# A new twist on excited dark matter: implications for INTEGRAL, PAMELA/ATIC/PPB-BETS, DAMA

###### Abstract

We show that the 511 keV gamma ray excess observed by INTEGRAL/SPI can be more robustly explained by exciting dark matter (DM) at the center of the galaxy, if there is a peculiar spectrum of DM states , and , with masses GeV, , and . The small mass splitting should be keV. In addition, we require at least two new gauge bosons (preferably three), with masses MeV. With this spectrum, is stable, but can be excited to by low-velocity DM scatterings near the galactic center, which are Sommerfeld-enhanced by two of the 100 MeV gauge boson exchanges. The excited state decays to and nonrelativistic , mediated by the third gauge boson, which mixes with the photon and . Although such a small 100 keV splitting has been independently proposed for explaining the DAMA annual modulation through the inelastic DM mechanism, the need for stability of (and hence seqestering it from the Standard Model) implies that our scenario cannot account for the DAMA signal. It can however address the PAMELA/ATIC positron excess via DM annihilation in the galaxy, and it offers the possibility of a sharper feature in the ATIC spectrum relative to previously proposed models. The data are consistent with three new gauge bosons, whose couplings fit naturally into a broken SU(2) gauge theory where the DM is a triplet of the SU(2). We propose a simple model in which the SU(2) is broken by new Higgs triplet and 5-plet VEV’s, giving rise to the right spectrum of DM, and mixing of one of the new gauge bosons with the photon and boson. A coupling of the DM to a heavy may also be necessary to get the right relic density and PAMELA/ATIC signals.

###### pacs:

98.80.Cq, 98.70.Rc, 95.35.+d, 12.60Cn## I Introduction

Dramatic developments in observational astronomy have started to alter our picture of dark matter (DM); instead of being a single state, observations have indirectly suggested that DM could be a multiplet with small mass splittings. In ref. AH () it was argued that such a scenario can explain not only the positron/electron excess recently indicated in the 10-100 GeV region by the PAMELA pamela () and (to some extent) HEAT heat () experiments, and a similar one in the 500-800 GeV region seen by ATIC atic () and PPB-BETS ppb-bets (), but also the 511 keV gamma rays observed by INTEGRAL/SPI integral (); integral2 (), the WMAP haze haze () and the annual modulation observed by DAMA/LIBRA dama (). The connections were further explored in cholis (). The common link between them is that they can be explained in terms of DM undergoing enhanced scattering and subsequent annihilation into light bosons which decay to secluded (), or else exciting a DM state with a small mass splitting above the ground state, which might decay back to the ground state and if the splitting is greater than . In the case of DAMA, the mass splitting provides the kinematics which would enable DAMA to be more sensitive than other experiments to a necessarily inelastic collision. The ideas of excited dark matter (XDM) FW () and inelastic dark matter (IDM) iDM () were proposed before ref. AH (), but the latter took the step of trying to unify them into an appealing theoretical framework, and to use it to also explain the excess positron/electron results of PAMELA/ATIC.

The unified description of dark matter has one shortcoming, however.
In ref. PR (), it was shown that the XDM mechanism falls short
of being able to reproduce the experimental observation by nearly three
orders of magnitude, even if the galactic DM scattering cross section
saturates the unitarity limit in the s-wave
contribution. The main loophole for circumventing this conclusion
was to hope that higher- partial waves could increase by a factor of 300. Such an enhancement was argued to be unlikely in ref. PR (), and we will show that this argument is borne out in the
class of models proposed by ref. AH (). Thus the XDM
explanation of the 511 keV excess remains unrealized.^{1}^{1}1 This
conclusion depends on the small error bars for the necessary value of
estimated by ref. ldm2 (). However there
seems to be a larger uncertainty in the DM density at the center of
the galaxy, . Since positron production in the XDM scenario
scales like , the original XDM idea with
might be salvaged if is greater by a factor of 17 (even more if
the unitarity bound is not saturated) than in the model found to be
preferred in the best fit to the INTEGRAL data by ref. ldm2 (). It would be worthwhile, though beyond the scope of the
present work, to further investigate this point.

