Neutrinos from Kaluza-Klein dark matter in the Sun
aa r X i v : . [ h e p - ph ] J a n MPP-2009-170
Neutrinos from Kaluza–Klein dark matter in the Sun
Mattias Blennow, ∗ Henrik Melb´eus, † and Tommy Ohlsson ‡ Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut),F¨ohringer Ring 6, 80805 M¨unchen, Germany Department of Theoretical Physics, School of Engineering Sciences,Royal Institute of Technology (KTH) – AlbaNova University Center,Roslagstullsbacken 21, 106 91 Stockholm, Sweden
Abstract
We investigate indirect neutrino signals from annihilations of Kaluza–Klein dark matter in theSun. Especially, we examine a five- as well as a six-dimensional model, and allow for the possibilitythat boundary localized terms could affect the spectrum to give different lightest Kaluza–Kleinparticles, which could constitute the dark matter. The dark matter candidates that are interestingfor the purpose of indirect detection of neutrinos are the first Kaluza–Klein mode of the U (1) gaugeboson and the neutral component of the SU (2) gauge bosons. Using the DarkSUSY and WimpSimpackages, we calculate muon fluxes at an Earth-based neutrino telescope, such as IceCube. For thefive-dimensional model, the results that we obtained agree reasonably well with the results thathave previously been presented in the literature, whereas for the six-dimensional model, we findthat, at tree-level, the results are the same as for the five-dimensional model. Finally, if the firstKaluza–Klein mode of the U (1) gauge boson constitutes the dark matter, IceCube can constrainthe parameter space. However, in the case that the neutral component of the SU (2) gauge bosonsis the LKP, the signal is too weak to be observed. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION Dark matter (DM) is one of the most active research areas in the interdisciplinary contextof particle physics, astroparticle physics, and cosmology today. In recent years, experimentsand observations have resulted in compelling evidence for the existence of non-baryonicDM. A good fit to experimental data is obtained if it is assumed that the DM is cold, i.e. ,non-relativistic at the epoch of matter-radiation equality, and that there is a cosmologicalconstant accounting for the observed present expansion of the Universe. An analysis com-bining the five-year WMAP data, baryon acoustic oscillations, and supernova observationsindicates that the DM has a density of Ω DM h = 0 . ± . . Hence, the search for a DM candidate is stronglyconnected to physics beyond the SM.An important class of DM candidates constitutes Weakly Interacting Massive Particles(WIMPs), which are weakly interacting particles with masses in the range O (1 − Although neutrinos only interact weakly, they are too light to be able to make up more than a smallfraction of the DM density. R -parity in supersymmetric models, and itensures that the lightest Kaluza–Klein particle (LKP) is stable. If neutral, the LKP couldbe a good DM candidate, analogously to the lightest supersymmetric particle (LSP) [11].In the UED models, the mass spectrum of the KK particles, and in particular, the identityof the LKP, depends on specific assumptions on the model at the cutoff scale, where thetheory breaks down. In the most widely studied five-dimensional model, known as theMinimal Universal Extra Dimensions (MUED) model, the LKP has been determined to bethe first KK mode of the U (1) gauge boson by calculating radiative corrections to the tree-level masses of the KK excitations [12]. However, as has explicitly been shown in Ref. [13],other assumptions could change this conclusion.The most stringent constraints to date set on the five-dimensional MUED model ofKKDM are given by the IceCube collaboration [14], using the 22-string configuration. Whencompleted with in total 80 strings, IceCube will have the ability to set even stronger con-straints on the model.In this paper, we calculate muon fluxes induced by neutrinos coming from annihilationof KKDM in the Sun. We study different LKP candidates in a five- as well as a six-dimensional model. We use the DarkSUSY [15] and WimpSim [16] packages to properlytreat the capture rates, the interactions of the annihilation products in the Sun, and thepropagation of neutrinos to the Earth, and to simulate the interactions of neutrinos in thedetector medium on Earth.Neutrinos from KKDM annihilations in the Sun have previously been studied in Ref. [17],and recently in Ref. [18]. In comparison to those works, we use a more careful treatment,and we find relatively small differences when comparing the results. To our knowledge, theonly study of neutrinos from KKDM annihilations in the Sun in a six-dimensional modelhas been performed in Ref. [19]. In that work, only the MUED model is investigated, and3t is concluded that no observable neutrino telescope signal is obtained.The rest of the paper is organized as follows: In Sec. II, we describe the formalism ofWIMP annihilations in the Sun, and how this is implemented in DarkSUSY and WimpSim.In Sec. III, we introduce the extra-dimensional models that we investigate. In Sec. IV, wepresent our results in the form of muon fluxes in a neutrino telescope on Earth and comparethese to the existing results in the literature. Finally, in Sec. V, we summarize and discussour results. In addition, technical details are provided in the appendices. II. NEUTRINOS FROM WIMP ANNIHILATIONSA. The annihilation rate
The evolution of the total number of WIMPs in the interior of the Sun, N ( t ), is governedby the equation d N d t = C c − C a N , (1)where C c is the capture rate of WIMPs in the Sun and C a = h σv i V V . (2)Here, σ is the total WIMP pair annihilation cross section, v is the relative WIMP velocity,and V j ≃ . · ( jm DM /
10 GeV) − / m are effective volumes [20, 21]. The solution toEq. (1) is N ( t ) = r C c C a tanh (cid:18) tτ (cid:19) , (3)where τ ≡ / √ C a C c sets the equilibration time scale for the process of capture and annihi-lation. The annihilation rate is given byΓ A = C a N = C c (cid:18) tτ (cid:19) , (4)which is the expression that is used in DarkSUSY. For times t ≫ τ , Γ A ≃ C c / i.e. , theannihilation rate is determined by the capture rate only. In that case, the only importantaspect of the WIMP-WIMP annihilation cross sections are the relative branching ratios intodifferent channels. 4 . Capture rate The capture rate of WIMPs in the Sun is governed by the elastic WIMP-proton scatteringcross section. In the non-relativistic limit, which is applicable for scattering of cold DM,the interactions between WIMPs and atoms can be characterized as either spin-dependentor spin-independent. Spin-independent cross sections are proportional to the square of theatomic number A , while spin-dependent cross sections are proportional to the square of theangular momentum, J ( J + 1) [22].For the WIMPs that we will consider, the spin-dependent cross section is much largerthan the spin-independent one. The capture rate of WIMPs in the Sun is then commonlyapproximated by the expression [23] C SD c ≃ . · s − (cid:18) ρ . / cm (cid:19) (cid:18)
