Isospin-Violating Dark Matter in the Sun

We consider the prospects for studying spin-independent isospin-violating dark matter-nucleon interactions with neutrinos from dark matter annihilation in the Sun, with a focus on IceCube/DeepCore (IC/DC). If dark matter-nucleon interactions are isospin violating, IC/DC's reach in the spin-independent cross section may be competitive with current direct detection experiments for a wide range of dark matter masses. We also compare IC/DC's sensitivity to that of next generation argon, germanium, neon and xenon based detectors.


I. INTRODUCTION
The IceCube Collaboration has recently completed installation of the DeepCore extension. An updated estimate of IceCube/DeepCore's sensitivity to spin-dependent dark matternucleus scattering [1] with 180 days of data indicates that its sensitivity may be much greater than previously expected [2]. It is therefore of interest to also consider IC/DC's sensitivity to spin-independent scattering (see also [3]). This interest is heightened by recent developments in dark matter model-building, which have emphasized that dark matter couplings to protons and neutrons may be different. In these models of isospin-violating dark matter (IVDM) [4,5], the cross section for dark matter to scatter off any isotope of an element is determined by the relative number of protons and neutrons in that isotope. This realization has been exploited to construct models that can match the data from DAMA [6], CoGeNT [7] and CRESST [8], while remaining consistent with constraints from other dark matter direct detection experiments [9,10]. In particular, if dark matter interactions with neutrons destructively interfere with those with protons at the ∼ 70% level, then much of the low-mass data can be made consistent.
For the case of partial destructive interference, direct detection experiments using materials with a high atomic mass number A can suffer great losses of sensitivity due to the degradation of the usual A 2 coherent scattering enhancement, as well as the fact that high-A materials usually have a large neutron fraction. Conversely, detectors utilizing low-A materials, such as helium, carbon, nitrogen, oxygen and fluorine exhibit less suppressed sensitivity due to destructive interference. Hydrogen has no neutrons to cause destructive interference. A good way to study IVDM may be through neutrino detectors [11,12], which search for the neutrino flux arising from dark matter annihilating in the Sun after capture by elastic scattering from solar nuclei. Since a significant fraction of dark matter captures arise from scattering off low-A nuclei, neutrino detector sensitivity suffers the least suppression as a result of isospin-violating interactions.
Although isospin violation has been used to understand low-mass dark matter data, these lessons generalize to all mass ranges. For dark matter with mass in the 30-5000 GeV range, the detection prospects from leading experiments, such as CDMS-II and XENON100, can be significantly weakened if dark matter interactions violate isospin. While isospin violation also weakens the sensitivity of neutrino detectors, these sensitivities will be much less suppressed than those of direct detection experiments. Among neutrino detectors, IC/DC will have the best sensitivity to dark matter in this mass range, making it worthwhile to consider its prospects, relative to other direct detection experiments, in probing IVDM.
In this Letter, we perform an analysis of IC/DC's sensitivity to the spin-independent cross section on protons σ p SI , including the effects of isospin violation.

II. INDIRECT DARK MATTER DETECTION VIA NEUTRINOS
Neutrino detectors search for dark matter which is gravitationally captured in the Sun. The dark matter settles to the core and annihilates to Standard Model products, which in turn produce neutrinos. We focus on the most studied case, where dark matter capture processes with rate Γ C and annihilation processes with rate Γ A are in equilibrium, such that Neutrino detectors search for the charged leptons which are created by incoming neutrinos through a charged-current interaction. For the IC/DC detector we divide the muon events into upward events (due to upward going neutrinos interacting outside the detector volume) and contained events (due to neutrinos that interact within the instrumented volume); see Refs. [13,14] for details.