We note that the 511 keV anomaly is not just the finding of the INTEGRAL experiment, but it was first observed in 1972, and has been seen in four subsequent balloon- and satellite-borne experiments integral2 (). The observation is thus quite credible, and so far is lacking any highly convincing astrophysical or particle physics explanation, although attempts have been made using positron emission from low-mass x-ray binaries (LMXB’s) weid (), or annihilation of light MeV-scale dark matter (see for example ldm (); ldm2 ()) as well as decaying relics PR (); decays (). The LMXB hypothesis is predicated on a supposed correlation between the asymmetry in the disk component of the 511 keV gamma rays and the distribution of bright LMXB’s, but this has been criticized on several grounds in ref. silk () (although the discrepancies might be ameliorated if positrons produced in the disk can be transported to the bulge before annihilating lmxb ()). Therefore it is still interesting to find a technically natural particle physics explanation for the 511 keV line, regardless of the other experimental anomalies. This was the primary motivation for the present work. The fact that our positive finding for the INTEGRAL anomaly is consistent with the general framework outlined in ref. AH (), for also explaining the other experiments (apart from DAMA), heightens its interest.

Our new twist for making XDM viable is to have a mass splitting which is much smaller than between the middle DM state and the heaviest one, as illustrated in figure 1. These are assumed to be Majorana fermions, . We will design the model so that cannot decay directly to the ground state ; thus both and are stable. undergoes Sommerfeld-enhanced scattering AH (); Hisano (); LS (); MR () at the galactic center (where the DM density is highest) through multiple exchange of a light gauge boson with off-diagonal coupling , as envisioned in ref. AH (). The produced states subsequently decay through another vector which mixes with the photon to produce . These processes are shown in figure 2. The key ingredient which makes the scattering efficient enough is the small mass splitting . Previous attempts to implement the XDM mechanism have failed because a larger splitting was assumed, and this makes the excitation rate too small. The figure of merit is the ratio of the predicted rate of production to that observed, in the th partial wave, assuming the unitarity bound is saturated PR ():

(1) |

It is suppressed by the Maxwell-Boltzmann distribution at the threshold for production of the excited state. Eq. (1) was predicated on the assumption that the kinetic energy of each DM particle must be sufficient to produce one electron or positron. Since the characteristic DM velocity is fixed, must be sufficiently large to avoid the Boltzmann suppression, but the rate also scales like . Even with the optimal value of , one would need to more than double the estimated value of to make . Such a large change seems to be well outside the range of uncertainty in the current understanding of the DM velocity distribution. However in our scenario, is replaced by the smaller , which can significantly ease this tension. We will show that can be 1 as required without changing , and keeping GeV as desired for PAMELA/ATIC, if keV. It is intriguing that the same splitting has been advocated previously to account for the DAMA signal. However we will show that the need for stability of means that it cannot interact with baryons or leptons at detectable levels; thus our proposal does not seem to be compatible with the IDM explanation for DAMA.

In the remainder of the paper we will give details of the computation of the excitation rate (section II), tighten the case against the large mass gap scenario in section III, and show how a smaller value keV can improve the situation in section IV. In section V we build a simple particle physics model of DM which can accommodate our findings, and also address the PAMELA/ATIC/PPB-BETS observations. Section VI gives a brief account of the cosmological implications of the model. Conclusions are given in section VII.

## Ii Methodology

To obtain the desired result for the 511 keV signal, it is important to nearly saturate the unitarity bound in at least one partial wave (or to have significant scattering up to high values of , but we will show that this does not seem to be possible in the present context). The Sommerfeld enhancement which can occur at low DM velocities is crucial for getting such strong scattering. We follow the quantum mechanical treatment of appendix A.4 of ref. AH () to compute this effect.

