Neutrinos at IceCube from Heavy Decaying Dark Matter
Brian Feldstein, Alexander Kusenko, Shigeki Matsumoto, Tsutomu T. Yanagida
aa r X i v : . [ h e p - ph ] J un IPMU-13-0069September 13, 2018
Neutrinos at IceCube from Heavy Decaying Dark Matter
Brian Feldstein ( a ) , Alexander Kusenko ( a,b ) , Shigeki Matsumoto ( a ) , and Tsutomu T. Yanagida ( a )( a ) Kavli Institute for the Physics and Mathematics of the Universe (WPI),University of Tokyo, Kashiwa, Chiba 277-8568, Japan ( b ) Department of Physics and Astronomy,University of California, Los Angeles, CA 90095-1547, USA
Abstract
A monochromatic line in the cosmic neutrino spectrum would be a smokinggun signature of dark matter. It is intriguing that the IceCube experiment hasrecently reported two PeV neutrino events with energies that may be equalup to experimental uncertainties, and which have a probability of being abackground fluctuation estimated to be less than a percent. Here we exploreprospects for these events to be the first indication of a monochromatic linesignal from dark matter. While measurable annihilation signatures wouldseem to be impossible at such energies, we discuss the dark matter quantumnumbers, effective operators, and lifetimes which could lead to an appropriatesignal from dark matter decays. We will show that the set of possible decayoperators is rather constrained, and will focus on several viable candidateswhich could explain the IceCube events; R-parity violating gravitinos, hiddensector gauge bosons, and singlet fermions in an extra dimension. In essentiallyall cases we find that a PeV neutrino line signal from dark matter would beaccompanied by a potentially observable continuum spectrum of neutrinosrising towards lower energies.
Introduction
The IceCube collaboration has very recently reported a detection of two neutrinoevents with energies of 1.1 PeV and 1.3 PeV in an energy range where no more than0.01 background events was expected from atmospheric neutrinos [1, 2, 3]. Theseare stated to be either electron neutrino charged current events, or neutral currentevents of any neutrino flavor. It is interesting that the two detected neutrinos havesuch similar energies, and indeed, most astrophysical sources are expected to pro-duce power-law spectra– in particular one might have expected to see additionalevents at around 6.8 PeV, where the detector sensitivity is enhanced by the Glashowresonance [4]. The data may thus suggest a peak, or falloff, in the neutrino spectrumaround 1 PeV. It is possible that such a spectrum could be produced by some astro-physical sources [5]–[7], including intergalactic interactions of cosmic rays producedby blazars [8]–[16], but these models rely on some assumptions about the propertiesand evolution of the sources, as well as the intergalactic magnetic fields.The IceCube observations raise a question of whether dark matter could be com-posed of relic particles whose decays or annihilations into neutrinos produce a featurein the neutrino spectrum at ∼ PeV energy. In this paper we will concentrate on thepossibility that this feature could actually be a monochromatic neutrino line. Sim-ilar to a line in the gamma ray spectrum, a line in the neutrino spectrum could beconsidered a “smoking gun” signature for dark matter. Such line-like neutrino sig-natures from dark matter have been considered before [17]–[19], but in this paper weconsider the possibility of obtaining such a signal at the PeV scale, where the darkmatter particle cannot be a simple thermal relic. As we will show, the possibilitiesfor obtaining a neutrino line signal from dark matter at such energies are highlyconstrained, but there are nevertheless various viable scenarios. We should notethat due to the low statistics in the present data, power law spectra from cascadeannihilations or decays of dark matter into neutrinos might also give reasonable fits.We limit ourselves here to the possibility of a line signature since this is the mostexciting case– with further data, a line signature would directly point towards a darkmatter explanation, whereas a power law signature might be difficult to disentanglefrom astrophysical sources.For dark matter with an annihilation cross section into monochromatic neutrinossaturating the unitarity limit, σ Ann ≤ π/ ( m v ), the event rate expected at aneutrino telescope of fiducial volume V and nucleon number density n N is of orderΓ Events ∼ V L MW n N σ N (cid:18) ρ DM m DM (cid:19) h σ Ann v i . , (1)1here we have taken the neutrino-nucleon scattering cross section to be σ N ∼ × − cm at E ν ≃ . n N ≃ n Ice ≃ × /cm . ρ DM , v , and L MW are the milky way dark matter density(taken near the Earth for the purpose of our estimate), the typical dark matterparticle velocity, and the rough linear dimension of our galaxy, where these are fixedto be 0.4 GeV/cm , 10 − , and 10 kpc, respectively. The fiducial volume V is set tobe 1 km , which is roughly the size of the IceCube detector. We see that obtaininga neutrino line signal from dark matter annihilations at the PeV scale is essentiallynot possible. In what follows we will therefore restrict ourselves to the possibility ofa signal from dark matter decays.For dark matter decays also, obtaining a neutrino line signal at the energies ofinterest here turns out to be challenging. Indeed, suppose one wishes to mediate anappropriate decay via a simple dimension 4 operator such as L ⊃ λ ¯ ψLH , where λ isa coupling constant, ψ is the dark matter particle, L is a lepton doublet, and H isthe Higgs doublet. Then the decay rate to neutrinos is Γ DM = λ π m DM . Similarly tothe annihilation case above, we may estimate the event rate at a neutrino detectorfor m DM ≃ . Events ∼ V L MW n N σ N ρ DM m DM Γ DM ∼ (cid:18) λ − (cid:19) / year . (2)Clearly an exceptionally tiny coupling is required to obtain an appropriate signal,and a certain amount of model building would appear necessary.We may also consider whether or not higher dimension operators, suppressedby some large mass scale, could give more naturally small event rates. For higherdimension operators, however, it is a nontrivial constraint that in order to obtaina line signal, the decay final state must be two-body. Indeed, for many interestingoperators, neutrinos appear in the gauge singlet combination LH , and althoughnaively this could lead to a neutrino decay with H replaced by its vacuum expectationvalue v , this tends not to be the dominant process due to the large dark matter massesunder consideration. For example, if one considers the operator L ⊃ φ ( LH ) / Λ fora scalar dark matter particle φ , and with Λ a heavy mass scale, then the square ofthe amplitude for a four-body decay with two neutrinos and two Higgses is largerthan that for a two-body neutrino decay by a factor of ∼ ( m DM /v ) . For heavy darkmatter masses, phase space suppressions for multi-body final states are not enoughto prevent the four-body decay from being by far dominant.In this paper we will comprehensively discuss effective operators which couldmediate the decays of heavy dark matter particles into monochromatic neutrino lines,2nd we will find that only a handful of operators are viable. Several of these standout as being particularly interesting, and we will discuss possible models for them indetail. These will include the cases of gravitino dark matter, with a mass motivatedby the recent 125 GeV Higgs discovery, a hidden gauge boson with an extremelysmall mixing with hypercharge, and a singlet fermion in an extra dimension. Ineach case we will discuss simple ways in which an appropriately long lifetime forthe dark matter particle may be obtained in order to explain the IceCube data. Inthe gravitino case, the decay operator may be naturally suppressed by the scalesof R-parity violation and lepton number violation. In the gauge boson case, thekinetic mixing with hypercharge may be suppressed by the scale of non-abeliangauge symmetry breaking in the hidden sector, as well as the breaking of grandunified symmetry in the visible sector. In the extra dimensional model, the requiredhighly suppressed coupling may be produced naturally by an exponentially smallwave-function factor.The outline of this paper is as follows: In section 2 we will review the nature ofthe PeV IceCube neutrino events, as well as discuss the lifetime and mass of darkmatter particles which may be able to explain them. In section 3 we will discuss ingeneral the effective operators which might be able to lead to an appropriate darkmatter decay. Models yielding some of these operators will be discussed in section4. An interesting conclusion of our analysis will be that in essentially all cases, amonochromatic neutrino line would be accompanied by an appreciable continuumspectrum of neutrinos rising towards lower energies. The prospects for detectingsuch a signature will be discussed in section 5. Before going on to discuss effective operators which could mediate the decays ofheavy dark matter particles into monochromatic neutrino lines, we summarize thesituation with the PeV neutrino events which have recently been reported by theIceCube collaboration. According to a plot in reference [2], the exposures at theenergies of the two events turn out to be 4 . × [m s sr] and 5 . × [m s sr]for the 1.1 and 1.3 PeV events, respectively, assuming that both the two events werecaused by electron neutrinos. It follows that the total flux may be estimated to
It is possible that one or both events could have been caused by neutral current interactionsof arbitrary flavor, but in such cases one would expect the event energies to be much more spreadout. F ≃ . × − [cm − s − sr − ] . (3)The observed neutrino event energies imply a mass for the dark matter particle ofabout 2.4 PeV, while the neutrino flux can be related to the lifetime for dark matterneutrino decays. When the mass of the decaying dark matter particle is assumed tobe 2.4 PeV, the predicted flux of line neutrinos is estimated to be E ν d F dE ν ≃ . × − N ν (cid:18) s τ DM (cid:19) δ ( E ν − m DM /
2) [GeV cm − s − sr − ] , (4)where E ν , m DM , τ DM , and N ν are the neutrino energy, the dark matter mass, itslifetime, and the number of neutrinos produced in each decay, respectively. Herethe NFW profile was used for the dark matter density in our galaxy, and we haveadopted profile parameters with a critical radius of r c = 20 kpc, and a density atthe solar-system of ρ ⊙ = 0.39 GeV/cm [21]. The total flux is then given by F ≃ . N ν × − (cid:18) s τ DM (cid:19) [cm − s − sr − ] . (5)By comparing this prediction with the flux in equation (3), we find that the thelifetime of the dark matter particle must have the following value in order to explainthe data: τ DM ≃ . N ν × s . (6)We have thus found that a decaying dark matter particle with a mass of about2.4 PeV and with a lifetime as given in equation (6) can explain the IceCube PeVneutrino events. Note that as a result of neutrino oscillations, all neutrino flavorswill contribute equally to the final signal, independent of the original flavor structureof the dark matter decays. Here we list all operators which might lead to a high energy monochromatic neutrinoline from dark matter decays. In table 1, we show possible dark matter candidates,defined by standard model SU(2) L and U(1) Y quantum numbers. We only list candi-dates which have a leading decay operator to two standard model particles, includingat least one neutrino. We exclude cases in which there is an alternate decay mode4 ase Spin SU(2) L U(1) Y Decay Operator Coefficient for IceCube Data1. 0 3 1 ¯ L c φL × −
