Neutrinos from Off-Shell Final States and the Indirect Detection of Dark Matter

# Neutrinos from Off-Shell Final States and the Indirect Detection of Dark Matter

John Kearney and Aaron Pierce Michigan Center for Theoretical Physics (MCTP)
Department of Physics, University of Michigan, Ann Arbor, MI 48109
July 29, 2019
###### Abstract

We revisit the annihilation of dark matter to neutrinos in the Sun near the and kinematic thresholds. We investigate the potential importance of annihilation to in a minimal dark matter model in which a Majorana singlet is mixed with a vector-like electroweak doublet, but many results generalize to other models of weakly-interacting dark matter. We re-evaluate the indirect detection constraints on this model and find that, once all annihilation channels are properly taken into account, the most stringent constraints on spin-dependent scattering for dark matter mass are derived from the results of the Super-Kamiokande experiment. Moreover, we establish the model-independent statement that Majorana dark matter whose thermal relic abundance and neutrino signals are both controlled by annihilation via an -channel boson is excluded for . In some models, annihilation to can affect indirect detection, notably by competing with annihilation to gauge boson final states and thereby weakening neutrino signals. However, in the minimal model, this final state is largely negligible, only allowing dark matter with mass a few GeV below the top quark mass to evade exclusion.

MCTP-12-02

## I Introduction

The accumulation and subsequent annihilation of dark matter in the Sun could lead to a significant flux of high-energy neutrinos discernible from background Press and Spergel (1985); Silk et al. (1985); Srednicki et al. (1987); Jungman et al. (1996). Alternately, a lack of signal can be used to place limits on the rate of solar dark matter annihilation. Because capture and annihilation are often in equilibrium, this approach typically probes the dark matter capture rate, or equivalently dark matter-nucleon scattering cross sections. As such, it gives information on some of the same couplings probed in underground direct detection experiments. Generally, constraints on spin-independent (SI) scattering from indirect detection of neutrinos are substantially weaker than those from direct detection experiments. However, in sizable portions of parameter space, indirect detection provides the most stringent constraints on spin-dependent (SD) scattering Kamionkowski et al. (1995); Wikstrom and Edsjo (2009). The promise of additional data from DeepCore DeYoung (2011); Ha (2012) motivates revisiting the calculation of neutrino fluxes.

Studies of neutrino spectra and signals from dark matter annihilations generally concentrate on 2-body final states – see, for instance, Cirelli et al. (2005); Blennow et al. (2008). However, there are cases in which 3-body final states may contribute significantly to neutrino signals in spite of the phase space suppression they suffer relative to 2-body final states. For example, if the dark matter mass is just below the energy required to open up annihilation to a new 2-body final state, the rate of annihilation to a corresponding 3-body final state can be sizable. This has been studied previously in the Minimal Supersymmetric Standard Model (MSSM) for dark matter annihilation to and Chen and Kamionkowski (1998). Another example arises wherein spin-1 electroweak boson emission is capable of lifting helicity suppression in certain processes, which can lead to sizable branching ratios for dark matter annihilation to 3-body final states consisting of the two original final-state particles and an additional spin-1 boson Bergstrom (1989); Bringmann et al. (2008); Ciafaloni et al. (2011a, b). Furthermore, the neutrinos produced by these 3-body final states can be more energetic than those from the dominant 2-body channels, enhancing the importance of 3-body final states.

In this paper, we revisit the importance of annihilation to 3-body states just below 2-body thresholds for weakly-interacting dark matter. We build upon and generalize some results from Chen and Kamionkowski (1998). In section II, we discuss dark matter annihilation to . As a representative candidate for weakly-interacting dark matter, we use the minimal model studied in Arkani-Hamed et al. (2005); Mahbubani and Senatore (2006); Enberg et al. (2007); D’Eramo (2007); Cohen et al. (2011), see also Fayet (1974). In this model, the dark matter is a Majorana fermion composed of an admixture of a weak doublet and a sterile singlet. As such, it is similar to a mixed Bino–Higgsinso state of the MSSM. Not only is the “singlet-doublet” model an interesting dark matter candidate in its own right, but results we obtain for dark matter annihilating to are also applicable to a range of weakly-interacting dark matter models. These results are presented in section II.1. To highlight the potential significance of 3-body final states, we assess the overall importance of annihilation to in the minimal model in section II.2. In section III, we turn our attention to , presenting general results for dark matter annihilation to via an -channel vector boson. Motivated by our findings in section II.2, we re-evaluate the indirect detection limits on the singlet-doublet model in section IV, including the effects of subdominant annihilation channels. The revised limits are significantly stronger than those originally quoted by these authors in Cohen et al. (2011). Conclusions are presented in section V.