Because the gauge coupling is assumed to be off-diagonal, we have two states , , whose interaction Hamiltonian has the matrix form

(2) |

where and is the mass of .
The wave function for the two-state system (with components labeled by
index ) in the CM frame
is , where is
the initial momentum. Defining ,
the Schrödinger equation is^{2}^{2}2For the numerical solution it is useful to rescale and define the dimensionless variables
, , , so that the Schrödinger
equation takes the form , with the dimensionless potential . We must have
for the initial state to have enough energy to
produce the heavier final state.

(3) |

The equation is solved by the shooting method, where near for some complex number , which is then adjusted so that there are only outgoing and not incoming waves in as .

To extract the scattering amplitudes, we decompose the numerical solution into incoming and outgoing waves, , and Partial wave unitarity implies the conservation of flux, where , which we use as a check on our numerics. The fraction of incoming states which gets converted to the final state is thus

(4) |

in the th partial wave. This must be integrated with the Maxwell-Boltzmann distribution (or some more sophisticated distribution function, as we discuss below) to find the thermally averaged cross section . Doing so modifies the unitarity-saturating estimate (1) to read

(5) |

where , is the threshold velocity for production and km/s is the escape velocity at kpc, the outer edge of the region where INTEGRAL sees excess emission FW (). Since km/s, the error in extending the upper limit of integration to is small.

## Iii Constraints on large model

We have manually scanned the parameter space of the model to try to
maximize the fraction of excited state particles, , first
starting with the original class of models with the “large” mass
gap, .^{3}^{3}3In terms of ref. AH (), this
actually requires looking at the scattering since would have
and be even more suppressed. Ref. PR ()
has by virtue of charged intermediate states,
which allows each incoming to be excited by only rather
than . The goal here was to see if any parameters
could be found such that high partial waves could contribute, thus
overcoming the small prefactor
at the optimal mass and km/s PR ().
We considered the limit in which the gauge boson mass
can be neglected. Otherwise the range of the interaction
is reduced; this can only decrease the contribution from higher-
partial waves, which correspond to scattering at large impact
parameter. In that case, the relevant dimensionless
parameter turns out to be

(6) |

(see footnote 2.) For , the -wave dominates the cross-section, while for , a range of partial waves contribute significantly. However, in the latter case even the largest contributions fall short of the needed unitarity limit, so the total cross section is not actually enhanced. Working at the optimal mass , we have computed

(7) |

[see eq. (5)] as a function of . This quantity would take the limiting value for all the partial waves which reach the unitarity limit . At the optimal mass , , and ref. PR () shows that should have the value in order to match the INTEGRAL observations. Fig. 3 shows that in the actual model, reaches a maximum value of near , far below what is needed.

We have thus established that the high- loophole for the large scenario does not work. Another way of enhancing the effect would be to take advantage of its strong dependence on . One would need to boost by the factor , giving km/s instead of 180 km/s. On the other hand, more sophisticated estimates of the dark matter distribution indicate that is smaller than the fiducial value, rather than larger, in the center of the galaxy ldm2 (); PR (). Different models of the DM distribution function give radially-dependent average velocities which decrease toward . Therefore using the constant value for which best describes the bulk of the galaxy already overestimates the efficiency of DM excitation near the center, and the large possibility seems to be ruled out (see however footnote 1).

## Iv The case of small

Now we turn to the main point, that smaller values of the DM mass splitting keV can overcome the problem of too small a signal, without any need for increasing the DM velocity . Redoing the analysis of ref. PR () for general , one finds that for the optimal DM mass, is enhanced by an extra factor of . We can therefore achieve the desired effect if unitarity is nearly saturated only in the -wave (or other low- contributions), with keV. This estimate applies for the optimal mass GeV. However we can make the mechanism work at larger , as desired for getting the unified explanation of the PAMELA/ATIC/PPB-BETS observations AH (), by making only moderately smaller, as we will now show.

