Constraints on Spin-Dependent Dark Matter Scattering with Long-Lived Mediators from TeV Observations of the Sun with HAWC

We analyze the Sun as a source for the indirect detection of dark matter through a search for gamma rays from the solar disk. Capture of dark matter by elastic interactions with the solar nuclei followed by annihilation to long-lived mediators can produce a detectable gamma-ray flux. We search three years of data from the High Altitude Water Cherenkov Observatory and find no statistically significant detection of TeV gamma-ray emission from the Sun. Using this, we constrain the spin-dependent elastic scattering cross section of dark matter with protons for dark matter masses above 1 TeV, assuming an unstable mediator with a favorable lifetime. The results complement constraints obtained from Fermi-LAT observations of the Sun and together cover WIMP masses between 4 GeV and $10^6$ GeV. The cross section constraints for mediator decays to gamma rays can be as strong as $\sim10^{-45}$ cm$^{-2}$, which is more than four orders of magnitude stronger than current direct-detection experiments for 1 TeV dark matter mass. The cross-section constraints at higher masses are even better, nearly 7 orders of magnitude better than the current direct-detection constraints for 100 TeV dark matter mass. This demonstration of sensitivity encourages detailed development of theoretical models in light of these powerful new constraints.

We search three years of data from the High Altitude Water Cherenkov (HAWC) observatory and find no statistically significant detection of TeV gamma-ray emission from the Sun. Using this, we constrain the spin-dependent elastic scattering cross section of dark matter with protons for dark matter masses above 1 TeV, assuming an unstable mediator with a favorable lifetime. The results complement constraints obtained from Fermi-LAT observations of the Sun and together cover WIMP masses between 4 and 10 6 GeV. The cross section constraints for mediator decays to gamma rays can be as strong as ∼ 10 −45 cm −2 , which is more than four orders of magnitude stronger than current direct-detection experiments for 1 TeV dark matter mass. The cross-section constraints at higher masses are even better, nearly 7 orders of magnitude better than the current direct-detection constraints for 100 TeV dark matter mass. This demonstration of sensitivity encourages detailed development of theoretical models in light of these powerful new constraints.

I. INTRODUCTION
A variety of astrophysical observations, including galaxy rotation curves, large scale structure and cosmic microwave background (CMB) measurements, point towards the existence of non-baryonic dark matter in the Universe [1][2][3][4][5]. Testing the particle nature of dark matter candidates through their interactions with baryonic matter is a key aspect of research in physics beyond the Standard Model (SM).
The scattering cross section of weakly interacting massive particle (WIMP) dark matter can be studied in astrophysical environments of high matter density, such as the Sun. WIMPs from the galactic dark matter halo can be gravitationally trapped by the Sun through scattering off solar nuclei, and settle into thermal equilibrium at the core [6][7][8][9][10][11]. The overdensity of dark matter in the core can result in the annihilation of dark matter into SM particles [12][13][14][15][16]. Once equilibrium has been reached, the flux of the annihilation products only depends on the capture rate, and therefore, the scattering cross section (see Sec. II).
If dark matter has only spin-dependent elastic scattering interactions, the best sensitivity from direct-detection experiments [17][18][19][20] is several orders of magnitude weaker than for spin-independent scattering [21][22][23][24][25][26]. For studying spin-dependent cross sections, indirect methods based on WIMP capture in the Sun (with abundant hydrogen targets) can be substantially more sensitive than directdetection techniques [27,28]. IceCube [29], ANTARES [30] and Super-K [31] have performed searches for the neutrino signatures of annihilating dark matter in the Sun, and constrained the cross sections up to an order of magnitude better than direct-detection experiments for dark matter masses above a few hundred GeV.
WIMP annihilations also produce gamma rays, though they are extinguished by solar matter. In typical WIMP scenarios, the probability of observing a gamma-ray signal from the Sun is extremely low. The thermalized dark matter profile is peaked at the Sun's core, with a very small annihilation rate outside the solar atmosphere [13,[33][34][35] gamma-ray flux that could be probed with ground or satellite-based detectors, as shown in Ref. [34]. A different scenario -with enhanced prospects of gamma-ray detection -comes from models in which dark matter annihilates into a long-lived mediator that could escape and decay outside the Sun to produce gamma rays, electrons or other SM particles [14-16, 32, 36-57], as illustrated in Fig. 1 and detailed further in Sec. II. A fairly minimal dark sector contains a dark matter candidate, along with a mediator, which allows interaction between the dark and SM sectors. Dark mediators appear naturally in many ultraviolet complete theories, and include examples such as dark photons, dark Higgs, and axions [38][39][40][41]58]. If the mediators are light or have small couplings, they can be long-lived, and can decay outside the Sun into detectable gamma rays.
The prospects for detecting TeV signals from the decay of long-lived mediators outside the Sun with HAWC were first studied in Ref. [32]. It was predicted that the solar gamma-ray channel can provide very strong sensitivity to the dark matter scattering cross sections in the spin-dependent parameter space. In this work, we follow up with observations of the TeV Sun. The High Altitude Water Cherenkov (HAWC) Observatory can search for gamma rays from the Sun in an energy range that was not accessible before. We discuss the analysis and the resulting constraints on gamma rays above 1 TeV obtained by HAWC in a companion paper [59]. Our search for gamma rays from the Sun falls within an active part of solar cycle 24 (2014-2017) which is important for dark matter searches from the Sun, as described in Sec. III.
The paper is structured as follows. Section II outlines the mechanism of dark matter scattering and annihilation in the Sun. Section III reviews the search for GeV-TeV gamma rays from the Sun and describes the HAWC detector. In Section IV, we calculate the constraints on spin-dependent scattering for various annihilation channels, providing strong new limits. Section V concludes the paper.

