Impeded Dark Matter

We consider dark matter models in which the mass splitting between the dark matter particles and their annihilation products is tiny. Compared to the previously proposed Forbidden Dark Matter scenario, the mass splittings we consider are much smaller, and are allowed to be either positive or negative. To emphasize this modification, we dub our scenario “Impeded Dark Matter”. We demonstrate that Impeded Dark Matter can be easily realized without requiring tuning of model parameters. For negative mass splitting, we demonstrate that the annihilation cross-section for Impeded Dark Matter depends linearly on the dark matter velocity or may even be kinematically forbidden, making this scenario almost insensitive to constraints from the cosmic microwave background and from observations of dwarf galaxies. Accordingly, it may be possible for Impeded Dark Matter to yield observable signals in clusters or the Galactic center, with no corresponding signal in dwarfs. For positive mass splitting, we show that the annihilation cross-section is suppressed by the small mass splitting, which helps light dark matter to survive increasingly stringent constraints from indirect searches. As specific realizations for Impeded Dark Matter, we introduce a model of vector dark matter from a hidden SU(2) sector, and a composite dark matter scenario based on a QCD-like dark sector.


Introduction
Many theories in particle physics live through an infancy in which they are carved out by a few pioneering masterminds, a youth characterized by wild enthusiasm in the broader community, an adulthood in which they become part of university curricula, and the sunset years during which lack of experimental evidence leads to disillusionment or at least fatigue in the community. Models of Weakly Interacting Massive Particles (WIMPs) may be approaching this last stage of their life cycle and may eventually fade away unless solid experimental evidence for WIMP dark matter (DM) is discovered soon. Nevertheless, this time has not come yet, and in fact WIMPs are experiencing an Indian summer with fresh ideas and models sprouting from the arXiv on a regular basis. Promising recent developments include Secluded DM [1,2], SIMP [3][4][5][6], Selfish DM [7], Forbidden DM [8,9], Cannibal DM [10][11][12][13][14][15], Co-decaying DM [16][17][18], Semi-annihilating DM [19], Boosted DM [20][21][22], and DM with late-time dilution [23]. These scenarios are characterized by a dark matter sector with non-minimal particle content and interesting, unconventional dynamics. This is also true for the scenarios we wish to consider in the present work. In particular, we consider situations in which the dynamics of DM in the early Universe is governed by a dominant annihilation channel DM DM → X X, with the special feature that the mass splitting ∆ ≡ m DM − m X between DM and X is very small, |∆| m DM . Our scenario

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is thus closely related to Forbidden DM [8,9] and Co-decaying DM [16][17][18]. We allow ∆ to be either positive or negative, and we assume that X couples also to SM particles. The dynamics of DM annihilation in the non-relativistic regime is governed by the phase space factor in the cross-section, therefore we call this scenario "Impeded Dark Matter". Explicitly, the velocity-averaged annihilation cross-section for Impeded DM has the form (see also [24]) We consider only scenarios in which σ 0 is independent of v rel at leading order, i.e. in which DM annihilation is an s-wave process. Around the time of DM freeze-out, when v 2 rel ∼ 0.26 is large enough to neglect the mass difference ∆, but low enough to treat DM and X as non-relativistic, we obtain σv rel σ 0 v rel /2. This linear dependence on v rel distinguishes Impeded DM from most other DM models, in which σv rel is either velocityindependent or proportional to v 2 rel . (Scenarios where phase space suppression leads to a linear dependence of σv rel on v rel have also been discussed recently for instance in refs. [16,17,24].) From a theorist's point of view, the small mass splitting ∆ can be explained easily, for instance if the DM and X are members of the same multiplet under a gauge symmetry, global symmetry, or supersymmetry (SUSY). Once the symmetry is broken by a small amount -as is desirable to allow X to decay to SM particles -m DM and m X become split, either at tree level or through loop effects. In the following, we investigate in particular small mass splittings arising from a dark sector gauge symmetry SU(2) d , or from an approximate global chiral symmetry in a composite hidden sector. The annihilation processes for both cases are illustrated in figure 1. We will not discuss SUSY here, but we remark that Impeded DM can be easily realized in stealth SUSY [25][26][27] under the condition that ∆ is larger than the gravitino mass.
We summarize the main phenomenological features that distinguish Impeded DM models from other scenarios.
• The linear dependence of the annihilation cross-section on the DM velocity is crucial for indirect detection: it leads to strong signals in regions of larger DM velocity such as galaxy clusters or the Galactic center, while signals from objects with low DM velocity dispersion, in particular dwarf galaxies, are suppressed.
• If ∆ < 0 (DM lighter than X), annihilation to XX becomes kinematically forbidden at too low DM velocity. This is phenomenologically relevant when v 2 rel < 8|∆|/m DM , in which case the cross-section σv rel is Boltzmann suppressed. Typical values for v 2 rel are 10 −9 in dwarf galaxies, 10 −6 in the Milky Way, 10 −5 in galaxy clusters, 0.26 at freeze-out, and < 10 −9 GeV/m DM at the epoch of last scattering relevant to the CMB limits. 1

