February 1, 2008 DARK MATTER IN SUSY MODELS R. Arnowitt, B. Dutta and Y. Santoso 2 Center For Theoretical Physics, Department of Physics, 0 Texas A&M University, College Station TX 77843-4242 0 2 n a J Direct detection experiments for neutralino dark matter in the Milky Way 1 are examined within the framework of SUGRA models with R-parity invari- 2 ance and grand unification at the GUT scale, M . Models of this type ap- G 1 ply to a large number of phenomena, and all existing bounds on the SUSY v parameter space due to current experimental constraints are included. For 4 9 models with universal soft breaking at MG (mSUGRA), the Higgs mass and 1 b → sγ constraints imply that the gaugino mass, m , obeys m >(300- 1 1/2 1/2 0 400)GeV putting most of the parameter space in the co-annihilation domain 2 where there is a relatively narrow band in the m − m plane. For µ > 0 0 1/2 0 h/ we find that the neutralino -proton cross section ∼> 10−10 pb for m1/2 < 1 p TeV, making almost all of this parameter space accessible to future planned - detectors. For µ < 0, however, there will be large regions of parameter space p e with cross sections < 10−12 pb, and hence unaccessible experimentally. If, h however, the muon magnetic moment anomaly is confirmed, then µ > 0 and : v < m ∼ 800 GeV. Models with non-universal soft breaking in the third gener- i 1/2 X ation and Higgs sector can allow for new effects arising from additional early r universe annihilationthroughthe Z-channelpole. Herecross sections thatwill a be accessible in the near future to the next generation of detectors can arise, and can even rise to the large values implied by the DAMA data. Thus dark matter detectors have the possibility of studying the the post-GUT physics that control the patterns of soft breaking. PRESENTED AT NON-ACCELERATOR NEW PHYSICS Dubna,Russia, June 19–23, 2001 1 Introduction The recent BOOMERanG, Maxima and DASI data has allowed a relatively precise de- termination of the mean amount of dark matter in the universe, and these results are consistent with other astronomicalobservations. Within the Milky Way itself, the amount of dark matter is estimated to be ρ =∼ (0.3−0.5)GeV/cm3 (1) DM Supersymmetry with R-parity invariance possesses a natural candidate for cold dark mat- ter (CDM), the lightest neutralino, χ˜0, and SUGRA models predict a relic density con- 1 sistent with the astronomical observations of dark matter. Several methods for detecting the Milky Way neutralinos exist: (1) Annihilation of χ˜0 in the halo of the Galaxy leading to anti-proton or positron 1 signals. There have been several interesting analyses of these possibilities [1, 2], but there are still uncertainties as to astronomical backgrounds. (2) Annihilation of the χ˜0 in the center of the Sun or Earth leading to neutrinos and 1 detection of the energetic ν by neutrino telescopes (AMANDA, Ice Cube, ANTARES). µ Recent analyses [3,4] indicate that these detectors can be sensitive to such signals, but for the Minimal Supersymmetric Standard Model (MSSM) one requires mχ˜01 > 200 GeV (i.e. m > 500 GeV) and tanβ > 10, and for SUGRA models one is restricted to tanβ > 35 1/2 [3]. (3) Direct detection by scattering of incident χ˜0 on nuclear targets of terrestrial de- 1 tectors. Current detectors are sensitive to such events for χ˜0 − p cross sections in the 1 range σχ˜01−p ∼> 1×10−6pb (2) with a possible improvement by a factor of 10 - 100 in the near future. Future detectors (GENIUS, Cryoarray, ZEPLIN IV) may be sensitive down to (10−9 −10−10) pb and we will see that this would besufficient to cover the parameter space of most SUGRAmodels. In the following we will consider SUGRA models with R-parity invariance based on ∼ grand unification at the GUT scale M = 2×1016 GeV. In particular, we will consider G two classes of models: Minimal supergravity models (mSUGRA [5, 6]) with universal soft breaking masses at M , and non-universal models with non universal soft breaking at M G G for the Higgs bosons and the third generation of squarks and sleptons . Here the gaugino masses (m ) and the cubic soft breaking masses (A ) at M are assumed universal. 