Decaying Dark Matter from Dark Instantons
aa r X i v : . [ h e p - ph ] S e p Decaying Dark Matter from Dark Instantons
Christopher D. Carone, ∗ Joshua Erlich, † and Reinard Primulando ‡ Particle Theory Group, Department of Physics,College of William and Mary, Williamsburg, VA 23187-8795 (Dated: August 2010)
Abstract
We construct an explicit, TeV-scale model of decaying dark matter in which the approximatestability of the dark matter candidate is a consequence of a global symmetry that is broken onlyby instanton-induced operators generated by a non-Abelian dark gauge group. The dominant darkmatter decay channels are to standard model leptons. Annihilation of the dark matter to standardmodel states occurs primarily through the Higgs portal. We show that the mass and lifetime ofthe dark matter candidate in this model can be chosen to be consistent with the values favored byfits to data from the PAMELA and Fermi LAT experiments. ∗ [email protected] † [email protected] ‡ [email protected] . INTRODUCTION Evidence has been accumulating for an electron and positron excess in cosmic rays com-pared with expectations from known galactic sources. Fermi LAT [1] and H.E.S.S. [2] havemeasured an excess in the flux of electrons and positrons up to a TeV or more. The PAMELAsatellite is sensitive to electrons and positrons up to a few hundred GeV in energy, and is ableto distinguish positrons from electrons and charged hadrons. PAMELA detects an upturnin the fraction of positron events beginning around 7 GeV [3]. This is in contrast to theexpected decline in the positron fraction from secondary production mechanisms. Curiously,no corresponding excess of protons or antiprotons has been detected [4].Although conventional astrophysical sources may ultimately prove the explanation of theanomalous cosmic ray data [5], an intriguing possibility is that dark matter annihilation ordecay provides the source of the excess leptons. If dark matter annihilation is responsible forthe excess leptons, then the annihilation cross section typically requires a large boost factor ∼ − ∼ seconds [14, 15]. From a model-building perspective, an intriguing issue is the origin ofthis long lifetime, and whether it can be explained with a minimum of theoretical contrivance.With this goal in mind, we present a new model of TeV-scale dark matter, one in which ananomalous global symmetry prevents dark matter decays except through instantons of a non-Abelian gauge field in the dark sector. Instanton-induced decays naturally produce the longrequired lifetime. Small mixings between standard model leptons and dark fermions givesrise to the leptonic final states observed in the cosmic ray data. Dark matter annihilationthrough the Higgs portal allows for the appropriate dark matter relic abundance, with darkmatter masses consistent with the range preferred by PAMELA and Fermi-LAT data.Superheavy dark matter decays through instantons have been considered before as apossible explanation for ultra-high energy cosmic ray signals, but those scenarios assumed2 X X ψ χ (1) (2)(3) χχ I e ν +- e FIG. 1: Dark matter decay vertex. The circle represents the instanton-induced interaction, whileX’s represent mass mixing between the χ fields and standard model leptons. Note that e and ν represent leptons of any generation. superheavy dark matter with a mass of 10 GeV or higher [16] which cannot simultane-ously explain the lower energy electron and positron flux being considered here. Models ofanomaly-induced dark matter decays without a dark gauge sector can also be constructed.For example, a supersymmetric extension of the radiative seesaw model of neutrino massescan explain the PAMELA data through dark matter decays via an anomalous discrete sym-metry [17]. The TeV-scale model we present, which is based on the smallest, continuousnon-Abelian dark gauge group and smallest set of exotic particles necessary to implementour idea, suggests a prototypical set of new particles and interactions that could perhaps beprobed at the LHC.In Section II we present the model and describe the leptonic decay mode via instantons. InSection III we consider dark matter annihilation channels and demonstrate that annihilationthrough the Higgs portal can lead to the measured dark matter relic density. In Section IVwe consider dark matter interactions with nuclei and find that our model is safely belowcurrent direct detection bounds. We conclude in Section V.3 L ( , − / ψ uR , ψ dR ( , − / χ (1) L ( , +1 / + χ (1) uR , χ (1) dR ( , +1 / + χ (2) L ( , +1 / χ (2) uR , χ (2) dR ( , +1 / χ (3) L ( , +1 / − χ (3) uR , χ (3) dR ( , +1 / − H D ( , η ( , / TABLE I: Particles charged under the dark gauge groups. The SU(2) D × U(1) D charge assignmentsare indicated in parentheses; the subscripts +, − and 0 represent the standard model hypercharges+1, − ψ and χ states are fermions, while the H D and η arecomplex scalars. II. THE MODEL
The gauge group of the dark sector is SU(2) D × U(1) D . The matter content consists offour sets of left-handed SU(2) D doublets and right-handed singlets: ψ L ≡ ψ u ψ d L ψ uR , ψ dR ; χ ( i ) L ≡ χ ( i ) u χ ( i ) d L χ ( i ) uR , χ ( i ) dR ( i = 1 . . .
