Cosmology and Astrophysics of Minimal Dark Matter
aa r X i v : . [ h e p - ph ] J u l IFUP–TH/2007-12 SACLAY–T07/052
Cosmology and Astrophysics ofMinimal Dark Matter
Marco Cirelli a , Alessandro Strumia b , Matteo Tamburini ca Service de Physique Th´eorique, CEA-Saclay, France and INFN, Italy b Dipartimento di Fisica dell’Universit`a di Pisa and INFN, Italia c Dipartimento di Fisica dell’Universit`a di Pisa
Abstract
We consider DM that only couples to SM gauge bosons and fills one gauge multiplet,e.g. a fermion 5-plet (which is automatically stable), or a wino-like 3-plet. We revisitthe computation of the cosmological relic abundance including non-perturbativecorrections. The predicted mass of e.g. the 5-plet increases from 4 . E > ∼ eV (possiblypresent among ultra-high energy cosmic rays) can cross the Earth exiting in thecharged state and may in principle be detected in neutrino telescopes. Contents ± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Introduction
Observations and experiments tell that:i) Dark Matter (DM) exists [1], with abundance Ω DM h = 0 . ± .
005 [2];ii) DM is cosmologically stable; it has no electric charge [3], no strong interactions [4] andthe DM coupling to the Z boson is smaller than ∼ − g [5].i) suggests that DM might be a weak-scale particle: indeed Ω DM ∼ M and coupling g : M/g ∼ √ T M Pl ∼ TeV. ii) can be satisfied in appropriate models, although these properties are not typical weakscale particles.We here want to study Minimal DM (MDM) models: we add to the Standard Model(SM) one new multiplet, that only has gauge interactions and a SU(2) L -invariant mass term M . In this context, ii) singles out scalars or fermions with zero hypercharge that fill a SU(2) L representation with odd dimension: n = 3 , , . . . [6]. The neutral DM component is then lighterthan the charged DM ± components, by only 166 MeV. DM interacts with ordinary mattervia one-loop exchange of W ± : future direct DM searches will hopefully reach the sensitivityneeded to detect it.The fermion 5-plet is particularly interesting: all interactions other than the gauge inter-actions are incompatible with gauge and Lorentz symmetries [6], so that it is automaticallystable thanks to an accidental symmetry, like the proton in the SM. Other candidates that wewill consider lack this feature (i.e. some extra symmetry is responsible for their stability) butare also interesting. The fermion 3-plet is well known from supersymmetry with matter-parity,as ‘pure wino’ DM; the fermion doublet case as ‘pure higgsino’ DM. We will also study scalarMDM candidates. In general, this scenario has one free parameter, the DM mass M , alreadyfixed by Ω DM , such that all DM signals are univocally predicted. We here address the followingissues.First, we complete the computation of the cosmological abundance performed in [6] takinginto account p -wave annihilations and renormalization of the gauge couplings up to the DMscale, and, more importantly, we include non-perturbative Sommerfeld corrections: the DMwave-functions get distorted by Coulomb-like forces mediated by SM vectors. Their relevancewas pointed out in [7], in the case of gluino and wino as lightest supersymmetric particle.In section 2 we discuss the basic physics of non-perturbative Sommerfeld corrections, and insection 3 we show our results for the cosmological DM abundance.Next, in section 4 we compute the DM DM annihilation rates relevant for indirect MDMsignals, including the sizable enhancement due to Sommerfeld corrections.Finally, in section 5 we study the possibility that the ultra-high-energy cosmic rays (UHECR) contain some DM particle (although this looks difficult within the standard CR accelerationmechanism), showing that MDM candidates can cross the earth arriving to the detector in thecharged state. We discuss how it could manifest in detectors such as neutrino telescopes.Section 6 presents our conclusions. If the DM mass is M ≈ M Z and if the DM coupling is g ≈ g , DM DM annihilations intoSM vectors have a too large cross section, and consequently give a too low freeze-out DM2bundance. Since Ω DM ∝∼ M for M > M Z , the observed DM abundance is obtained for a valueof M sufficiently larger than M Z . Since this value turns out to be in the TeV or multi-TeVrange, we can ignore SM particle masses when computing the annihilation rates. Using SU(2)algebra, ref. [6] obtained a single expression for the cosmological abundance of any MDMcandidate, taking into account all (co-)annihilations in s -wave approximation. We here add p -wave annihilations and renormalization of the gauge couplings up to the DM mass M : eachone of these effects gives a O (5%) correction to Ω DM . Precise formulæ are given in appendix A.Furthermore, we take into account non-perturbative electroweak Sommerfeld corrections.Their relevance was pointed out in [7] and might look surprising, so we start with a generalsemi-quantitative discussion before proceeding with the detailed computations. Scatteringsamong charged particles due to point-like interactions are distorted by the Coulomb force,when the kinetic energy is low enough that the electrostatic potential energy is relevant. Thisleads e.g. to significant enhancements of the µ − µ + annihilation cross section (attractive force)or to significant suppressions of various nuclear processes (repulsive force). These effects can becomputed with a formalism developed by Sommerfeld [9]. In the language of Feynman graphs,these effects are described by multi-loop photon ladder diagrams. Perturbative computationswould include only the first few diagrams: the Sommerfeld enhancement is non-perturbativebecause a resummation of all ladder diagrams is needed. The generalization of the Sommerfeldformalism to the case of DM DM annihilations, that involves non abelian massive vectors, waspresented in [10].Let us first discuss Sommerfeld corrections due to one abelian vector with mass M V andgauge coupling α : this case already contains the relevant physics and can be analyzed in asimpler way. Cross sections at low energies are dominated by s -wave scattering, with p -wavegiving corrections of relative order T /M . Since the DM DM annihilation is local (i.e. it occurswhen the distance between the two DM particles is r ≃ R = | ψ ( ∞ ) /ψ (0) | , where ψ ( r ) is the (reduced) s -wave-function for the two-body DMDM state with energy K , that in the non-relativistic limit satisfies the Schr¨odinger equation − M d ψdr + V · ψ = Kψ V = ± αr e − M V r (1)with outgoing boundary condition ψ ′ ( ∞ ) /ψ ( ∞ ) ≃ iM β . Here K = M β is the kinetic energyof the two DM particles in the center-of-mass frame, where each DM particle has velocity β .The − sign corresponds to an attractive potential and the + sign to a repulsive potential. Wedefine ǫ ≡ M V /M , the adimensional ratio between the vector mass and the DM mass.Eq. (1) is the prototype of the equations that describe corrections to DM DM annihilationsmediated by a Z boson, or DM + DM − or DM + DM + co-annihilations mediated by a γ or W or Z , or corrections mediated by a gluon (in models where coannihilations with and betweencolored particles are relevant). In the three cases, the coupling α that appears in eq. (1) wouldbe α , α em and α s respectively. In the non abelian case, eq. (1) becomes a matrix equation and Another solution exists for M sufficiently lower than M Z , such that annihilation into vectors are kinemati-cally suppressed. In the minimal scenario that we consider, this solution is excluded by LEP2 and other colliderdata, but in general it is still allowed in some part of the parameter range of non-minimal scenarios [8].A DM mass M ≈ M Z can of course also be obtained by reducing g , i.e. by assuming that DM is any SU(2) L multiplet appropriately mixed with a singlet. Sometimes in the literature ‘non-perturbative’ is used with a different meaning: to denote effects thatvanish in the perturbative limit because suppressed e.g. by e − /α factors. -1 α / β -1 α / ε = M D M / ( M V / α ) Attractive potentialstrongTeVweak1.3 2 3 5 10 30 100 1000 10 -1 α / β -1 α / ε = M D M / ( M V / α ) Repulsive potentialstrongTeVweak 0.80.50.30.10.010.001