2. 1/2 0 0 ¯ LH c ψ × −
3. 1/2 3 0 ¯ Lψ a τ a H c × −
4. 1/2 2 − / LF ψ × − (PeV − )5. 1/2 3 − Lψ a τ a H × −
6. 1 0 0 ¯
L /V L × −
7. 3/2 0 0 ( ¯
LiD µ H c ) γ ν γ µ ψ ν × − (PeV − ) Table 1:
Dark matter candidates and the decay operators that may lead to a mono-chromatic neutrino line signature. Here, L and H represent the SM lepton and Higgsdoublets, respectively, while the dark matter particle is labeled by φ , ψ , V µ or ψ µ , de-pending on whether it has spin 0, 1/2, 1, or 3/2. The notation F in case 4 denotes either B µν σ µν , ˜ B µν σ µν , W aµν τ a σ µν or ˜ W aµν τ a σ µν with B µν and W aµν being the field strength ten-sors of the SM U(1) Y and SU(2) L gauge fields. In the final column, we give the coefficientfor the operator required in order to explain the two anomalous neutrino events reportedby the IceCube collaboration, assuming a dark matter particle mass of . × GeV. through an operator of lower dimension, or which require decays to additionalnon-standard model particles. Note that case 1 is a slight exception to this rule,since a decay through a lower dimension operator H † φH c is possible, but we includethis case since the two Higgs decay may be forbidden by lepton number. In the finalcolumn of the table, we give the coefficient for the operator required to explain thetwo IceCube events based on the flux in the previous section.Cases in which the dark matter particle carries electric or color charge have notbeen included in the table. Electrically charged dark matter is severely constrainedby several observations and experiments, and is required to be heavier than about10 GeV [22, 23, 24], primarily by difficulties with structure formation. Colored darkmatter, similarly, must be heaver than about 10 GeV [25, 26], with the primaryconstraint coming from the possibility of overheating the Earth’s core. We have,on the other hand, included cases with non-zero hypercharge, which naively haveexcluded tree level Z -boson exchange signatures at dark matter direct detection ex-periments. These constraints can be avoided, however, if there is a higher-dimensionoperator which induces a splitting among the components of the dark matter field In cases in which the dark matter particle carries hypercharge (case 1, 4, and 5 in table 1), aDirac mass partner is required. We only include in the table operators of lowest dimension whenconsidering all operators allowed for either member of the Dirac pair.
5n such a way that the lightest state becomes a Majorana particle. Such a splittingis then required to be larger than the recoil energies produced at direct detectionexperiments. This is in fact what occurs for the case of higgsino dark matter insupersymmetric models– the mixing with Majorana gauginos causes the splitting.As discussed in the introduction, all decay operators we consider in table 1 containonly three fields. This was done in order to ensure that a monochromatic neutrinoline signal dominates over other decay modes. Operators requiring extra insertionsof Higgs vacuum expectation values to yield a monochromatic neutrino decay arenot allowed, since multi-body decays with extra Higgs particles would give overlylarge alternate cosmic ray signatures.
Our first example model comes from the operator listed as case 7 in table 1. Thisoperator requires the dark matter particle to have spin 3/2– namely, to be a gravitino.Here we will show that, in an R-parity violating context, it is straightforward toobtain a monochromatic neutrino line from gravitino dark matter decays, with amass and lifetime appropriate for explaining the PeV IceCube events.We begin by considering the mass of the gravitino. Both the ATLAS and CMScollaborations at the LHC have reported the discovery of a Higgs boson with massaround 126 GeV [27, 28]. The mass is somewhat heavier than one could expect in theminimal supersymmetric standard model (MSSM), but large radiative corrections tothe Higgs quartic coupling [29]-[32] can lead to a heavier lightest Higgs mass in theMSSM . When the left-right mixing of scalar top quarks is negligible and tan β ≃ O (1) PeV assuming simple gravitymediated SUSY breaking. Based on this observation, several concrete models havebeen proposed [34]-[36], which are attractive from the viewpoint of the SUSY-Flavorand CP problems because all dangerous flavor changing processes are suppressed byheavy sfermion masses. Gravitino dark matter with a PeV mass is, therefore, quiteconsistent with the observed Higgs mass under the assumption that the gravitino isthe lightest supersymmetric particle (LSP).Let us now consider the lifetime of the gravitino, whose decay must be induced bysome form of R-parity violation. Here