3) (2.1)We include an SU(2) D doublet and singlet Higgs field, H D and η , respectively, that areresponsible for completely breaking the dark gauge group. In addition, the Higgs field H D is responsible for giving Dirac masses to the ψ and χ fields. The model is constructedso that ψ number corresponds to an anomalous global symmetry that is violated by the ψχ (1) χ (2) χ (3) vertex generated via SU(2) D instantons, as indicated in Fig. 1. The χ fieldsare assigned hypercharges so that they mix with standard model leptons, leading to thedecay ψ → ℓ + ℓ − ν . The required lifetime ( ∼ s) and the appropriate dark matter relicdensity (Ω D h ∼ .
1) constrain the free parameters of the model.The charge assignments for these fields are summarized in Table I. Let us first discussthe consistency of the charge assignments. Cancellation of the SU(2) D U(1) anomaliesrequires that the sum of the U(1) charges over all the dark doublet fermion fields mustvanish. As one can see from Table I, this is clearly the case for the U(1) D and U(1) Y charges of the left-handed doublet ψ and χ fields. Since SU(2) is an anomaly free groupand has traceless generators, all other SU(2) D anomalies vanish trivially. Now consider theU(1) pD U(1) qY anomalies (where p and q are non-negative integers satisfying p + q = 3). For4ach field in Table I with a given U(1) D × U(1) Y charge assignment, one notes that there isanother with the same charge assignment but opposite chirality. As far as the Abelian groupsare concerned, the theory is vector-like and the corresponding anomalies vanish. Finally, wenote that the theory has precisely four SU(2) D doublets and is free of a Witten anomaly.The gauge symmetries of the model lead to a global U(1) ψ symmetry that prevents thedecay of the lightest ψ mass eigenstate at any order in perturbation theory. To confirm thisstatement, we need to show that all renormalizable interactions that violate this symmetryare forbidden by the dark-sector gauge symmetry. The possible problematic interactionsthat could violate this global symmetry fall into the following categories:1. Terms involving ψ c ψ . Here the superscript indicates charge conjugation, ψ c ≡ iγ γ ψ T .This combination has U(1) ψ charge +2. However, it also has U(1) D charge −
1. Since wehave no Higgs field with the U(1) D charge ±
1, there are no renormalizable interactions thatviolate ψ number by two units.2. Terms involving a χ fermion and ψ or ψ c . Such terms violate ψ number by ± ψ and any χ have U(1) D charges ± / ± / D charge to form a renormalizablegauge invariant term of this type.3. Terms involving a standard model fermion and ψ or ψ c . Such an interaction wouldviolate ψ number by ±
1, but would have U(1) D charge ± /
2. Again, we have no Higgsfields with charge ± / ψ symmetry, noperturbative process involving these interactions will violate the global symmetry. However,since the SU(2) D U(1) ψ anomaly is non-zero, non-perturbative interactions due to instantonswill generate operators that violate the U(1) ψ symmetry.Instantons are gauge field configurations which stationarize the Euclidean action but havea nontrivial winding number around the three-sphere at infinity. Following ’t Hooft [18,19], if there are N f Dirac pairs of chiral fermions which transform in the fundamentalrepresentation of a gauge group, then due to the chiral anomaly a one-instanton configurationviolates the axial U(1) A charge by 2 N f units. The non-Abelian, SU( N f ) × SU( N f ) chiralsymmetry is non-anomalous, so the instanton process must involve the 2 N f chiral fermionsin a symmetric fashion. Fig. 1 shows the effective ψχ (1) χ (2) χ (3) interaction induced by the5nstanton configuration in our model. Given the hypercharge assignments of the χ fields,these states have electric charges +1, 0 and −