We have done a preliminary exploration of the parameter space, to see what can be achieved in the concrete framework at hand. We defer a more comprehensive analysis to the future; here we will just present a working example. To get a large enough effect, it is important to vary the mass of the exchanged gauge boson . Physically, this is due to resonant scattering when a bound state of nearly zero energy forms AH (); Hisano (); MR (). This effect is only possible for a finite-range potential such as the Yukawa type. Generally, we find enhanced scattering for larger values of , which is not surprising since the interaction strength is , and there is an optimal (though not sharply peaked) value of the dimensionless parameter

(8) |

giving the resonant effect.

As an example, we present the case where , . We consider the quantity rather than of eq. (7), where . In the large scenario, the best one could do was to optimize such that , but for small , we can obtain much larger results at smaller values of . In contrast to the large case, we can hold fixed at a a value which is larger than optimum explaining for the 511 keV line, but more interesting for simultaneously explaining the other potentially DM-related anomalies. From eq. (5) it follows that at GeV and , one only needs to explain the INTEGRAL observations.

The enhancement factor is plotted as a function in figure 4, where the convergence of the successive partial wave contributions is shown. The needed value of can be obtained for . For GeV and , this implies the mass splitting keV. From and , we infer that the fine-structure constant of the new gauge coupling is , and the mass of the gauge boson is MeV.

So far, we have assumed that the initial state has a particular number density, namely that which was used in the analysis of ldm2 () to determine the cross section cm needed to explain the INTEGRAL anomaly. However, with three nearly degenerate DM states, one expects the density to be this value, given that decayed into and not . The signal is proportional to the integral of along our line of sight, so this would require the cross section to be nine times higher than we have assumed. The additional suppression can be counteracted if the DM density in the central region kpc is three times higher than assumed in the analysis of ref. ldm2 (). Even apart from any uncertainties in the shape of , we note that it is normalized to the local energy density kpc) = GeV/cm, which is estimated to be uncertain by a factor of 2 in the upward direction KK (). If this uncertainty works in our favor, we only need an additional factor of enhancement in the central region. This is a modest shortfall, since in the four models considered by ref. ldm2 (), kpc) varies by a factor of 27. Thus the smaller density does not seem to pose a serious problem for the model. On the other hand, the missing factor of 300 in the large case looks more daunting.

We have also neglected the probable -dependence of discussed at the end of the previous section, which also tends to reduce the predicted signal. There is not yet a consensus on the precise form of , but if it proves to give a significant reduction, this can be compensated to some extent by taking more optimal (smaller) values of the DM mass. It is also possible that more favorable examples at large exist, which would make it interesting to perform a wider and more systematic search of the model parameter space than we have been able to do so far.

## V ATIC and model building

The simplest and most theoretically appealing way of getting three DM states and several gauge bosons is to assume that the gauge symmetry is SU(2) and that transforms as a triplet. We will now explore the consequences of this hypothesis, with particular attention to its implications for the ATIC/PPB-BETS excess at 300-800 GeV. Of course we will also maintain all ingredients needed for our successful explanation of the 511 keV anomaly.

Already to explain INTEGRAL, we required two gauge bosons, which couples to to facilitate , and which mixes with the photon and mediates the decay . However, with only these states, it is difficult to produce a rather sharply-peaked excess of high-energy leptons on top of a lower energy continuum, which is suggested by the ATIC data and which we have reproduced in figure 5. (The PPB-BETS data are consistent, but not so strongly suggestive of the peak.) The process specified by ref. AH () in this regard is shown in fig. 6, where the final state bosons subsequently decay to . The annihilation initially produces back-to-back , each carrying energy , which should give a leptonic spectrum that is rather uniformly distributed in energy. It would be desirable to have an additional channel which produces a single nearly monoenergetic pair, each lepton having energy . Such a spike would be significantly broadened by Coulomb scattering of the primary particles in the galactic medium gil ()-entropic (), possibly giving the peak-like shape in the ATIC data. We will come back to this issue in the next section.