we will consider a simple set of assumptions It is also possible that large supersymmetry breaking terms cause some squarks to form Higgs-like bound states, hence relaxing the MSSM limits on the mass of the lightest Higgs boson [33]. LH u , with a coefficient of order m / /M pl , where L is a leptondoublet of arbitrary flavor, m / is the gravitino mass, and M pl is the Planck scale.There are several assumptions required: The first is that the R-charges of all MSSMmatter fields are equal to 1, while those of the Higgses are equal to 0. Next we sup-pose that it is actually a Z subgroup of U (1) R which is a symmetry of the theory, andnot the full continuous U (1) R . Since the gravitino mass is a spurion for R-symmetrybreaking with R-charge 2, and Z R-symmetry requires that superpotential termshave R-charge equal to 2 mod 3, we find that the R-parity violating operator LH u appears with a coefficient of m / /M pl as promised [37]. Note that other R-parity vi-olating operators, U DD , LLE and
QLD all appear at order m / /M pl , and thereforealso with one suppression by the Planck scale. These will lead to continuum neutrinodecay spectra in addition to the monochromatic line (plus continuum) obtained fromthe R-parity violating operator LH u . Note, however that the decay rates from theseother R-parity violating operators will have additional phase space suppressions dueto extra final state particles, and are thus naively expected to be sub-dominant.As a result of the LH u operator, the lifetime of the gravitino to decay into aneutrino plus a Higgs or a neutrino plus a Z boson is estimated to be [38] τ / ≃ π ( M pl /m / ) m − / ≃ s , (7)where for illustration the sneutrino mass is assumed to be the same as the gravitinomass, namely 2.4 PeV, and M pl ≃ . × GeV. This lifetime of order 10 s istoo short to be consistent with the two IceCube PeV neutrino events.We now note, however, the discussion leading to equation (7) potentially missesan important point. Indeed, the operator LH u carries B − L charge −
1. In anycase, B − L symmetry must be broken to allow for Majorana masses for right handedneutrinos N R [39]-[41]. Therefore, as is standard we may introduce B and ¯ B fieldscarrying B − L charges +1 and −
1, respectively in order to break this symmetry. Nowone can write
W ⊃ N R N R h ¯ B i /M pl , which provides the Majorana mass term. If oneassumes M N ∼ GeV, the expectation value of the ¯ B field is h ¯ B i ∼ − M pl .Then the R-parity violating operator becomes of order W = ( m / h ¯ B i /M ) LH u . (8)Because of this modification, the lifetime of the gravitino (decaying into a neutrinoand either a Higgs boson or Z boson) is now about 10 times longer than the lifetime Note that B and ¯ B have R charge 0 according to our charge assignments, as required.
7n equation (7). To be more precise, the lifetime is then estimated to be τ / ≃ π ( M pl / h ¯ B i ) ( M pl /m / ) m − / ≃ s , (9)where the Majorana mass has been set to 10 GeV. This lifetime is fully consistentwith the one implied by the IceCube PeV neutrino flux in equation (6).Finally, let us discuss gravitino production in the early universe and the darkmatter abundance. The next-to-lightest-superpartner (NLSP) in this model willgenerically decay to the gravitino (plus its standard model partner) with a lifetimeof order M /m . With an NLSP mass at the PeV scale this is roughly of order10 − seconds. The NLSP decays thus do not disrupt the successful predictions of bigbang nucleosynthesis. On the other hand, the NLSP freeze-out abundance, whichwill then be converted into the gravitino relic abundance, will be too high by perhapsa factor of ∼ due to the large NLSP mass. If the reheating temperature is abovethe NLSP mass, entropy production by a factor of ∼ will thus be required todilute the dark matter abundance. If the reheating temperature is below the NLSPmass on the other hand, then it is possible to produce an appropriate gravitinoabundance through a small branching fraction of the inflaton into the gravitino. Another interesting possibility for a dark matter particle which could give a neutrinoline signature at IceCube comes from case 6 in table 1. Here we require a new gaugeboson V µ with a very small coupling ∼ − to at least one standard model lepton.What is very interesting about this case is that a coupling of this size may beobtained in a very simple and natural way. In particular, let us suppose thatthe visible sector is part of a standard grand unified theory, with a unification scaleof M GUT ∼ GeV. We may take a minimal SU(5) theory with GUT symmetrybroken by the vacuum expectation value of an adjoint scalar field Σ for illustration.Now, we consider the possibility that there is a completely hidden sector with anew non-abelian gauge symmetry, broken at the PeV scale. Let us take this gaugesymmetry to be SU(2) for simplicity, and suppose that it is completely broken by thevacuum expectation value (vev) of a scalar field Φ in the fundamental representation.Because both the visible and hidden sector gauge symmetries are fundamentallynon-abelian, dimension four kinetic mixing between their respective field strengthscoming from an operator ∼ F µν V µν is forbidden, where now F µν and V µν are takento be the SU (5) and hidden sector field strengths, respectively. However, after gauge For another model which may be used to give a similar resulting decay operator please see [42].