II. DARK MATTER IN THE SUN
We briefly review WIMPs from the dark matter halo that are captured by the Sun. WIMPs can lose kinetic energy via scattering and settle into thermal equilibrium in the core of the Sun [6-8, 12, 15, 60-63]. The overdensity of dark matter in the core can result in dark matter annihilation into SM particles. Evaporation is not important for dark matter masses above a few GeV [64,65]. Ignoring self-interactions [66], the number of dark matter particles N in the Sun, at a time t, can be written as a function of the capture and annihilation rates [8,32], where Γ cap is the capture rate, and C ann is a factor accounting for the annihilation cross section and the dark matter number density. Initially, when the Sun was formed, the capture rate far exceeded the number of annihilation events per unit time, Γ ann . Eventually, when capture and annihilation reach equilibrium (dN/dt = 0), the annihilation rate becomes, The factor of 1/2 accounts for two dark matter particles being depleted in each annihilation event. The annihilation rate in equilibrium is independent of the annihilation cross section σ A v , and is set by Γ cap , which depends on the scattering cross section and the local halo mass density, among other things [35,67]. Observed signals of annihilation would be a direct probe of the WIMP capture rate and therefore, the spin-dependent cross section σ SD [13,32,68]. In addition, it may be possible to determine the WIMP mass m χ through a cutoff in the spectrum of its annihilation products. The angular profile of the region where annihilation is concentrated is narrow and embedded deep within the Sun [35]. Detecting a dark matter signal in gamma rays, therefore, is only possible in models in which the annihilation proceeds via long-lived mediators, as shown in Fig. 1. In the Sun's core, the dark matter first annihilates into a boosted long-lived mediator particle. The mediator can escape the Sun, decaying outside through observable SM channels. For a discussion of the various fields that can mediate the interaction of dark matter to photons, see Refs. [38,69]. For mediators that decay outside the Sun, the energy flux from dark matter annihilation is given by, where Γ ann is the rate of annihilation, R i is the branching ratio into the ith channel, D is the distance between Sun and Earth, and L is the decay length of the mediator. An important pre-requisite for an observable signal is that the mediator has a sufficiently long lifetime τ or decay length L, exceeding the solar radius R , so that the gamma rays are not extinguished [14,32,38,57]. The decay length is related to the mass m χ of dark matter particle, the mass m Y of the mediator, and the mediator lifetime by Observations of the Sun can therefore jointly constrain the mediator lifetime and the WIMP-proton scattering cross section [32]. In this work we consider the optimal case where L ∼ R , such that the mediator decays just outside the Sun, producing a gamma-ray signal that would be correlated with the center of the solar disk.