Let us show how a simple SU(2) model could account for the observations. If is a real triplet, its gauge interactions are

(9) |

Letting , then in the previous notation, , , . A simple way to get kinetic mixing of the photon with is by including a Higgs triplet in the dark sector, and the dimension-5 operator

(10) |

where is the field strength of the SM weak hypercharge gauge field. If we assume that only gets a VEV, this generates the required mixing of with the photon and the boson. It is straightforward to work out the transformation that diagonalizes the kinetic term of the , (photon) and boson. If is the mass of the , we find that the flavor states are related to the mass eigenstates by

(11) |

where and is the Weinberg angle. It is important for this class of models that does not mix with the photon at [in fact neither does it mix at higher orders], because otherwise would acquire a coupling to which would be just as strong as the coupling of to . In that case, INTEGRAL would observe a narrow line from at energy 1 MeV, in addition to that at 511 keV, but this of course has not been seen. The - mixing implies a decay rate of for , and a corresponding lifetime of order s if . For and GeV as we will find below, the scale is TeV.

With the breaking pattern needed in eq. (10), the kinetic term of the triplet only gives mass terms for and ,

(12) |

(where is the dark SU(2) gauge coupling), so we must also include symmetry breaking from another new Higgs field. One simple possibility is a symmetric traceless tensor, , the 5D representation, whose VEV is only in the or components. In fact, a global rotation around the direction (which is the subgroup of SU(2) left unbroken by our choice of as the triplet VEV) can conveniently put the 5-plet VEV in the diagonal components alone. The group generators in the 5D representation can be written as . The VEV generates a mass term proportional to

(13) |

The resulting gauge boson mass spectrum is then and . To get the desired radiative mass splittings of the ’s below, we will need to assume that , so that .

An important property of the interaction (9) (and indeed all the interactions in our model) is that it preserves the symmetry , , which is needed to keep stable, if we assume that and are also charged under the . The discrete symmetry is thus unbroken by the VEV of . Note that the nonabelian cubic and quartic interactions of the gauge bosons have the schematic form and , which also respect the symmetry. A notable consequence of the symmetry is that do not acquire any couplings to SM matter which would allow them to decay into , nor to cause nuclear recoil in direct DM detection experiments. Therefore even though the excitation has the right kinematics for the IDM explanation of DAMA, the putatively exchanged - boson cannot interact with the detector (nor can work, both for this reason and because of the larger mass splitting). The remaining excitation is possible from the point of view of the interactions of the exchanged boson, but the mass splitting is too large for it to proceed at a detectable rate.

Next we consider the spectrum of the DM triplet.
The tree level mass
gets split by cheung () by the diagrams of fig. 7.^{4}^{4}4It is important to put the external DM states on shell
to get the correct result. We are assuming that since, as we will show,
this is what gives the desired
DM spectrum. Figure 7 indicates that
the radiative correction gives
a mass which is larger than that of
and by , but
and
but remain degenerate with each other. We can break the remaining
degeneracy using the same VEV of the 5-plet as in eq. (13) by including the Yukawa
interaction

(14) |

This splits by . The resulting spectrum has the form

(15) |

We should choose to allow the decay of , and to get the small splitting . Using the value suggested by our analysis of the INTEGRAL signal, and assuming that for example, we get the required spectrum with GeV, GeV, .

We can easily construct a potential for the Higgs sector which leads to the desired symmetry breaking pattern:

(16) | |||||

To see that this can work, first consider the limit . The VEV of can always be rotated into the direction by a global SU(2) transformation, while the VEV of has no preferred orientation. When is turned on, prefers to have the elements in the first row and column vanish, , so that the term remains zero. We can still perform a global rotation around the 1 axis to make vanish; this rotation leaves invariant.