Figure 1:
Iso-contours of the non-perturbative Sommerfeld correction to the DM DM anni-hilation. Here α is the coupling constant, β is the DM velocity, ǫ is the ratio between thevector mass and the DM mass. The labels indicate where some classes of DM candidates liein this plane: ‘weak’ indicates weak-scale DM particles, ‘TeV’ indicates DM with multi-TeVmass, and ‘strong’ indicates strongly-interacting particles that in some models give dominantco-annihilations. Within the shaded region thermal masses dominate over masses, effectivelyshifting the value of α/ǫ as indicated by the arrow. V is the sum of the various contributions mediated by all SM vectors. Higgs exchange can alsobe relevant, in models where DM sizably couples to the Higgs.For ǫ = 0 (massless vector) the Schr¨odinger equation has the same form as the one thatdescribes e.g. the hydrogen atom, and it can be analytically solved: the Sommerfeld factor R that multiplies the perturbative cross section is R = − πx − e πx x = ± αβ (for M V = 0) (2)This shows that R sizably differs from 1 at β < ∼ πα . DM DM annihilations into SM particlesfreeze-out when the temperature T cools down below T /M ∼ / ln( M Pl /M ) ∼ /
26. Thishappens to be numerically comparable to the SM gauge couplings, i.e. α ≈ /
30. ConsequentlyDM freeze-out occurs when β ∼ . πx ∼
1: the Sommerfeld correction is significant, R ∼ O (1).Of course we must take into account that the relevant W, Z vectors are massive: since R must now be computed numerically, it is convenient to first identify on which parameters R depends. We notice that R only depends on the two ratios α/β and α/ǫ , so that R can beplotted on a plane. This can be proofed noticing that R is adimensional and that physics isinvariant under r → λr , M → M/λ , M V → M V /λ . Fig. 1 shows iso-contours of R as functionof α/β and α/ǫ . We can distinguish various regions:4) Non-perturbative effects are negligible (i.e. | R − | ≪
1) if πα ≪ β and/or if α ≪ ǫ . Wehave seen that πα ∼ β , so non-perturbative effects are relevant when M > ∼ M V /α , where M V are the SM vector masses and α are their gauge couplings [10].b) If β < ∼ ǫ < ∼ α (‘lower triangle region’) β is so low that its value is no longer relevant: R depends only on α/ǫ showing, in the attractive case, a series of resonances. Indeed, byincreasing α/ǫ the potential develops more and more bound states: resonant enhancementhappens when a bound state first appears with energy E B just below zero: in this case,at low energy R depends on K as dictated by a Breit-Wigner factor 1 / ( K − E B ).c) If ǫ < ∼ β < ∼ α (‘upper triangle region’) non-perturbative effects depend almost only on α/β :vector masses have a minor effect and R reduces to eq. (2). Its value does not depend onwhether a loosely bound resonance is present or not.When DM DM bound states are relevant, the Sommerfeld correction might not encode the fulldynamics: one should compute the density and life-time of DM DM bound states.We include one more effect, not discussed in [7]. At finite temperature T , the γ, Z, W masses and the mass splitting ∆ M between the components of the DM multiplet are differentthan at zero temperature. First, because these masses are proportional to the SU(2)-breakingHiggs vev v , that depends on T and vanishes at T > T c when SU(2) invariance gets restored bythermal effects, via a second-order phase transition. This effect can be roughly approximatedas [11] v ( T ) = v Re(1 − T /T c ) / . (3)The critical temperature T c depends on the unknown Higgs mass ( T c ≃ m H for m H ≫ v ): wehere assume T c = 200 GeV.Second, the squared masses of all SM vectors W, Z, γ, g get an extra contribution known asthermal mass. More precisely, even in the non abelian case, the Coulomb force gets screenedby the thermal plasma: this can be described by a vector Debye mass, equal to [12] m = 116 g Y T , m = 116 g T , m = 2 g T (4)in the SM at T ≫ M W,Z . We approximate vector masses by summing the squared massesgenerated from v ( T ) with the SU(2)-invariant Debye squared masses of eq. (4).Let us now estimate the relevance of these thermal effects. For the W and Z bosons, thermalmasses dominate at T > ∼ v , and for the photon and the gluons thermal masses dominate always.At temperature T DM particles have a typical energy K ≈ T , so that thermal masses act likea contribution to ǫ of order ǫ T ∼ √ παβ . When this thermal effect is relevant, in fig. 1 onehas to shift α/ǫ vertically down to the diagonal of the shaded triangle: we see that (dependingon the precise value of α/β ) this shift has a mild or negligible effect on R , since thermal effectsare relevant when R does not have a significant dependence on ǫ . Following [7, 10], we now give a operative summary about how to compute Sommerfeld cor-rections to s -wave DM DM annihilation rates. The set of two-body DM i DM j states that mixamong them is labeled as i, j = { , . . . N } . For example, when studying the wino triplet one5ncounters the { DM − DM + , DM DM } system. The dynamics is encoded in the N × N poten-tial matrix V and in another N × N matrix Γ, that describe the tree-level (co)-annihilationrates. The strategy is the same as in the abelian case of eq. (1), plus a careful book-keeping ofindices. For each j one solves the following set of N coupled differential equations for ψ ( j ) i ( r ): − M ∂ ψ ( j ) i ∂r + N X i ′ =1 V ii ′ ψ ( j ) i ′ = Kψ ( j ) i (5)with boundary conditions: ψ ( j ) i (0) = δ ij , ∂ψ ( j ) i ( ∞ ) ∂r = q M ( K − V ( ∞ ) ii ) ψ ( j ) i ( ∞ ) (6)chosen such that each wave has the same normalization at r = 0, where the annihilation occursas described by the Γ matrices. The annihilation cross section σ i with given initial state i at r = ∞ is obtained by factorizing out the oscillating phase, A ij ≡ lim r →∞ ψ ( j ) i ( r ) /e i ℜ √ M ( K − V ( ∞ )) r ,and contracting with the annihilation matrix: σ i = c i ( A · Γ · A † ) ii (7)The factor c i is given by c i = 2 if the initial state i has two equal DM particles (e.g. as inthe DM DM state) and c i = 1 otherwise (e.g. as in the DM − DM + state). This formalismautomatically takes into account that some states cannot exist as free particles at r ≫ /M W ,when the kinetic energy K is below their mass.We next need to compute the Coulomb-like potential matrices V and the tree-level annihi-lation matrices Γ for all DM components. They split into sub-systems with given values of the L (orbital angular momentum), S (total spin) and Q (total electric charge) quantum numbers.Recalling that we neglect SM particle masses and that we consider s -wave (i.e. L = 0) tree-levelannihilations into two SM particles, annihilations obey the following selection rules:0) two-body DM DM states with spin S = 0 can annihilate into two SM vectors, that canhave electric charge Q = 0 , ± , ± S = 1 can annihilate into two SM fermions and intotwo SM Higgses, that can have electric charge