upper bound on m DM . To obtainthose bounds precisely, we include threshold corrections to λ ( ˜ m ) in the next section. The full prediction for λ ( ˜ m ) in ISS is λ ( ˜ m ) = λ ( ˜ m ) tree + δλ ( ˜ m ) , (4.1)where λ tree is the the tree-level result, (2.3), and δλ the quantum corrections that ariseon integrating out heavy sparticles. The largest contributions arise from sparticles with thelargest couplings to the light Higgs; hence the most important mass parameters are the massesof the third generation doublet squark m ˜ q , the third generation up-type squark m ˜¯ u , the bino M , the wino M , 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 m ˜ q and m ˜¯ u , which we denote as m − .Since quantum corrections are greater than λ tree only for tan β (cid:39)
1, we neglect correctionswhich vanish in this limit. Using the results in [26], the corrections are given by32 π δλ ( m − ) = 3 y t (cid:32) ln m q m − + ln m u m − + 2 X t F (cid:18) m ˜ q m ˜¯ u (cid:19) − X t G (cid:18) m ˜ q m ˜¯ u (cid:19)(cid:33) − (cid:18) g (cid:48) + 2 g (cid:48) g + 163 g (cid:19) − g (cid:48) f (cid:18) M µ (cid:19) − g f (cid:18) M µ (cid:19) − g (cid:48) g f (cid:18) M µ , M µ (cid:19) − ( g (cid:48) + 2 g (cid:48) g + 3 g ) ln (cid:18) µm − (cid:19) + 18 (cid:16) g (cid:48) + 2 g (cid:48) g + 3 g (cid:17) ln m A m − . (4.2)Here, X t ≡ ( A t − µ ) /m ˜¯ u m ˜¯ q , and the functions F, G, f , f are given by F ( x ) = 2 x ln xx − , G ( x ) = 12 x (1 − x + (1 + x ) ln x )( x − ,f ( x ) = 3( x + 1) x − + 3( x − x ln x x − ,f ( x, y ) = 3(1 + x + y − xy )8( x − y −
1) + 3 x ln x x − ( x − y ) − y ln y y − ( x − y ) . (4.3)– 8 – igure 4 . Threshold corrections to the Higgs quartic coupling as a function of sparticle mass param-eters. The six curves correspond to m = ( A t , µ, m + , m A , M , M ) with the remaining five parametersfixed at m − = min( m ˜ q , m ˜¯ u ). The Higgsino can be the LSP on the solid curves, but is not the LSPon the dashed part of the curves for µ, M and M . Left A t >
0. Vacuum instability occurs when A t , µ (cid:38) . m − . Right A t <
0. Vacuum instability occurs when | A t | , µ (cid:38) . m − . They are normalized so that they are unity when the arguments are unity. For a degeneratemass spectrum and negligible X t , δλ ( m − ) (cid:39) − . δλ varies as a function of sparticle masses.The left and right panels correspond to A t positive and negative, respectively. Each curvecorresponds to varying one of ( A t , µ, m + , m A , M , M ), while keeping all the others fixed at m − . With all these parameters near m − , the correction is δλ ( m − ) (cid:39) − .
002 for A t > .
002 for A t <
0. For | X t | (cid:38) m − , the electroweak vacuum is unstable, as shown bythe sudden discontinuation of the A t and µ curves. The bound on X t from the instability isderived in Appendix A. The Higgsino can be the LSP on the solid curves, but is not the LSPon the dashed part of the curves for µ, M and M . The slepton mass parameter m ˜ l may betaken 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 thestrong coupling constant varied within ± σ uncertainty from its central value in the top andbottom panels. The right axis shows cos2 β corresponding to λ ( m − ) when δλ (cid:28) λ tree . Thelower bound on the dark matter mass from XENON1T is shown in green, and the lowerbound on threshold corrections to λ ( m − ) is shown in red. Together, these bounds require m t (cid:46) . m t (cid:46) . × 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– 9 – igure 5 . Prediction for the top quark mass as a function of m − = min(m ˜q , m ˜¯u ) and the Higgsquartic coupling at m − . Contours of m t span 3 σ above and below the current central value for m t ,(172 . ± .