To summarize, this model is extremely simple: it needs a new SU(2) (not SU(2)U(1) cheung ()) in the dark sector, broken by a new Higgs triplet and 5-plet, and a dimension-5 coupling which induces mixing between the photon and the new gauge boson. These ingredients easily give us the desired mass spectrum, fig. 1(a), and coupling of one of the gauge bosons to leptons. The stability of the middle state is guaranteed by an unbroken symmetry, which also keeps the 0.1 GeV-scale and gauge bosons stable, but these (as we will show presently) are cosmologically harmless and not subject to accelerator constraints.

## Vi Cosmological implications

It is interesting to note that the model we have put forward based primarily on the INTEGRAL and ATIC observations happens to predict a relic density for the DM which is not so far from the required value. To match the WMAP value , one needs pb PDG (); for three colors of nearly degenerate DM, this becomes pb. Take , where is the momentum of one of the incoming ’s in the center of mass frame. At the freeze-out temperature , . The Mandelstam variable ranges between the values , while . A somewhat detailed computation gives us an estimate for the matrix element for the annihilation process shown in fig. 8(a) as (however we have not been careful enough to determine whether the three diagrams interfere constructively or destructively—this estimate assumes the former). Then

(17) |

For the values we have favored thus far, and Gev, (17) comes out too small by a factor of . We should thus hope to find an example of sufficient positron production for INTEGRAL at a value of approximately 10 times higher, . This is preferable to achieving the right relic density by making smaller, since that would force us to give up our explanation for ATIC. As we mentioned before, the numerical scattering computation becomes prohibitively slow at these larger values of ; work is in progress to explore this region of parameter space.

However, it is possible to add extra interactions to the model to adjust the relic density without the need for changing our preferred value of the gauge coupling or DM mass. If there is an extra gauge boson with the couplings to the DM and to right-handed electrons, it can provide extra annihilation channels which easily bring up to the required value; we just need . If the for some reason couples mainly to and not other SM particles, this can maintain the preference for annihilation into but not heavier charged particles, as indicated by PAMELA/ATIC. Such a would have to correspond to a U(1) symmetry broken at the scale since the masses do not conserve the current to which couples. If such a solves the relic density problem, then its contribution to annihilation in the galaxy will also dominate the PAMELA/ATIC signals, giving nearly monoenergetic via the process of fig. 9.

The analogous freeze-out computation can be done for the diagrams of fig. 8(b) to determine the relic density of the MeV stable gauge bosons -. The estimate for is similar to to eq. (17), except that the relevant mass scale is rather than . This makes the cross section larger by a factor of . Since scales like , the cosmological energy density of the stable ’s is negligible, . They can annihilate into because of their coupling to , but this would probably be very difficult to detect due to the low density of the ’s. The stable gauge bosons thus seem to be rather innocuous cosmological relics.

Note added. After completing the first version of this work, M. Pospelov pointed out a potentially serious difficulty: the process can also proceed through Sommerfeld-enhanced scatterings in the early universe, exponentially depleting the density of states which we assumed to be as numerous as the ’s. This would take place well below the initial freeze-out temperature, when the kinetic energy of the ’s becomes comparable to the mass splitting , and their velocity is of order , close to the value in the galaxy today.

