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Asymmetric Higgsino Dark Matter Kfir Blum,1,∗ Aielet Efrati,2,† Yuval Grossman,3,‡ Yosef Nir,2,§ and Antonio Riotto4,¶ 1Institute for Advanced Study, Princeton 08540, USA 2Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel 3Institute for High Energy Phenomenology, Newman Laboratory of Elementary Particle Physics, Cornell University, Ithaca, NY 14853, USA 4Département de Physique Théorique and Centre for Astroparticle Physics (CAP), 24 quai E. Ansermet, CH-1211 Genève, Suisse Inthesupersymmetricframework,ahiggsinoasymmetryexistsintheuniversebeforetheelectroweak phasetransition. Weinvestigatewhetherthehiggsinoisaviableasymmetricdarkmattercandidate. Wefindthatthisisindeedpossible. Thegauginos,squarksandsleptonsmustallbeveryheavy,such 2 thattheonlyelectroweak-scalesuperpartnersarethehiggsinos. Thetemperatureoftheelectroweak 1 phase transition must be in the (1−10) GeV range. 0 2 n Introduction. The matter content of our universe is 2) before the EWPT, higgsino number-changing inter- a made of two main components: the dark matter (DM), actions become slow enough that the higgsino asym- J with Ω ∼0.20, and baryons, with Ω ∼0.04. Intrigu- metry remains constant; 3) after the EWPT, higgsino- DM b 2 ingly, neither of these numbers can be explained within antihiggsino oscillations are slow or, if they are fast, the 1 the Standard Model (SM) of particle physics. The most higgsino-antihiggsino annihilation rate is slow. Then, a intensively studied scenarios involve very different mech- rather large relic higgsino density can survive. In this ] h anisms to explain these two numbers. The DM relic work, we study the constraints on the supersymmetric p abundance is explained by a freeze-out of a weakly in- spectrum and on cosmological parameters that follow - p teracting massive particle number density that occurs from imposing this set of conditions. e when its annihilation rate becomes slower than the ex- The higgsino asymmetry. The asymmetry in a par- h pansionrateoftheuniverse. Thebaryonrelicabundance ticlexnumberdensity,∆n ≡(n −n ),isrelatedtoits [ x x x¯ isexplainedbyanasymmetrybetweenbaryonsandanti- chemical potential, µx, via (for µx (cid:28)T) 1 baryons. Underthesecircumstances,thereisnoexplana- v tion of the fact that the energy densities of the DM and ∆neq = gxT2µxK(z ), (1) 9 the baryons are surprisingly close to each other, which is x 6 x 9 6 then just a coincidence. It would be more satisfying if it where gx is the number of internal degrees of freedom of 2 could be naturally explained. This can be the case if the theparticlex,z ≡m /T,K(z (cid:28)1)=2(1)forbosons x x x . DM density were also the result of an asymmetry, rather (fermions) and K(z (cid:29) 1) is exponentially suppressed. 1 x 0 than of freeze-out [1–12]. This type of scenarios comes We are interested in relating the higgsino asymmetry to 2 under the name of asymmetric dark matter. the baryon asymmetry. Under the conditions that will 1 One of the best motivated extensions of the SM is the be of interest to us (see below), all of the sfermions are : v Minimal Supersymmetric Standard Model (MSSM). Ex- muchheavierthanthehiggsinosandthesfermionnumber i tending this framework, there are several ways to gener- densitiesarenegligiblebecausetherelevantK(z)factors X ate the baryon asymmetry. Perhaps the most plausible areexponentiallysuppressed. Ontheotherhand,forthe r a oneisvialeptogenesis(forareview,see[13]). Priortothe quarks and leptons K(z) = 1. With these assumptions, ElectroWeak Phase Transition (EWPT), when the hig- the baryon and lepton asymmetries are given by gsinoisofDiracnature,leptogenesisgeneratesalsoHiggs T2 T2 andhiggsinoasymmetriesthatareinitiallyofsimilarsize ∆Y = (2µ +µ +µ ), ∆Y = (2µ +µ ), (2) to the lepton asymmetry (see e.g. [14]). Regardless of B 6 Q u d L 6 L e thesourceofthebaryonandhiggsinoasymmetries,how- whereµ ≡(cid:80) µ (i=1,2,3isagenerationindex). We ψ i ψi ever, the conditions of chemical equilibrium imply that, definethecomovingasymmetryvia∆Y ≡∆n /s,where x x at temperatures above the EWPT, a non-zero baryon sistheentropydensity. WefollowthederivationofRefs. asymmetryrequiresthatthereisalsoanon-zerohiggsino [15, 16]. Imposing the conditions of fast gauge, Yukawa, asymmetry. Could such a higgsino asymmetry survive and sphaleron interactions, the Majorana nature of the and lead to asymmetric higgsino dark matter after the gauginos, the Dirac nature of the higgsinos (prior to the EWPT,whenthehiggsinobecomesofMajorananature? EWPT), and hypercharge neutrality, we obtain This is the question that we address in this work. Imagine that the following set of conditions applies: ∆Yh˜ =− 2K(h˜) . (3) 1) the higgsino is the lightest supersymmetric particle; ∆YB 12+3[K(hu)+K(hd)+2K(h˜)] 2 Eq. (3) relies on super-equilibrium (SE), that is chem- Finally, decoupling the decays and inverse decays of ical equilibrium between Higgs particles and higgsinos. sfermions to higgsino-fermion [16] at T > µ requires NSE At the end of the epoch of SE, a higgsino asymmetry is all of the sfermions to be heavy, m >(10−40)µ. Here, f˜ ∼ conserved until the EWPT. the stronger bound refers to top squarks and the weaker Aslongasthehiggsinosarerelativistic,therighthand bound refers to the electron super-partners. side of Eq. (3) is (in absolute value) in the range (0.10− We note that since, as shown in the next section, 0.15). If the higgsinos become non-relativistic while SE µ>T ,theabovespectrumguaranteesthatSEends EWPT persists, then ∆Y is quenched by the K(h˜) factor in before the EWPT. h˜ Eq.(3). Suchquenchingof∆Y isnotallowedifhiggsinos The spectrum. The higgsino spectrum includes two h˜ are to provide the DM. Thus, we are led to impose that neutral mass eigenstates, with mass splitting ∆m [18], 0 SE must be broken while higgsinos are still relativistic, and a charged higgsino, split by ∆m from the lightest + freezing ∆Y at a value neutral higgsino [19, 20]. As concerns ∆m , this quan- h˜ + tity does not violate higgsino number and in our sce- ∆Y ≈−10−11. (4) h˜ nario it is therefore dominated by gauge loops, ∆m ≈ + In the next section we find the conditions on the super- 12αWs2WmZ ∼ 350 MeV. The results of the previous symmetricspectrumthatwouldleadtoearlybreakdown section have, however, interesting implications for ∆m0. of SE. The operators of Eq. (5) lead to Non-Super-Equilibrium (NSE). Our goal is to find ∆m = ∆ +∆ , (10) 0 u d the conditions on the particle spectrum such that NSE ∆ = (cid:104)h (cid:105)2/Λ , ∆ =(cid:104)h (cid:105)2/Λ . occurswhilethehiggsinosarerelativistic, T >µ. We u u u d d d NSE use Γ < H as the rough criterion for an interaction to Eq. (6) implies ∆ < 10 keV. An analogous bound, u ∼ be out of equilibrium, where the Hubble expansion rate ∆ < (10keV)/tan2β, applies unless the second Higgs d ∼ is given by H ≈ 10T2/m and the interaction rate is Pl doublet, h , ismuchheavierthanthehiggsinos. Wenow d givenbyΓ=n(cid:104)σv(cid:105),nbeingthenumberdensityoftarget showthatexperimentalconstraintsleadustochoosethe particles. second possibility, namely m (cid:29)µ. Higgsino number is violated in Higgs-Higgs scattering hd Constraints on the higgsino spectrum in our scenario (hh → h˜h˜) and Higgs-antihiggsino scattering (hh˜c → come from direct and indirect DM searches. The dom- hch˜). TheseprocessesarisefromtheeffectiveLagrangian inant higgsino-nucleus interactions are spin-independent 1 1 inelastic interactions (see, e.g., [20–22]). The cross sec- −L = h˜ h˜ h∗h∗ + h˜ h˜ h∗h∗, (5) eff Λ u u u u Λ d d d d tionwithasinglenucleon,inthelimitofzeromasssplit- u d ting, isσ ≈10−38 (10−36)cm2 forµ=100(1000)GeV. n generated at tree level by gaugino and at one loop by This is orders of magnitude above current bounds. The quark-squark diagrams. Requiring that h h →h˜ h˜ is u u u u only way to evade these bounds (and maintain higgsino not in equilibrium at T ∼µ puts a lower bound on Λ , u LSP) is by a large enough mass splitting that will make (cid:16) µ (cid:17)1/2 the inelastic scattering kinematically forbidden [23–25]. Λ >3×109 GeV . (6) u ∼ 1TeV