Q = 0 , ± The initial DM DM state with S = 1 of case 1) does not exist if DM is a scalar. We now give explicit expressions for the annihilation matrices Γ ii ′ ,jj ′ and potential matrices V ii ′ ,jj ′ . Their off-diagonal entries do not have an intuitive physical meaning, and are precisely Non-perturbative corrections are mostly relevant when SU(2) L is unbroken: in this limit one could performa simplified analysis by replacing the electric charge quantum number with the SU(2) L isospin quantum number I . Then, the only DM DM states that can annihilate into SM particles are those with S = 0 and I = 1 and5 (that annihilate into SM vectors; I = 1 indicates the singlet) and the state with S = 1 and I = 3 (thatannihilates into SM fermions and Higgses.). These selection rules also apply to DM DM annihilations relevant for indirect astrophysical DM signals,and reproduce well known results. E.g. if DM is a Majorana fermion the spin-statistics relation forbids S = 1,so that annihilations into vectors are not possible. A non vanishing amplitude for DM DM → f ¯ f , proportionalto m f , is allowed when one takes into account the small mass m f of the SM fermions f . ii ′ → jj ′ ,where the indices ii ′ denote the two DM component in the initial state, and jj ′ denote the twoDM components in the final state.For MDM scalars with Y = 0 one has (all states have S = 0)Γ ii ′ ,jj ′ = N ii ′ N jj ′ πng DM M X A,B { T A , T B } ii ′ { T A , T B } jj ′ (8a) V ii ′ ; jj ′ ( r ) = ( M i + M j − M ) δ ii ′ δ jj ′ + N ii ′ N jj ′ X AB K AB ( T Aij T Bi ′ j ′ + T Aij ′ T Bi ′ j ) e − m A r r (8b)where N ij = 1 if i = j and 1 / √ i = j . The indices A, B run over the SM vectors { γ, Z, W + , W − } with generators T A in the DM representation. The gauge couplings are in-cluded in the generators, such that the gauge-covariant derivative is D µ = ∂ µ + iA Aµ T A . Weemphasize that the (off-diagonal) entries of T ± are defined up to an arbitrary phase normal-ization for each component of the multiplet: one needs to choose any convention such that theSU(2)-invariant DM mass term has the same sign for all DM components. The matrices K AB and C AB respectively describe SM vector propagation and the gauge content of all other SMparticles, and are given by: K AB = , C AB = 25 g s sc sc c + 414 g Y c − sc − sc s . For MDM fermions with Y = 0 one hasΓ S =0 ii ′ ,jj ′ = N ii ′ N jj ′ πng DM M X A,B { T A , T B } ii ′ { T A , T B } jj ′ (9a)Γ S =1 ii ′ ,jj ′ = N ii ′ N jj ′ πng DM M X A,B T Aii ′ T Bjj ′ C AB (9b)Γ S =1 is already summed over the 3 spin components. The potentials V remain the same as in thescalar case of eq. (8b), up to an extra − sign that must be taken into account when computingelements of V mediated by the W , such as V − , − = − V − , − , because an extra − sign appearsin the definition of two body fermion states: | DM DM − i = −| DM − DM i . Furthermore, Fermistatistics forbids the existence of some states, such as | DM DM i with S = 1.Similar formulæ apply to MDM candidates with Y = 0, after taking into account that DM now becomes a complex particle (its normalization factor changes from N = 1 / N = 1)and that one is only interested in DM DM annihilations and two body states. Summing Γ ii ′ ,ii ′ over all components one reproduces the total co-annihilation rates of [6], here reported andextended in appendix A. Explicit expressions for V and Γ are given in the next sections. To include non-perturbative corrections a dedicated lengthy analysis is needed for each MDMcandidate. 7 Ω D M h Fermion triplet with Y = 0 (‘wino’) z = M / T -4 -3 -2 -1 σ β i n1 / T e V M = 2.7 TeV -27 -26 -25 -24 -23 -22 σ β i n c m / s ec Indirect signal at β = 10 -3 DM DM →γγ DM DM → W + W - z = M / T -13 -12 Y = n / s M = 2.7 TeV Figure 2:
The fermion triplet with zero hypercharge (‘wino’) . Fig. 2a (upper left):our result for its cosmological freeze-out DM abundance as function of the DM mass. Fig. 2c(upper right) show an example (for the indicated mass M ) of the temperature dependence of the DM DM annihilation cross section, and fig. 2d (lower right) shows the resulting cosmologicalevolution of the DM abundance. Fig. 2b (lower left) shows our result for DM DM annihilationcross section relevant for indirect DM detection, as discussed in section 4. In each case, thecontinuous line is our full result, while the dashed line is the result obtained without includingnon-perturbative effects.The fermion triplet with Y = 0 (‘wino’). It is not automatically stable: one needs to impose a suitable symmetry. Since Y = 0 thelightest component of this multiplet is neutral under the γ and under the Z , and therefore itsatisfies bounds from direct DM searches [13, 6]. This multiplet appears in supersymmetricmodels as SU(2) L gauginos, and behaves as MDM in limiting cases where it is much lighterthan other sparticles.We include p -wave DM DM annihilations, but we only include non perturbative correctionsto s -wave annihilations, that are splitted into four sectors with L = 0, total charge Q = { , } and total spin S = { , } . All sectors involve one component, except the sector with Q = 0 and S = 0, that involves two components { DM − DM + , DM DM } . In this case and in the followingwe indicate the basis we employ by writing the charges of the DM components around the Γand V matrices, and by indicating Q and S as pedices and apices on Γ and V . Using eq. (9),for the fermion triplet we get:Γ S =0 Q =0 = πα M + 0 − √ √ ! , V S =0 Q =0 = + 0 − − A −√ B −√ B ! , (10)8 S =1 Q =0 = 25 πα M V S =1 Q =0 = 2∆ − A (11)Γ S =0 Q =1 = πα M , Γ S =1 Q =1 = 25 πα M , V S =0 , Q =1 = ∆ − B (12)Γ S =0 Q =2 = πα M , V S =0 Q =2 = 2∆ + A. (13)where ∆ = 166 MeV, A = α em /r + α c e − M Z r /r and B = α e − M W r /r . These results areequivalent to those of [7].We numerically solve the Schr¨odinger equations use the finite-temperature values for ∆ andvector masses (e.g. ∆ vanishes when SU(2) L -invariance is restored, etc.). Fig. 2a shows ourresult for its freeze-out cosmological abundance: as previously noticed in [7] non-perturbativecorrections are relevant, and significantly increase the multi-TeV value of M that reproducesthe measured cosmological abundance. In the supersymmetric case, a multi-TeV wino lightestsupersymmetric particle implies a fine-tuning in the Higgs mass above the 10 level, so thatthis scenario seems not motivated by the Higgs mass hierarchy problem.A bound state first appears for M > M ∗ ≈ . Q = 0 and total spin S = 0. Fig. 2c and d show more details of the computation for M = 2 . M is just above M ∗ , the bound state is loosely bound, E B ≈ −
67 MeV,and gives rise to a significant non-perturbative enhancement R of the annihilation cross section,apparent in fig. 2c as a dip at M = M ∗ . The result is reliable, because temperatures T < ∼ | E B | (at which a dedicated treatment of bound states would be necessary) do not significantly affectthe final cosmological abundance, as indicated by fig. 2d and by the fact that the dip is not themain effect. The scalar triplet with Y = 0 . This MDM candidate is not automatically stable: one needs to impose a suitable symmetry.Having Y = 0, this candidate is compatible with bounds from direct DM searches. Any SU(2) L multiplet of scalars can have a quartic interaction with the Higgs, − λ H ( X † T a X )( H † τ a H ), thatgenerates a tree-level mass splitting within the multiplet. Since this mass splitting is suppressedby the DM mass M [6] it is not unbelievable that scalars behave as MDM because λ H is smallenough, λ H ≪ .