1, the same as standard model leptons, ofany generation. After the dark and standard model gauge symmetries are spontaneouslybroken, there is no symmetry which prevents the χ states and the standard model leptonsfrom mixing. By including a single vector-like lepton pair, we now show that mixing leadingto the decay ψ → ℓ + ℓ − ν can arise via purely renormalizable interactions.We introduce a vector-like lepton pair, E L , E R , with mass M E and the same quantumnumbers as a right-handed electron; in the notation of Table I: E L ∼ E R ∼ ( , − . (2.2)In addition, we assume in this model that standard model neutrinos have purely Diracmasses. If the Higgs vacuum expectation values (vevs) are smaller than the masses of theheavy states, then the mixing to standard model leptons shown in Fig. 1 can be estimated viathe diagram in Fig. 2. Otherwise, one has to diagonalize the appropriate fermion mass ma-trices. We discuss the exact diagonalization in an appendix for the reader who is interestedin the details. Here, the diagrammatic approach is sufficient to establish that the mixing ispresent, and is no larger than order h η i /M χ , h η i /M χ , and h η ih H i / ( M χ M E ), where H is thestandard model Higgs, for the χ (1) L − e cR , χ (2) L − ν cR and χ (3) L − e L mixing angles, respectively.We take each mixing angle to be 0 .
01 in the estimates that follow, and demonstrate in theappendix how this choice can be easily obtained. Further, we assume that decays to theheavy eigenstates are not kinematically allowed, as is also illustrated in the appendix. Dueto the mixing, the χ ( i ) particles decay quickly to standard model particles via couplingsto the Higgs bosons and standard model electroweak gauge bosons. The heavier ψ masseigenstate decays to lighter states via SU(2) D gauge-boson-exchange interactions.The instanton-induced vertex in Fig. 1 follows from an interaction of the form L I = C g D exp (cid:18) − π g D (cid:19) (cid:18) m ψ v D (cid:19) / v D (2 δ αβ δ γσ − δ ασ δ βγ ) · h ( χ (2) cL β ψ αL )( χ (1) cL σ χ (3) γL ) − ( χ (1) cL β ψ αL )( χ (2) cL σ χ (3) γL ) i + h.c. , (2.3)where α , β , γ and σ are SU(2) D indices [19, 20]. The dimensionless coefficient C canbe computed using the results in Ref. [19] and one finds C ≈ × . The operators in In this model, Planck-suppressed operators of this form, if they are present, are negligible compared tothe instanton-induced effects. (1)(2)(3) χχ I e ν e XXX X cc E (cid:13)
H< >< h >< h >< h > FIG. 2: Diagrammatic interpretation of mixing from χ states to standard model fermions, corre-sponding to the right-hand-side of Fig. 1. Here E represents the vector-like lepton described in thetext, and H is the standard model Higgs. Eq. (2.3) lead, via mixing, to operators of the form ¯ ν R ψ L ¯ e R e L and ¯ e R ψ L ¯ ν R e L . Assumingthat the product of mixing angles is ≈ − , as discussed earlier, one may estimate the decaywidth: Γ( ψ → ℓ + ℓ − ν ) ≈ g D exp( − π /g D ) (cid:18) m ψ v D (cid:19) / m ψ . (2.4)For example, for m ψ = 3 . v D = 4 TeV, one obtains a dark matter lifetime of 10 sfor g D ≈ . , (2.5)where g D is defined in dimensional regularization and renormalized at the scale m ψ [19].For similar parameter choices, one can slightly adjust g D to maintain the desired life-time. As mentioned earlier, dimension-six Planck-suppressed operators are much smallerthan the operators in Eq. (2.3). Sphaleron-induced interactions are suppressed by ∼ exp[ − πv D / ( g D T )] ∼ exp( −
44 TeV / T ), and become negligible well before the tempera-ture at which dark matter freeze out occurs.Finally, let us consider whether the choice v D = 4 TeV conflicts with other meaningfulconstraints on the heavy particle content of the model. In short, a spectrum of ∼ χ and E fermions with order 0 .