## Ii WW∗

Electroweakly-interacting dark matter with mass will annihilate to 2-body final states consisting of pairs of Standard Model fermions, . The dark matter may also annihilate to final states of the form , where denote Standard Model fermions that arise from an off-shell . Near threshold, annihilation to may contribute to neutrino signals.

This is particularly true in the case of Majorana dark matter for two reasons. First, dark matter particles in the Sun have velocities , so the static limit applies. In this limit, only the -wave component of cross sections survives; for annihilation to fermion pairs, helicity arguments require a suppression of . Second, the mass dependence favors the process (with subdominant contributions from and ). Neutrinos from ’s tend to be significantly harder than those from ’s, and the presence of additional, particularly energetic neutrinos from the final state could enhance neutrino signals.

The singlet-doublet model consists of a gauge singlet fermion and a pair of fermionic doublets. The doublets have a vector-like mass term, and the neutral components of the doublets mix with the gauge singlet through renormalizable couplings to the Higgs field, . The new fields are odd under a symmetry, ensuring the stability of the lightest state. We denote the singlet as and the doublets as :

 D=(νE) Dc=(−Ecνc), (1)

with hypercharges and respectively. Mass terms and interactions for this model are given by:

 ΔL=−λDHN−λ′~HDcN−mDDDc−12mNN2+ h.c., (2)

where doublets are contracted with the Levi-Civita symbol and . When the Higgs field attains its vacuum expectation value, the singlet and the neutral components and of the doublets mix, such that the spectrum consists of a charged Dirac fermion of mass , which we denote , and three neutral Majorana fermions (, ), the lightest of which is the dark matter. Details can be found in Cohen et al. (2011). This dark matter candidate couples only to the bosons of the electroweak theory, and not additional exotic states, so in this sense is minimal. The singlet-doublet mixing produces Majorana dark matter, evading the much too large -mediated spin-independent cross sections that Dirac dark matter exhibits. We view this model as a useful bellwether – it provides an indication of the status of very simple dark matter models without special features (such as stau or stop co-annihilation in the MSSM).

We implemented the singlet-doublet model in MadGraph Alwall et al. (2007), and simulated dark matter annihilations at corresponding to , approximately reproducing the conditions of dark matter annihilations in the Sun. We then decayed unstable particles using the MadGraph DECAY package. The unweighted event output was modified111ID codes for the incoming dark matter particles were changed to those corresponding to annihilation. such that it could be passed to Pythia for showering and hadronization Sjostrand et al. (2006). Relevant data about neutrinos and their parents was extracted for each event, and fed to a modified version of WimpSim in order to simulate neutrino interaction and propagation to a detector (either IceCube/DeepCore or Super-K) Edsjo ().222For propagation, we use the default WimpSim parameters: , , , , and Maltoni et al. (2004), but our results are not particularly sensitive to the exact choice of these parameters. To validate this method, we confirmed that the cross sections and branching ratios given by MadGraph agreed with those from analytic expressions for annihilations in the static limit (e.g., from Dreiner et al. (2010)). Furthermore, we confirmed that the spectra given for annihilations were the same as those given by the unmodified version of WimpSim and Cirelli et al. (2005).

### ii.1 General Results for WW∗ Neutrino and Muon Spectra

Injection spectra for dark matter annihilation to , and based on a simulation of events are shown in figure 1 for . The corresponding spectra of muons at DeepCore for each annihilation channel is shown in figure 2. As anticipated, the neutrino and muon spectra from annihilation to are significantly harder than those from annihilation to , although they are softer than those from . Recall, decays produce energetic that can oscillate to give at a detector.