05, that the ∆ = 166 MeV mass splitting induced by gauge couplings isdominant. This is particularly plausible in the case of scalars with Y = 0, because λ H is notgenerated by one-loop RGE corrections. Indeed RGE corrections that induce a λ H proportionalto g happen to vanish, because the Higgs H is a doublet, and the doublet representation ofSU(2) L has vanishing symmetric tensor: Tr T aH { T bH , T cH } = 0.To include non-perturbative corrections to DM DM annihilations we split them into threesectors, with L = 0, S = 0 and total charge Q = { , , } . The V and Γ matrices areΓ S =0 Q =0 = 2 πα M + 0 − √ √ ! , V S =0 Q =0 = + 0 − − A −√ B −√ B ! , (14)Γ S =0 Q =1 = 2 πα M , V S =0 Q =1 = ∆ + B, Γ S =0 Q =2 = 2 πα M , V S =0 Q =2 = 2∆ + A. (15)Fig. 3a shows our result for its freeze-out cosmological abundance: non-perturbative correc-tions are again important. Fig. 3c and d show details of the computation for M = 2 . Ω D M h Scalar triplet with Y = 0 z = M / T -4 -3 -2 -1 σ β i n1 / T e V M = 2.5 TeV -27 -26 -25 -24 -23 -22 σ β i n c m / s ec Indirect signal at β = 10 -3 DM DM →γγ DM DM → W + W - z = M / T -13 -12 -11 Y = n / s M = 2.5 TeV Figure 3:
The scalar triplet with zero hypercharge. Plots have the same meaning as in fig. 2. mass close to the critical mass M ∗ ≈ . Q = 0.We here do not study scalar and fermion triplets with | Y | = 1; the neutral component has T = Y = 0 and consequently couples to the Z , giving a too large direct detection rate. Anon-minimal mixing with a singlet is needed to avoid this problem. One expects significantnon-perturbative corrections, similarly to the Y = 0 case, as SU(2) L gauge interactions aremore significant than U(1) Y the extra gauge interaction. The fermion quintuplet with Y = 0 . This is the smallest multiplet whose lightest component is neutral under the γ and underthe Z and is automatically stable on the cosmological timescale. Indeed gauge couplings arethe only renormalizable interactions (compatible with gauge and Lorentz invariance) that thismultiplet can have with other SM particles. Its phenomenology is univocally dictated by asingle parameter: its Majorana mass M .To include non-perturbative corrections to DM DM annihilations we split these scatteringsinto four L = 0 sectors, with total spin S = { , } and total charge Q = { , , } . The Q = 0and S = 0 system is composed by 3 states: V S =0 Q =0 = −− − − A − B − B − A − √ B − √ B , Γ S =0 Q =0 = 3 πα M ++ + 0 −−
12 6 2 √ − √
20 2 √ √ , (16)10 Ω D M h Fermion quintuplet with Y = 0perturbative non-perturbative z = M / T -5 -4 -3 -2 -1 σ β i n1 / T e V M = 10 TeV -26 -25 -24 -23 -22 σ β i n c m / s ec Indirect signals at β = 10 -3 WW γγ z = M / T -14 -13 -12 -11 Y = n / s M = 10 TeV Figure 4:
The fermion quintuplet with zero hypercharge. Plots have the same meaning as infig. 2.
Due to Fermi statistics, the Q = 0, S = 1 system has no DM DM state: V S =1 Q =0 = −− − ++ 8∆ − A − B + − B − A ! , Γ S =1 Q =0 = πα M ++ + −− − ! , (17)The systems with Q = 1 and S = { , } have two states each: V S =0 , Q =1 = ++ + − − A −√ B −√ B ∆ − B ! , (18)Γ S =0 Q =1 = 3 πα M ++ + − √ √ ! , Γ S =1 Q =1 = πα M ++ + − √ √ ! . (19)Finally, in the systems with Q = 2 the tree-level annihilation rate Γ is non vanishing only for S = 0: V S =0 Q =2 = ++ +0 4∆ − √ B + − √ B
2∆ + A ! , Γ S =0 Q =2 = 3 πα M ++ +0 4 −√ −√
12 3 ! . (20)Fig. 4a shows our result for its freeze-out cosmological abundance: the value of M that re-produces the measured DM abundance was M ≈ . Ω D M h Fermion doublet with | Y | = 1/2 (‘higgsino’) 0 0.5 1 1.5 2 2.5 3DM mass in TeV10 -28 -27 -26 -25 -24 σ β i n c m / s ec Indirect signal at β = 10 -3 DM DM →γγ DM DM → W + W - Figure 5:
Cosmological freeze-out abundance of the fermion doublet with Y = 1 / (‘Higgsino’).Plots have the same meaning as in fig. 2, except that since non-perturbative corrections negligiblyaffect the cosmological abundance we do not show details of the computations. M ≈ . p -wave and RGE corrections (perturbative result in fig. 4a),and increases up to almost M ≈
10 TeV after including the non-perturbative Sommerfeld cor-rections. Fig. 4d shows that Sommerfeld corrections are mostly relevant when T ≫ ∆ M , suchthat bound states are not expected to play a relevant rˆole. The scalar quintuplet with Y = 0 . As in the scalar triplet case, is plausible to assume that the quartic coupling with the Higgsis negligibly small, because it is not generated by RGE effects. The V and Γ matrices arerelated to those relevant for the fermion 5-plet as described in section 2.1. Non-perturbativecorrections increase the value of the mass M that reproduces the cosmological abundance from M ≈ . M ≈ . The scalar eptaplet with Y = 0 . This is the smallest scalar multiplet that is automatically stable (its cubic scalar couplingidentically vanishes) and contains a MDM candidate compatible with all bounds. Again, it isplausible to assume that the quartic coupling with the Higgs is negligibly small. Although we donot show dedicated plots nor the V and Γ matrices, we computed non-perturbative correctionsfinding that they increase the value of the mass M that reproduces the cosmological abundancefrom M ≈ . M ≈