III. SOLAR GAMMA-RAY OBSERVATIONS
In this section, we describe the dominant astrophysical foreground for solar dark matter gamma-ray searches, and why the time window for our search is ideally situated to reduce this foreground. We also describe the GeV-TeV data sets used to set limits on the dark matterproton spin-dependent elastic scattering cross section.
For solar dark matter searches, the sensitivity to gamma rays is accompanied by a challenge: significant foregrounds that are not well understood [70][71][72][73][74][75]. These foreground gamma rays are due to cosmic-ray interactions with solar matter and photons. The Sun has been observed in MeV-GeV gamma rays by satellite detectors, leading to the identification of two distinct components [74,[76][77][78][79][80]: emission from the solar disk due to hadronic cosmic rays producing pions in collisions with solar gas, and a spatially extended ∼ 20 • halo due to the inverse-Compton upscattering of solar photons by electron cosmic rays.
A dark matter signal would be distinguishable from a cosmic-ray induced flux by its hard spectrum and a cutoff at the dark matter mass (see Fig. 2). Moreover, the flux of GeV gamma rays detected by the Fermi-LAT from the solar disk shows a distinct variability in time [74,75]. Energy Flux [ TeV cm FIG. 2. HAWC 95% C.L. limits on gamma-ray flux from the Sun using three years of data [59]. The observed flux and the 90% upper limits by the Fermi-LAT spanning the same period are shown in red (2014-2017) [70]. The Fermi-LAT solar min. flux was measured during the solar minimum of the solar cycle 24. The dashed line shows the theoretical maximum flux produced by hadronic interactions [70]. Also shown are a subset of predictions for various dark matter masses obtained using nominal annihilation rates allowed by the Fermi measurements, showing the power of the new HAWC limits.
The measured flux is anti-correlated with the solar activity, whereas gamma rays from dark matter annihilation should be steady in time. Thus, a search for dark matter signals from the Sun is most viable during periods of high solar activity when the foregrounds are low. As noted in Refs. [70,74,75,81], the periods of relatively high solar activity correspond to a lower gamma-ray flux and a softer spectrum from the solar disk. The three-year time window considered here is expected to have a lower gamma-ray flux than during solar minimum, and hence can give stronger constraints on dark matter.

A. HAWC Search for TeV Gamma Rays
The HAWC observatory is located at an altitude of 4100 m above sea-level in the state of Puebla, Mexico. With an area of 22,000 m 2 and an instantaneous fieldof-view of 2 sr, HAWC continuously surveys the sky for gamma rays in an energy range from ∼1 TeV to more than 100 TeV. HAWC consists of an array of 300 tanks; each tank contains about 200,000 liters of purified water and four photomultiplier tubes attached to its floor. Cosmic rays and gamma rays initiate particle showers in the atmosphere and produce Cherenkov light in the tanks as the particle shower passes through the water. The detection technique allows for continuous operation and gives HAWC the unique ability to observe the Sun. A complete description of the detector, data-selection and reconstruction procedure is given in Refs. [82][83][84][85].
We analyzed data collected by HAWC between November 2014 and December 2017 [59]. The HAWC period of observation falls in the second half of solar cycle 24, when the Sun is slowly becoming less active over time as it approaches the upcoming solar minimum. In a companion paper, we present the details of the measurement and the sensitivity of HAWC to TeV gamma rays from the Sun [59]. We focus on gamma rays from the solar disk and, with no significant detection, our analysis rules out a gamma-ray flux above a few times 10 −12 TeV −1 cm −2 s −1 at near 1 TeV, approaching a sensitivity near 10% of the flux from the Crab nebula. Figure 2 shows constraints on the energy flux obtained by HAWC.

B. Fermi Search for GeV Gamma Rays
We also use GeV data from Fermi-LAT observations of the Sun, covering the same time period as HAWC data (Fig. 2). The observed gamma-ray flux and upper limits from Fermi-LAT up to 400 GeV outside the solar minimum [70,75] allow us to further constrain the annihilation rates studied in Ref. [32]. With the updated GeV results from Fermi-LAT and TeV limits from HAWC, we are able to compute cross section limits for dark matter masses between 4 and 10 6 GeV.