For both types of events, the rate depends on dark matter interactions only through Γ C , and the choice of annihilation channel. The experimental sensitivity reflects the capture rate necessary for IC/DC to distinguish the neutrino flux due to dark matter annihilation from the atmospheric neutrino background. The capture rate is proportional to the dark matter-nucleon scattering cross section [15], so we may parameterize the capture rate by a capture coefficient C 0 . If dark matter-nucleon scattering is spin-dependent, the capture coefficient is given by On the other hand, if dark matter-nucleon scattering is spin-independent, then the effect of coherent scattering against heavy solar nuclei depends non-trivially on the relative strength of the dark matter couplings to neutrons and protons, f n and f p , respectively. Then, The background event rate determines the dark matter-initiated charged lepton event rate Γ event to which the detector is sensitive. This in turn implies that the detector has sensitivity to σ p SI > σ . We calculate C 0 using DarkSUSY [16] assuming a local halo density ρ = 0.3 GeV/cm 3 and a Maxwellian velocity distribution with dispersion v = 270 km/s; see the appendix.
IC/DC reports its sensitivity to the spin-dependent scattering cross section σ p SD . From this, one can easily determine IC/DC's sensitivity to σ p SI for any choice of f n /f p simply by rescaling the projected limit by an appropriate ratio of capture coefficients: . ( The IceCube Collaboration has recently presented an updated estimate for the sensitivity of the completed IC/DC configuration to σ p SD with 180 live days of data. This estimate assumes dark matter annihilation to the "hard" channel, that is, to ττ for m X ≤ 80 GeV and to W + W − for m X > 80 GeV. The choice of annihilation channel affects the neutrino spectrum, which in turn affects the muon event rate, and thus the detector's sensitivity. A more conservative assumption would be dark matter annihilation to bb, which is referred to as the "soft" channel. The relative sensitivity in the soft channel can be obtained by determining the ratio of muon rates at the detector (assuming a fixed dark matter-nucleon scattering cross-section) from different annihilation channels. This procedure is a valid approximation when the dark matter mass is much larger than the detector threshold since the dependence on the shapes of the neutrino spectra is weakened by integrating over a wide energy range. For dark matter masses close to the detector threshold, our results should be viewed as simply indicative of the relative sensitivities. The ratio of sensitivities for the τ + τ − , W + W − and bb channels (for various m X ) are given in Table I.

A. IVDM
It is often assumed that dark matter couples identically to protons and neutrons. Under this assumption, dark matter will scatter coherently off nucleons in a nucleus, leading to IceCube up (> 70 GeV) IceCube con. (> 70 GeV) DeepCore (> 35 GeV)  I. Relative sensitivities to the W + W − , bb and τ + τ − channels at IC/DC obtained from the integrated muon event rates for each of these channels. The τ + τ − channel proves to be most favorable provided it has a sizable branching fraction. Due to b−hadron absorption by the solar medium, a DM mass significantly above the muon energy detector threshold is necessary for the bb channel to be visible. Both upward and contained events at IceCube assume a half-year observation time and an optimistic threshold of 70 GeV. Experimental selection cuts are not included in the IceCube contained rate for which we assume a km 3 volume. The effective area for upward events is given in Ref. [17]. For the DeepCore effective volume we adopt the parameterization of Ref. [14].
an A 2 enhancement in the scattering cross section for heavy nuclei. This enhancement is the reason why the solar spin-independent capture rate is dominated by heavier nuclei, even though the Sun is largely composed of hydrogen [15]. While this assumption of isospinconserving interactions is a valid approximation for neutralinos, it need not be true more generally.
Although isospin-violating dark matter [4,5] has been used as an explanation of the DAMA and CoGeNT data, it is really a more general scenario in which dark matter couples differently to protons than to neutrons. The dark matter-nucleus spin-independent scattering cross section is given by where µ A is the reduced mass of the dark matter-nucleus system. The non-trivial dependence of σ A on f n /f p is the reason for the dependence of C SI 0 on f n /f p .