1/2 0 G SUGRA models apply to a wide range of phenomena, and data from different ex- periments interact with each other to greatly sharpen the predictions. We list here the important experimental constraints: Higgsmass: m >114GeV[7]. Thetheoreticalcalculationofm stillhasananerrorof h h ∼ 3 GeV, and so we will (conservatively) interpret this bound to mean m (theory) > 111 h GeV. 1 b → sγ branching ratio. We take a 2σ range around the central CLEO value [8]: 1.8×10−4 ≤ B(B → X γ) ≤ 4.5×10−4 (3) s χ˜0 relic density: We assume here 1 0.02 ≤ Ω h2 ≤ 0.25 (4) DM The lower bound takes into account of the possibility that there is more than one species of DM. However, results are insensitive to raising it to 0.05 or 0.10. 500 400 ) V e G 300 ( 0 m 200 100 300 400 500 600 700 800 900 1000 m (GeV) 1_ 2 Figure 1: Corridors in the m −m plane allowed by the relic density constraints for 0 1/2 (bottom to top) tanβ = 10, 30, 40, A = 0 and µ > 0. The lower bound on m is due to 0 1/2 the m lower bound for tanβ = 10, due to the b → sγ bound for tanβ = 40, while both h these contribute equally for tanβ = 30. The short lines cutting the channels represent upper bound from the g −2 experiment. [17] µ Muon a = (g −2)/2 anomaly. The Brookhaven E821 experiment [9] reported a 2.6σ µ µ deviation from the Standard Model value in their measurement of the muon magnetic moment. Recently a sign error in the theoretical calculation [10, 11] has reduced this to a 1.6σ anomaly, though recent measurements [12] used to calculate the hadronic contri- bution may have raised the deviation. Since there is a great deal of more data currently being analyzed (with results due this spring) that will reduce the errors by a factor of ∼2.5, we will assume here that there is a deviation in a due to SUGRA of amount µ 11×10−10 ≤ aSUGRA ≤ 75×10−10 (5) µ We will, however, state our results with and without including this anomaly. To illustrate how the different experimental constraints affect the SUSY parameter space, we consider the mSUGRA example: 2 1000 ) 800 V e G ( 600 0 m 400 200 400 500 600 700 800 900 1000 m (GeV) 1_ 2 Figure 2: Corridors in the m − m plane allowed by the relic density constraint for 0 1/2 tanβ = 40, µ > 0 and (bottom to top) A = 0, −2m , 4m . the curves terminate at 0 1/2 1/2 the lower end due to the b → sγ constraint except forA = 4m which terminates due 0 1/2 to the m constraint. The short lines cutting the corridors represent the upper bound on h m due to the g −2 experiment. [17] 1/2 µ (1) The m and b → sγ constraints put a lower bound on m : h 1/2 > m ∼ (300−400)GeV (6) 1/2 > ∼ which means mχ˜01 ∼ (120 − 160) GeV (since mχ˜01 = 0.4m1/2). (2) Eq.(6) now means that most of the parameter space is in the τ˜ − χ˜0 co-annihilation domain in the relic 1 1 densitycalculation. Thenm (thesquarkandsleptonsoftbreakingmass)isapproximately 0 determined by m ascanbeseen inFigs. 1and2. (3) Ifwe include thea anomaly, since 1/2 µ aSUGRA is a decreasing function of m and m , the lower bound of Eq.(5) produces an µ 1/2 0 upper bound on m and the positive sign of a implies that the µ parameter is positive. 1/2 µ In addition one gets a lower bound on tanbeta of tanβ > 5. Thus the parameter space has begun to be strongly constrained, allowing for more precise predictions. In order to carry out detailed calculations, however, it is necessary to include a number of analyses to obtain accurate results. We list some of these here: Two loopgaugeandoneloopYukawarenormalizationgroupequations(RGE)areused in going from M to the electroweak weak scale M , and QCD RGE are used below G EW M for the light quark contributions. Two loop and pole mass corrections are included EW in the calculation of m . One loop corrections to m and m [13, 14] are included which