IV. DARK MATTER CONSTRAINTS
Here we present our results for dark matter scenarios with long-lived mediators, in light of the gamma-ray data described in Sec. III. We calculate new constraints on the spin-dependent cross section of dark matter with protons.

A. Calculated Dark Matter Signals
To set limits on the WIMP-proton scattering cross section σ SD , we use Eq. (3) and the HAWC constraints on the gamma-ray flux, for a given mediator lifetime, dark matter mass and branching ratio. We note that this is a conservative calculation based on letting the signal be 100% of the data or limits.
Assuming equilibrium has been reached, the annihilation rate Γ ann of solar dark matter is related to the capture rate as Γ ann = 1 2 Γ cap . We note that the standard thresholds of masses and cross sections for which equilibrium is reached [86] can be substantially extended by including Sommerfeld enhancement [87] of the dark matter annihilation rate in the Sun [50]. This is naturally present in the case that the mediator is sufficiently light, which is a common property of long-lived mediators. We use an implementation of DarkSUSY [88,89] to compute the annihilation rate under this assumption, for a given dark matter scattering cross section and mass.
We generate the dark matter energy spectra using Pythia8 [90]. We simulate an effective resonance with twice the dark matter energy, which decays to two mediators Y Y , which then decay to two SM final states. These SM states are either gamma rays themselves (χχ → Y Y → 4γ), or produce gamma rays via radiation, or hadronic decays. We simulate the fully decayed spectra in vacuum. We assume 100% branching fractions into individual SM final states. The limits on individual states can be rescaled individually, however for a full model with combinations of varying final states, there would be cross-pollution and the limits would vary.
Our approach is independent of mediator mass, provided that the mediator has sufficient boost factor γ = m χ /m Y to escape the Sun, i.e., L = γβτ γcτ > R . (5) There is also the possibility that the mediator is produced at an angle away from the Earth, and the gamma rays are absorbed by the Sun. Such mediator gamma rays will contribute only to the lower energy part of the dark matter energy spectrum. Our limits are set primarily by the high-energy part of the dark matter spectrum, so such effects do not affect the results. We emphasize that we are studying the most optimistic scenario. We assume the mediator decays just outside the Sun, i.e., L R . The gamma-ray flux varies by only a factor of ∼ 2 across the target decay lengths between the solar surface and the Earth (see Ref. [32]), when only considering the exponential decay factors. This also means that the Sun is considered as a point source. While the angular resolution of HAWC [91] at high energies, is better than the 0.5 • angular diameter of the Sun, our analysis conservatively uses a larger region of interest to account for Sun-shadow effects, effectively studying it as a point source [59]. So long as the mediator is highly boosted (as we assume throughout) the decay products will move radially from the Sun, producing point-source emission as observed from Earth. If the mediator is not highly boosted, however, the size of the emission region depends on the mediator lifetime. Longer-lived, slowly moving mediators would produce more diffuse signals that would decrease HAWC's sensitivity to the solar gamma-ray signal [36]. As such, we assume as per the optimal scenario that the signal strength only depends on γcτ .
We assume the dominant dark matter annihilation mode is two mediators that decay to SM states. This generically produces the same dark matter energy spectra in Pythia8, regardless of the model properties such as mass and spin, provided the mediator is sufficiently boosted. However, in some specific dark matter models, different topologies may dominate. For example, if the long-lived mediator is a pseudoscalar, the two-mediator annihilation mode is p-wave suppressed, and instead the s-wave χχ → Y Y Y may dominate [92][93][94], leading to a different spectral energy distribution. In such scenarios, the upper limits will be different from what we found for the optimal case.
The limits would also be weaker if the assumed decay length was much shorter or longer than the optimal case. For L < R , there would be attenuation inside the Sun, with essentially any depth under the surface extinguishing the gamma rays, so that the reduced signal would just be the portion decaying outside the Sun. For L R , the sensitivity declines with the flux loss as per Eq. 3.
Our goal is not the detailed exploration of specific models. Instead, we demonstrate the power of the first strong constraints on solar TeV gamma rays as a probe of spindependent dark matter-proton scattering. With the op- The dark matter-proton spin-dependent cross section σ SD for annihilation into pairs ofbb, e + e, τ + τ − , and γγ, assuming an optimal mediator decay length equal to the solar radius; in less favorable models, which remain to be explored, the limits would be weaker. The Fermi-LAT constraints are updated from Ref. [32] using gamma-ray data from the Sun in the solar maximum (2014-2017). Also shown are the strongest direct detection constraints, obtained from PICO-60 [22].
timal scenario considered here, we go several orders of magnitude below what is presently constrained by direct searches.