25 TeV.
The fermion doublet with Y = 1 / (‘Higgsino’). It is not automatically stable, and, having Y = 0, is excluded by direct DM searches. Nev-ertheless, we consider it because n = 2 is the smallest multiplet, and because it is present insupersymmetric models. The first problem can be solved by imposing a suitable symmetry, andincompatibility with direct detection experiments can be avoided by assuming a mass mixingwith a neutral Majorana singlet (automatically present in supersymmetric models and knownas ‘bino’). We assume that this mass mixing is small enough not to affect the observableswe compute. An almost-pure Higgsino LSP is realized in corners of the MSSM parameterspace: it behaves as Minimal Dark Matter in the sense that its properties are not dictated by12upersymmetry but only by gauge invariance. In this limit, the µ -term is the only relevantsupersymmetric parameter, and coincides with our DM mass parameter M .To include non-perturbative corrections to DM DM annihilations we split these scatteringsinto four L = 0 sectors, with total charge Q = { , } and total spin S = { , } . The Γ and V matrices that describe each sector areΓ S =0 Q =0 = πα M + ¯0 − t + t − t + t − t + t t + t ! , Γ S =1 Q =0 = πα M + ¯0 − t + 25 41 t −
250 41 t −
25 41 t + 25 ! ,V S =0 , Q =0 = + ¯0 − − α em /r − (2 c − e − M Z r / rc − α e − M W r / r − α e − M W r / r − e − M Z r α / rc ! (21)Γ S =0 Q =1 = πα t M , Γ S =1 Q =1 = 25 πα M , V S =0 , Q =1 = + α e − M Z r r c − c (22)where t = tan θ W , c = cos θ W and ∆ = 341 MeV is the mass splitting among charged andneutral components generated by loop effects. Fig. 5a shows our result for its cosmologicalabundance: in agreement with previous studies (e.g. [6]) we see that the observed DM abun-dance is reproduced for M = 1 TeV. Non perturbative corrections are negligible. This alsoholds for the scalar doublet with | Y | = 1 /
2, so we do not study it.The Higgs potential in the Minimal Supersymmetric Standard Model depends on the µ parameter: a | µ | = 1 TeV gives a contribution to the squared Z -mass which is 2 µ /M Z = 240times too large: therefore the Higgsino as Minimal Dark Matter can only be realized at theprice of a sizable fine tuning. We now study the usual “indirect DM signals”, generated by DM DM annihilations intoSM particles. In the MDM case one only has tree-level annihilations into W + W − , while themost interesting final states, γγ and γZ (both detectable as γ with energy practically equalto E γ = M ), arise at loop level. Assuming Y = 0 the cross sections (averaged over DMpolarizations) are, in the fermion DM case σ (DM DM → W + W − ) β = ( n − πα M , σ (DM DM → γγ ) β = ( n − πα α M W . (23)and 2 times higher in the scalar DM case. These cross-sections can be significantly affected bynon-perturbative Sommerfeld corrections [10], because in our galaxy DM has a non-relativisticvelocity β ≈ − . The computation employs the same tools already described in the compu-tation of DM DM annihilations relevant for cosmology. Non-perturbative effects can enhancethe cross section by orders of magnitude for specific values of M : the values at which thepotential develops a new bound state with Q, L, S = 0. In the triplet and quintuplet casesthis enhancement is sizable, since the DM mass suggested by cosmology is close to one of suchcritical values. We show our results for the fermion triplet in fig. 2b, for the scalar triplet infig. 3b, for the fermion quintuplet in fig. 4b, and for the fermion doublet in fig. 5b. The vertical13 -2 -1 Ω ˙ d N γ / d E i n1 / m y r T e V H . E . S . S . J = 1300, Ω = 10 -3 n = 3 M = 2.7 TeV n = 5 M = 10 TeV Figure 6:
Spectrum of γ from DM DM annihilations around the galactic center, in a region ofsize Ω = 10 − , as seen by a detector with σ E /E = 0 . . The total rate is computed assumingthe NFW profile, J = 1300 , and for fermion MDM with n = 3 and M = 2 . , and for n = 5 and M = 10 TeV . The continuous line is the H.E.S.S. result [14]. bands are the 3 σ range of M suggested by cosmology. We have not plotted annihilation crosssections into γZ and ZZ since all MDM candidates with Y = 0 predict σ γZ = 2 σ γγ / tan θ W = 6 . σ γγ , σ ZZ = σ γγ / tan θ W = 10 . σ γγ . (24)These results allow to compute the energy spectra of annihilation products, such as photons,antiprotons, antideutrons and positrons. Note however that, precisely for the proximity to oneof the critical values in mass, the cross sections depend strongly on the DM mass M andtherefore the overall fluxes cannot be accurately predicted. Moreover, such total rates areaffected by sizable astrophysical uncertainties. For example, the number of detected photonswith energy E = M is [15] N γ (at E = M ) ≈ yr sr J Ω (cid:18) TeV M (cid:19) σ γγ β + σ γZ β − cm / s . (25)where Ω is the angular size of the observed region and the quantity J ranges between a few and10 , depending on the unknown DM density profile in our galaxy. Keeping these limitations inmind, fig. 6 shows the predicted energy spectrum of detected photons from MDM annihilations.We have assumed realistic detector parameters (Ω = 10 − and an energy resolution of 15%). Thecontribution from annihilations into W + W − is included using the spectral functions computedin [15, 10]. The total rate has an uncertainty of about two orders of magnitude and is herefixed assuming the indicated value of M and the Navarro-Frenk-White DM density profile, J = 1300 [16, 1], which makes this signal at the level of the present sensitivity. Indeed thecontinuous line shows the spectrum of diffuse γ , dN γ /dE ≈ ( TeV /E ) . / m yr TeV sr, asobserved (in a region of angular size Ω ≈ − around the galactic center) by H.E.S.S [14] upto energies of about 10 TeV, where the number of events drops below one. Its spectral indexsuggests that the observed photons are generated by astrophysical mechanisms, rather than byDM DM annihilations. In view of the predicted value of σ γγ /σ W W , the best MDM signal is thepeak at E γ = M . With the chosen parameters one gets N γ = 0 . . / m · yr for the fermion3-plet (5-plet). 14 Minimal Dark Matter at Ultra High Energies
In this section we discuss the possibility that Minimal Dark Matter might be detected viathe tracks left by its charged partners in experiments mainly devoted to Cosmic Ray (CR)detection, such as IceCUBE [18] and km3net [17]. Indeed MDM behaves differently from otherneutral particles: • Neutrinos with energy
E < ∼ eV can cross the whole Earth [19] and, impinging on therock or ice below the detector, produce a charged partner (a muon, a tau) that cruisesthe instrumented volume. • Neutral particles much heavier than neutrinos (e.g. the speculative neutralinos) interactrarely enough that they can cross the Earth even at Ultra High Energies (UHE), E ≫ eV [20]. If they do interact before the end of their journey, they do produce a chargedpartner (e.g. the speculative chargino), that however usually decays almost immediatelyback to the neutral particle. For such reasons, it is generically difficult to detect them inthis way.MDM is both heavy and quasi-degenerate with a charged partner, leading to the followingbehavior. To be quantitative, we focus on MDM multiplets with Y = 0. They have a stableneutral DM particle, DM , and a charged DM ± component which is 166 MeV heavier. The latterdominantly decays via DM ± → DM π ± with a macroscopic life-time τ = 44 cm / ( n −
1) [6].At Ultra High Energy Lorentz dilatation makes the DM ± life-time long enough that a sizablefraction of the travel is done as DM ± , leading to detectable charged tracks. This opens in principle an interesting and distinctive channel for the detection of DM. Inorder to assess its feasibility in practice, we need to study the system of MDM from productionto detection. More precisely, first we study the issue of how a MDM system behaves whencrossing the Earth at UHE, having assumed that the flux of UHE CR (observed up to E ∼ GeV) does contain some DM particles. Mechanisms to produce such UHE DM are discussedlater in section 5.2. We finally discuss detection signatures in section 5.3.