01 mixing angles with standard model leptons presents nophenomenological problems. These states are above all direct detection bounds; they are7ector-like under the standard model gauge group so that the S parameter is small; they mixweakly enough with standard model leptons so that other precision observables are negligiblyaffected. On this last point, we note that the correction to the muon and Z -boson decaywidths due to the fermion mixing is a factor of 10 − smaller than the widths predicted inthe standard model, which is within the current experimental uncertainties. The dark sectorgauge bosons are also phenomenologically safe. They do not have couplings that distinguishstandard model lepton flavor (since they do not couple directly to standard model leptons)so that tree-level lepton-flavor violating processes are absent. The effective four-standard-model-fermion operators that are induced by dark gauge boson exchanges are suppressed by ∼ (0 . /v D ∼ / (40 ,
000 TeV) , which is consistent with the existing contact interactionbounds [25].We now turn to the question of whether the model provides for the appropriate darkmatter relic density. III. RELIC DENSITY
For the regions of model parameter space considered in this section, dark matter annihi-lations to standard model particles proceed via mixing between the dark and ordinary Higgsbosons, often described as the Higgs portal [21]. We take into account mixing between thedoublet Higgs fields, H D and H , in our discussion below. This is consistent with a simplify-ing assumption that the η Higgs does not mix with the others in the scalar potential. Suchan assumption is adequate for our purposes since we aim only to show that some parameterregion exists in which the correct dark matter relic density is obtained. Consideration of amore general potential would likely provide additional solutions in a much larger parameterspace, but would not alter the conclusion that the desired relic density can be achieved.In this section, ψ will refer to the dark matter mass eigenstate, i.e. , the lightest masseigenstate of the ψ u - ψ d mass matrix, which we take as diagonal, for convenience. Thepotential for the doublet fields has the form: V = − µ H † H + λ ( H † H ) − µ D H † D H D + λ D ( H † D H D ) + λ mix ( H † H )( H † D H D ) . (3.1)8n unitary gauge, H and H D are given by H = 1 √ v + h , H D = 1 √ v D + h D , (3.2)where v and v D are the H and H D vevs, respectively. At the extrema of this potential, v ( − µ + λ v + 12 λ mix v D ) = 0 v D ( − µ D + λ D v D + 12 λ mix v ) = 0 . (3.3)The h - h D mass matrix follows from Eq. (3.1), M H = λ v λ mix v v D λ mix v v D λ D v D . (3.4)Diagonalizing the mass matrix, one finds the mass eigenvalues m , = ( λ D v D + λ v ) ∓ ( λ D v D − λv ) p y , (3.5)where y = λ mix v v D λ D v D − λ v . (3.6)The mass eigenstates h and h are related to h and h D by a mixing angle h = h cos θ − h D sin θh = h sin θ + h D cos θ, (3.7)where tan 2 θ = y . (3.8)Dark matter annihilations proceed via exchanges of the physical Higgs states h and h . We take into account the final states W + W − , ZZ , h h and t ¯ t , where t representsthe top quark. For the parameter choices considered later, final states involving h will besubleading. The relevant annihilation cross sections are given by σ W + W − = g m ψ sin θ cos θ πm W v D s (cid:12)(cid:12)(cid:12)(cid:12) s − m + im Γ − s − m + im Γ (cid:12)(cid:12)(cid:12)(cid:12) × s − m ψ s r − m W s (cid:18) − m W s + 12 m W s (cid:19) , (3.9)9 ZZ = g m ψ sin θ cos θ πm W v D s (cid:12)(cid:12)(cid:12)(cid:12) s − m + im Γ − s − m + im Γ (cid:12)(cid:12)(cid:12)(cid:12) × s − m ψ s r − m Z s (cid:18) − m Z s + 12 m Z s (cid:19) , (3.10) σ h h = m ψ πv D (cid:12)(cid:12)(cid:12)(cid:12) g sin θs − m + im Γ + g cos θs − m + im Γ (cid:12)(cid:12)(cid:12)(cid:12) × s − m ψ s s − m h s , (3.11) σ t ¯ t = 3 m ψ m t sin θ cos θ πv D v s (cid:12)(cid:12)(cid:12)(cid:12) s − m + im Γ − s − m + im Γ (cid:12)(cid:12)(cid:12)(cid:12) × (cid:18) − m t s (cid:19) (cid:18) − m ψ s (cid:19) . (3.12)In Eqs. (3.9) and (3.10), g is the standard model SU(2) gauge coupling. In Eq. (3.11), g and g represent the h and h h couplings, respectively: g = ( λ cos θ + 12 λ mix cos θ sin θ ) v − ( λ D sin θ + 12 λ mix sin θ cos θ ) v D ,g = [3 λ cos θ sin θ − λ mix (cos θ sin θ −