B. Limits on Spin-Dependent Dark Matter Scattering
The limits we present on spin-dependent dark matter scattering require the presence of a sufficiently longlived dark mediator, for the produced gamma rays to escape the solar surface. Dark matter captured in the Upper limits on the spin-dependent dark matter-proton scattering cross section σ SD , from various gamma-ray and neutrino experiments. The thick solid lines show the limits obtained from HAWC in this work forbb, e + e, τ + τ − and γγ channels. The dashed lines following the same channel-wise color scheme show the limits from Fermi-LAT forbb, e + e, τ + τ − and γγ, updated from Ref. [32]. The HAWC and the Fermi-LAT limits are for scenarios where the mediator has a decay length L = R . The 2011 Fermi-LAT results are for mediators decaying into electrons, with decay lengths between 0.1 and 5 AU [56]. The thin solid lines show the results from IceCube for thebb, W + W − , τ + τ − channels [29], and are for scenarios with short-lived mediators (results for long-lived mediators would be better [14,32], but IceCube results are not yet available). The grey region indicates the parameter space excluded by PICO-60 [22].
Sun predominantly annihilates to two on-shell mediators (annihilation to only one mediator is phase-space suppressed). Each of these mediators decays to two SM final states, leading to a total of four SM final states. We chose four representative cases for SM final states, due to their varying spectral energy distributions: 4γ (box spectrum, hardest), 4τ (hard spectrum), 4e and 4b (softer spectra). Figure 3 shows the constraints on the spin-dependent scattering cross section obtained here for gamma rays from mediator decays to the γγ, e + e − , τ + τ − , and bb channels. For any given channel, the constraints provided by HAWC are at least three to four orders of magnitude stronger than the strongest limits provided by direct-detection methods, for dark matter masses above 1 TeV (at high masses, the gain is much more than that). Compared to a previous study of Milagro sensitivity for constraining long-lived mediator scenarios with TeV solar gamma rays, the HAWC constraints are about three orders of magnitude stronger [38].
Previously, Fermi-LAT has also searched for long-lived mediators decaying into electrons outside the Sun, and the resulting constraints for dark matter masses between 70 and 2000 GeV are stronger than the limits from solar gamma rays [56]. While the limits in Ref. [56] are set by a non-observation of electrons from the Sun, our analysis uses the observed gamma rays from Fermi-LAT, and requires 100% of the flux to contribute to the expected DM signal, which yields a less constraining but conservative result. Moreover, in this work we only consider the gamma rays produced from dark matter annihilation. For the electron final state, the gamma rays are subdominant because they are produced through bremsstrahlung. In principle, the HAWC measurement may also include a flux of electrons from the Sun which cannot be differentiated from the gamma rays. Adding the electron contribution would only improve our results, and could make the χχ → 4e channel as strong as that of χχ → 4γ. Doing so would require an estimate of the electron deflection angle from the Sun to the Earth, which is beyond the scope of the current work.
For a given channel, the constraints become weaker at higher dark matter mass. This is because the rate of capture and hence annihilation decreases for m χ 100 GeV [95]. The differences between the limits for different channels at the highest energies, depend on their spectral energy distributions relative to the HAWC sensitivity. Note that for a generic WIMP, unitarity and bound state effects become important around 100 TeV [96][97][98][99][100][101]. For model-dependent choices, such constraints and effects should be taken into account for the heaviest masses we consider.
We show constraints on the spin-dependent scattering cross section from Fermi-LAT, updated from Ref. [32] to match the time period with HAWC (away from the solar minimum). Fermi's GeV measurements are complementary to HAWC's TeV measurements, providing the strongest constraints for m χ < 1 TeV, and becoming less sensitive at higher dark matter masses. Together, Fermi and HAWC measurements provide powerful bounds for dark matter masses between 4 GeV and 10 6 GeV. Figure 4 shows the constraints on the spin-dependent scattering cross section obtained with HAWC and Fermi-LAT compared to other sensitive direct and indirect detection experiments. Among the direct-detection experiments, the most stringent constraints are provided by PICO 60 -C 3 F 8 [22]. Unlike the Xe-based detectors that have only a fraction of isotopes sensitive to spindependent scattering, PICO uses fluorine as the target nucleus which, due to its odd number of nucleons, is more sensitive to spin-dependent scattering. Also shown are the complementary neutrino channel limits from IceCube for general WIMP scenarios, where the dark matter neutrino signal could originate from the Sun's core without an intermediate long-lived mediator [29]. With long-lived mediators, there is less neutrino attenuation and the resulting constraints from neutrino telescopes can be much stronger above 1 TeV [14,32,53].
For TeV-scale dark matter with long-lived mediators, both gamma-ray and neutrino searches are prone to foregrounds due to the astrophysical emission from the Sun. This astrophysical emission includes solar atmospheric neutrinos [102][103][104][105] and gamma rays from cosmic-ray interactions (Sec. III). Once the sensitivity is good enough to detect the astrophysical flux, further improvements become more difficult, i.e., a (soft) sensitivity floor has been reached. For neutrinos, the astrophysical foreground may soon be detectable [106] and be indistinguishable from a dark matter signal due to the limited energy resolution of neutrino telescopes at the relevant energies [11,[102][103][104][107][108][109]. For gamma rays, the astrophysical foreground at TeV energies is still unknown [59]. The minimum flux of gamma rays from cosmic-ray interactions with the solar limb [73,110] is three orders of magnitude below the upper bound shown in Fig. 2. There is significant room for improvement in sensitivity to gamma rays from the Sun before the floor is reached. Even then, gamma rays from cosmic-ray interactions could be distinguished from dark matter based on their unique spectrum and time-variability.
Collider searches for missing-momentum attributed to dark matter production can also be translated to limits on the dark matter-proton scattering cross section, by making some model dependent choices [111,112]. Both ATLAS [113] and CMS [114] have recast their limits in the case where dark matter-quark interactions are mediated by an axial-vector, with coupling 0.25 to quarks and 1 to dark matter, obtaining limits on the spin-dependent scattering cross section up to about 10 −42 cm 2 for dark matter masses less than around a few hundred GeV. While these limits hold only for this specific model and parameter choices, in general, collider limits are complementary to those we obtain in this work from HAWC, which is most sensitive to higher dark matter masses.