Consider a flux of DM particles that is crossing the Earth at UHE. Via Charged Current(CC) interactions with nucleons of Earth’s matter, DM ± particles are produced. Being chargedparticles traveling in a medium, these loose a part of their energy and eventually decay backto DM particles. This chain of production and decay is analogous to the process that tauneutrinos undergo in matter (“ ν τ -regeneration”). In the n = 5 case DM ±± also play a role, butwe will write explicit equations for the n = 3 (wino-like) case. Such a system is described bya pair of coupled integro-differential equations for the evolution with the position ℓ of fluxes n = dN /dE and n ± = d ( N + + N − ) /dE of DM and (DM + + DM − ). They read: dn ( ℓ, E ) dℓ = + ME n ± ( E ) τ + N N (cid:20) − n ( E ) σ CC ( E ) + 12 Z ∞ E dE ′ n ± ( E ′ ) dσ CC ( E ′ , E ) dE (cid:21) , (26a) Supersymmetric models with gravitino LSP may have a long-lived, charged next-to-lightest particle thatbehaves in a way qualitatively similar to MDM [21]. Furthermore, fermionic MDM with n = 3 and Y = 0 arisesin supersymmetry in the limit of pure-wino lightest supersymmetric particle. Similar equations have been of course used in the analogous problem for neutrinos [19] and for high energyneutralinos [20]. n ± ( ℓ, E ) dℓ = − ME n ± ( E ) τ + N N (cid:20) − n ± ( E ) σ CC ( E ) + Z ∞ E dE ′ n ( E ′ ) dσ CC ( E ′ , E ) dE (cid:21) + − ddE ( n ± dE loss dℓ ) , (26b)where N N ( ℓ ) is the number density profile of nucleons in the Earth [22], and E is the DMenergy. The first terms account for DM ± decays. The integro-differential terms within squarebrackets account for DM ↔ DM ± CC scatterings. The last term accounts for the energy lossesundergone by DM ± , due to electromagnetic and Neutral Current (NC) scatterings: dE loss dℓ = dE loss , em dℓ + dE loss , NC dℓ . (27)We will later verify that the NC energy losses are subdominant and can be approximated ascontinuous, like electromagnetic energy losses.The cross section for the CC partonic scatterings DM d → DM − u , DM ¯ d → DM + ¯ u ,DM u → DM + d and DM ¯ u → DM − d are all given by, for fermionic MDM: d ˆ σ CC d ˆ t = E (ˆ s − M ) d ˆ σ CC dE = ! g ( n − π M + 2ˆ s + 2ˆ s ˆ t + ˆ t − M ˆ s (ˆ s − M ) (ˆ t − M W ) (28)where ˆ s and ˆ t are the usual partonic Mandelstam variables. The total cross section for DM N → DM + N ′ , DM − N ′ is σ CC ( s ) = Z dx q ( x ) Z − (ˆ s − M ) / ˆ s d ˆ t d ˆ σd ˆ t , (29)where q ( x ) is the parton distribution function [23], summed over all quarks and anti-quarks. Wehere approximated the neutron/proton fraction in Earth matter as unity. The nucleon N is atrest with mass m N , and DM has energy E , so that s = M + 2 m N E and ˆ s = M + 2 xm N E .A similar expression holds for NC scatterings of DM ± particles DM ± q → DM ± q (whereas DM has no NC interactions due to Y = 0): d ˆ σ NC d ˆ t = g ( g Lq + g Rq )16 π M + 2ˆ s + 2ˆ s ˆ t + ˆ t − M ˆ s (ˆ s − M ) (ˆ t − M Z ) (30)where g Lq = T q − Q q s , g Rq = − Q q s and q denotes any quark or anti-quark. The same crosssections, up to terms suppressed by powers of M W,Z /M , hold for scalar MDM.In fig. 7a we plot the MDM total CC cross section. We can understand its main features asfollows. The total partonic cross section approximatively isˆ σ CC ≃ g ( n − πM W · ( xE/E ∗ ) at xE ≪ E ∗ xE ≫ E ∗ (31)where E ∗ ≡ M M W /m N ∼ eV (the transition between the two regimes happens when t ∼ − M W ). Therefore one has a constant ˆ σ CC ≈ − cm at xE ≫ E ∗ . Upon integrationover the parton densities, the nucleonic σ CC becomes a slowly growing function of E becauseat higher E partons with smaller x > ∼ E ∗ /E contribute, and q ( x ) grows as x →
0. This isanalogous to what happens for neutrinos [24]. Indeed, at high energies, the MDM CC crosssection in fig. 7 approaches the dotted red line of the analogous result for ν µ,τ .16 Energy in eV10 -36 -35 -34 -33 -32 -31 c r o sss ec ti on i n c m ν µ , τ CC 3 5 10 Energy in eV10 -8 -7 -6 -5 -4 -3 E n e r gy l o ss β i n1 / k m w e ν µ , τ CC em3 5 CC3 5 NC3 5
Figure 7:
Left plot: CC cross section for fermionic MDM (black solid) with n = 3 , on nucleons.The same cross section for neutrinos (green dot-dashed) is reported for comparison. Right plot:average energy loss parameters β = − d ln E/dℓ due to CC interactions (black solid lines), NCinteractions (blue dashed), and electromagnetic interactions (red dotted). For comparison, thegreen dot-dashed line show the CC energy loss of ν µ,τ . The horizontal lines show the thicknessof the Earth, crossed vertically. For the electromagnetic energy losses of DM ± one has the standard parameterization [25] − dE loss , em dℓ = α + β em E (32)where α ≈ .