270 km / s¯ v (cid:19) σ SDWIMP , p − pb ! (cid:18) m WIMP (cid:19) , (5)where ρ is the local DM density, ¯ v ≡ h v i / is the root-mean-square of the DM velocitydispersion, σ SDWIMP , p is the spin-dependent elastic WIMP-proton scattering cross section, and m WIMP is the mass of the WIMP.In DarkSUSY, a more accurate result for the capture rate is obtained by dividing the Suninto shells and numerically integrating over these shells, as well as the velocity distributionof the WIMPs. We use the DarkSUSY default values for the WIMP population, which is aGaussian velocity distribution with root-mean-square velocity dispersion ¯ v = 270 km / s, andlocal density ρ = 0 . / cm . For more details on the calculation of the capture rate, seeRef. [24].In Fig. 1, we show the ratio of the result obtained from Eq. (5) to the one obtained fromDarkSUSY. For this figure, we use σ SDWIMP , p = 1 . · − pb (1 TeV /m WIMP ) , which is thespin-dependent cross section for a B (1) WIMP with r q R = 0 . −
27 %, with the largerdifference at the lowest mass, i.e. , at the larger cross section.Since the capture rate is proportional to the WIMP-nucleon scattering cross section,which is also the quantity that is measured in direct detection experiments, the results ofneutrino telescope searches and direct detection experiments are expected to be correlated[25]. In particular, as stronger limits are set on the WIMP-nucleon cross sections by di-rect detection experiments, the prospects of detecting neutrinos from WIMP annihilations5
00 400 600 800 1000m
WIMP [GeV]1.21.221.241.261.281.3 C a pp r ox / C D a r k S U S Y FIG. 1: The ratio of the capture rate obtained from Eq. (5) to that obtained from DarkSUSY asa function of the WIMP mass m WIMP . We have used the spin-dependent elastic scattering crosssection σ SDWIMP , p = 1 . · − pb (1 TeV /m WIMP ) . decrease.In direct detection experiments, the spin-independent interactions are easier to measure,since a large scattering cross section can be obtained by using heavy target nuclei [22].Hence, the limits on the cross sections are stronger for spin-independent interactions thanfor spin-dependent ones. The current strongest limits on spin-independent cross sectionsare set by the CDMS [26] and XENON10 [27] experiments. The CDMS upper limit on thescattering cross section for a 60 GeV WIMP is 4 . · − pb at 90 % C.L. These constraintsare strong enough to rule out an observation of neutrinos from annihilations of DM in anyrunning or planned neutrino telescope, if the DM only interacts spin-independently [25].The most stringent constraints on the spin-dependent cross sections from direct detectionexperiments are set by XENON10 [28], COUPP [29], and KIMS [30] for scattering on protonsand by XENON10 [28], CDMS [26], and ZEPLIN III [31] for scattering on neutrons. TheXENON10 results constrain the WIMP-neutron spin-dependent cross section to be less than5 · − pb for a 30 GeV WIMP, whereas the KIMS results constrain the corresponding crosssection for protons to be less than 2 · − pb for an 80 GeV WIMP. In contrast to the limits6n the spin-independent cross sections, these constraints are weak enough to allow for asufficiently strong signal in neutrino telescopes. It is interesting to note that for the case ofspin-dependent scattering on protons, IceCube [3] and Super-Kamiokande [7] have alreadyobtained stronger constraints in some regions of parameter space through indirect detection.In conclusion, it is especially interesting to use neutrino telescopes to search for DMcandidates, which mainly have spin-dependent interactions with nuclei. In particular, scalarDM cannot have spin-dependent interactions, and hence, that class of DM candidates is notinteresting in this context. C. Propagation to the detector
Once the annihilation rate and branching ratios of the DM annihilations are known,we can proceed by computing the resulting neutrino signals in an Earth based detector.In addition to the geometric scaling of the flux by 1 /r , the neutrino flux is subject tointeractions with the solar medium as well as matter-enhanced flavor oscillations within theSun, while the propagation from the solar surface to the Earth is only affected by neutrinooscillations in vacuum.The consistent theoretical treatment of simultaneous neutrino interactions and oscilla-tions is through the introduction of the neutrino density matrix ρ ( E, r ), where E is theneutrino energy. This matrix describes the energy-dependent neutrino flavor composition ata distance r from the center of the Sun and follows the evolution equationd ρ d r = [ H, ρ ] − i2 { Γ , ρ } + d ρ d r (cid:12)(cid:12)(cid:12)(cid:12) NC + d ρ d r (cid:12)(cid:12)(cid:12)(cid:12) CC , (6)where H is the neutrino oscillation Hamiltonian, Γ is a diagonal matrix containing theinteraction rates of the different flavors with the medium, and the two last terms are regen-eration terms due to neutral- and charged-current interactions, respectively. In general, theneutral-current regeneration term is simply a decrease of the neutrino energy E , since theneutral current is flavor blind, while the charged-current regeneration term projects out thetau-neutrino component of the scattering, since only taus decay before stopping in the Sun.For the specific form of the regeneration terms, see Ref. [32].In our analysis, we use the results of the WimpSim package [16], which provides an event-based Monte Carlo simulation that allows us to compute the neutrino signal in a detector,7ncluding all of the effects mentioned above. The oscillations and interactions in the Sun aretreated in a consistent framework which can be easily shown to have the same properties asthe density matrix evolution described above. Furthermore, the results from WimpSim havebeen tabulated and are included in DarkSUSY, thus greatly simplifying our implementation.Finally, we compute the resulting muon-antimuon flux at the detector through the standardDarkSUSY routines. This flux can then be compared to the experimental limits in orderto rule out models or to see what kind of sensitivity that would be needed to discover theindirect neutrino signal [33]. III. THE MODELS