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SU (2) Composite (a) (b) Figure 1. Dark matter annihilation in two Impeded DM models: (a) a dark SU(2) model with ∆ ≡ m DM − m X < 0, where K 1,2,3 are the gauge bosons associated with the new gauge symmetry. K 1 and K 2 are degenerate in mass, while K 3 is slightly heavier thanks to a higher-dimensional coupling. The same coupling also mixes K 3 with the SM hypercharge boson B. (b) a QCD+QEDlike composite dark sector with ∆ > 0, in which the mass-degenerate charged dark pions π ± H act as the dark matter, while their neutral partner π 0 H can decay to two dark photons A through the chiral anomaly.
• For ∆ > 0, the parametric dependence of the annihilation cross-section changes at very low velocity, making σv rel dominated by the mass splitting and independent of v rel . DM annihilation is never kinematically forbidden in this case.
The remainder of this paper is organized as follows: in section 2, we introduce a dark SU(2) model as a promising example for Impeded DM with ∆ < 0. The setup of this model is discussed in section 2.1. We calculate the relic abundance in section 2.2, and discuss the direct detection, CMB and indirect constraints in section 2.3, section 2.4 and section 2.5 respectively. As an example for ∆ > 0, we then study a dark pion model in section 3. Once again, we commence by introducing the model in section 3.1, and then investigate its freeze-out dynamics and detection prospects in section 3.2. In section 4, we conclude.

Model
Impeded Dark Matter is realized most easily when the DM particle and its annihilation partner are members of the same multiplet under a symmetry group, so that, for unbroken symmetry, their masses are exactly equal. Let us consider in particular a dark sector governed by a dark SU(2) d symmetry, with the associated gauge bosons accounting for the DM and its annihilation products. SU(2) d is broken by a scalar doublet Φ = ( is the vacuum expectation value (vev), and φ is a physical dark Higgs boson. The dark sector Lagrangian is then,

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with the potential The three SU(2) d gauge fields K 1 µ , K 2 µ and K 3 µ initially obtain equal masses due to a residual global SO(3) symmetry. A dark sector with gauge boson DM can couple to the SM in various different ways. In most models considered in the literature, the dark and visible sectors are connected through Higgs portal interactions [28][29][30][31][32][33][34][35][36][37][38][38][39][40][41]. Some models instead feature particles from the dark or visible sector that are charged under both SM and hidden gauge symmetries [42][43][44][45], use Abelian kinetic mixing between a hidden U(1) gauge boson and the SM hypercharge boson B µ [46], or invoke loop processes and higher dimensional operators [16,47] to connect the two sectors. In order to avoid introducing extra particles, we will here consider only the renormalizable Higgs portal interaction and non-Abelian kinetic mixing of the form at the non-renormalizable level. 2 Here, B µν is the field strength tensor of SM hypercharge. Non-Abelian kinetic mixing allows K 3 to couple to, and decay into, SM particles. The operator in eq. (2.5) could arise for instance from a box loop involving heavy vector-like SU(2) d doublet fermions charged under SM hypercharge and another SU(2) d singlet heavy fermion carrying the same SM hypercharge. We will assume kinetic mixing between φ and the SM Higgs boson h (eq. (2.4)) to be small compared to the mixing between gauge bosons from eq. (2.5). As long as m φ > 2m k , an assumption we will make in the following, eq. (2.4) is not needed to allow φ to decay. Instead, the dominant φ decay will be φ → K i K i (i = 1, 2, 3).
After SU(2) d breaking, the non-Abelian kinetic mixing term takes the form , where θ w is the Weinberg angle. We see that mixing affects the kinetic terms of K 3 µ and B µ , while K 1,2 µ are unaffected. To move to the physical field 2 Other operators like 1 can also contribute to mixing [16], but if the heavy fermions generating eq. (2.5) do not carry SU(2)L quantum numbers, these operators will not be generated.