To estimate the severity of the problem, we compute the rate of scatterings over the Hubble rate as a function of temperature. We take the value of the cross section required for matching the INTEGRAL observation, cm(where is the typical velocity in the galaxy), and the density with corresponding to the value . The DM fell out of equilibrium at GeV; at this time its momentum was , and thereafter it redshifted like . We can write , where if the DM underwent standard freeze-out. However we will be interested in more general, nonthermal ways of generating the DM, so we keep unspecified for now. Taking , we find

(18) |

This is independent of except through the number of species . From (18) it is clear that the -depleting process will be in equilibrium in the standard scenario, where . We need to avoid the depletion of . This could happen if the ’s were produced nonthermally. A straightforward way would be through the late decays of a heavy scalar . For example if and the decay occurs at MeV, then and . This nonstandard production mechanism requires that the standard one be subdominant, i.e., we need larger values of or of the couplings (as discussed above) to suppress the relic density at freeze-out of the perturbative annihilations. It is straightforward to verify that the ’s will not reestablish kinetic equilibrium with the SM particles after this point. By design, the strongest interaction with the SM is the scattering via -channel exchange of the boson. The rate for this is 8 orders of magnitude smaller than the Hubble rate at MeV.

One question which arises with respect to out-of-equilibrium production of DM through decays is how to get a sufficiently small coupling between and so that the decay happens at such low temperatures. However, this can be naturally explained in the context of a Grand Unified Theory (GUT), with a heavy gauge boson that couples to and to . Integrating out the boson gives rise to an interaction of the form , where is the GUT gauge coupling. This interaction does not allow for to decay, but if we suppose that gets a VEV at the TeV scale, a coupling which leads to decays is generated, . The decay rate must equal the Hubble rate at MeV, leading to for TeV. Taking TeV, we find GeV, the GUT scale. Of course would need to have stronger couplings to other particles to freeze out at the right density.

## Vii Conclusions

We have demonstrated a modification of the original XDM scenario which makes the effect strong enough to explain the 511 keV anomaly, without any radical change in our understanding of the DM distribution in the galaxy, and within a theoretically sound class of particle physics models. The key idea is to have two nearly degenerate states of stable DM, such that the heavier one must first be excited by the small energy keV into a third state, in order to decay to the lowest state (fig. 1). Moreover, we need at least two kinds of new 0.1 GeV-scale gauge bosons, and (with a theoretical preference for a third, ), such that and mediate the excitation , while is responsible for the decay . A novelty of this scenario is that the first excited DM state is stable, in addition to the ground state.

Although our primary interest was to address the INTEGRAL observation, we have also been motivated by the suggestion AH () of a class of models which can simultaneously explain the other anomalies; thus we have focused on parameters where is at least 500 GeV. Since the INTEGRAL signal scales like , we can more easily accommodate it by relaxing this requirement. To make contact with ATIC/PPB-BETS, it may be desirable to find a model in which is somewhat higher than the 500 GeV value we were able to obtain thus far, since the ATIC data indicate an excess going up to 800 GeV. The fact that we did not yet find an example with GeV may just be due to the numerical challenges of our method, where the integration time to solve eq. (3) (subject to the appropriate boundary conditions) becomes increasingly long as (eq. (6)) and are increased. It would be desirable to find an approximate analytic solution in the large regime to make further progress.

Our proposal for modifying the excited dark matter spectrum has also motivated us to construct a model which is simpler than others that have been suggested along these lines cheung (); for example the new gauge group in the dark sector is SU(2) rather than SU(2)U(1). A bonus of our model is the possibility of a new signal due to annihilation of DM states into a single pair instead of two pairs, giving a sharper feature in the lepton spectrum at energies near the DM mass. In this paper we have not tried to predict the PAMELA/ATIC/PPB-BETS signals in as much detail as we have treated that of INTEGRAL; doing so would clearly be a worthwhile next step.

We thank Nima Arkani-Hamed for interactions which revived our interest in explaining the INTEGRAL anomaly, Gilles Gerbier for enlightening insights about DAMA, and Gil Holder for discussions on DM density and electron energy loss, Maxim Pospelov for disscussions related to the “note added,” and Douglas Finkbeiner and Neal Weiner for discussions about the LMXB hypothesis. We also thank Itay Yavin for pointing out a sign error in the radiative mass shifts in our first version. FC is supported by a Schulich Fellowship at McGill University. Our work is supported by NSERC.

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