30) GeV. The red shaded region requires unrealistically large negative supersymmetricthreshold corrections to the quartic coupling. The green shaded region and the green dotted lines areas in Fig. 3. Values of m t are experimentally disfavored in the dark blue region. – 10 – igure 6 . Upper bound on the dark matter mass m DM as a function of the top quark mass m t for a range of typical threshold corrections. The blue curve shows the bound when the thresholdcorrections are zero, the orange curve when the sparticle spectra are degenerate m − , and in green ,when M , = √ m − . Equivalently, the figure shows an upper bound on m t as a function of m DM . sparticle masses. As shown in Fig. 4, large wino or bino masses give negative thresholdcorrections to the quartic coupling, thereby relaxing the upper bounds on the top quark massand the dark matter mass. In Fig. 6, we show the upper bound on the dark matter mass asa function of the top quark mass or, equivalently, the upper bound on the top quark massas a function of the dark matter mass. The blue curve is without threshold corrections, theorange curve has threshold corrections for a degenerate mass spectrum with A t (cid:39) µ , and onthe green curve, the degeneracy is lifted by taking M , = √ m − . With this hierarchy, theupper bound on the dark matter mass is relaxed by a factor of 2, and that on the top quarkmass 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 scalarmasses.)In Fig. 7, the upper bound on the dark matter mass or, equivalently, the upper boundon the top quark mass or the lower bound on the strong coupling constant, is shown. Herewe impose δλ ( m − ) > − . σ uncertainty of m t and α s ( m Z ) are shown bywide bands. The uncertainty of α s ( m Z ) can be reduced by a factor of 10 by measurementsat 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 [17–20]. At this stage, thetheoretical computation of the running of the Higgs quartic coupling should be improved; themost recent computation [6] has a theoretical uncertainty equivalent to the shift of the topquark mass by 100 MeV. – 11 – igure 7 . Upper bound on the dark matter mass m DM as a function of the top quark mass m t andthe strong coupling constant α s ( m Z ) shown in blue . Equivalently, the figure shows an upper boundon m t as a function of α s ( m Z ) and m DM , and a lower bound on α s ( m Z ) as a function of m t and m DM .The wider gray bands show the current 2 σ uncertainties of m t and α s ( m Z ), and the narrower bandsshow the expected future uncertainties. Dark matter direct detection bounds are shown in green . In this section, we discuss the quartic coupling at the supersymmetry breaking scale, ˜ m ,starting from boundary conditions at the unification scale ∼ GeV. We show that thetree-level quartic coupling is typically 0 . − . m is small when m H u ∼ m H d . A relation m H u = m 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 scalarmasses. The relation is necessarily destabilized by quantum correction from the top quarkYukawa coupling, ddln µ m H u = 3 y t π (cid:0) m H u + m q + m u + A t (cid:1) + · · · , (5.1)where the ellipsis denotes terms independent of the top Yukawa. We compute the renormal-ization group running of the MSSM from a scale 10 GeV down to ˜ m with a UV boundarycondition motivated from SU (5) unification, m H u = m H d = m H , m q = m u = m e = m , m d = m (cid:96) = m ,M = M = M = m / , A t = A t,G . (5.2)The SM top yukawa coupling is matched to the MSSM top yukawa coupling at ˜ m assumingtan β (cid:39) y t, MSSM = √ y t, SM . The soft masses m H u and m H d at the renormalization scale– 12 – igure 8 . Prediction for the tree-level quartic coupling with a UV boundary condition m H u = m H d .In the blue shaded region, reproducing λ ( m − ) requires m t < .
86 GeV, 3 σ away from the centralvalue. Here we impose α s ( m Z ) < . δλ ( m − ) > − . – 13 –10 , ) GeV are given by the analytic results m H u (10 GeV) =0 . m H − . m − . A t,G +0 . m / + 0 . m / A t,G ,m H d (10 GeV) =1 . m H +0 . m / , (5.3) m H u (10 GeV) =0 . m H − . m − . A t,G +0 . m / + 0 . m / A t,G ,m H d (10 GeV) =1 . m H +0 . m / . (5.4)In Fig. 8, we show the tree-level quartic coupling as a function of m / /m H for sev-eral representative boundary conditions; the left (right) panels have m H = 10 GeV (10 GeV). We fix the renormalization scale to be the matching scale used in the previous sec-tion, m − , the lighter of m ˜ q and m ˜¯ u . The boundary condition for m is not specified as itdoes not affect the renormalization group running of m H u . Note that the bino, ˜ b , is thelightest gaugino and the right-handed slepton, ˜ e , is the lightest scalar in the matter ten-plet. We define m (˜ b, ˜ e ) to be the smaller of m ˜ b and m ˜ e . On the five lines, µ is fixed to be( (cid:28) m (˜ b, ˜ e ) , m (˜ b, ˜ e ) / , m (˜ b, ˜ e ) , m H d , m H d ). As µ is increased, the tree-level quartic couplingdecreases rapidly, as expected from (2.3), (2.4) and Fig. 2. For large values of m the Hig-gsino is the LSP above the green dot-dashed line, and the region below the line is excludedbecause at low (high) m / the LSP is the bino (a charged right-handed slepton). For smallvalues of m the tau sneutrino can be the LSP throughout the plane, although at low µ theHiggsino LSP is also possible. In the blue shaded region, the top quark mass must be below171 .