Integration gives the total flux of muons per annihilation above a threshold energy , , which can be used to determine the relative signal at a detector from each final state. The fluxes for two different threshold energies, (projected for DeepCore DeYoung (2011)) and (a more conservative value quoted in Barger et al. (2011)), are given in table 1. For , the ratios of flux from to the fluxes from the 2-body final states (again, normalized to a single annihilation of each type) are

 ΦWf¯f′μΦτ+τ−μ(Eμ≥10 GeV)=0.46,ΦWf¯f′μΦb¯bμ(Eμ≥10 GeV)=25. (3)

If we increase the threshold to , the ratios become

 ΦWf¯f′μΦτ+τ−μ(Eμ≥35 GeV)=0.46,ΦWf¯f′μΦb¯bμ(Eμ≥35 GeV)=670. (4)

These values are fairly constant over the mass range .333Fixing , for fluxes are approximately 0.8 times those given, and for about 1.2 times those given, with ratios remaining largely constant. The flux from annihilation to is substantially larger than that from annihilation to , and is comparable to that from annihilation to . Thus, if the branching ratio is not too small, can conceivably contribute significantly to indirect detection signals. For reference, muon fluxes at Super-K () are also given in the table.

Strictly-speaking, these values are model-dependent, as they depend on the polarization of the final state ’s. For singlet-doublet dark matter in the static limit, annihilation to occurs via - and -channel exchange of the .444The sum of the contributions to the final state from -channel exchange – both where the couples to and where the couples to a pair of fermions, one of which radiates a – is small. Annihilation to ’s via exchange of a fermion in the - or -channel produces only transversely-polarized bosons (with the two polarizations in equal proportions). In many models of electroweak dark matter (particularly in the 0 limit), it is precisely via - and -channel fermion exchange that annihilation to bosons occurs, so our results are certainly valid for these models. But, in addition, the neutrino spectrum produced by annihilation to equal proportions of the three polarizations is similar to that from annihilation to equal proportions of the two transverse polarizations, and the differences between the overall results obtained from each are small. Thus, the values given in table 1 are applicable to a range of models in which states may be important and can be used to calculate the magnitude of the effect of these states in a given model – all that must be known is the relevant annihilation branching ratios.

### ii.2 WW∗ in the Singlet-Doublet Model

As a concrete example, we evaluate the importance of annihilation to in the singlet-doublet model. A sample point for which annihilation to is important has and , and . The values of and are fixed via the following reasoning.

In Cohen et al. (2011), it was shown that for a light Higgs boson (), the dark matter-Higgs boson coupling must be relatively small to avoid SI direct detection constraints. There exists a value Cohen et al. (2011) where the dark matter-Higgs boson coupling cancels completely, a value which we define as . While we choose , this choice of is for simplicity; if the dark matter-Higgs boson coupling is sufficiently small to avoid direct detection constraints, it is not important for setting the relic density.555An exception occurs if – the enhancement to annihilation due to a small -channel Higgs boson propagator allows the correct relic density to be achieved with a small dark matter-Higgs boson coupling, which generates spin-independent cross sections below experimental bounds. Moreover, dark matter annihilation through a scalar vanishes in the static limit, so a non-zero dark matter-Higgs boson coupling would not affect neutrino signals.666This also requires the spin-independent solar capture rate to be negligible for these points, which it is. We then fix the overall size of by requiring the relic density (calculated using micrOmegas Belanger et al. (2011)), consistent with the determination by the seven-year Wilkinson Microwave Anisotropy Probe (WMAP) and other data on large scale structure Jarosik et al. (2011). Note, if , gauge invariance sets .

For this choice of parameters, the correct relic density is achieved by -wave annihilation via an -channel boson to Standard Model fermions in the early universe. The annihilation is largely democratic amongst light fermions, taking advantage of the relatively large velocities () to avoid the significant -wave suppression. Today in the Sun, dark matter velocities are sufficiently small that -wave annihilation is negligible, and so annihilation occurs dominantly to . This process is mass suppressed, permitting a non-trivial branching ratio for annihilation to . Branching ratios for the dominant solar annihilation channels are shown in table 2, for both the case where only 2-body final states are considered and for the case where the final state is included. While we give the branching ratio to , its contribution to the muon flux is negligible. These branching ratios can be used in conjunction with values of from the previous subsection to determine the model-specific indirect detection limits with and without annihilation to for the sample point.