We investigate the signals of a five- as well as a six-dimensional UED model. For bothcases, we choose the models most commonly studied in the literature, though we allow fora more general mass spectrum than that of the MUED models. Each model is based on theSM gauge group SU (3) × SU (2) × U (1) and includes the minimal particle content necessaryto reproduce the SM in the low-energy limit.For simplicity, we follow the usual practice of ignoring electroweak symmetry breaking(EWSB) effects. In particular, we treat all the SM particles as massless. We also neglectthe Yukawa couplings, which are small compared to the gauge couplings. Since we ignoreEWSB effects, we use a gauge where all four components of the Higgs field appear as physicalstates. Finally, none of the processes that we study involve the self-coupling of the Higgsboson, and hence, we ignore this interaction.It should be noted that any extra-dimensional field theory is non-renormalizable, andhence, it can only be viewed as an effective theory. Thus, the models that we study areonly valid up to some cutoff scale Λ, at which some more fundamental ultraviolet (UV)completion of the model is needed. The possible effects from physics at the cutoff scale arediscussed below. A. Five dimensions
In five dimensions, spinors are four-component objects, as in four dimensions. However,there is no chirality operator [34], and hence, the Dirac representation is irreducible, i.e. ,8ll fermions are four-component Dirac fermions. This implies that the simplest choice ofgeometry for the fifth dimension, the circle S , does not reproduce the SM in the low-energylimit. This problem can be solved by replacing the circle by the orbifold S / Z , which isobtained from S by identifying the points y and − y , where y is the coordinate along thecircle. For the action to be invariant under this transformation, the five-dimensional fieldshave to be either even or odd in y . The KK expansion for an even field is A (even) ( x µ , y ) = 1 √ πR " A (0) ( x µ ) + √ ∞ X n =1 cos (cid:16) nyR (cid:17) A ( n ) ( x µ ) , (7)and for an odd field A (odd) ( x µ , y ) = r πR ∞ X n =1 sin (cid:16) nyR (cid:17) A ( n ) ( x µ ) . (8)In particular, an odd field does not have a zero mode, and hence, the field is not presentin the low-energy theory, which only contains the zero modes. The left- and right-handedparts of a five-dimensional Dirac field can be given opposite transformation properties, inwhich case it will have a definite chirality in the low-energy limit. The model can then bemade compatible with the SM by introducing such a Dirac field for each chiral fermion inthe SM. For a SM fermion f , we denote the Dirac fermion corresponding to the left-handedpart by f D and that corresponding to the right-handed part by f S . In general, the weakeigenstates f S and f D mix to form massive eigenstates, but the mixing is proportional tothe mass of the corresponding SM fermion, and hence, we neglect it.The orbifolding procedure introduces additional complications to the model. The orbifold S / Z has two fixed points, y = 0 and y = πR , where R is the radius of the circle. In general,terms localized to these points, so-called boundary localized terms (BLTs), can be includedin the Lagrangian. These terms affect the spectrum and the coupling constants of the modelalready at tree level. It has been shown that BLTs are generated by radiative corrections[12, 35], and hence, it is inconsistent not to include them.The BLTs break translational invariance in the fifth dimension, and hence, the fifth com-ponent of the momentum vector is not conserved. However, if the BLTs appear symmetri-cally between the fixed points, there is a remaining mirror symmetry, and correspondingly,a multiplicatively conserved quantum number, called KK parity [10]. It is defined as ( − n ,where n is the KK number, and it is analogous to R -parity in supersymmetric theories [11].9n particular, it ensures that the LKP is stable. In this work, we always assume that KKparity is conserved.In contrast to the bulk terms, the BLTs are not determined by the SM parameters. Inprinciple, all such terms that are allowed by the SM gauge symmetry should be included, and a priori , nothing can be said about these terms, except what constraints can be put on themfrom experiments [13]. Hence, their existence decreases the predictivity of the model. Thecommon solution to this problem is to assume that the BLTs vanish at the cut-off scale Λ ofthe theory. In the resulting model, usually referred to as the MUED model, there are onlytwo free parameters in addition to the SM parameters, namely the compactification radius R and the cut-off scale Λ. The procedure is similar to the fixing of supersymmetry breakingparameters by making an ansatz at the scale of grand unification, such as mSUGRA [11].However, as long as nothing is known regarding the UV completion of the UED model, theMUED ansatz is arbitrary and only made for simplicity.Our approach to the BLTs is similar to that given in Ref. [36]. We do not investigatethe effects of individual terms, but assume that the spectrum is affected in such a way as togive different LKPs. For simplicity, the possible effects on the couplings are not taken intoaccount.Gauge fields in higher-dimensional theories have an extra component for each additionaldimension. Thus, in the five-dimensional case, each gauge field has a single extra component, A . From the four-dimensional point of view, the zero mode of this field would appear as amassless scalar, which would be in conflict with experimental results, since no such scalarhas been found. Hence, the only phenomenologically viable option is to take A to be oddin y , so that it does not have a zero mode. On the other hand, the four components A µ haveto be even in order to reproduce the SM at low energies.For the Higgs doublet, we use the conventionΦ = i h +1 √ ( h + i h ) , (9)and the conjugate of h + is denoted by h − . As in the SM, the zero modes h and h ± (0) areeaten by the Z and W ± bosons, respectively. For n >
0, on the other hand, h n ) and h ± ( n ) mix with Z ( n )5 and W ± ( n )5 , and for each pair, one linear combination is eaten to give mass tothe corresponding KK mode of a gauge boson, while the orthogonal combination survives as10 physical scalar. However, the mixing is proportional to v R , where v is the Higgs vacuumexpectation value, and since we ignore EWSB effects, there is no mixing. This means thatthe scalar that is eaten is the fifth component of the gauge boson, while all four componentsof the Higgs boson appear as physical scalars at each non-zero KK level.At each KK level, the electroweak gauge bosons B ( n ) and W n ) mix to form two massiveeigenstates, γ ( n ) and Z ( n ) , in the same way as in the SM. However, in the UED model, themass matrix in flavor basis at the n th level is given by [12] n R + δm B ( n ) + g v g g v g g v n R + δm W n ) + g v , (10)where δm B ( n ) and δm W n ) are corrections to the tree-level masses. For n >
0, there arelarge contributions to the diagonal entries, and the mass matrix is almost diagonal. For R − ≥