24 TeV / kmwe approximates the energy loss due to ionization effects [25] andthe β em term accounts for the radiative losses: it receives contributions from bremsstrahlung, e + e − pair production and photonuclear scattering (i.e. inelastic electromagnetic collisions onnuclei). The latter two effects have comparable importance, while bremsstrahlung is subdomi-nant for a very heavy particle such as MDM [26]. We follow the approach of [26] and adopt aparameterization for β em with a mild dependance on the energy: β em ≃ . − M " (cid:18) E eV (cid:19) . . (33)Before proceeding to the numerical solution of the equations in (26), we can gain under-standing on the expected results by simplifying the treatment of the CC energy losses. Inanalogy with the standard parameterization, we can define an average energy loss suffered byDM particles due to CC interactions as − β CC ( E ) = 1 E dE loss , CC dℓ = N N Z dx q ( x ) Z − (ˆ s − M ) / ˆ s d ˆ t ˆ t ˆ s − M d ˆ σ CC d ˆ t . (34)A completely analogous expression holds for β NC = − /E dE loss , NC /dℓ . Both are plotted infig. 7b, in units of 1 / kmwe = 10 − cm / gram. The horizontal line indicates the value abovewhich a particle looses a significant amount of energy while vertically crossing the Earth (itsthickness being 1 . kmwe). Energy losses can be understood analogously to the total crosssection, and are dominated by partons with xE ∼ E ∗ . Parton distributions have been measured17t x > ∼ − and theoretical extrapolations to smaller x can have O (1) uncertainties, that affectour results.The fraction of energy lost in one scattering is small, ∼ β/N N σ < ∼ − at the UHE energiesthat will be relevant for us, so that all energy losses can be approximated as continuous: − dE dℓ = β CC E , − dE ± dℓ = α + ( β CC β em + β NC ) E ± (35)where E ( E ± ) is the energy of DM (DM ± ). Therefore, we do not need to study the evolutionof the DM energy spectrum (dictated by eq. (26)) but we just need to follow how any initialenergy decreases while crossing the Earth. This process is well approximated by the followingsystem of two ordinary differential equations: dN dℓ = − N N σ CC N + 1 − N /τ + 1 /N N σ CC (36a) dEdℓ = N dE dℓ + (1 − N ) dE ± dℓ . (36b)where 0 ≤ N ≤ − N is the fraction ofDM present in charged state. The initial conditions are N (0) = 1 and E (0) = E in .Fig. 8a shows the numerical relevance of the various effects, and allows to understandtheir interplay. The DM interaction length starts to be smaller than the Earth thickness at E > ∼ eV, but these interactions have no effects, since the produced DM ± regenerates DM in a negligible length and with negligible energy losses. At E > ∼ eV energy losses start tobe moderately relevant, and at roughly the same energy the DM , ± mean free-path in Earthmatter becomes comparable to the DM ± decay length: the Earth is crossed loosing an orderone fraction of the initial energy, and spending an order one fraction of the path as DM ± ratherthan as DM .Fig. 8b shows the numerical results: the black continuous line shows the DM ± fraction1 − N as function of the initial energy, after vertically crossing the Earth. The dashed lineshows the same fraction as function of the final energy. Actually, we have two dashed linesobtained by solving the full and simplified system of evolution equations: their agreementconfirms the validity of the continuum energy loss approximation. (In the full case we assumedan initial DM energy spectrum proportional to the CR energy spectrum). These DM ± fractionis roughly given by 1 − N ≈ (1 + N N σ CC /τ ) − , and depends on the local nucleon density N N ,averaged over a typical scale of tens of km. This figure allows to compute the DM ± rate forany initial energy spectrum of DM. The upper red dot-dashed line shows the ratio E out /E in between the final and initial energy.Fig. 9 shows the analogous result for the fermion 5-plet, which has a DM ±± component witha much shorter life-time ≈ .
05 cm. Due to its fast decay, the new DM ±± component is notpresent with a significant abundance at the UHE CR energies E < ∼ eV. Minimal DM particles may exist at ultra high energies if they are a component of the primarycosmic rays, if they manage to be produced by standard UHE CR in the earth’s atmosphere The fraction of energy lost in one DM ± → DM decay is ∆ M/M ∼ − and can be neglected. Energy in eV10 P a t h i nk m MDM with n = 3 τ DM ± DM ± range DM d L /dln E D M ↔ D M ± Energy in eV10 -4 -3 -2 -1 fr ac ti on MDM with n = 3DM ± /DM E out / E in Figure 8:
We consider the wino-like MDM: a fermion with n = 3 and M = 2 . . Leftplot:
The dotted lines show the length scales that characterize energy losses in matter with ρ = 6 . / cm . More precisely: with the upper blue dotted line we show the effective /β CC experienced by a DM as if it never transformed in the charged state, and with the red dotted linethe range of a stable DM ± . The solid lines show the effects of neutral to charge transformationsand vice-versa: the solid ascending red line is the DM ± life-time while the solid descending pinkline is the mean free path for DM → DM ± CC interactions.
Right plot: we consider a DM which crosses the Earth vertically with initial energy E in and we show E out /E in (upper red dot-dashed line) and the fraction of DM ± over the total number of DM particles ( DM + DM ± ).These fractions are shown as a function of E out in the simplified numerical approach (blackdashed line) and in the full numerical approach (blue dashed line, almost superimposed to theformer) as well as a function of E in (black solid line). or if more exotic CR scenarios will prove to be motivated. Let us look at these opportunitiesin turn.Although the mechanism that accelerates the UHE primary CR has not yet been estab-lished, the plausible astrophysical standard mechanism (first-order Fermi acceleration in mag-netic shock waves around various objects, such as gamma-ray bursts and supernova remnants)accelerates stable charged particles. In the MDM scenario, the neutral DM is accompaniedby a charged partner with life-time τ ∼ cm: this value is not macroscopic enough to leadto acceleration of DM ± particles. Moreover, the standard mechanism accelerates protons orcharged nuclei in regions that are believed to be transparent: i.e. the local density is so smallthese primary CR particles negligibly hit on the surrounding DM accelerating it, or on thesurrounding material producing DM pairs.Production of DM pairs might be relevant when UHE CR particles hit the Earth. Thenumber of DM particles generated by one UHE proton with energy E > ∼ (2 M ) /m p ∼ eV(which is the energy range that leads to a sizable DM ± fraction) is r ∼ σ/σ pp ∼ − , where σ pp ∼ /m π is the total hadronic cross section, and σ is the pp → DM DM cross section,computed in [6]. A UHE electron neutrino produces a higher fraction of DM, r ∼ σ/σ ν ∼ − ,but only at higher energies E > ∼ (2 M ) /m e . These processes lead to a double DM signature,but their rate looks too small for present detectors (see e.g. [20, 26]).A UHE DM rate detectable in forthcoming experiments can arise in more speculative sce-narios: e.g. they may be a component of the UHE CR generated by decays of ultra-heavyparticle, in the so called ‘top-down’ scenarios [27].19 Final energy E out in eV10 -5 -4 -3 -2 -1 fr ac ti on MDM with n = 5 D M ± / D M D M ±± / D M E out / E in Figure 9:
We consider the MDM fermion with n = 5 and M = 10 TeV which crosses the earthvertically with initial energy E in and final energy E out . We show E out /E in (upper red dot-dashedline) and the fraction of DM ± and of DM ±± over the total number of DM particles as functionof E out . In terms of absolute numbers, if DM constitutes a number fraction r of CR with energyabove 10 eV (10 eV), one expects a flux of ≈ r ( ≈ r ) DM ± per km · yr, in the case n = 3. Atmospheric neutrinos generate a flux of ≈ ( ≈
10) up-going muons per km · yr with E > eV (10 eV): as discussed in the next section, this is a background to DM ± searches. DM ± Let us study how UHE DM ± can be searched for. In a detector like IceCUBE or Antares DM ± roughly look like muons with fake energy E µ = E ± β CC /β µ ∼ − E ± , because muon energylosses are approximatively given by − dE µ /dℓ = α + β µ E µ with β µ ≈ . / kmwe [25]: indeed β is roughly inversely proportional to the particle mass.One therefore needs to carefully study charged tracks to see the difference between a DM ± with E ± ∼ eV and a muon with E µ ∼ eV. The muon would loose all its energy in about2 kmwe (with a characteristic energy loss profile) while DM ± have a much longer range in matter.On the contrary, the muon is essentially stable, while DM ± decays in about 10 km (less at lowerenergy: see fig. 8a), suddenly disappearing into a DM and a π ± with energy E π ∼ m π E ± /M .Furthermore, the characteristic scale for the DM ↔ DM ± transformation is about tens of km.Unfortunately it seems very difficult to exploit these differences to discriminate between a µ ± and a DM ± in the forthcoming IceCUBE km-size detector [18], because it only has a 1 km sizeand modest granularity. We considered DM that fills one SU(2) multiplet and only interacts with SM vectors (‘MinimalDark Matter’ [6]). This model has one free parameter, the DM mass M , which is fixed fromthe observed DM abundance assuming that DM is a thermal relic. We have reconsidered thiscomputation including non-perturbative corrections due to Coulomb-like forces mediated by SMvectors, that significantly increase M . For example, a fermion 5-plet automatically behaves asMDM (and in particular is automatically stable): its mass increases from M = 4 . DM mass in GeV10 -46 -45 -44 -43 -42 σ S I ( D M N ) i n c m Xenon boundSuperCDMS CXenon 1ton 2 S F S,F S,F
Figure 10:
Predicted mass and predicted spin-independent cross sections per nucleon of MDMcandidates. S calar ( F ermionic) SU(2) n -tuplets are denoted as n S ( n F ). The shaded region isexcluded by [13] and the dashed lines indicate the planned sensitivity of future experiments [28].The prediction for doublets with Y = 0 hypercharge holds up to the caveats discussed in the text;furthermore we assumed the nuclear matrix element f = 1 / and the Higgs mass m h = 115 GeV .