III. ICECUBE/DEEPCORE VERSUS DIRECT DETECTION EXPERIMENTS
The assumption of isospin-conserving interactions i.e., f n = f p , is commonly made in normalizing the dark matter-nucleus scattering cross section for a nucleus with Z protons to that of dark matter scattering against a single nucleon. This normalized cross section σ Z N is given by [5] where σ p SI is the spin-independent cross section for dark matter to scatter off a single proton. The summation is over the different isotopes with atomic number Z, and η i is the natural abundance of each isotope. As expected, We may also define the quantity which, in analogy to F Z , is the factor by which a neutrino detector's sensitivity will be suppressed if dark matter interactions violate isospin. In particular, if σ p SI is the actual dark matter-proton spin-independent scattering cross section, then σ N = σ p SI /F is the "normalized to nucleon" scattering cross section which would be inferred from neutrino detector data, if one assumes isospin-conserving interactions.
We can define the quantity R[ , Z](m X , f n /f p ): For a fixed m X , the maximum of R[ , Z] (varying over f n /f p ) is the maximum factor by which a detector with atomic number Z must exclude a signal from a neutrino detector (assuming isospin conservation) such that the signal is still excluded even if isospin violation is allowed. Similarly, the minimum of R[ , Z] (equivalently, the maximum of R[Z, ]) is the maximum factor by which a neutrino detector must exclude a signal from a detector with atomic number Z (assuming isospin conservation) such that the signal is still excluded even if isospin violation is allowed. dramatic for direct detection experiments using heavy nuclei, such as xenon or tungsten. Since heavy atoms tend to have many more neutrons than protons, partial destructive interference between neutron and proton interactions can strikingly reduce their sensitivity. Partial destructive interference has less of an effect on nuclei such as carbon, nitrogen and oxygen, which dominate the solar capture rate, and has no effect at all on hydrogen.
On the other hand, from Table III we see that neutrino detectors can never be disadvantaged by isospin violation (relative to other direct detection experiments) by more than a factor of ∼ 17 within the mass range considered. This "worst-case scenario" occurs when there is almost complete destructive interference (f n = −f p ) in the limit of large dark matter mass (when dark matter capture through scattering off hydrogen is very inefficient). In this case, the detectors which benefit the most relative to neutrino detectors are the ones with large atomic number (and thus a mismatch between the number of protons and neutrons).
Using Eq. (3), one can rescale a neutrino detector's reported sensitivity to σ p SD for any annihilation channel, and determine its sensitivity to σ p SI for any choice of f n /f p and the same annihilation channel. Using Eq. (6), one can rescale the sensitivity to σ Z N reported by a direct detection experiment, and obtain its actual sensitivity to σ p SI for any choice of f n /f p . In Fig. 1, we plot IC/DC's sensitivity (in the hard channel) to σ p SI for a variety of choices of f n /f p , assuming 180 live days of data. (In our figures, all curves are at the 90% C. L.) We also plot the CDMS-II bound, the current bound from XENON100, and the expected sensitivity of XENON100 with 6000 kg · days exposure. In the case of isospinconserving interactions (f n /f p = 1), IC/DC's reach is comparable with that of XENON100 in the range 260 GeV m X 800 GeV. For complete destructive interference (f n /f p = −1), current bounds are stronger than what IC/DC can achieve over the entire mass range considered. However, for m X ∼ 400 GeV (m X ∼ 1000 GeV), the 180-day sensitivity of IC/DC will exceed current XENON100 bounds for −0.84 f n /f p 1 (−0.82 f n /f p 0.28). Moreover, for m X ∼ 100 GeV, the sensitivity of IC/DC will exceed current CDMS bounds for −0.87 f n /f p −0.46. We thus see that for wide ranges of m X and f n /f p , IC/DC's sensitivity with 180 days of data (assuming hard channel annihilation) will exceed current bounds on dark matter-nucleon spin-independent scattering. Note that for f n /f p = −0.7 (the value for which the sensitivity of a xenon detector is maximally suppressed by isospin violation), the sensitivity of IC/DC exceeds current bounds as well as the expected sensitivity of XENON100 over the 50 − 5000 GeV mass range; see Fig. 2.  [18] (blue) and XENON100 [10] (black), expected sensitivity for XENON100 [19] (green), and IceCube (80 strings) with the DeepCore extension (6 strings) in the hard channel (red), are shown. The hard channel is annihilation to ττ for m X < 80 GeV, and annihilation to W + W − for m X ≥ 80 GeV.