300 GeV, the first-level Weinberg angle is bounded as sin θ (1) W . .
05, and we canmake the approximation B (1) ≃ γ (1) and W ≃ Z (1) . We use the basis ( B (1) , W ),although the ( γ (1) , Z (1) ) basis is sometimes used in the literature.We also ignore the effects of the KK modes above the first level. The relevant parts ofthe five-dimensional Lagrangian, written in terms of the KK modes, are then [37, 38] L gauge = − g f abc F (0) ,aµν A (1) ,bµ A (1) ,cν − g f abc ( ∂ µ A (1) ,aν − ∂ ν A (1) ,aµ )( A (0) ,bµ A (1) ,cν + A (0) ,cν A (1) ,bµ ) − g (cid:2) f abc ( A (0) ,bµ A (1) ,cν + A (0) ,cν A (1) ,bµ ) (cid:3) , (11) L fermion = i g (cid:16) f (0) γ µ A (1) µ P L f (1) D + P L f (1) D γ µ A (1) µ f (0) (cid:17) +i g (cid:16) P L f (1) D γ µ A (0) µ P L f (1) D + P R f (1) S γ µ A (0) µ P R f (1) S (cid:17) , (12) L scalar = i g (cid:2) A (0) µ (cid:0) Φ (1) ∂ µ Φ (1) † − Φ (1) † ∂ µ Φ (1) (cid:1) + A (1) µ (cid:0) Φ (0) ∂ µ Φ (1) † − Φ (1) † ∂ µ Φ (0) + Φ (0) † ∂ µ Φ (1) − Φ (1) ∂ µ Φ (0) † (cid:1)(cid:3) + g (cid:2) A (1) µ A (0) µ (cid:0) Φ (1) Φ (0) † + Φ (1) † Φ (0) (cid:1) + A (1) µ A (1) µ | Φ (0) | + A (0) µ A (0) µ | Φ (1) | (cid:3) . (13)The symbols f abc are the structure constants for the gauge group, and in Eqs. (12) and(13), A ≡ A a T a , where T a are the generators of the gauge group. For completeness, thecorresponding Feynman rules are presented in Appendix A 1.11 . Dark matter candidates Once the boundary terms are known, the spectrum of the model can be worked out, andthe identity of the LKP can be resolved. In general, the LKP could be the first KK mode ofany particle in the SM, or of the graviton. The particle content of the first KK level is equalto that of the SM, except that the left- and right-handed Weyl fermions are replaced bytwo Dirac fermions with the corresponding quantum numbers and that all the componentsof the Higgs doublet are physical. Since the LKP has to be neutral with respect to theelectromagnetic as well as the color interactions in order to be a good DM candidate, thelist of interesting possibilities reduces to the neutrinos, the two neutral components of theHiggs doublet, and the B and W bosons. As mentioned above, scalar DM is not interestingfor the purpose of neutrino telescope detection, and hence, we will not consider the Higgsbosons as DM candidates. In the context of the MUED model, the spin-independent crosssection for KK neutrinos scattering on protons [39] is larger than the limits set by directdetection experiments, and hence, this particle is ruled out as a DM candidate. The sameconclusion holds in our model, provided that we do not modify the couplings. We also notethat although the first KK mode of the graviton could be the LKP, this particle interactsvery weakly, and its phenomenology is very different from that of WIMPs [40]. Thus, weare left with the first KK modes of the B and W gauge bosons.In the five-dimensional MUED model, the B (1) is the LKP [12]. Detailed calculations ofthe relic density, including the effects of coannihilations, have shown that the B (1) could giverise to the relic density measured by WMAP if its mass is roughly in the range 500 GeV − B (1) wereconsidered in Ref. [17] and signals from B (1) and W were considered in Ref. [18]. We willcompare the results of these works to our results in Sec. IV. B. Six dimensions
In six dimensions, spinors are eight component objects, and as in four dimensions, thereis a chirality operator, i.e. , an 8 × fulfilling Γ = 1 and { Γ , Γ M } = 0, whereΓ M are the six gamma matrices in six dimensions [34]. This operator, which is analogous to γ in four dimensions, makes it possible to define a six-dimensional chirality. The operators12rojecting onto the states of definite chirality (which are usually referred to as states of +/ − chirality) are P ± = (1 ± Γ ) /
2. The states of definite chirality are four-component Diracspinors. Thus, the fermions can be defined as Dirac spinors, and the situation is then thesame as in five dimensions.The condition of anomaly cancellation in the six-dimensional model places constraintson the chirality assignments for the fermions [42]. Most notably, this condition forces thenumber of generations to be an integer multiple of three, motivating the existence of threegenerations in the SM. It also has implications for the six-dimensional chirality structure ofthe SU (2) doublets and singlets. The fermion content of the model is [43] Q + = ( U + , D + ), U − , D − , L + = ( N + , E + ), and E − , where the SU (2) doublets have left-handed zero-modes,and the singlets have right-handed zero-modes.In this work, we study a six-dimensional UED model, where the extra dimensions arecompactified on the so-called chiral square, T / Z [44–46]. This orbifold can be obtained bystarting from a square with side length L and identifying the points ( y,
0) and (0 , y ) as wellas the points ( y, L ) and (
L, y ) for 0 ≤ y ≤ L . In this model, there are four possible choicesfor the boundary conditions of a field [44]. The KK decomposition is given by A ( x µ , x , x ) = 1 L " δ n, A (0 , ( x µ ) + X j ≥ X k ≥ f ( j,k ) n ( x , x ) A ( j,k ) ( x µ ) , (14)where f ( j,k ) n ( x , x ) = 11 + δ j, (cid:20) e − i nπ/ cos (cid:18) jx + kx R + nπ (cid:19) ± cos (cid:18) kx − jx R + nπ (cid:19)(cid:21) . (15)Here, the index n = 0 , , , R = L/π . The squared mass for a KK mode with indices ( j, k ) is m j,k = ( j + k ) /R . Note that only the n = 0 functions have a zero mode and that there isno (0 ,