12 sin θ )] v + [3 λ D sin θ cos θ − λ mix (sin θ cos θ −
12 cos θ )] v D . (3.13)Finally, in all our annihilation cross sections, Γ (Γ ) represents the decay width of theHiggs field h ( h ). The width Γ is comparable to that of a standard model Higgs bosonand can be neglected without noticeably affecting our numerical results. However, since oureventual parameter choices will place the mass of the heavier Higgs field around 2 m ψ , wemust retain Γ ; the leading contributions to Γ come from the same final states relevant tothe ψ annihilation cross section:Γ h → W + W − = g m πm W sin θ s − m W m (cid:18) − m W m + 12 m W m (cid:19) Γ h → ZZ = g m πm W sin θ s − m Z m (cid:18) − m Z m + 12 m Z m (cid:19) Γ h → h h = g πm s − m m Γ h → t ¯ t = 3 m m t πv sin θ (cid:18) − m t m (cid:19) / . (3.14)10 ψ (TeV) √ λv (TeV) q λ D v D (TeV) λ mix m (GeV) m (TeV)1.0 0.19 1.98 0.21 158 1.981.5 0.22 2.98 0.28 199 2.982.0 0.26 3.97 0.39 241 3.972.5 0.27 4.97 0.42 257 4.973.0 0.29 5.96 0.52 277 5.963.5 0.31 6.96 0.57 299 6.964.0 0.35 7.95 0.70 339 7.95TABLE II: Examples of viable parameter sets for v D = 4 TeV. For each point listed, Ω D h ≈ . The evolution of the ψ number density, n ψ , is governed by the Boltzmann equation dn ψ dt + 3 H ( t ) n ψ = −h σv i [ n ψ − ( n EQψ ) ] , (3.15)where H ( t ) is the Hubble parameter and n EQψ is the equilibrium number density. Thethermally-averaged annihilation cross section times relative velocity h σv i is given by [22] h σv i = 18 m ψ T K ( m ψ /T ) Z ∞ m ψ ( σ tot ) ( s − m ψ ) √ s K ( √ s/T ) ds , (3.16)where σ tot is the total annihilation cross section, and the K i are modified Bessel functionsof order i . We evaluate the freeze-out condition [23]Γ H ( t F ) ≡ n EQψ h σv i H ( t F ) ≈ , (3.17)to find the freeze-out temperature T f , or equivalently x f ≡ m ψ /T f . We assume the non-relativistic equilibrium number density n EQψ = 2 (cid:18) m ψ T π (cid:19) / e − m ψ /T , (3.18)and the Hubble parameter H = 1 . g / ∗ T /m P l , appropriate to a radiation-dominateduniverse. The symbol g ∗ represents the number of relativistic degrees of freedom and m P l =1 . × GeV is the Planck mass. For the parameter choices in Tables II and III, we find x f ∼ Y = 1 Y f + r π m P l m ψ Z x x f g / ∗ x h σv i dx (3.19)11 ψ (TeV) √ λv (TeV) q λ D v D (TeV) λ mix m (GeV) m (TeV)1.0 0.16 1.98 0.21 121 1.981.5 0.15 2.98 0.28 118 2.982.0 0.16 3.97 0.39 127 3.972.5 0.15 4.97 0.42 124 4.973.0 0.15 5.96 0.52 122 5.963.5 0.15 6.96 0.57 127 6.964.0 0.15 7.95 0.70 122 7.95TABLE III: Examples of viable parameter sets for v D = 4 TeV, with m below 130 GeV. For eachpoint listed, Ω D h ≈ . where Y is the ratio of the number to entropy density and the subscript 0 indicates thepresent time. The ratio of the dark matter relic density to the critical density ρ c is given byΩ D = Y s m ψ /ρ c , where s is the present entropy density, or equivalentlyΩ D h ≈ . × GeV − Y m ψ . (3.20)In our numerical analysis, we assume that the heavy states are sufficiently nondegenerate,so that we do not have to consider co-annihilation processes [24]. In Tables II and III, weshow representative points in the model’s parameter space, spanning a range of ψ masses,in which we obtain the correct dark matter relic abundance, Ω D h ≈ .