In Fig. 3, we show IC/DC's estimated sensitivity with 180 days of data compared to the projected sensitivity of several future experiments, such as XENON1T, SuperCDMS, MiniCLEAN, DEAP-3600 and CLEAN (using both neon and depleted argon). We plot these for f n /f p = 1, −0.7, −0.82. f n /f p = −0.82 is the value for which an argon detector's sensitivity is maximally suppressed. IC/DC with 1800 days of data will have roughly three times the sensitivity estimated with 180 days of data.
We see that for some ranges of f n /f p , IC/DC will be competitive with XENON1T, DEAP-3600 and CLEAN (depleted Ar). However, CLEAN (Ne) typically will have better sensitivity  Fig. 1, for f n /f p = −0.7 (dashed curves), including current bounds from CDMS-II (blue) and the expected sensitivity for XENON100 (green) and IC/DC (red). Solid curves show the isospin-conserving (f n /f p = 1) bounds and sensitivities for comparison. The gray dotted curve is the expected IC/DC sensitivity to σ SD for the hard channel, which is translated into a σ p SI sensitivity by assuming all captures are due to SI scattering.
than IC/DC can achieve. This result is not unexpected, as neon has about as many neutrons as protons, and thus a neutrino detector will see very little relative gain in sensitivity compared to a neon detector; see Table II.

IV. CONCLUSIONS
We studied the prospects for spin-independent isospin-violating dark matter-nucleon scattering searches at neutrino detectors, with a focus on IceCube/DeepCore using the latest estimates of its sensitivity.
We found that isospin violation can have a very dramatic effect on the sensitivity of neutrino detectors relative to direct detection experiments. The "worst-case scenario" for neutrino detectors is complete destructive interference between proton and neutron interactions. But even in this case, neutrino detectors cannot be disadvantaged by more than a factor ∼ 17. On the other hand, isospin violation can disadvantage direct detection experiments relative to neutrino experiments by up to three orders of magnitude. This difference is largely due to the many different nuclei in the Sun, including the presence of hydrogen, which is immune to the effects of destructive interference.
We plotted the expected limits from IC/DC (assuming that dark matter annihilates to the "hard" channel) and XENON100 and current limits from CDMS-II and XENON100 to σ p SI , for a variety of choices of f n /f p . For the standard assumption of isospin-conserving interactions, IC/DC's projected sensitivity after 180 live days is comparable with that of  [21] as labelled. IC/DC's sensitivity with 1800 days of data will be roughly three times better than that with 180 days of data.
other detectors in the mass range 260 GeV m X 800 GeV. For complete destructive interference f n /f p = −1, current bounds exceed IC/DC's sensitivity. But the most optimistic scenario for the relative sensitivity of a neutrino detector is for f n /f p ∼ −0.7; for this scenario, IC/DC's sensitivity exceeds that of XENON100 over the entire 50 − 5000 GeV range. It thus appears that for this class of IVDM models, IC/DC may indeed provide the best current prospect for dark matter detection for a wide range of parameters.
We also compared IC/DC's detection prospects with 180 days of data (its sensitivity improves by ∼ 3 with 1800 days of data) to that possible with upcoming direct detection experiments, like XENON1T, SuperCDMS, and the CLEAN family of neon/argon detectors. Although IC/DC would not be able to compete with a neon-based CLEAN detector, it could (depending on the nature of isospin violation) provide sensitivity competitive with the next generation of argon, germanium and xenon-based detectors.
In Table IV we present C SI 0 (m X , f n /f p ) = Γ SI C (m X , f n /f p )/σ p SI for several values of f n /f p between −1 and 1. For values of f n /f p outside this range, we instead definẽ Table V presentsC 0 in the range −1 ≤ f p /f n ≤ 1. Finally, in Table VI we