L ⊃ M Σ F µν Φ † V µν Φ . (10)This leads to a kinetic mixing between hypercharge and the lightest new gauge bosonof order h Φ i h Σ i M ∼ PeV M GUT M ∼ − . The hidden gauge boson will then obtain acoupling to the standard model leptons of the right order to explain the IceCubedata! Of course, we are assuming here that there are no light hidden sector particlesinto which the hidden gauge boson may rapidly decay.Note that in the absence of supersymmetry, this model introduces a new hierarchyproblem for the mass of the scalar field Φ. There is also a dangerous allowed quarticcoupling between Φ and the visible sector Higgs boson, which would lead to veryrapid V µ decays to Higgs bosons. There is, however, no obstacle to implementingthe model in a supersymmetric framework, and doing so can prevent the quadraticdivergence of the Φ mass, as well as forbid the Φ / Higgs quartic coupling. On theother hand, there is one additional type of dangerous operator which supersymmetrycannot forbid. This is an operator of the form1 M Φ † V µ Φ H † ∂ µ H, (11)which may be generated by a Kahler potential term M Φ † Φ H † H and which results in V µ decays to two Higgses. Note that the operator (10) which leads to the monochro-matic neutrino line is suppressed by an additional factor of M GUT M pl compared to (11).We thus require that the new operator be suppressed by a factor of about 100-1000beyond the naive estimate in (11) of M in order that the monochromatic neutrinoline is the dominant cosmic ray signature. Note that by gauge invariance (11) isnecessarily accompanied by a factor of the hidden gauge coupling constant, whilethe operator (10) may not be, so that a somewhat small gauge coupling may be ableto account for some or all of the required suppression.Similarly to the gravitino case, an appropriate dark matter relic abundance forthe hidden gauge boson may be obtained through non-thermal production. Forexample, we may suppose that the inflaton decays with an appropriate branchingfraction into the hidden sector, while also reheating the visible sector. Operators with different combinations of Φ and Φ † are similarly allowed. .3 A singlet fermion in an extra dimension Here we point out that in the context of an extra dimension, it is straightforwardto obtain a highly suppressed coupling such as that needed for a monochromaticneutrino line from dark matter decay. In particular, we may consider a scenario toproduce the operator of case 2 in table 1.Suppose that there exists an S /Z orbifolded fifth dimension separating twobranes. One of these branes, at y = 0, hosts all of the standard model fields, whilethe other, at y = ℓ , hosts a Majorana mass term for the right handed part of asinglet Dirac fermion Ψ which propagates in the bulk. In addition, Ψ has a massterm in the bulk and a Yukawa coupling to a lepton and the Higgs on the standardmodel brane. The zero mode of the right handed part of this bulk fermion may thenbe exponentially suppressed on the standard model brane, taking the form Ψ (0) R ( y, x ) = r me mℓ − √ M ∗ e my ψ (4 D ) R ( x ) ≡ εe my ψ ( x ) , (12)where M ∗ is the fundamental scale related to the four-dimensional Planck scale by M = M ∗ ℓ. Here we have written the action for the zero mode of Ψ as S = Z d x dy (cid:8) M ∗ (cid:0) i ¯Ψ (0) Γ A ∂ A Ψ (0) + m ¯Ψ (0) Ψ (0) (cid:1) + h δ ( ℓ − y ) M R ¯Ψ (0) cR Ψ (0) R + δ ( y ) λ ¯Ψ (0) R L H + h . c . io . (13)It is then straightforward to choose M R to be 2.4 PeV to explain the energies of theIceCube events, while due to exponential suppression ε may be taken to be of order10 − to yield an appropriate ψ dark matter lifetime even if λ is of order 1.As in the previous examples we have discussed, inflaton decays may yield anappropriate dark matter relic abundance. In this case there is also another interestingpossibility- namely, we may take the dark matter particle to carry gauged B − L symmetry (along with the standard model fermions, and two more right handedneutrino-like states for anomaly cancellation), so that the mass M R is only producedafter spontaneous B − L breaking. In this case, B − L interactions in the earlyuniverse may be used to produce the needed ψ relic density. The correct abundanceof dark matter can be attained if the reheat temperature T R is below the scale atwhich the U(1) B − L gauge symmetry is restored, and also below the temperature atwhich ψ particles would come into thermal equilibrium through gauge interactions.The population of ψ particles can be produced in processes ll → ψψ mediated by the The zero mode for the left handed part of Ψ is set to zero by choosing it to be odd under the Z orbifold symmetry as usual. B − L gauge boson. The resulting density to entropy ratio can be estimatedas in Ref. [44]: Y ψ ≡ n ψ s ∼ h σv i n f / ˜ H π g ∗ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T = T R ∼ − (cid:16) g ∗ (cid:17) (cid:18) M B − L GeV (cid:19) − (cid:18) T R × GeV (cid:19) , (14)where ˜ H is the Hubble parameter, g ∗ is the number of relativistic degrees of freedomat the time of reheating, h σv i ∼ T /M B − L is the production cross section, n f ∼ T isthe number density of standard model fermions in the plasma, and the first equalityis evaluated at reheating. Numerical solution of the Boltzmann equation gives avalue consistent with this result [45]. The dark matter mass density is thenΩ dark = 0 . × (cid:18) m ψ . × GeV (cid:19) (cid:18) Y ψ . × − (cid:19) . (15) In this paper we have catalogued all of the operators which may lead to decaysof PeV dark matter particles into monochromatic neutrino lines, and have givenexamples of models which may lead to appropriate decay rates to explain the twoanomalous events recently reported by the IceCube collaboration. Here we wouldlike to highlight an interesting feature of our analysis: For all of the operators thatwe have discussed in this paper one actually obtains also a lower energy continuumof cosmic ray neutrinos in addition to the monochromatic neutrino line. These areproduced since in every case there are necessarily alternate primary decay products–in addition to the primary neutrinos– which include Higgses, W-bosons, Z-bosonsand charged leptons, whose decays in turn produce neutrinos at lower energies. Forexample, in the hidden gauge boson model discussed in section 4.2, the vector darkmatter particle decays into all standard model particles carrying hypercharge. Inparticular, we will obtain decays to muons and tau leptons leading to a continuumneutrino signature. In the gravitino model of section 4.1 and the singlet fermionmodel in section 4.3, there are necessarily decays to W-boson + charged leptonwhich produce continuum neutrinos, in addition to those produced from the Higgsand Z-boson final state particles in the primary neutrino decays. For all cases wehave considered in this paper, these final states leading to continuum neutrino signalshave a similar branching fraction to the monochromatic neutrino events which havebeen our primary interest. We therefore have the important result that if the IceCube11igure 1:
Line and continuum neutrino signals from PeV dark matter decays.