Theminimummasssplittingrequiredisafunctionofthe higgsino mass, but in the entire range of interest for µ If h is not heavy and decoupled, then a similar bound d it is smaller than 400 keV. We learn that the sbottom holds for Λ . d contributionto∆m mustbesubstantial,∆ >400keV. The gaugino contributions to (5) are given by 0 d ∼ As a result, the only configuration consistent with early 1 1 g(cid:48)2 g2 breakdown of SE, T > µ, is one with a very heavy = = + . (7) NSE Λ Λ 8M 8M second doublet, m (cid:29)µ. u d 1 2 hd As can be seen from Eqs. (8) and (10), requiring The stop and sbottom contributions are given by 1 = 3αW2 m3t sin2θ lnm2t˜2 , ∆m0 >∼1MeV (11) Λ 2m4 s4 t m2 constrains the sbottom sector to satisfy, u W β t˜1 Λ1 = 32αmW24 mc43b sin2θb lnmm˜2b22 . (8) sin2θb >∼2×10−2 (cid:18)ta2n0β(cid:19)−2 . (12) d W β ˜b1 Finally,with∆m (cid:29)10eV,thepresenthiggsinopopu- Eq. (6), therefore, implies 0 lationissymmetric(seebelow),andthereforeannihilates (cid:16) µ (cid:17)1/2 andmayprovidesignalsinindirectsearchesforDM.The M > 108 GeV , i ∼ 1TeV annihilationcrosssectionintoWW andZZ pairsisgiven sin2θ lnm2t˜2 < 10−6(cid:18)1TeV(cid:19)1/2 . (9) by [26] t m2 ∼ µ (cid:104)σannv(cid:105)≈10−26 cm3 sec−1 (1 TeV/µ)2. (13) t˜1 3 TheboundonthecrosssectionfromtheFermi-LATdata [27], when compared with Eq. (13), requires 0.01 0.001 µ>190 GeV. (14) V 10(cid:45)(cid:76)4 ∼ e k 10(cid:45)(cid:72)5 0 Oscillations, damping and expansion. At the m 10(cid:45)6 (cid:68) EWPT,theHiggsacquiresaVEV,andthehiggsinosmix 10(cid:45)7 withthegauginos. Theresultingpropagationeigenstates 10(cid:45)8 changefromDiractoMajoranafermions,andoscillations 200 300 500 700 1000 begin. On the other hand, incoherent interactions with Μ GeV the plasma continue and damp the oscillations. In this section, we study the time evolution of the higgsino sys- FIG.1: Therequired∆m0 asaf(cid:72)uncti(cid:76)onofµ,withTEWPT = tem (h˜,h¯˜) under the simultaneous effects of oscillations, 100 GeV. annihilations,dampingandtheexpansionoftheuniverse. Todoso,weemploytheformalismofthedensitymatrix. asymmetry. In particular, to provide Ω h2 ≈0.11, our DM Thisformalismwasoriginallydevelopedtostudyneutri- scenario needs to fit nos [28–31] and we adapt it to our case. Our equations are consistent with those of Ref. [32] which deals with Yobs ≈7.6×10−13(1 TeV/µ). (16) 0 closely related issues (see also [33]). The quantum rate We consider three free parameters: µ, ∆m and T equations for Y ≡n /s (where n ±n are the number 0 EWPT µ µ 0 3 and ask whether there is a range of these parame- densitiesofthehiggsinosandanti-higgsinos,andn ∓in 1 2 ters where such a fit is achieved. If the asymmetry is are the off diagonal elements of the density matrix) read washed out before the symmetric decoupling tempera-   ture, Tsym ≈ µ/25, then the present DM relic density is D V 0 dec d 1 the standard symmetric one, as if an initial asymmetry Y = − V D ∆m Y, dlogz H  0 was never generated: 0 −∆m 0 0 Ysym ≈5.9×10−13(µ/1 TeV). (17) d s 0 Y = (cid:104)σannv(cid:105) dlogz 0 H Comparing Eqs. (16) and (17) we learn that for µ∼1.1 (cid:20) (cid:21) 1 1 × 2YeqY¯eq− Y2+ Y2+G(Y2+Y2) , TeV, higgsinos can account for DM without asymmetry. 2 0 2 3 1 2 Since in the presence of an asymmetry the total number density is always larger than in the symmetric case, this where z ≡ µ/T. The damping (or decoherence) factor puts an upper bound of 1.1 TeV on the higgsino mass. D, and the effective matter potential V, are given by Thus, the range of interest is 190 GeV ≤ µ ≤ 1.1 TeV. (cid:88) Withinthisrange,thelighterthehiggsino,thelargerthe D =2 n (cid:104)σ v(cid:105),V =8ζ(3)α η T3/(πm2 ). f h˜+f→h˜+f W B W asymmetry that is required in order to satisfy (16). f We can obtain Y (∞) > Ysym if either of the follow- 0 0 ing two conditions applies: 1) ∆m is small enough that While D is proportional to the elastic scattering cross 0 theoscillationsareslow,andatleastpartofthehiggsino section and to the total number density of the massless asymmetry survives down to T < Tsym; 2) The EWPT fermions in the