Recent results from Super-K give a model-independent limit on the total flux from dark matter annihilations of for Tanaka et al. (2011). This can be converted to a limit on by assuming the dark matter annihilates to a particular final state – generally, “soft” limits are given by assuming annihilation to . Using the conversion factor of

 σpSD/Φμ=7.6×1011 cm2 s pb (5)

for annihilation exclusively to given in Wikstrom and Edsjo (2009), the “soft” bound from Super-K is (as given in Tanaka et al. (2011)).

This bound can be adapted to get the appropriate, model-dependent bound for the sample point by rescaling by the ratio of the average flux per dark matter annihilation for the sample point to the flux per annihilation exclusively to . Using the branching ratios from table 2 and the fluxes from table 1, we find these ratios to be

 Φave, 2-body onlyμΦb¯bμ(Eμ≥2 GeV)=2.2,Φave, incl. 3-bodyμΦb¯bμ(Eμ≥2 GeV)=3.2. (6)

The numerators incorporate the flux from all relevant final states, appropriately weighted by branching ratios. Thus, the model-dependent bounds for the sample point are

 σp,2-body onlySD<2.4×10−3 pb,σp,incl. 3-bodySD<1.6×10−3 pb. (7)

So including the effects of the final state improves the bounds by a factor of

 Φave, incl. 3-bodyμΦave, 2-body % onlyμ(Eμ≥2 GeV)=1.5. (8)

We note that both of these bounds are stronger than the bound of given in Tanaka et al. (2011). This arises in part from appreciating the importance of the subdominant but hard channel. Using standard assumptions, we find that these revised Super-K limits exclude the spin-dependent cross section for this particular point, . We can express the limits as what the local density of dark matter would have to be for this point to not be excluded. We find the values to be

 ρ2-body only<0.23 GeV cm−3,ρincl. 3-% body<0.15 GeV cm−3. (9)

Thus, this point is very tightly constrained by indirect detection experiments, if not excluded. We quote the limits in terms of , with the understanding that this is a placeholder for other astrophysical uncertainties. For instance, the bound on goes as , where represents the local dark matter velocity dispersion Jungman et al. (1996). So, while a decrease in could weaken the bound sufficiently for the sample point to avoid exclusion, so too could an increase in . Recent analyses suggest that lies somewhere in the interval Weber and de Boer (2010); Catena and Ullio (2010); Widrow et al. (2008) and may vary by up to McMillan and Binney (2009); Bovy et al. (2009); Reid et al. (2009).777For a thorough, up-to-date review of these uncertainties, see Green (2011). The effect of on this point is sufficient to push it to a regime where there is substantial tension, even taking these uncertainties into account.

To give a broader sense of how the importance of annihilation to can vary, we calculate the ratio

 R32≡Φave, incl. 3-bodyμΦave% , 2-body onlyμ(Eμ≥Ethreshμ) (10)

at DeepCore/IceCube for a variety of points. Again, this represents the factor by which signals (and hence constraints) are enhanced by including annihilation to . A contour plot showing the as a function of and is shown in figure 3 for , subject to the same requirements as the benchmark point that and is fixed by requiring . The majority of the variation in the plot arises from changes in the branching ratio to . In general, points of this type (, suppressed and set by annihilation via an -channel boson) exhibit the largest values of , comparable to current spin-dependent constraints. Thus, including the effect of will push neutrino signals above exclusion limits for certain points.

Given the impact of annihilation to on neutrino signals, it is reasonable to wonder whether it may also have a significant effect on the relic density. This would not have been taken into account by our calculations of , which considered only 2-body final states. The potential importance of 3-body final states on relic density calculations has been highlighted previously in Yaguna (2010). However, for the points considered here, the fact that the cross section for -wave annihilation to is comparable to the mass-suppressed cross section for -wave annihilation to fermions suggests that in the early universe -wave annihilation to fermions will still dominate. This intuition is confirmed by numerical tests. Comparisons of branching ratios to in the early universe for given by micrOmegas indicate the relic density calculation is wrong by at most for points shown in the contour plot, and that for most points it is much more accurate, to .888For , both ’s can be on-shell, so the branching ratio to calculated by considering only 2-body final states will be accurate. Thus, the difference the branching ratio to for and gives the approximate error introduced by neglecting annihilation to . So, the effect of the final state on neutrino signals must be considered even in cases where its effect on the relic density is negligible.