86 GeV, more than 3 σ away from the central value, in order for λ ( m − ) to be consis-tent with the running of the Higgs quartic coupling. To derive a conservative bound, wetake α s ( m Z ) = 0 . σ above the central value, and δλ = − . λ tree result for larger m H , which gives less running, largervalues of µ/m H and smaller values of m /m H and A t,G /m H . For m H = 10 GeV, λ tree < .
003 over much of the parameter space. Including threshold corrections, Fig. 5 shows thatthis is ideal for consistency with the observed Higgs mass, and requires a low value of the topquark mass. For m H = 10 GeV, λ tree < .
01 over much of the parameter space, except atlow values of µ , which from Fig. 5 again shows excellent consistency with the observed Higgsmass, and leads to the expectation that Higgsino/sneutrino dark matter will be discovered atplanned experiments. In this section, we discuss how the heavy LSP dark matter can be populated in the earlyuniverse. Most of the discussion in this section is applicable to generic heavy dark matterwith electroweak interactions. Standard freeze-out during the radiation dominated era over-produces the LSP because of its large mass. To avoid this, the reheating temperature of theuniverse must be smaller than the LSP mass, and the LSP must be produced during the– 14 –eheating process. We discuss reheating by the inflaton φ , but, if the LSPs produced duringinflaton reheating are subdominant, the following discussion also applies to the case wheresome other particle or condensate dominates the energy density of the universe. 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 ρ LSP s (cid:39) N LSP m DM T RH m φ = 10 eV m DM GeV 10 GeV m φ T RH MeV N LSP , (6.1)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 (cid:29) m DM and the inflaton dominantly decays into SMcharged particles, showering leads to N LSP (cid:29) T RH > We first derive the evolution of the temperature of the universe. We consider the case wherethe dissipation of the inflaton occurs by perturbative processes, with dissipation rates givenby Γ = Γ : T < m φ Γ (cid:16) Tm φ (cid:17) n : m φ < T . (6.2)For T < m φ , dissipation is governed by the zero-temperature decay rate Γ , while for m φ < T ,thermal effects should be taken into account. n = 1 arises when dissipation is caused by adimensionless coupling, while n = − φhh † .The dependence of the temperature on the Hubble scale is given by T RH < m φ : T = T RH (cid:16) HH RH (cid:17) / : T RH < T < m φ m φ (cid:18) HT H RH m φ (cid:19) / (4 − n ) : m φ < T, (6.3) m φ < T RH : T = T RH (cid:18) HH RH (cid:19) / (4 − n ) , (6.4)where H RH = (cid:112) π g ∗ / T /M Pl is the Hubble scale at the completion of reheating. Weimplicitly 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.– 15 – ase 1: T FO < m φ < m DM During freeze-out, when T FO = m φ /x FO < m φ , radiation is created from the zero-temperaturedecay 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 n LSP ρ φ (cid:39) H FO (cid:104) σv (cid:105) ρ φ = 13 (cid:104) σv (cid:105) H FO M . (6.5)Using ρ φ /s (cid:39) T RH / ρ LSP s (cid:39) x (cid:114) π g ∗ πα T M Pl m DM πα /m (cid:104) σv (cid:105) . (6.6)Here, we assume that radiation thermalizes around the freeze-out temperature. Thisassumption is valid if 4 πα T FO > H FO , requiring T RH > (cid:20) α (cid:114) g ∗
90 ( m DM /x FO ) M Pl (cid:21) / ≡ T RH , th . (6.7)If this condition is violated, the radiation produced from the inflaton does not reach thermalequilibrium by the would-be freeze-out. We expect that the distribution of radiation in thiscase 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 thewould-be temperature and the radiation is in an over-occupied state. The energy distributionhas a cutoff, above which the scattering is inefficient and the distribution is exponentiallysuppressed.For large m DM , the reheating temperature to reproduce the observed abundance fromEq. (6.6) is in fact smaller than T RH , th . Then the LSP abundance is exponentially suppressedand LSPs are under-produced. As T RH approaches T RH , th , the LSP production is not sup-pressed, and the freeze-out picture is applicable. Since T RH ∼ T RH , th is larger than thatto 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 byLSP production during reheating is shown in Fig. 9. Above the black dashed line, T FO