V. CONCLUSIONS
Dark matter capture and annihilation in the Sun provides a probe of the dark matter-proton scattering cross section. In the presence of sufficiently long-lived dark mediators, the gamma rays produced can also escape the Sun, providing new ways to detect dark matter.
We have demonstrated that gamma-ray measurements of the Sun are exceptionally sensitive to the dark matter scattering cross section. Using new data from HAWC's search for gamma rays from the solar disk, we place the strongest existing constraints on TeV dark matter and its spin-dependent scattering rate, assuming an optimal long-lived mediator lifetime. We also update limits from Fermi-LAT with data during the same period of observation, placing the strongest existing constraints on spin-dependent scattering of GeV dark matter. Together, Fermi and HAWC place severe bounds on spin-dependent dark matter scattering, for dark matter masses between 4 GeV and 10 6 GeV. Under optimal model assumptions, this reaches down to scattering cross sections of about 10 −45 cm 2 , outperforming leading direct detection experiments by many orders of magnitude, especially for large dark matter masses.
Long-lived mediators are present naturally in many new physics models, especially if a dark mediator has small couplings or a low mass. Our first strong constraints on solar TeV gamma rays provide a new, powerful way to probe theoretical models of long-lived mediators, along with the nature of dark matter. The constraints are the strongest across a wide range of dark matter masses, and will be important for future studies of new physics. These new bounds invite detailed exploration of which model-specific realizations of the long-lived mediator scenario are not eliminated.