1) mode. Hence, for each field, there is a single lightest KK mode, with indices (1 , /R . We call this mode the first KK mode.As the S / Z orbifold in five dimensions, the chiral square T / Z has fixed points, (0 , L, L ), and (0 , L ). These fixed points induce BLTs, and break momentum conservation inthe extra dimensions to the conservation of KK parity, which in this model is defined as( − j + k . In the same way as in the five-dimensional model, the conservation of KK parityensures the stability of the LKP. 13n six dimensions, the gauge fields have two additional components. Similar to the five-dimensional case, one linear combination of these fields is eaten at each KK level to givemass to the corresponding KK mode of the four-dimensional components. However, theother linear combination appears as a physical scalar, known as an adjoint scalar, at eachnon-zero KK level. The adjoint scalar corresponding to the gauge field A is denoted A H .Note that the index H is not a running index.The KK-decomposed Lagrangian density has been derived in Ref. [46]. We are onlyinterested in interactions involving zero modes and the first KK modes. Neglecting all othermodes simplifies the Lagrangian dramatically. Since k = 0 in all the modes that are left, wesuppress this index. We use a gauge in which the linear combination of the extra componentsof the gauge fields that are eaten vanishes, and the four-dimensional part is massive. Hence,the fields that are denoted A G in Ref. [46] vanish. As in the five-dimensional model, weignore EWSB effects and Yukawa couplings.The modifications to the couplings from the integration over the extra dimensions areencoded in the δ symbols, which are defined as δ j ,...,j r n ,...,n r ≡ L Z L d x Z L d x f j n · · · f j r n r . (16)The order of the indices does not matter, as long as the upper and lower indices are shifted inthe same way. Only the n = 0 functions have zero modes, and as we only consider j r ∈ { , } ,this means that n r > j r = 1. Finally, it holds that δ j ,j ,j , n ,n , , = δ j ,j ,j n ,n , .Using the properties summarized above, we obtain the interaction Lagrangian L gauge = − gf abc δ j ,j ,j , , A ( j ) ,aµ A ( j ) ,bν ∂ µ A ( j ) ,cν + (cid:16) g f abc A (1) ,aH ( ∂ µ A (1) ,bH ) A (0) ,cµ + h . c . (cid:17) − g f abc f ade δ j ,j ,j ,j , , , A ( j ) ,bµ A ( j ) ,cν A ( j ) ,dµ A ( j ) ,eν − g f abc f ade A (1) ,cH A (1) ,eH A (0) ,bµ A (0) ,dµ , (17) L fermion , + = gδ j ,j ,j , , f ( j ) D A ( j ) µ γ µ f ( j ) D + gf (1) S A (0) µ γ µ f (1) S + (cid:16) i gf (0) D A (1) H f (1) S + h . c . (cid:17) , (18) L fermion , − = gδ j ,j ,j , , f ( j ) S A ( j ) µ γ µ f ( j ) S + gf (1) D A (0) µ γ µ f (1) D + (cid:16) i gf (0) S A (1) H f (1) D + h . c . (cid:17) , (19) L scalar = (cid:16) i gδ j ,j ,j , , Φ ( j ) † A ( j ) µ ∂ µ Φ ( j ) + h . c . (cid:17) + g δ j ,j ,j ,j , , , Φ ( j ) † A ( j ) µ A ( j ) ,µ Φ ( j ) − g Φ (0) † A (1) H A (1) H Φ (0) , (20)where j i ∈ { , } , i = 1 , , ,
4, and in Eqs. (18), (19), and (20), A ≡ A a T a , where T a arethe generators of the gauge group. The Feynman rules for this model are the same as in the14ve-dimensional case, with the addition of new interactions involving the adjoint scalars.These new rules are listed in Appendix A 2.
1. Dark matter candidates
In comparison to the five-dimensional model, each KK level in the six-dimensional modelalso includes an adjoint scalar for each gauge boson. In particular, the first-level adjointscalars B (1) H and W H are neutral, and therefore, they are possible DM candidates. However,being scalars, these candidates are not interesting for our purposes. It has been shown thatin the MUED version of the model, the adjoint scalar B (1) H is the LKP [45]. Thus, it isnecessary to go beyond the MUED scenario in order to possibly obtain a neutrino telescopesignal. We conclude that the interesting LKP candidates in the six-dimensional model arethe same as in five dimensions, i.e. , the B (1) and the W . IV. RESULTS
The elastic WIMP-proton scattering cross sections for B (1) and W LKPs in the five-dimensional model have been calculated in Refs. [9] and [36], respectively. The results are σ SD B (1) , p ≃ . · − pb (cid:18) . r q R (cid:19) (cid:18) m B (1) (cid:19) , (21) σ SD W , p ≃ . · − pb (cid:18) . r q L (cid:19) (cid:18) m W (cid:19) , (22)where r q L/R ≡ ( m q (1) R/L − m LKP ) /m LKP . In Eq. (21), the contributions from s - and t -channelexchange of left-handed KK quarks has been neglected. These contributions would increasethe cross section by about 3 %, while introducing an additional free parameter, r q L . On theother hand, in Eq. (22), there is no contribution from the right-handed KK quarks, since the W boson couples only to left-handed chiral fermions. In the six-dimensional model, thereare no tree-level contributions to these cross sections from the adjoint scalars, and hence,Eqs. (21) and (22) are valid also for that model.The relic DM density was calculated in Ref. [36] for both B (1) and W in the five-dimensional model. These calculations include coannihilations in all channels, and wereperformed for a relative mass splitting r q R/L between the LKP and the KK quarks in the15 inal state B (1) W r q uu dd νν l + l − hh ZZ W + W − B (1) and W LKPs. Except for the LKPand the KK quarks, the masses of the KK particles are assumed to be larger than the LKP massby 10 %, except for the W ± (1) , which are assumed to be degenerate with W . region from 0.01 to 0.5. For the B (1) calculations, the MUED spectrum was used for therest of the first-level KK particles, and for the W , it was assumed that the W ± (1) aredegenerate with W , that the KK gluons are heavier by 20 %, and that all other first-levelKK particles are heavier by 10 %. The conclusion is, that in order to obtain the measuredrelic density, the mass of the B (1) should be roughly in the range 500 GeV − r q R , and the mass of the W should be roughly inthe range 1800 GeV − B (1) , the preferred mass decreases with increasingmass splitting, while for the W , the opposite is true. Since the interactions in the six-dimensional model are the same as those in the five-dimensional one, the only possibledifference in the relic density would come from coannihilations involving adjoint scalars. Inthis work, we ignore that contribution, and assume that the relevant mass intervals for therespective candidates are approximately the same as for the five-dimensional model.Using the annihilation cross sections presented in Appendix B, we calculate the branchingratios into different final states, which are presented in Tab. I. The branching ratios intoa pair of Higgs bosons are small, and give only a small contribution to the muon-antimuonfluxes. Since this channel would introduce additional model dependence, we choose to neglectit in our calculations.We present our final results as fluxes of muons and anti-muons through an imaginary16lane in an Earth-based detector. The detector medium is assumed to be water, with thedensity 1 . / cm . The IceCube collaboration has placed limits on this flux [3] for