Consistent with the findings in Wikstrom and Edsjo (2009), we find that the strongest constraints on SD scattering of singlet-doublet dark matter with arise from indirect detection experiments. For instance, both bounds given in Eq. (7) for are more stringent than the direct detection limits from SIMPLE of Felizardo et al. (2011) and than the preliminary limits from COUPP of Dahl (2011). This motivates a re-evaluation of the constraints on in the singlet-doublet model – the previous analysis performed in Cohen et al. (2011) used direct detection limits from Felizardo et al. (2011) for . We perform such a re-evaluation in section IV, fully taking into account subdominant annihilation channels.

## Iii tt∗

In this section, we present general results for dark matter annihilating to . We do not include a discussion of the importance of annihilation to a final state in the singlet-doublet model (as we did for ) simply because the effects of annihilation to are small in the majority of the singlet-doublet model parameter space. We comment briefly on this in section IV. That said, it is not difficult to construct a model in which annihilation could have a significant effect on neutrino signals. The heaviness of the top quark obviates the static-limit suppression. As a result, near threshold, the cross section for annihilation to a final state can readily compete with , and . Furthermore, the neutrino and muon spectra from top quarks and bosons are significantly harder than those from bottom quarks. Consequently, as in the case of annihilation to , if the dark matter annihilates predominantly to , the combination of these effects might lead to significant enhancement to neutrino signals from annihilation to . For instance, in models with a new , the freedom to control the couplings of quarks to the allows one to essentially make a final state arbitrarily important. Alternatively, an MSSM model with light stops could allow the final state to dominate, although in such models one must also ensure that the capture rate in the Sun is large enough to give a measurable signal. Rather than contrive a model to emphasize the potential importance of a final state, we instead present general spectra for . These results can be applied for your favorite model, requiring only a calculation of the relevant branching ratios.

The neutrino injection spectra from Majorana dark matter with annihilating to a final state via an -channel massive, neutral, vector boson are shown in figure 4. Also shown are spectra from , and 2-body annihilation final states from WimpSim. Spectra from annihilation to are comparable to those from , so are omitted for clarity. The corresponding muon spectra at DeepCore/IceCube are shown in figure 5. The integrated number of muons above threshold per annihilation for two different thresholds, and , are given in table 3, and ratios of muon flux from to fluxes from the various 2-body final states are given in table 4. Ratios of muon fluxes (and neutrino and muon spectra) are largely constant over the range , although variation in neutrino energy with dark matter mass can lead to changes of in integrated fluxes at either end of the range.

Once again, the hardest neutrino and muon spectra are produced by annihilation to . Muons from are also slightly softer than those from , but are significantly harder than those from . Thus, annihilation to may enhance indirect detection signals if the dominant 2-body annihilation mode is to or other light fermions (which tend to have even softer spectra than , again with the exception of ). It may also be the case that the possibility of annihilation to degrades indirect detection signals for models in which the dominant contributions to the muon flux arise from annihilation to or (as annihilation to may decrease the branching ratio to these final states).

## Iv Revised Indirect Detection Constraints and Discovery Prospects for the Singlet-Doublet Model

In section II.2, we found that the indirect detection limits from Super-K on singlet-doublet dark matter with were significantly more stringent than those quoted in Tanaka et al. (2011), and stronger than the direct detection limits from COUPP and SIMPLE. Indirect detection limits are also stronger than those from collider experiments.999ATLAS mono-jet and missing energy searches ATL (2011) can place robust bounds on dark matter-quark interactions (see, for instance, Goodman et al. (2010); Fox et al. (2011); Rajaraman et al. (2011)) but, because the mediator of singlet-doublet dark matter-quark interactions is the (relatively) light boson, collider limits are substantially weakened in this model Goodman and Shepherd (2011). This merits a revision of the indirect detection limits on the singlet-doublet model presented in Cohen et al. (2011).

### iv.1 mχ<mW

Figure 3 demonstrates that, for , annihilation to can enhance neutrino signals for specific points in the singlet-doublet model. However, even if the looser bound for singlet-doublet dark matter with is applied, we find that this minimal model is more tightly constrained than suggested in Cohen et al. (2011), which used the SIMPLE bound of . In fact, this bound has significant implications for Majorana dark matter in general. For Majorana dark matter with and thermal relic abundance set by annihilation via an -channel boson, is fixed for a particular value of . The reason is simple: the dark matter- boson coupling is set by requiring the correct relic density – see, for example, figure 5 of Cohen et al. (2011) and discussion thereof. Dark matter annihilation in the Sun also occurs via -channel boson exchange. So, for masses somewhat below , the “2-body only” branching ratios given in table 2, and thus the bound, apply. We find that this bound excludes Majorana dark matter with whose annihilations are controlled by an -channel Z.