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basis, we redefine thus removing kinetic mixing and properly normalizing the kinetic terms. We then apply a unitary transformation to diagonalize the gauge boson mass matrix. Henceforth, we will use the notation K 3 µ , Z µ and A µ to refer to the physical neutral gauge bosons. The mass of the physical K 3 µ is shifted by a term proportional to ε 2 relative to eq. (2.3): and thus In this expression, θ w is the weak mixing angle in the ε → 0 limit and m Z,SM is the Z boson mass in that limit. We see that ∆ > 0 is possible only in a narrow mass window in which m W,SM < m k < m Z,SM . For ε = 0, the Z boson mass is shifted to . The coupling of K 3 µ to the SM electromagnetic and neutral weak currents J µ em and J Z is given by (2.10) Note that eq. (2.6) implies a derivative coupling between K 3 , φ, and the photon, as well as couplings of K 1 µ , K 2 µ to the photon and the Z. The K 1 K 2 γ coupling can be interpreted as a DM magnetic dipole moment. These operators lead to the annihilation processes K 1 K 1 , K 2 K 2 → K 3 γ, K 1 K 2 → φγ, and K 1 K 1 , K 2 K 2 → γγ, which are phenomenologically interesting as they feature mono-energetic photons (see also ref. [48]).

Relic density
In the following, we investigate the DM relic density, in the SU(2) d model introduced in section 2.1 as a function of the model parameters. To do so, we need to solve the Boltzmann equations describing K 1,2 annihilation and K 3 decay in the early Universe. The most relevant DM annihilation process is K 1 K 1 , K 2 K 2 → K 3 K 3 , the main properties of which are summarized in the first row of table 1. Other annihilation channels, in particular K 1 K 1 , K 2 K 2 → K 3 γ, K 1 K 2 → φγ, and K 1 K 2 → ff , W + W − (see table 1) are all suppressed by ε 2 . We do not consider K 1 K 1 , K 2 K 2 → γγ in the calculation because it is suppressed by a factor ε 4 , but we still list it in table 1. We will also disregard K 1 K 2 → φγ in the following, assuming m φ > 2m k . Finally, we neglect three-body annihilation processes JHEP12(2016)033 Table 1. The dominant DM annihilation processes of the DM particles K 1,2 in the SU(2) d model. Note that the channel K 1 K 2 → φγ is kinematically not accessible for m φ 2m k . We list (from left to right), the Feynman diagrams contributing to a given process, its dependence on the relative velocity v rel of the annihilating DM particles, its possible suppression by powers of the kinetic mixing parameter ε, and its relevance for DM freeze-out, CMB constraints, indirect and direct detection.
Their cross-sectionss are suppressed by ε 2 and by three-body phase space, and are therefore expected to be even smaller than those for annihilation to monoenergetic photon, K 3 γ and φγ.
The K 3 particles produced in K 1,2 annihilation decay to SM particles through their kinetic mixing. We will assume that this decay is faster than the Hubble rate at T m k , which is the case if where g * is the total number of relativistic degrees of freedom in the Universe, α em is the electromagnetic fine structure constant, and M Pl is the Planck mass. If eq. (2.11) is fulfilled, the number density of K 3 follows its equilibrium value most of the time during