thetwo annihilation channels W + W − and ¯ bb . The W + W − channel, which has been chosen torepresent models resulting in hard neutrino energy spectra, can be used to constrain ourmodels. The reason for this is, that in the case that the W is the LKP, the branchingratio into W + W − is around 90 %, whereas for the B (1) , the majority of the annihilationsresult in pairs of charged leptons, which also give a hard spectrum. In addition, the limitsfor the B (1) have been investigated by the IceCube collaboration [14], and it has been foundthat they differ from the W + W − limits by at most 10 %.In Figs. 2 and 3, we present the muon-antimuon fluxes for the case that the LKP is the B (1) and the W , respectively. We also include the IceCube limits, and we use the samemuon energy threshold as in Ref. [3], i.e. , E th µ = 1 GeV. We give the fluxes as functionsof the LKP mass, and present results for a number of different values for the relative masssplitting r q between the LKP and the first-level KK quarks. For all plots, we indicate themass range for the LKP giving the correct relic density. The masses of the rest of thefirst-level KK particles only affect the results through the branching ratios, and are lessimportant than the masses of the LKP and the first-level KK quarks. We assume that the W ± (1) are degenerate with the LKP and that all other particles are heavier than the LKPby 10 %.The fluxes for the five- and the six-dimensional models are equal for both LKP candidates.The reason for this is that, for our purposes, the contributions from the adjoint scalarsconstitute the only difference between the models, and at tree-level, these do not enter theannihilation processes or the scattering of LKPs on nucleons. Hence, each set of fluxes isvalid for both of our models.The fluxes decrease rapidly with increasing LKP mass, an effect which is partly due tothe behavior of the scattering cross sections, Eqs. (21) and (22), which fall off as m − , andpartly due to the kinematics of the capture of WIMPs in the Sun, which can be seen fromEq. (5) to be approximately proportional to m − σ sdWIMP , p . Taken together, this means thatthe capture rate falls off approximately as m − . The relative mass splitting r q mainly affectsthe fluxes through the scattering cross sections, Eqs. (21) and (22), which both behave as r − q . Except for this effect, the mass splitting also affects the branching ratios into differentfinal states, though this is a less important effect.17
00 1000 1500 2000 2500 3000m
LKP [GeV]10 -2 -1 M uon - a n ti m uon f l ux [ k m - y r - ] r q = 0.01r q = 0.05r q = 0.1r q = 0.5IceCube FIG. 2: The muon-antimuon flux in a detector at Earth as a function of the WIMP mass m WIMP forthe case of a B (1) LKP and muon energy threshold E th µ = 1 GeV. The dashed (blue), dash-dotted(red), double dash-dotted (green), and dotted (brown) curves represent the relative mass splittings r q = 0.01, 0.05, 0.1, and 0.5, respectively. Also plotted as the solid (black) curve is the IceCubelimit for the hard channel in Ref. [3]. For each curve, the LKP mass range giving the correct relicabundance is indicated by a thicker segment. Comparing the flux from B (1) to that from W , we find that the respective behaviorswith mass and KK quark mass splitting are similar, but that the absolute scale of the latterflux is always lower by a factor varying between 1 −
5. One of the reasons for this is thelower scattering cross section for the W , which is smaller than that for B (1) by a factorof approximately 5. In addition, the fluxes are affected by the different distribution of finalstates, which can be seen in Tab. I. While pairs of B (1) particles annihilate mainly intocharged leptons, with only a negligible fraction into gauge bosons, the W annihilates to W ± pairs with a very large fraction, and only a small fraction goes to charged leptons. Directneutrino production has a small but non-zero branching ratio for both LKP candidates. Thedifferent annihilation channels will result in two effects. First, the shape of the neutrinospectrum can be very different depending on the channel and a channel with a hard energyspectrum will be easier to distinguish from the background in a neutrino telescope. Thus,for hard spectra, a more stringent bound on the spin-dependent cross section can be put18
00 1000 1500 2000 2500 3000m
LKP [GeV]10 -8 -6 -4 -2 M uon - a n ti m uon f l ux [ k m - y r - ] r q = 0.01r q = 0.05r q = 0.1r q = 0.5IceCube FIG. 3: The muon-antimuon flux in a detector at Earth as a function of the WIMP mass m WIMP for the case of a W LKP and muon energy threshold E th µ = 1 GeV. The dashed (blue), dash-dotted (red), double dash-dotted (green), and dotted (brown) curves represent the relative masssplittings r q = 0.01, 0.05, 0.1, and 0.5, respectively. Also plotted as the solid (black) curve is theIceCube limit for the hard channel in Ref. [3]. For each curve, the LKP mass range giving thecorrect relic abundance is indicated by a thicker segment. given the same resulting muon flux. The second effect is that of neutrino oscillations,which can increase or decrease the muon flux depending on the annihilation channel. Sinceoscillations mix the fluxes of different neutrino flavors, a channel giving rise to a muonneutrino dominated spectrum will have its signal decreased by oscillations, and vice versa[16, 32].Our results should be compared to the results for the five-dimensional model that havepreviously been presented in the literature. In Ref. [17], the muon-antimuon flux was calcu-lated for the case that the B (1) is the LKP, and in Ref. [18], these results were updated toinclude the effects of neutrino oscillations, and the corresponding results for the case thatthe W is the LKP were also calculated. In Ref. [18], the results are presented as rates ofmuon events per effective detector area for a large volume detector such as IceCube, using adetector depth of 1 km. Due to this depth, the event rate per area is expected to be largerthan the flux through a plane in the detector, and therefore, those results are not directly19omparable to the results presented in Figs. 2 and 3. In order to enable such a comparison,we also calculate the event rates per area using DarkSUSY, and the event rates that weobtained