For , the proximity of to causes dark matter annihilation via an -channel boson in the early universe to be enhanced due to the smaller propagator. As a result, the correct relic density can be achieved with a smaller dark matter- boson coupling, leading to suppressed values of that still evade this bound. However, in the singlet-doublet model specifically, the enhancement to neutrino signals from annihilation to can still lead to the exclusion of some points with that exhibit a non-negligible branching ratio to . This effect is most relevant for small .

### iv.2 mW<mχ<mt

For , the main contribution to neutrino signals in the singlet-doublet model generally comes from annihilation to on-shell electroweak boson pairs. In the static limit, the dark matter can annihilate to via - and -channel exchange of the charged -partner of the dark matter or, for , to via - and -channel exchange of the dark matter itself and the heavier neutral states, and . If , annihilation to a final state, both via - and -channel exchange of the neutral states and via an -channel boson, will also occur. Assuming annihilation exclusively to would correspond to the “hard” limit quoted in Tanaka et al. (2011), which is

 σpSD<3.0×10−4 pb(hard,W+W−orZZ) (11)

for . This is a factor of stronger than the corresponding previous bound quoted in Desai et al. (2004) (used in Cohen et al. (2011)). For larger values of , the decrease in the local number density weakens the bound, whereas the increase in strengthens it. These effects are comparable, such that the hard limit is over the entire range .

How does this compare to the typical values of in the singlet-doublet model? Singlet-doublet dark matter with mass , which does not undergo co-annihilation in the early universe (and evades bounds on from XENON100 Aprile et al. (2009)), achieves the correct relic density predominantly by annihilation via an -channel to , light fermions and (if possible) , with additional contributions from annihilation to electroweak boson pairs via - and -channel fermion exchange. As such, achieving the correct relic density requires a small range of dark matter- boson coupling. This fixes for singlet-doublet dark matter of this type (the upper horizontal band of figure 4 in Cohen et al. (2011)). The hard limit of Eq. (11) is a factor of below this, which would exclude all such points. For a light Higgs boson ( – favored by recent ATLAS and CMS results atl (2011); cms (2011)) and , the situation is becoming squeezed for singlet-doublet dark matter: to avoid the combination of SI and SD experimental constraints, dark sector masses must be tuned to permit early universe co-annihilation. Only in this case can the correct relic density be achieved without the corresponding generation of large dark matter-nucleon scattering cross sections.

While these conclusions are based on the hard limit given in Eq. (11), whereas exact bounds depend on the specific annihilation branching ratios, we nonetheless find them to be robust. The and final states yield sufficiently comparable spectra that annihilation to instead of would not significantly alter the bound. Annihilation to would weaken the bound by at worst a factor of (assuming the Higgs boson is sufficiently light that it decays overwhelmingly to , which would contribute negligibly to indirect detection signals relative to the single ). The bound would weaken more drastically if annihilation to light fermions could be made to dominate. However, this is not easy to do. While annihilation to is suppressed in the static limit for large , the same cannot be said of annihilation to , for which the mass of one of the exchanged particles is fixed to be . Thus, if the dark matter- boson coupling is large enough to generate a sizable , there will generically be a sizable cross section for dark matter annihilation to . Furthermore, potentially competing cross sections for annihilation to light fermions are suppressed by . Consequently, the branching ratio for annihilation to boson pairs invariably dominates, and we find that overall neutrino signals are at most degraded by a factor of . As the bounds are significantly lower (a factor of ) than the general of interest in the singlet-doublet model, such a degradation would not affect the conclusion that the majority of points with large and are excluded. In fact, the bound is sufficiently strong that this holds even if the bound is weakened by taking more pessimistic (but reasonable) choices of astrophysical parameters.