1, and in which themasses m and m are consistent with the LEP bound m , > . D h in our model should be larger than desirable. The reason this is not the case is thatwe have chosen parameters for which the heavier Higgs h is within 1% of 2 m ψ , leadingto a resonant enhancement in the annihilation rate. While we would be happier withoutthis tuning, it is no larger than tuning that exists in, for example, the Higgs sector of theminimal supersymmetric standard model. It is also worth pointing out that this tuningis related to the portal that connects the dark to standard model sectors of the theoryand is not strictly tied to the mechanism that we have proposed for dark matter decay.12ther portals are possible. For example, one might study the limit of the model in whichthe U(1) D gauge boson is lighter and kinetically mixes with hypercharge, a possibility thatwould lead to other annihilation channels. Finally, we point out that Tables II and IIIincludes m ψ = 3 . ℓ + ℓ − ν [15]. However, other masses should not be discounted since astrophysical sources mayalso contribute to the observed positron excess [5]. IV. DIRECT DETECTION
We now consider whether the parameter choices described in the previous section areconsistent with the current bounds from direct detection experiments. The most relevantconstraints come from experiments that search for spin-independent, elastic scattering ofdark matter off target nuclei. The relevant low-energy effective interaction from t -channelexchanges of the Higgs mass eigenstates is given by L int = X q α q ¯ ψψ ¯ qq , (4.1)where α q = m q m ψ sin θ cos θv v D (cid:18) m − m (cid:19) . (4.2)This interaction is valid for momentum exchanges that are small compared to m , , whichis always the case given that typical dark matter velocities are non-relativistic. Followingthe approach of Ref. [27], Eq. (4.1) leads to an effective interaction with nucleons L eff = f p ¯ ψψ ¯ pp + f n ¯ ψψ ¯ nn , (4.3)where f p and f n are related to α q through the relation [27] f p,n m p,n = X q = u,d,s f ( p,n ) T q α q m q + 227 f ( p,n ) T g X q = c,b,t α q m q , (4.4)where h n | m q ¯ qq | n i = m n f nT q . Numerically, the f ( p,n ) T q are given by [28] f pT u = 0 . ± . , f pT d = 0 . ± . , f pT s = 0 . ± .
062 (4.5)and f nT u = 0 . ± . , f nT d = 0 . ± . , f nT s = 0 . ± . , (4.6)13 −48 −47 −46 −45 −44 −43 −42 m ψ (TeV) C r o ss s e c t i on pe r nu c l eon ( c m ) LUX LZ20TSuperCDMSCDMS
FIG. 3: Dark matter-nucleon elastic scattering cross section for the parameter sets in Table II(stars) and Table III (triangles). The solid line is the current bound from CDMS Soudan 2004-2009 Ge [26]. The dashed line represents the projected bound from SuperCDMS Phase A. Thedotted line represents the projected reach of the LUX LZ20T experiment, assuming 1 event sen-sitivity and 13 ton-kilodays. The graph is obtained using the DM Tools software available athttp://dmtools.brown.edu. while f ( p,n ) T g is defined by f ( p,n ) T g = 1 − X q = u,d,s f ( p,n ) T q . (4.7)We can approximate f p ≈ f n since f T s is larger than other f T q ’s and f T g . For the purposeof comparing the predicted cross section with existing bounds, we evaluate the cross sectionfor scattering off a single nucleon, which can be approximated σ n ≈ m r f p π (4.8)where m r is nucleon-dark matter reduced mass 1 /m r = 1 /m n + 1 /m ψ . Our results areshown in Fig. 3, for the parameter sets given in Tables II and III. The predicted crosssections are far below the current CDMS bounds [26] for dark matter masses between 1 and4 TeV. However, there is hope that the model can be probed by the future LUX LZ20Texperiment [29]. 