plasma, V is proportional to the elas- dec occurs late enough that annihilations are already slow tic scattering amplitude and to the fermion-antifermion when oscillations begin, T < Tsym. To study the asymmetry. In the limit of large damping, the effective EWPT dec rate of oscillation is Γ ∼ (∆m )2/D, as obtained in first possibility, we fix TEWPT = 100 GeV, and solve for osc 0 Y (∞) as a function of µ and ∆m . We find that the [12]. The G factor measures the ratio between the an- 0 0 lighter the higgsinos are, the smaller the mass splitting nihilation cross section with and without including co- that is required to provide the DM abundance. The rea- annihilations. sonisthatsmallerµleadstosmallerTsym,andtheasym- Theinitialconditions,attheEWPT,arethefollowing: dec metry is required to survive to later times. The required Y0 =nh˜/s, Y1 =Y2 =0, Y3 =−10−11, (15) ∆m0 as a function of µ is shown in Fig. 1. It ranges between (10−2−10−8) keV for µ in the range (1000−200) where n is the solution of the Boltzmann equations (at GeV. Such a small mass splitting is excluded by direct h˜ the EWPT) with constant asymmetry, and the value of DM searches. We conclude that it is impossible for the Y =∆Y is taken from Eq. (4). asymmetry to survive once the EWPT takes place. 3 h˜ Asymmetry-assisted higgsino DM. Weaimtosolve Second, we fix ∆m = 1 MeV (the results are not 0 for Y (∞) which gives the final total number density in sensitive to changes in ∆m in the range where it is 0 0 higgsinos, and for Y (∞) which gives the final higgsino not excluded by direct searches), and solve for Y (∞) 3 0 4 In addition, the temperature of the electroweak phase 50.0 transition must be somewhat low, of order (1-10) GeV. V 10.0(cid:76) The supersymmetric spectrum is somewhat reminis- Ge 5.0 cent of split supersymmetry models [34, 35]. The super- (cid:72) PT symmetric flavor problem is solved. Grand Unification W 1.0 E remains a viable possibility. Supersymmetry does not T 0.5 solve the fine tuning problem. It does however explain both the baryon asymmetry and the dark matter abun- 200 300 500 700 1000 dance, and relates the two. The initial source of both Μ GeV asymmetries could be leptogenesis. FIG.2: TherequiredTEWPT as(cid:72)afun(cid:76)ctionofµ,with∆m0 = Thisscenario,wheretheonlynewparticlesattheelec- 1 MeV. The approximate analytic solution (18) is shown in troweak scale are the higgsinos, poses a challenge to the dashed line. LHC. Work on experimental and observational signals is in progress. as a function of µ and T . Our numerical result for Acknowledgments. We thank Nima Arkani- EWPT the required T as a function of µ is shown as the Hamed, Rouven Essig, Yonit Hochberg, Jesse Thaler EWPT smoothlineinFig.2. Abovetheline,thefinalDMabun- and Tomer Volansky for useful discussions. KB is sup- dance is too low. Below the line, the final abundance ported by DOE grant DE-FG02-90ER40542. YG is sup- is too high. The required temperature ranges between ported by NSF grant PHY-0757868 and by a grant from (30−0.1) GeV for µ in the range (1000−200) GeV. For the BSF. YN is the Amos de-Shalit chair of theoretical T > 3 GeV(µ/1 TeV)2, we can obtain an approximate physics and supported by the Israel Science Foundation ∼ analytic solution for the required T , (grant #377/07), and by the German-Israeli foundation EWPT for scientific research and development (GIF). (cid:16) µ (cid:17)3 T =33 GeV , (18) EWPT 1 TeV shown as the dashed line in Fig. 2. We conclude that a viable scenario of asymmetric hig- ∗ Electronic address: [email protected] gsino dark matter could have occurred as follows: a hig- † Electronic address: [email protected] gsinonumberasymmetryofasizethatisaboutafactorof ‡ Electronic address: [email protected] 10 smaller than the baryon asymmetry exists before the § Electronic address: [email protected] ¶ Electronic address: [email protected] EWPT.AttheEWPT,theasymmetryisquicklywashed out due to higgsino-antihiggsino oscillations. The result- [1] S. Nussinov, Phys. Lett. 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