h b τ are important at large tanβ. Large tanβ NLO SUSY corrections to b → sγ [15, 16] are included. In calculating the relic density, all stau-neutralino co- annihilation channels are included, and this calculation is done in a fashion valid for both small and large tanβ. 3 1·10-4 ) b p 1·10-5 -60 1 (p 1·10-6 s ~0c-1 1·10-7 400 500 600 700 800 900 1000 m (GeV) _1 2 Figure 3: σχ˜01−p for mSUGRA for µ < 0, A0 = 1500 GeV, for tanβ = 6 (short dash), tanβ = 8 (dotted), tanβ = 10 (solid), tanβ = 20 (dot-dash), tanβ = 25 (dashed). Note that the tanβ = 6 curve terminates at low m due to the Higgs mass constraint, and 1/2 the other curves terminate at low m due to the b → sγ constraint [18]. 1/2 We do not include Yukawa unification or proton decay constraints, since these depend sensitively on post-GUT physics, about which little is known. 2 mSUGRA MODEL ThemSUGRAmodelisthesimplest, andhencemostpredictiveofthesupergravitymodels in that it depends on only four new parameters and one sign (in addition to the usual SM parameters). We take these new parameters to be m and m (the universal soft 0 1/2 breaking scalar and gaugino masses at M ), A (the universal cubic soft breaking mass G 0 at M ) , tanβ =< H > / < H > at the electroweak scale (where < H > gives rise to G 2 1 2 up quark masses and < H > to down quark masses) and the sign of µ (the Higgs mixing 1 parameter which appears in the superpotential as µH H ). We examine these parameters 1 2 over the range m ,m ≤ 1 TeV, 2 < tanβ < 50, |A | ≤ 4m . The bound on m 0 1/2 0 1/2 1/2 corresponds to the gluino mass bound of m < 2.5 GeV which is also the reach of the g˜ LHC. The relic density analysis involves calculating the annihilation cross section for neu- tralinos in the early universe. This characteristically proceeds through Z and Higgs s- channel poles (Z, h, H, A where H and A are heavy CP even and CP odd Higgs bosons) and through t-channel sfermion poles. However, if there is a second particle which be- comes nearly degenerate with the neutralino, one must include it in the early universe annihilation processes, which then leads to the co-annihilation phenomena. In mSUGRA models, this accidental near degeneracy occurs naturally for the light stau, τ˜ . One can 1 understand this semi-quantitatively by considering the low and intermediate tanβ region 4 0.03 0.01 ) b p -60 0.005 1 (p s ~0c-1 0.002 0.001 150 200 250 300 350 m~c 0 (GeV) 1 Figure 4: σχ˜01−p as a function of the neutralino mass mχ˜01 for tanβ = 40, µ > 0 for A0 = −2m1/2,4m1/2,0 from bottom to top. The curves terminate at small mχ˜01 due to the b → sγ constraint for A = 0 and −2m and due to the Higgs mass bound (m > 114 0 1/2 h GeV) for A0 = 4m1/2. The curves terminate at large mχ˜01 due to the lower bound on aµ of Eq. (5)[17]. where the RGE give for the right selectron, e˜ , and the neutralino the following masses R at the electroweak scale: m2 = m2 +0.15m2 −sin2θ M2 cos2β (7) e˜R 0 1/2 W W m2 = 0.16m2 (8) χ˜01 1/2 ∼ the numerics coming from the RGE analysis. The last term in Eq. (7) = (40GeV)2. ∼ Thus for m = 0 the e˜ will become degenerate with the χ˜0 at m = 400 GeV, and 0 R 1 1/2 ∼ co-annihilation thus begins at m = (350− 400) GeV. As m increases, m must be 1/2 1/2 0 raised in lock step (to keep me˜R > mχ˜01). More precisely, it is the light stau, which is the lightest slepton that dominates the co-annihilation phenomena. However, one ends up with corridors in the m −m plane for allowed relic density with m closely correlated 0 1/2 0 with m increasing as m does, as seen in Figs. 1 and 2. 1/2 1/2 For dark matter detectors with heavy nuclei targets, the spin independent neutralino - nucleus cross section dominates, which allows one to extract the χ˜0 -proton cross section, 1 σχ˜01−p. The basic quark diagrams for this scattering go through s-channel squark poles and t-channel Higgs (h, H) poles. The general features of σχ˜01−p that explain its properties are the following: σχ˜01−p increases with increasing tanbeta (9) σχ˜01−p decreases with increasing m1/2 and increasing m0 (10) Since co-annihilation generally correlates m and m , if m increases so does m (at 0 1/2 1/2 0 fixed tanbeta and A ). 