14 . CONCLUSIONS We have presented a new TeV-scale model of decaying dark matter. The approximatestability of the dark matter candidate, ψ , is a consequence of a global U(1) symmetry thatis exact at the perturbative level, but is violated by instanton-induced interactions of a non-Abelian dark gauge group. The instanton-induced vertex couples the dark matter candidateto heavy, exotic states that mix with standard model leptons; the dark matter then decaysto ℓ + ℓ − ν final states, where the leptons can be of any generation desired. We have shownthat a lifetime of ∼ s, which is desirable in decaying dark matter scenarios, can beobtained for perturbative values of the non-Abelian dark gauge coupling. In addition, bystudying dark matter annihilations through the Higgs portal, we have provided examplesof parameter regions in which the appropriate dark matter relic density may be obtained,assuming dark matter masses that are consistent with fits to the results from the PAMELAand Fermi-LAT experiments. The nucleon-dark matter cross section in our model is lowerthan the present bound from CDMS, but may be probed in future experiments. It mightalso be possible to probe the spectrum of our model at the LHC.The model in this paper provides a concrete, TeV-scale scenario in which dark matterdecay is mediated by instantons, and gives a new motivation for the study of non-Abeliandark gauge groups [30]. However, it is by no means the only possible model of this type. Onemight study variations of the model in which different annihilation channels are dominant,or the dark matter is lighter, or the standard model leptons are directly charged underthe new non-Abelian gauge group. It may also be worthwhile to consider how low-scaleleptogenesis and baryogenesis might be accommodated in this type of scenario. While wehave assumed parameter choices motivated by the observed cosmic ray positron excess, onemight incorporate the present model in a multi-component dark matter scenario if this wererequired to explain new results from ongoing and future direct detection experiments. Acknowledgments
This work was supported by the NSF under Grant PHY-0757481. We thank Will Detmoldand Marc Sher for useful comments. 15 ppendix A: Mass mixing example
In Sec. II, we presented a diagrammatic representation of the mixing that takes the χ states to standard model leptons. Here we study the numerical diagonalization of thecorresponding fermion mass matrices, to demonstrate that mixing angles of the size assumedin our analysis are easily obtained. To simplify the discussion, we focus on mixing withstandard model leptons of a single generation, which we denote by e and ν . We include (1)Dirac masses for the χ fields: L ⊃ X i h a i χ ( i ) L h H D i χ ( i ) uR + b i χ ( i ) L h H D i χ ( i ) dR + c i χ ( i ) L h e H D i χ ( i ) uR + d i χ ( i ) L h e H D i χ ( i ) dR i + h.c. , (A1)where e H D ≡ iσ H ∗ D . These terms generate a completely general two-by-two Dirac massmatrix for the χ fermions. (2) Mixing between the χ fields and standard model leptons: L ⊃ g h η i χ (1) dR e cR + g h η i χ (1) uR e cR + λ e L h H i e R + g h η i χ (2) dR ν cR + g h η i χ (2) uR ν cR + λ ν L h ˜ H i ν R + h.c. (A2)(3) Mixing involving the vector-like leptons E L and E R : L ⊃ g h η i χ (3) dR E L + g h η i χ (3) uR E L + M E E L E R + g L h H i E R + h.c. (A3)We now write down the mass matrices which follow from Eqs. (A1,A2,A3). For the neutralstates, we work in the basis f L = ( χ (2) uL , χ (2) dL , ν cR ) and f R = ( χ (2) uR , χ (2) dR , ν cL ). The neutral massterms can be written as f L M f R + h.c. , where M = 1 √ c v D d v D a v D b v D g v η g v η √ m ν , (A4)assuming, for simplicity, that the vevs and couplings are real. Similarly, the mass termsfor the charged states may be written f − L M c f − R + h.c. , where we assume the basis f − L =( χ (1) cuR , χ (1) cdR , χ (3) uL , χ (3) dL , E L , e L ) and f − R = ( χ (1) cuL , χ (1) cdL , χ (3) uR , χ (3) dR , E R , e R ). In this case, M c = 1 √ c v D a v D g v η d v D b v D g v η c v D d v D a v D b v D g v η g v η √ M E
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