### iv.3 mχ>mt

In this region of parameter space, unsuppressed -wave annihilation to via an -channel boson allows the correct relic density to be achieved with a significantly lower value of – singlet-doublet dark matter with that evades bounds on from XENON100 typically exhibits (the lower horizontal band of figure 4 in Cohen et al. (2011)). As a result, current bounds do not yet constrain for singlet-doublet dark matter with . The situation is more optimistic for DeepCore; the projected DeepCore limits are for annihilation of dark matter with to DeYoung (2011). The hardest neutrinos from the final state arise from the ’s produced in decay, which will be softer than the ’s produced in direct annihilations. Consequently, the spectra are softer than those from the final state and the comparable final state (as one would have expected from the results of section III). Thus, for singlet-doublet dark matter with and large values of (which habitually annihilates to via an -channel boson in the static limit), table 4 indicates that the projected hard DeepCore limits will be degraded by a factor of for a threshold of . However, supposing limits comparable to those projected are achieved, DeepCore will still be sufficiently sensitive to probe much of the remaining singlet-doublet parameter space with , consistent with the claims of Cohen et al. (2011).

As limits derived from assuming annihilation to are weaker than those for annihilation to , one might worry that annihilation to a final state could degrade the indirect detection limits just discussed for . However, for Super-K the degradation would be at worst a factor of (if the dark matter annihilated exclusively to ) due to Super-K’s low . As discussed in the last subsection, this degradation would not change the conclusion that the majority of points with large and are excluded. Assuming that that the hard limits are degraded by a factor of , we can ask at what point the contribution from annihilation to in the early universe is sufficiently large to permit the right relic density to be achieved with a value of that is small enough to avoid the (degraded) bound. In other words, there may be exceptional points with for which SI constraints are evaded and a thermal relic density is achieved, which are not excluded by Super-K due to a combination of two factors:

1. The correct thermal relic density is achieved with a smaller dark matter- boson coupling (and hence a smaller ) due to the significant contribution from annihilation to in the early universe.

2. Hard indirect detection limits are degraded by a factor of due to the dark matter annihilating predominantly to in the Sun. For , the hard limit from Super-K is Tanaka et al. (2011), leading to a degraded limit of .

Numerically integrating expressions for and (from Gondolo and Gelmini (1991)), we find that these conditions are only fulfilled for approximately less than . This indicates that Super-K, when taken in concert with XENON100, indeed excludes much of the singlet-doublet parameter space from GeV all the way up to .

## V Conclusion

In this work, we have explored the potential importance of dark matter annihilation to 3-body final states near threshold, and have highlighted situations in which certain such final states can significantly impact the reach of indirect detection experiments. In particular, we have shown that for the minimal singlet-doublet model of dark matter, consideration of the final state can improve indirect detection bounds by up to a factor of for . Meanwhile, annihilation to will frequently degrade bounds by competing with annihilation to electroweak boson pairs, but this turns out to be largely unimportant in the singlet-doublet model except for very close to .

We have also demonstrated that indirect detection searches (performed by Super-K and IceCube) still provide the most extensive probe of spin-dependent dark matter-nucleon scattering over much of the singlet-doublet model parameter space, though direct detection is becoming competitive. In fact, for , bounds from direct and indirect detection experiments are extremely similar for dark matter annihilating exclusively to in the Sun. Indirect detection bounds are more stringent for singlet-doublet dark matter with because of the sizable contribution to neutrino signals from the subdominant but hard channel, which doubles the strength of indirect detection bounds in spite of its small () branching ratio. Data from DeepCore will prove conclusive for the singlet-doublet model with a thermal history, except perhaps in the exceptional cases – the dark matter sits on resonance, or co-annihilation is crucial in setting the relic density.

These findings reinforce the sensitivity of neutrino signals to dark matter annihilation branching ratios. As such, signals may encode information about other Beyond the Standard Model particles – for instance, the effect of the final state in the singlet-doublet model is sensitive to the mass of the charged fermion. Thus, an understanding of 3-body final states may be important to maximizing the utility of indirect detection experiments.

### Acknowledgments

We thank Timothy Cohen and David Tucker-Smith for useful conversations and comments on the draft. J.K. is particularly thankful to Ran Lu for introduction to and assistance with several pieces of software used in this analysis. The work of A.P. was supported in part by NSF Career Grant NSF-PHY-0743315 and by DOE Grant #DE-FG02-95ER40899.

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