PeV events are due to dark matter decays, then there should also be a continuum ofexcess lower energy events that can also be discovered in the sub-PeV region.
While the precise size and shape of this continuum is model-dependent, qualita-tively it always has a similar form. In figure 1 we show both line and continuumsignals assuming that the partial decay width of the continuum signal is twice thatof the line signal. This corresponds to the cases of either the gravitino model or thesinglet fermion model discussed in the text. The combined atmospheric neutrinobackground (including those from prompt decays) [2] is also shown for comparison.The continuum flux was calculated using the method adopted in reference [47], and isbased on the contribution from hadronic cascade decays of SM particles. In addition,we can also expect another contribution from leptonic decays, but this is not includedin the figure for simplicity. Note that in both the gravitino and singlet fermion caseswe also have direct decays into a W-boson plus a charged lepton l , with the flavorof the lepton being model dependent. Error introduced by our approximation ofdropping leptonic decays will be negligible for the cases of l = e or l = τ , while if Note that for the dominantly monochromatic neutrino spectra which we are considering in thispaper, one necessarily also obtains a continuum of soft neutrinos via electroweak bremsstrahlung,independent of any model building considerations. However, such bremsstrahlung induced neutrinoshave a spectrum which is too soft to be observable at IceCube. In particular, they only contributeto the continuum spectrum in an appreciable way at low energies where they are dwarfed by theatmospheric background. The decays of primary decay products thus give the most importantcontribution to the neutrino continuum. = µ , the continuum spectrum in the sub-PeV region will be somewhat enhanced.Let us note one special case in which the prediction of appreciable continuumneutrinos may be avoided– namely, one may consider the possibility that dark mat-ter decays produce neutrinos along with a new hidden sector particle, rather thanadditional standard model ones. For an interesting example, we may take a scalardark matter particle φ which decays into two hidden sector singlet fermions ψ . If ψ actually mixes with standard model neutrinos, then this will lead to decays of φ to ψ plus a neutrino, without an appreciable continuum neutrino signal. We will givedetails of a split seesaw model which realizes this scenario in appendix A.One might wonder in addition about the possibility of other types of cosmic raysignatures from decay products in our models, such as gamma rays or antiprotons.Unfortunately these are unlikely to be detectable in the foreseeable future. Thereason is the following: First, backgrounds of diffuse gamma rays and cosmic rayantiprotons have fluxes whose energy spectra are softer than 1 /E , due to theirproduction by cosmic-ray protons. On the other hand, gamma rays and antiprotonsfrom dark matter decays have fluxes whose energy spectra are harder than 1 /E (typically going as 1 /E ). This is because the signal spectra are essentially determinedby the fragmentation functions of dark matter decays and these must be harder than1 /E , otherwise their integrals over energy will diverge. As a result, the ratio of thesignal flux to the background flux becomes smaller at smaller energies. Moreover,both gamma rays and antiprotons are now observed at most up to 1 TeV in energy,making detection difficult. This situation may be clearly seen in reference [48] forthe gamma-ray case, where it was shown that near future gamma-ray observationscan cover dark matter lifetimes at most up to 10 seconds.Note Added: After this work was completed a paper by the IceCube collaborationdiscussing these events was released [43]. The event energies were adjusted slightlycompared to those used here, but there is no significant impact on our results. Acknowledgments