90 ( m DM /x FO ) − n m nφ M Pl (cid:35) / ≡ T RH , th . (6.9)The reheating temperature required to produce the observed dark matter abundance is shownin Fig. 9. The above analysis is applicable between the dashed and dotted lines. Case 3: m φ < T RH For the inflaton mass below the reheating temperature, the temperature during freeze-out isgiven by Eq. (6.4). The LSP density is given by ρ LSP s (cid:39) x − n FO (cid:114) π g ∗ πα T − n RH M Pl m − n DM πα /m (cid:104) σv (cid:105) . (6.10)Radiation thermalizes before freeze-out if T RH > (cid:20) α (cid:114) g ∗
90 ( m DM /x FO ) − n M Pl (cid:21) − n ≡ T RH , th . (6.11)– 17 –he reheating temperature required to produce the observed dark matter abundance is shownin Fig. 9. This analysis is applicable below the dotted line. Is is possible that the maximal temperature of the universe is the reheating temperature. Thisoccurs when reheating is instantaneous, the dissipation rate of the inflaton increases towardsthe end of inflation [39], or a kinematically available decay channel opens suddenly [40]. In thiscase, the correct LSP abundance is obtained if the reheating temperature is about m DM / m DM , the PBHs emitLSPs only after they lose most of their mass by Hawking radiation into light particles, andthe LSP abundance is suppressed. As a result the correct LSP abundance can be obtainedfor sufficiently large initial PBH masses [41, 42]. In recent decades, the theoretical and experimental investigations of supersymmetry werefocused on weak scale supersymmetry. The discovery of the Higgs with a mass of 125 GeVhas revealed a new scale of the SM, the Higgs quartic scale µ λ = 10 − GeV, at which theSM Higgs quartic coupling vanishes. In this paper, we focused on Intermediate Scale Super-symmetry 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 rangeof supersymmetry breaking parameters. The LSP is a dark matter candidate, and we studiedthe cases of Higgsino and sneutrino LSP, which scatter with nuclei via tree-level Z bosonexchange. Direct detection experiments have already excluded the LSP mass below 3 × GeV, and will probe it up to 10 GeV.The Higgs quartic scale is sensitive to SM parameters. Currently, the uncertainty of thescale is dominated by the top quark mass and the strong coupling constant. We derived anupper bound on the LSP mass as a function of the top quark mass and the strong couplingconstant shown in Fig. 7. Around the central value of SM parameters, dark matter signalsshould be discovered by near future experiments. Conversely, the figure shows an upper boundon the top quark mass and a lower bound on the strong coupling constant as a function ofthe 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. Wefind 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 effectsduring LSP production. We determined the required reheating temperature as a function of– 18 –he inflaton mass and the LSP mass. Once the LSP mass is fixed by the signal rate at directdetection experiments, the reheating temperature is predicted from the inflaton mass.
Acknowledgement
This work was supported in part by the Director, Office of Science, Office of High Energyand Nuclear Physics, of the US Department of Energy under Contracts DE-AC02-05CH11231(LJH), by the National Science Foundation under grant PHY-1915314 (LJH), as well as byFriends of the Institute for Advanced Study (KH).
A Stability bound on a trilinear coupling
In this appendix, we derive an upper bound on the trilinear coupling between between theHiggs and stops from the stability of the electroweak vacuum. We consider the case oftan β (cid:39) H u → (cid:32) h − H (cid:33) , H d → (cid:32) h + H (cid:33) , q → √ u
00 00 0 , ¯ u → √ ¯ u , (A.1)where h , H , u , and ¯ u are real fields with potential V ( h, H, q, ¯ u ) = 12 m A H + 12 m q q + 12 m u u − √ y t h ( A − µ ) u ¯ u − √ y t H ( A + µ ) u ¯ u + y t (cid:18) u ¯ u + 14 ( h + H ) (cid:0) u + ¯ u (cid:1)(cid:19) + g (cid:48) (cid:18) hH + 112 u −
13 ¯ u (cid:19) + g (cid:18) hH − u (cid:19) + g (cid:0) u − ¯ u (cid:1) . (A.2)The renormalization scale of the coupling constants is taken to be around the sparticle massscale.The tunneling rate per volume is given by [43]Γ V = M exp − S B , (A.3)where S B is a bounce action and M is a pre-factor as large as the typical energy scaleassociated with the tunneling, which we take to be the sparticle mass scale. To avoid tunnelinginto another vacuum, we require that Γ /V × H <
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