are smaller by a factor of approximately 20 % - 30 %. This difference between theresults is expected, since we use DarkSUSY to calculate the capture rate, while in Ref. [18],the approximation given in Eq. (5) is used. As the event rate is proportional to the cap-ture rate, the difference between the event rates is explained by the difference between thecapture rates plotted in Fig. 1. It should also be noted that the branching ratios that weobtain for the W differ from those given in Ref. [18]. We obtain a larger branching ratiointo W + W − pairs, while the branching ratios into fermions are smaller. However, we havecalculated the muon flux using the branching ratios of Ref. [18], and have found that the dif-ference compared to our results due to this is not larger than about 10 %. Finally, the fluxeshave been calculated for the five-dimensional MUED model by the IceCube collaboration[14], and their results are also similar to our results. V. SUMMARY AND CONCLUSIONS
In this work, we have studied neutrinos from annihilations of KKDM in the Sun. Usingthe DarkSUSY and WimpSim packages, we have calculated the flux of neutrino-inducedmuons and antimuons in an Earth-based neutrino telescope, such as IceCube. We haveinvestigated KKDM in a five-dimensional model, based on the orbifold S / Z , and in asix-dimensional model, based on the chiral square T / Z . Rather than restricting ourselvesto the MUED models, where the LKP is uniquely determined, we have allowed for thepossibility that BLTs might affect the spectrum in such a way as to change the identity ofthe LKP. The possible DM candidates are then the first KK modes of the neutrinos, theneutral components of the Higgs field, and the B and W gauge bosons. In addition, in thesix-dimensional model, the adjoint scalars B (1) H and W H are possible candidates. Amongthese particles, neutrinos are already ruled out by direct detection experiments, while theinteractions of scalars with nucleons are too strongly constrained to allow for an observablesignature in neutrino telescopes. In five as well as six dimensions, only the B (1) and the W are left as interesting DM candidates.For a given LKP mass, the flux is somewhat lower for the case of the W as the LKPthan for the case of the B (1) . However, it is important to note that a B (1) LKP is not expected20o have the same mass as a W LKP. Relic density calculations show that the mass of the B (1) should lie in the range from about 500 GeV − W should befar more massive, in the range from about 1800 GeV − m − , this implies that the fluxes from the W are expected to bemuch smaller. In the relevant range, the flux is predicted to be smaller than 0 . − yr − .This means that, even assuming a perfect 1 km detector, no observable muon fluxes willbe generated in IceCube. We wish to emphasize this fact, which has previously not beendiscussed in the literature. In contrast, in the case that the B (1) is the LKP, IceCube willbe able to put constraints on the relevant parts of the parameter space.Another novel conclusion is that, since the fluxes are equal for the five- and the six-dimensional model for each LKP, it is not possible to distinguish these two models usingthis indirect detection method.In conclusion, in the case that the B (1) is the LKP, neutrino telescopes such as IceCubehave the ability to set constraints on both the five- and the six-dimensional model that wehave studied. On the other hand, if W is the LKP, the prospects for indirect detectionthrough neutrinos are much worse. Note also, that in this case, the scattering cross sectionis far below the sensitivity of any currently running or planned direct detection experiments. Acknowledgments
We would like to thank Joakim Edsj¨o and Matthias Danninger for useful discussions, andKyoungchul Kong for providing data from relic density calculations.This work was supported by the European Community through the European Commis-sions Marie Curie Actions Framework Programme 7 Intra-European Fellowship: NeutrinoEvolution [M.B.], the Swedish Research Council (Vetenskapsr˚adet), contract nos. 623-2007-8066 [M.B.], 621-2005-3588, and 621-2008-4210 [T.O.], and the Royal Swedish Academy ofSciences (KVA) [T.O.].
Appendix A: Feynman rules
In this appendix, we present the relevant Feynman rules for our models. We do not includeQCD interactions or Higgs boson self-interactions, since these do not enter our calculations.21he conventions for the components of the Higgs multiplet are given in Eq. (9). We usethe hypercharge normalization Q = T + Y . The U (1) and SU (2) coupling constants aredenoted g and g , respectively. All momenta are defined to point inwards. Double linesare used to represent non-zero KK modes, or KK modes with a general index n . We onlyinclude interactions between zero modes and first level KK modes.
1. Five dimensions
Gauge boson self-interactions (cid:1) p p p A (0) µ W − (1) λ W +(1) ν = e [ g µν ( p − p ) σ + g νσ ( p − p ) µ + g σµ ( p − p ) ν ] (cid:1) p p p Z (0) µ W − (1) λ W +(1) ν = c w g [ g µν ( p − p ) σ + g νσ ( p − p ) µ + g σµ ( p − p ) ν ] (cid:1) p p p W µ W − (1) λ W +(0) ν = g [ g µν ( p − p ) σ + g νσ ( p − p ) µ + g σµ ( p − p ) ν ] A ( n ) ν A ( n ) µ A ( n ) λ A ( n ) ρ = − i C (cid:0) g µν g λρ − g µλ g νρ − g µρ g νλ (cid:1) (A1)22 ertex C Vertex C A (0) µ A (0) ν W +(1) λ W − (1) ρ e W +(0) µ W − (0) ν W λ W ρ g Z (0) µ Z (0) ν W +(1) λ W − (1) ρ c w g A (0) µ W ν W ± (0) λ W ∓ (1) ρ s w g A (0) µ Z (0) ν W +(1) λ W − (1) ρ c w s w g W ± (0) µ W ± (0) ν W ∓ (1) λ W ∓ (1) ρ g Z (0) µ W ν W ± (0) λ W ∓ (1) ρ c w g W +(0) µ W +(1) ν W − (0) λ W − (1) ρ g TABLE II: The expressions for the coefficient C defined in Eq. (A1) for all possible interactionsof four gauge bosons. This diagram represents all possible interactions of four gauge bosons, and A stands for anygauge boson. The expressions for the coefficient C for the different vertices are given inTab. II. Gauge boson-scalar interactions
These interactions are independent of the KK indices of the interacting modes, exceptthat the vertices have to obey conservation of KK parity. Also, the basis (