The authors thank F. Halzen and A. Ishihara for very helpful discussions about theIceCube events. A.K. was supported by DOE Grant DE-FG03-91ER40662. S.M. andT.T.Y. were supported by the Grant-in-Aid for Scientific research from the Ministryof Education, Science, Sports, and Culture (MEXT), Japan (No. 23740169 for S.M.and No. 22244021 for S.M. & T.T.Y.). This work was also supported by the WorldPremier International Research Center Initiative (WPI Initiative), MEXT, Japan.13
A Split Seesaw Model
Here we will discuss a model which is outside of the main line of argument in thetext for two reasons: the first is that there is an additional hidden sector particle inthe decay final state, and the second is that the decay may not be thought of as dueto a single effective operator, since it is a result of a mixing between two low massparticles. As mentioned in the discussion section, the basic idea is to have a scalardark matter particle φ decaying to two light hidden sector fermions ψ through a(highly suppressed) φψψ interaction, and also require that ψ has some mixing withstandard model neutrinos. We will now show that such a situation may be obtainedin a split seesaw framework in an extra dimension [44], in which the fermion ψ canliterally be a right handed neutrino in the sense that it leads to a seesaw neutrinomass in the standard model [39, 40, 41], even though ψ itself will be very light. The basic setup is similar to the one used in section 4.3. We again put standardmodel fields on a y = 0 brane in an extra dimension, with a Ψ field propagatingin the bulk as in that section, and with a zero mode wavefunction peaked on thebrane at y = ℓ . Again we also put a Yukawa coupling between Ψ L and H on thestandard model brane leading to an interaction ελψLH , where we are continuing touse the notation of section 4.3. A difference here however, is that we will now put theMajorana mass M R for Ψ on the standard model brane rather than the y = ℓ brane.As a result, ψ will obtain a highly suppressed mass of ε M R . An interesting result–and the original motivation for the split seesaw framework– is that a seesaw mass isthen obtained for a standard model neutrino which is interestingly independent ofthe wavefunction suppression factor ε , with m ν = λ v /M R .Finally, we introduce a new scalar field φ living on the standard model brane,with a Yukawa coupling to Ψ resulting in an interaction of size gε φψψ . φ will beour dark matter particle, and thus we choose its mass to be 2 . ψ and a neutrino ν of λvεM R , where v is the standard model Higgs vev. Whilethe primary decay mode of φ will be to two ψ particles, as a result of the mixing, φ may also decay to ψν with a lifetime of order (cid:16) − gε (cid:17) × s, where we havetaken the mixing angle to be of order 1. Obtaining an appropriate neutrino mass m ν with λ also of order 1 requires M R to be of order 10 GeV as usual. Finallylet us point out that, as was discussed in section 4.3, an interesting possibility for
We of course need more than one non-zero neutrino mass in the standard model sector, andthus require more than one right handed neutrino. This will not concern us here as a single ψ fieldis sufficient for our present purpose. φ carries gauged U (1) B − L charge, so that high temperature B − L interactions produce the relic φ particles. The estimate for the resulting relic density is analogous to that in section4.3. References [1] A. Ishihara, talk at the Institute for Cosmic Ray Research neutrino workshop,University of Tokyo March 15, 2013, Kashiwanoha, Japan.[2] A. Ishihara, talk at the
Neutrino 2012 conference, June, 2012, Kyoto, Japan.[3] F. Halzen, talk at
Neutrino Oscillations Workshop (NOW– 2012) , September9–16, 2012, Otranto, Lecce, Italy.[4] A. Bhattacharya, R. Gandhi, W. Rodejohann and A. Watanabe, JCAP (2011) 017, arXiv:1108.3163.[5] I. Cholis and D. Hooper, arXiv:1211.1974.[6] M. D. Kistler, T. Stanev and H. Yuksel, arXiv:1301.1703.[7] R. -Y. Liu and X. -Y. Wang, arXiv:1212.1260.[8] W. Essey and A. Kusenko, Astropart. Phys. , 81 (2010), arXiv:0905.1162.[9] W. Essey, O. Kalashev, A. Kusenko and J. F. Beacom, Astrophys. J. , 51(2011), arXiv:1011.6340.[10] W. Essey and A. Kusenko, Astrophys. J. , L11 (2012), arXiv:1111.0815.[11] K. Murase, C. D. Dermer, H. Takami and G. Migliori, Astrophys. J. , 63(2012), arXiv:1107.5576.[12] S. Razzaque, C. D. Dermer and J. D. Finke, Astrophys. J. , 196 (2012),arXiv:1110.0853.