Neutrinos from OffShell Final States and the Indirect Detection of Dark Matter
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 vectorlike electroweak doublet, but many results generalize to other models of weaklyinteracting dark matter. We reevaluate the indirect detection constraints on this model and find that, once all annihilation channels are properly taken into account, the most stringent constraints on spindependent scattering for dark matter mass are derived from the results of the SuperKamiokande experiment. Moreover, we establish the modelindependent 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.
MCTP1202
I Introduction
The accumulation and subsequent annihilation of dark matter in the Sun could lead to a significant flux of highenergy 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 matternucleon scattering cross sections. As such, it gives information on some of the same couplings probed in underground direct detection experiments. Generally, constraints on spinindependent (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 spindependent (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 2body final states – see, for instance, Cirelli et al. (2005); Blennow et al. (2008). However, there are cases in which 3body final states may contribute significantly to neutrino signals in spite of the phase space suppression they suffer relative to 2body final states. For example, if the dark matter mass is just below the energy required to open up annihilation to a new 2body final state, the rate of annihilation to a corresponding 3body 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 spin1 electroweak boson emission is capable of lifting helicity suppression in certain processes, which can lead to sizable branching ratios for dark matter annihilation to 3body final states consisting of the two original finalstate particles and an additional spin1 boson Bergstrom (1989); Bringmann et al. (2008); Ciafaloni et al. (2011a, b). Furthermore, the neutrinos produced by these 3body final states can be more energetic than those from the dominant 2body channels, enhancing the importance of 3body final states.
In this paper, we revisit the importance of annihilation to 3body states just below 2body thresholds for weaklyinteracting 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 weaklyinteracting dark matter, we use the minimal model studied in ArkaniHamed 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 “singletdoublet” 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 weaklyinteracting dark matter models. These results are presented in section II.1. To highlight the potential significance of 3body 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 reevaluate the indirect detection limits on the singletdoublet 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
Electroweaklyinteracting dark matter with mass will annihilate to 2body 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 offshell . 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 singletdoublet model consists of a gauge singlet fermion and a pair of fermionic doublets. The doublets have a vectorlike 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 :
(1) 
with hypercharges and respectively. Mass terms and interactions for this model are given by:
(2) 
where doublets are contracted with the LeviCivita 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 singletdoublet mixing produces Majorana dark matter, evading the much too large mediated spinindependent 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 coannihilation in the MSSM).
We implemented the singletdoublet 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 modified^{1}^{1}1ID 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 SuperK) Edsjo ().^{2}^{2}2For 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 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 2body final states (again, normalized to a single annihilation of each type) are
(3) 
If we increase the threshold to , the ratios become
(4) 
These values are fairly constant over the mass range .^{3}^{3}3Fixing , 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 SuperK () are also given in the table.
1.5
Final State  

1
Strictlyspeaking, these values are modeldependent, as they depend on the polarization of the final state ’s. For singletdoublet dark matter in the static limit, annihilation to occurs via  and channel exchange of the .^{4}^{4}4The 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 transverselypolarized 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 in the SingletDoublet Model
As a concrete example, we evaluate the importance of annihilation to in the singletdoublet 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 matterHiggs boson coupling must be relatively small to avoid SI direct detection constraints. There exists a value Cohen et al. (2011) where the dark matterHiggs boson coupling cancels completely, a value which we define as . While we choose , this choice of is for simplicity; if the dark matterHiggs boson coupling is sufficiently small to avoid direct detection constraints, it is not important for setting the relic density.^{5}^{5}5An 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 matterHiggs boson coupling, which generates spinindependent cross sections below experimental bounds. Moreover, dark matter annihilation through a scalar vanishes in the static limit, so a nonzero dark matterHiggs boson coupling would not affect neutrino signals.^{6}^{6}6This also requires the spinindependent 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 sevenyear 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 nontrivial branching ratio for annihilation to . Branching ratios for the dominant solar annihilation channels are shown in table 2, for both the case where only 2body 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 modelspecific indirect detection limits with and without annihilation to for the sample point.
1.5
2body only  86.9%  7.9%  5.1%  – 