A, Z ) is used for n = 0, while the basis ( B, W ) is used for n = 1. A ( n ) µ h ( n ) h ( n ) = i C P µ (A2)This diagram represents all interactions of one gauge boson and two scalars. For all possibleinteractions of this kind, the expressions for the coefficient C and the momentum P aregiven in Tab. III. 23 ertex C P Vertex C PA µ h + h − e p + − p − W µ h h g p − p B µ h + h − g p + − p − W − µ h + h g p + − p − B µ h h g p − p W − µ h + h g p − p + Z µ h + h − − s w c w g p + − p − W + µ h − h g p − − p W µ h + h − g p + − p − W + µ h − h g p − − p Z µ h h c w g p − p TABLE III: The expressions for the coefficient C and the momentum P defined in Eq. (A2) forall possible interactions of one gauge boson and two scalars. The interactions are independent ofthe KK indices of the interacting modes, except that the vertices have to obey conservation of KKparity. The basis ( A, Z ) is used for n = 0, while the basis ( B, W ) is used for n = 1, i.e. , A = A (0) and Z = Z (0) , while B = B (1) and W = W . A ( n ) ν A ( n ) µ h ( n ) h ( n ) = i C g µν (A3)This diagram represents all interactions involving KK modes of two gauge bosons and twoscalars. For all possible interactions of this kind, the expressions for the coefficient C aregiven in Tab. IV. Gauge boson-fermion interactions A (0) µ f (1) f (1) = i Q f eγ µ Z (0) µ f (1) f (1) = i g c w (cid:0) c w T f − s w Y f (cid:1) γ µ ertex C Vertex C Vertex C A µ A ν h + h − e W µ W ν h h g A µ W − ν h + h eg A µ B ν h + h − eg A µ Z ν h + h − − s w c w eg B µ W − ν h + h g g B µ B ν h + h − g A µ W ν h + h − eg A µ W − ν h + h − eg B µ B ν h h g B µ Z ν h + h − s w − s w c w g B µ W − ν h + h − g g B µ B ν h h g B µ W ν h + h − g g Z µ W + ν h − h eg Z µ Z ν h + h − (1 − s w ) c w g B µ Z ν h h − c w g g Z µ W + ν h − h eg Z µ W ν h + h − − s w c w g B µ W ν h h − g g Z µ W − ν h + h − i2 eg W µ W ν h + h − g B µ Z ν h h − c w g g Z µ W − ν h + h eg Z µ Z ν h h c w g B µ W ν h h − g g W + µ W − ν h + h − g Z µ W ν h h c w g A µ W + ν h − h − i2 eg W + µ W − ν h h g W µ W ν h h g B µ W + ν h − h − i2 g g W + µ W − ν h h g Z µ Z ν h h c w g A µ W + ν h − h − eg Z µ W ν h h c w g B µ W + ν h − h − g g TABLE IV: The expressions for the coefficient C defined in Eq. (A3) for all possible interactionsof two gauge bosons and two scalars. The interactions are independent of the KK indices of theinteracting modes, except that the vertices have to obey conservation of KK parity. The basis( A, Z ) is used for n = 0, while the basis ( B, W ) is used for n = 1, i.e. , A = A (0) and Z = Z (0) ,while B = B (1) and W = W . W ± (0) µ f (1) D f ′ (1) D = i g √ γ µ B (1) µ f (1) D f (0) = i g Y f γ µ P L (1) µ f (1) S f (0) = i g Y f γ µ P R W µ f (1) D f (0) = i g T f γ µ P L W ± (1) µ f ′ (1) D f (0) = i g √ γ µ P L
2. Six dimensions
The only difference between the interactions in the five- and the six-dimensional modelis that the six-dimensional model includes additional interactions due to adjoint scalars.Hence, we only present the Feynman rules for these additional interactions in this section.
Gauge boson-adjoint scalar interactions A (0) µ W − (1) H W +(1) H = e ( p + − p − ) µ Z (0) µ W − (1) H W +(1) H = c w e ( p + − p − ) µ W ± (0) µ W ∓ (1) H W H = ± g ( p ∓ − p ) µ ertex C Vertex C A µ A ν W + H W − H e A µ W ± ν W ∓ H W H − c w g A µ Z ν W + H W − H c w eg W + µ W − µ W H W H g A µ W ± ν W ∓ H W H − eg W ± µ W ± ν W ∓ H W ∓ H − g Z µ Z ν W + H W − H c w g W + µ W − ν W + H W − H g TABLE V: The expressions for the coefficient C defined in Eq. (A4) for all possible interactionsof two gauge bosons with two adjoint scalars. A (0) µ A (0) ν A (1) H A (1) H = C g µν (A4)This diagram represents all possible interactions of two gauge bosons with two adjointscalars, and A stands for any gauge boson. The expressions for the coefficient C for thedifferent vertices are given in Tab. V. Scalar-adjoint scalar interactions A (1) H A (1) H h (0) h (0) = i C (A5)This diagram represents all possible interactions of two adjoint scalars with two scalars, andhere A H stands for any adjoint scalar and h stands for any component of the Higgs field.The expressions for the coefficient C for the different vertices are given in Tab. VI.27 ertex C Vertex C B H B H h + h − − g B H W H h h g g B H B H h h − g B H W H h h g g B H B H h h − g W + H W − H h + h − − g B H W + H h − h g g W + H W − H h h − g B H W + H h − h g g W + H W − H h h − g B H W − H h + h − i2 g g W H W H h + h − − g B H W − H h + h g g W H W H h h − g B H W H h + h − − g g W H W H h h − g TABLE VI: The expressions for the coefficient C defined in Eq. (A5) for all possible interactionsof two adjoint scalars with two scalars. Adjoint scalar-fermion interactions B (1) H f (1) D/S f (0) = − i g Y f P L/R W H f (1) D f (0) = − i g P L W +(1) H f (1) D f ′ (0) = − i g √ P L Appendix B: Annihilation cross sections
In this appendix, we present the relevant WIMP-WIMP annihilation cross sections forthe models that we consider. The relevant quantity for our calculations is the cross sectiontimes relative velocity, σv , in the static limit s → m . We refer to this quantity as the28ross section. We let the masses of all first-level KK particles be free parameters. Since weignore EWSB effects, all the Higgs components appear as physical particles and the symbols W ± and Z refer only to the transversal components of those gauge bosons. To calculate thescattering rate into a pair of physical (massive) gauge bosons, the two contributions shouldbe added. For example, the cross section for the annihilation of two W bosons into a W + W − pair is given by the sum σ ( W W → h + h − ) + σ ( W W → W + W − ).Since we ignore all SM masses, we simplify the notation by suppressing the KK-indices, i.e. , m B (1) ≡ m B , etc . In order to further simplify the notation, we also introduce thefunctions f ( x ) = 1(1 + x ) , (B1) f ( x ) = 11 − x + 3 x (1 + x ) , (B2)and f ( x ) = 1 + 30 x + 35 x + 4 x + 6 x x (1 + x ) . (B3)The annihilation cross sections for the B (1) are σv ( B (1) B (1) → ¯ f f ) = 2 N c Y f g πm B f (cid:0) m f /m B (cid:1) , (B4)where N c is the number of colors of the fermion f , σv ( B (1) B (1) → h h ) = σv ( B (1) B (1) → h h )= 12 σv ( B (1) B (1) → h + h − ) = g πm B f (cid:0) m h /m B (cid:1) . In the case of degenerate first-level KK masses, the total annihilation cross section is σv ( B (1) B (1) ) tot = g πm B X f N fc Y f + 1576 ! ≃ .
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