0 5 The smallest cross sections occur for the case µ < 0. This is because a special can- cellation can occur over a fairly wide range of tanβ and m [19, 18] driving the cross 1/2 section below 10−13 pb. This is illustrated in Fig. 3. In these regions, there would be no hope for currently planned dark matter detectors to be able to detect Milky Way neu- tralinos. However, if the a anomaly is confirmed by the new BNL E821 data (currently µ being analyzed), then µ < 0 is forbidden, and the special cancelations do not occur for µ > 0. Large cross sections can then occur for large tanbeta. Thus is seen in Fig. 4 for tanβ = 40, with m > 114 GeV. If the Higgs mass bound were to rise, the lower bounds h on m1/2 would increase. Thus for mh > 120 GeV, one has mχ˜01 > (200,215,246) GeV for A = (−2,0,4)m . 0 1/2 The lowest cross sections for µ > 0 are expected to occur for small tanbeta and large m . This is seen in Fig. 5 for tanβ = 10 where one also sees that decreasing A gives 1/2 0 smaller cross sections. In general one finds σχ˜01−p ∼> 10−10 pb for µ > 0, m1/2 < 1TeV (11) Such cross sections are within the reach of future planned detectors. 0.005 ) b p -60 0.002 1 (p s ~0c-1 0.001 0.0005 120 130 140 150 160 170 180 190 m~c 0 (GeV) 1 Figure 5: σχ˜01−p as a function of mχ˜01 for tanβ = 10, µ > 0, mh > 114 GeV for A0 = 0 (upper curve), A0 = −4m1/2 (lower curve). The termination at low mχ˜01 is due to the mh bound for A0 = 0, and the b → sγ bound for A0 = −4m1/2. The termination at high mχ˜01 is due to the lower bound on a of Eq. (5)[17]. µ 3 NON-UNIVERSAL MODELS New results can occur if we relax the universality of the squark, slepton and soft breaking Higgs masses at M . To maintain the flavor changing neutral current bounds, we do this G 6 1000 ) V 800 e G ( 600 0 m 400 200 400 450 500 550 600 650 700 750 m (GeV) 1_ 2 Figure 6: Effect of a nonuniversal Higgs soft breaking mass enhancing the Z0 s-channel pole contribution in the early universe annihilation, for the case of δ =1, tanβ = 40, 2 A = m , µ > 0. The lower band is the usual τ˜ coannihilation region. The upper 0 1/2 1 band is an additional region satisfying the relic density constraint arising from increased annihilation via the Z0 pole due to the decrease in µ2 increasing the higgsino content of the neutralino[18]. only in the third generation and for the Higgs bosons. One may parameterize the soft breaking masses at M as follows: G m 2 = m2(1+δ ); m 2 = m2(1+δ ); H1 0 1 H2 0 2 m 2 = m2(1+δ ); m 2 = m2(1+δ ); m 2 = m2(1+δ ); qL 0 3 tR 0 4 τR 0 5 m 2 = m2(1+δ ); m 2 = m2(1+δ ). (12) bR 0 6 lL 0 7 with −1 ≤ δ ≤ +1. While the non-universal models introduce a number of new parame- i ters, it is possible to understand qualitatively what effects they produce on dark matter detection rates, since the parameter µ2 governs much of the physics. Thus as µ2 decreases (increases), the higgsino content of the neutralino increases (decreases), and then σχ˜01−p increases (decreases). One can further see semi-quanitatively the dependence of µ2 on the non-universal parameters for low and intermediate tanβ where the RGE may be solved analytically [20]: t2 1−3D 1 1−D µ2 = ( 0 + )+ 0(δ +δ ) t2 −1 2 t2 2 3 4 (cid:20) 1+D δ − 0δ + 1 m2 +universalparts + loopcorrections. (13) 2 2 t2# 0 ∼ ∼ where t = tanβ and D = 1−(m /200sinβ)2. In general D is small i.e. D = 0.25, and 0 t 0 0 one sees that the universal part of the m2 contribution is quite small, and it does not take 0 7 1 0.1 ) b p -60 1 0.01 (p s ~0c-1 0.001 400 500 600 700 800 900 1000 m (GeV) _1 2 ∼ Figure 7: σχ˜01−p as a function of m1/2 (mχ˜01 = 0.4m1/2) for tanβ = 40, µ > 0, mh > 114 GeV, A = m for δ = 1. The lower curve is for the τ˜ − χ˜0 co-annihilation channel, 0 1/2 2 1 1 and the dashed band is for the Z s-channel annihilation allowed by non-universal soft breaking. The curves terminate at low m due to the b → sγ constraint. The vertical 1/2 lines show the termination at high m due to the lower bound on a of Eq. (5)[21]. 