[13] A. Prosekin, W. Essey, A. Kusenko and F. Aharonian, Astrophys. J. , 183(2012), arXiv:1203.3787.[14] F. Aharonian, W. Essey, A. Kusenko and A. Prosekin, arXiv:1206.6715. In the original split-seesaw papers ψ was a dark matter candidate with a keV mass whichcould explain pulsar kicks [49, 50]. However, the keV mass scale was not a definitive predictionof the model, which essentially creates a “democracy of scales”: Majorana masses of differentorders of magnitude can arise from the exponential suppression employed in this model. Forexample, two degenerate right-handed neutrinos with GeV masses as used in the ν MSM [51] canbe accommodated in this scenario as well. (2010) 141102, arXiv:0912.3976.[16] O. E. Kalashev, A. Kusenko, W. Essey and , arXiv:1303.0300.[17] R. Allahverdi, S. Bornhauser, B. Dutta and K. Richardson-McDaniel, Phys.Rev. D , 055026 (2009), arXiv:0907.1486.[18] M. Blennow, H. Melbeus and T. Ohlsson, JCAP , 018 (2010),arXiv:0910.1588.[19] M. Lindner, A. Merle and V. Niro, Phys. Rev. D , 123529 (2010),arXiv:1005.3116.[20] R. Gandhi, C. Quigg, M. H. Reno and I. Sarcevic, Phys. Rev. D , 093009(1998), hep-ph/9807264.[21] A. Esmaili, A. Ibarra and O. L. G. Peres, JCAP , 034 (2012),arXiv:1205.5281.[22] A. Gould, B. T. Draine, R. W. Romani and S. Nussinov, Phys. Lett. B ,337 (1990).[23] K. Kohri and T. Takahashi, Phys. Lett. B , 337 (2010), arXiv:0909.4610.[24] A. Kamada, N. Yoshida, K. Kohri and T. Takahashi, arXiv:1301.2744 [astro-ph.CO].[25] M. Kawasaki, H. Murayama and T. Yanagida, Prog. Theor. Phys. , 685(1992).[26] G. D. Mack, J. F. Beacom and G. Bertone, Phys. Rev. D , 043523 (2007),arXiv:0705.4298.[27] https://twiki.cern.ch/twiki/bin/view/AtlasPublic .[28] https://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResults .[29] Y. Okada, M. Yamaguchi and T. Yanagida, Prog. Theor. Phys. , 1 (1991).[30] J. R. Ellis, G. Ridolfi and F. Zwirner, Phys. Lett. B , 83 (1991).[31] H. E. Haber and R. Hempfling, Phys. Rev. Lett. , 1815 (1991).[32] A. Arvanitaki, N. Craig, S. Dimopoulos and G. Villadoro, arXiv:1210.0555 [hep-ph].[33] J. M. Cornwall, A. Kusenko, L. Pearce and R. D. Peccei, Phys. Lett. B ,951 (2013), arXiv:1210.6433.[34] M. Ibe and T. T. Yanagida, Phys. Lett. B , 374 (2012), arXiv:1112.2462.1635] M. Ibe, S. Matsumoto and T. T. Yanagida, Phys. Rev. D , 095011 (2012),arXiv:1202.2253.[36] See also, N. Arkani-Hamed, SavasFest: Celebration of the Life and Work ofSavas Dimopoulos (2012), .[37] S. Shirai, F. Takahashi and T. T. Yanagida, Phys. Lett. B , 485 (2009),arXiv:0905.0388.[38] K. Ishiwata, S. Matsumoto and T. Moroi, Phys. Rev. D , 063505 (2008),arXiv:0805.1133.[39] T. Yangida, in Proceedings of the “Workshop on the Unified Theory and theBaryon Number in the Universe” , Tsukuba, Japan, Feb. 13-14, 1979, edited byO. Sawada and A. Sugamoto, KEK report KEK-79-18, p. 95, and ”HorizontalSymmetry And Masses Of Neutrinos” , Prog. Theor. Phys. (1980) 1103.[40] M. Gell-Mann, P. Ramond and R. Slansky, in ”Supergravity” (North-Holland,Amsterdam, 1979) eds . D. Z. Freedom and P. van Nieuwenhuizen, Print-80-0576(CERN).[41] See also, P. Minkowski, Phys. Lett. B , 421 (1977).[42] C. -R. Chen, F. Takahashi and T. T. Yanagida, Phys. Lett. B , 255 (2009),arXiv:0811.0477.[43] M. G. Aartsen et al. [IceCube Collaboration], arXiv:1304.5356 [astro-ph.HE].[44] A. Kusenko, F. Takahashi and T. T. Yanagida, Phys. Lett. B , 144 (2010),arXiv:1006.1731.[45] S. Khalil and O. Seto, JCAP , 024 (2008), arXiv:0804.0336; G. Gelmini,S. Palomares-Ruiz and S. Pascoli, Phys. Rev. Lett. , 081302 (2004),astro-ph/0403323; G. Gelmini, E. Osoba, S. Palomares-Ruiz and S. Pascoli,JCAP , 029 (2008), arXiv:0803.2735.[46] V. Berezinsky, M. Kachelriess and S. Ostapchenko, Phys. Rev. Lett. , 171802(2002) [hep-ph/0205218].[47] M. Birkel and S. Sarkar, Astropart. Phys. , 297 (1998), hep-ph/9804285.[48] K. Murase and J. F. Beacom, “Constraining Very Heavy Dark Matter UsingDiffuse Backgrounds of Neutrinos and Cascaded Gamma Rays,” JCAP ,043 (2012), arXiv:1206.2595.[49] A. Kusenko, G. Segre and , Phys. Rev. Lett. , 4872 (1996) [hep-ph/9606428].1750] A. Kusenko, Phys. Rept. , 1 (2009), arXiv:0906.2968.[51] T. Asaka, S. Blanchet and M. Shaposhnikov, Phys. Lett. B631