10 TeV. The cosmological freeze-out abundance is plotted as function of M in fig. 4a (fermion5-plet), 2a (wino-like fermion 3-plet, M ≈ . M ≈ . M ≈ . M ≈ . M ≈
25 TeV).Having fixed M , we studied the MDM signals.Concerning the signals at colliders (see [6, 29]), we here point out that the charged compo-nents DM ± in the MDM multiplet manifest as non-relativistic cm-scale ionizing charged tracksnegligibly bent by the B ∼ Tesla magnetic field present around the interaction region. DM ± decays with 97 .
7% branching ratio into π ± , giving a relativistic track bent by the magneticfields. Everything happens in the inner portion of the detector and this signature is free fromSM backgrounds. However, at a hadron collider like LHC it seems not possible to trigger on thissignature: level-1 triggers are located far from the interaction point and MDM candidates areso heavy that the event rate is too low. Unlike in other scenarios we cannot choose a favorablebenchmark point in a vast parameter space.Astrophysics offers more promising detection prospects.The cross section for direct DM detection negligibly depends on M , and remains the sameas in [6]: fig. 10 summarizes the situation. Since MDM is much heavier than a typical nucleus,direct DM searches can precisely reconstruct only the combination σ SI /M .Next, we considered indirect DM detection , computing how non-perturbative effects enhancethe DM DM annihilations into W + W − , γγ , γZ , ZZ . Results are shown in fig. 2b (wino-likefermion 3-plet [10]), 3b (scalar 3-plet), 4b (5-plet), and 5b (Higgsino-like fermion 2-plet [10]).Signal rates can now be computed by combining this particle physics with (uncertain) astro-physics: for example fig. 6 shows the predicted spectrum of galactic γ .Finally, we assumed that some DM particles are present among the cosmic rays at UltraHigh Energies and identified an unusual signal, characteristic of heavy and quasi-degenerate21ultiplets containing neutral and charged components: a DM crosses the Earth without loosingmuch energy and, at E > ∼ eV, a sizable fraction of the travel is done as DM ± , leading tocharged tracks, that might be detectable in neutrino telescopes such as IceCUBE or km3net [17].Fig.s 8 and 9 quantitatively show how frequent this phenomenon is. Acknowledgements
We thank Masato Senami for several clarifications about [7]. We alsothank Gianfranco Bertone, J¨urgen Brunner, Giacomo Cacciapaglia, Paschal Coyle, MicheleFrigerio, Dario Grasso, John Gunion, Teresa Montaruli, Slava Rychkov, G¨unter Sigl, IgorSokalski and Igor Tkachev for useful discussions. The work of M.C. is supported in partby INFN under the postdoctoral grant 11067/05 and in part by CEA/Saclay.
A Annihilation cross sections
We here give results for the tree-level DM DM annihilation cross sections. We define theadimensional ‘reduced cross section’ ˆ σ ( s ) = Z − s dt X | A | πs (37)where s, t are the Madelstam variables and the sum runs over all DM components and over allSM vectors, fermions and scalars, assuming that all SM masses are negligibly small.The DM abundance is computed by solving the Boltzmann equation sZHz dYdz = − (cid:18) Y Y − (cid:19) γ A , γ A = T π Z ∞ M ds s / K (cid:18) √ sT (cid:19) ˆ σ A ( s ) (38)where z = M/T , K is a Bessel function, Z = (1 − zg s dg s dz ) − , the entropy density of SMparticles is s = 2 π g ∗ s T / Y = n DM /s where n DM is the number density of DM particlesplus anti-particles, and Y eq is the value that Y would have in thermal equilibrium. We can writea single equation for the total DM density because DM scatterings with SM particles maintainthermal equilibrium within and between the single components. We ignored the Bose-Einsteinand Fermi-Dirac factors as they are negligible at the temperature T ∼ M/
26 relevant for DMfreeze-out.We assume that DM fills one SU(2) multiplet with dimension n and hypercharge Y ; when Y = 0 the conjugate multiplet is also added in order to allow for a gauge-invariant DM massterm. For example, the Higgsino has n = 2 and Y = 1 /
2; the wino has n = 3 and Y = 0. Wedefine g X as the number of DM degrees of freedom: g X = n for scalar DM with Y = 0; g X = 2 n for scalar DM with Y = 0 and for spin-1/2 (Majorana) DM with Y = 0; g X = 4 n for spin-1/2(Dirac) DM with Y = 0. For fermion DM we getˆ σ A = g X πn " (9 C − C ) β + (11 C − C ) β − (cid:18) C ( β −
2) + C ( β − (cid:19) ln 1 + β − β + g X (cid:18) g ( n −
1) + 20 g Y Y π + g ( n −
1) + 4 g Y Y π (cid:19)(cid:18) β − β (cid:19) (39)and for scalar DM we getˆ σ A = g X πn " (15 C − C ) β + (5 C − C ) β + 3( β − (cid:18) C + C ( β − (cid:19) ln 1 + β − β g X (cid:18) g ( n −
1) + 20 g Y Y π + g ( n −
1) + 4 g Y Y π (cid:19) · β (40)where x = s/M and β = q − /x is the DM velocity in the DM DM center-of-mass frame.The first line gives the contribution of annihilation into vectors, the second line contains thesum of the contributions of annihilations into SM fermions and vectors respectively. The gaugegroup factors are defined as C = X A,B Tr T A T A T B T B = g Y nY + g g Y Y n ( n − g n ( n −
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