Including 3body  76.9%  7.0%  4.5%  11.5% 
1
Recent results from SuperK give a modelindependent 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
(5) 
for annihilation exclusively to given in Wikstrom and Edsjo (2009), the “soft” bound from SuperK is (as given in Tanaka et al. (2011)).
This bound can be adapted to get the appropriate, modeldependent 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
(6) 
The numerators incorporate the flux from all relevant final states, appropriately weighted by branching ratios. Thus, the modeldependent bounds for the sample point are
(7) 
So including the effects of the final state improves the bounds by a factor of
(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 SuperK limits exclude the spindependent 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
(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).^{7}^{7}7For a thorough, uptodate 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
(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 spindependent 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 2body final states. The potential importance of 3body 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 masssuppressed 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 .^{8}^{8}8For , both ’s can be onshell, so the branching ratio to calculated by considering only 2body 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 singletdoublet 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 reevaluation of the constraints on in the singletdoublet model – the previous analysis performed in Cohen et al. (2011) used direct detection limits from Felizardo et al. (2011) for . We perform such a reevaluation in section IV, fully taking into account subdominant annihilation channels.
Iii
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 singletdoublet model (as we did for ) simply because the effects of annihilation to are small in the majority of the singletdoublet 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 staticlimit 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 2body 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 2body 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.
1.5
Final State  

1
1.5
0.22  0.15  
0.48  0.28  
9.6  22 
1
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 2body 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 SingletDoublet Model
In section II.2, we found that the indirect detection limits from SuperK on singletdoublet 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.^{9}^{9}9ATLAS monojet and missing energy searches ATL (2011) can place robust bounds on dark matterquark interactions (see, for instance, Goodman et al. (2010); Fox et al. (2011); Rajaraman et al. (2011)) but, because the mediator of singletdoublet dark matterquark 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 singletdoublet model presented in Cohen et al. (2011).
iv.1
Figure 3 demonstrates that, for , annihilation to can enhance neutrino signals for specific points in the singletdoublet model. However, even if the looser bound for singletdoublet 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 “2body 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 singletdoublet model specifically, the enhancement to neutrino signals from annihilation to can still lead to the exclusion of some points with that exhibit a nonnegligible branching ratio to . This effect is most relevant for small .
iv.2
For , the main contribution to neutrino signals in the singletdoublet model generally comes from annihilation to onshell 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
(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 singletdoublet model? Singletdoublet dark matter with mass , which does not undergo coannihilation 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 singletdoublet 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 singletdoublet dark matter: to avoid the combination of SI and SD experimental constraints, dark sector masses must be tuned to permit early universe coannihilation. Only in this case can the correct relic density be achieved without the corresponding generation of large dark matternucleon 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 singletdoublet 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
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 – singletdoublet 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 singletdoublet 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 singletdoublet 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 singletdoublet 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 SuperK the degradation would be at worst a factor of (if the dark matter annihilated exclusively to ) due to SuperK’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 SuperK due to a combination of two factors:

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.

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 SuperK 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 SuperK, when taken in concert with XENON100, indeed excludes much of the singletdoublet 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 3body 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 singletdoublet 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 singletdoublet model except for very close to .
We have also demonstrated that indirect detection searches (performed by SuperK and IceCube) still provide the most extensive probe of spindependent dark matternucleon scattering over much of the singletdoublet 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 singletdoublet 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 singletdoublet model with a thermal history, except perhaps in the exceptional cases – the dark matter sits on resonance, or coannihilation 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 singletdoublet model is sensitive to the mass of the charged fermion. Thus, an understanding of 3body final states may be important to maximizing the utility of indirect detection experiments.
Acknowledgments
We thank Timothy Cohen and David TuckerSmith 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 NSFPHY0743315 and by DOE Grant #DEFG0295ER40899.
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