1/2 µ a great deal of non-universal contribution to produce additional effects. Most interesting things happen when µ2 is decreased, since the increased Higgsino content of the neutralino increases the χ˜0 −χ˜0 −Z coupling, and this coupling opens a 1 1 new annihilation channel through the Z-pole in the relic density calculations. As a simple example we consider the case where only the H soft breaking mass is affected i. e. δ = 1 2 2 and all other δ = 0. Fig. 6 shows the new allowed region in the m − m plane for i 0 1/2 tanβ = 40,A = m ,µ > 0,andFig. 7showsthecorresponding effectontheneutralino 0 1/2 - proton cross section. On sees that the co-annihilation corridor is significantly raised and widened due to the new Z-channel annihilation, and the cross section is significantly increased. The next round of upgraded dark matter detectors should be able to reach parts of this parameter space if such a non-universality were to occur. As a second example we consider a soft breaking pattern consistent with an SU(5) invariant model with δ (= δ = δ = δ ) = −0.7, and all other δ = 0. Here the τ˜ soft 10 3 4 5 i R breaking mass is reduced , i.e m2 = m2(1+δ ) < m2. Thus the τ˜ −χ˜0 co-annihilation τ˜R 0 5 0 1 1 occurs at a larger value of m than in mSUGRA. In addition again a new Z-channel 0 neutralino annihilation channel occurs since µ2 is reduced. The effects are shown in Figs. 8 and 9 for tanβ = 40, A = m , µ > 0. Again the cross sections are larger, and should 0 1/2 be accessible to CDMS when it moves to the Soudan mine and to GENIUS. The maximum value of σχ˜01−p for fixed tanβ and A0 occurs when we chose the non- universalities to minimize µ2. This occurs when δ < 0 and δ > 0. This is shown 1,3,4 2 in Fig. 10 where the maximum cross section is plotted for A = 0, tanβ = 12 (upper 0 curve), tanβ = 7 (lower curve). The bound that m > 114 GeV, eliminates the region h 8 1200 1000 ) V e 800 G ( 0 m 600 400 450 500 550 600 650 700 750 m (GeV) 1_ 2 Figure 8: Allowed regions in the m − m plane for the case tanβ = 40, A = m , 0 1/2 0 1/2 µ > 0. The bottom curve is the mSUGRA τ˜ coannihilation band of Fig. 1 (shown for 1 reference). The middle band is the actual τ˜ coannihilation band when δ = −0.7. The 1 10 top band is an additional allowed region due to the enhancement of the Z0 s-channel annihilation arising from the nonuniversality lowering the value of µ2 and hence raising < the higgsino content of the neutralino. For m ∼ 500 GeV, the two bands overlap [18]. 1/2 with mχ˜01 < 100 GeV. However, one sees for this case that it is possible to have detection cross sections in the region of the DAMA data. 4 CONCLUSIONS Wehave discussed heredirect detectionofMilky WayneutralinosforSUGRAtypemodels with R-parity invariance and grand unification at the GUT scale. By combining data from a variety of sources, e.g. Higgs mass bound, b → sγ branching ratio, relic density constraints and the possible new muon magnetic moment anomaly of the BNL E821 experiment, one can greatly sharpen predictions. For the mSUGRA model, the m and b → sγ bounds create a lower bound on m h 1/2 > > of m1/2 ∼ (300 − 400)GeV (i. e. mχ˜01 ∼ (120 − 140)GeV). Thus puts the parameter space mostly in the τ˜ − χ˜0 co-annihilation domain, which strongly correlates m with 1 1 0 m1/2. For µ > 0 and m1/2 < 1TeV, one finds σχ˜01−p ∼< 10−10 pb which is within the upper reach of future planned dark matter detectors, while for µ < 0 there will be large regions unaccessible to such detectors. If the a anomaly is confirmed, then µ > 0 and m < 800 µ 1/2 GeV. Non-universal soft breaking models allow one to raise σχ˜01−p by a factor as large as 10 - 100, which could account for the large cross sections of the DAMA data. They can 9