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DM freeze-out, but will deviate from equilibrium when n K 3 is very small and its decay is balanced by residual DM annihilation. If this is not the case, freeze-out can be significantly delayed and requires a significant increase in annihilation cross-section [16,17]. 3 → 2 or 4 → 2 processes can also play an important role in reducing the DM abundance if the coupling g d is large [16]. In this regime, where eq. (2.11) is violated, the temperature of the dark sector deviates significantly from that of the SM sector and evolves as ∼ log a, where a is the scale factor of the Universe. In our calculation, we focus on the parameter region where eq. (2.11) is fulfilled. The decoupling of DM from the thermal bath is described by the following coupled Boltzmann equations: where n 12 is the total number density of DM particles (K 1 and K 2 combined), n 3 is the number density of K 3 , Γ K 3 is the K 3 decay rate, and the thermally averaged annihilation cross-sections σv 11→33 , σv 11→3γ , and σv 12→ff ,W + W − correspond to the processes where we have introduced the notation x φ ≡ m 2 φ /(4m 2 k ). The annihilation cross-sections for the processes K 1 K 2 → ff and K 1 K 2 → W + W − are listed in appendix A. Note that we include only final state species lighter than m k . The thermally averaged crosssections σv rel are obtained from these expressions along the lines of [49]. The decay rate Γ K 3 receives contributions from K 3 → ff and from K 3 → W + W − . The corresponding expressions are listed in appendix B.
The dark sector and SM sector are kept in equilibrium dominantly through K 3 → ff decay and its inverse. Other 2-to-2 scattering or annihilation processes like and of the DM annihilation product K 3 for a particular choice of parameters in the SU(2) d model. Ωh 2 is obtained by scaling the instantaneous number density by the subsequent expansion of the Universe and normalizing to the critical density today. We see that the density of K 1 , K 2 (red) begins to deviate from its equilibrium value (turquoise) around x ≡ m k /T ∼ 20, while K 3 (brown) stays in equilibrium until x ∼ 50. Note that for much smaller kinetic mixing ε, the K 3 decay rate can become lower than the Hubble rate, substantially delaying DM freeze-out. (This situation was dubbed "co-decaying DM" in ref. [16]).
In figure 2, we plot the solution to the Boltzmann equations, eqs. (2.12) and (2.13), for a specific set of model parameters as indicated in the plot. We see that DM freezes out at around x f ≡ m k /T f ∼ 20, similar to a conventional Weakly Interacting Massive Particle (WIMP). (This is only true because the K 3 decay rate is faster than the Hubble rate.) At late times, around x ∼ 50, the number density n 3 of K 3 begins to deviate from its equilibrium value. At that time, n 3 is so small that K 3 production in residual DM annihilation comes into equilibrium with K 3 decay.
We study the parameter dependence of the DM relic density in figure 3. The left panel in this figure shows the value of the SU(2) d gauge coupling g d required to obtain the correct DM relic density as a function of the DM mass m k and the kinetic mixing parameter ε (dashed black contours). We see that for large DM mass larger g d is required to compensate for smaller annihilation cross-sections. At 10 −6 ε 10 −2 , the relic density is independent of ε because the dominant annihilation process in this regime, K 1 K 1 , K 2 K 2 → K 3 K 3 happens entirely in the dark sector. At ε 10 −2 , the ε 2 -suppressed annihilation channels to K 3 γ, ff , and W + W − also become significant, leading to distortion of the contours in figure 3 (a). At very small ε, on the other hand, the K 3 decay rate drops below the Hubble rate. The lingering K 3 can annihilate back to K 1,2 K 1,2 , thus reducing the net DM  [55], collider searches (ATLAS dilepton, dark green) [56], electroweak precision data (EWPD, yellow) [57], and dark photon searches (gray) . In the region below the dot-dashed gray line, Γ K3 < H(T = m k ) and the model is in the "co-decaying" regime [16]. Constraints labeled with "-line" correspond to bounds on a monoenergetic gamma ray flux from : Constraints in the m k vs. g d plane in the region 10 −6 ε 10 −3 , in which the DM relic density is independent of ε. In the gray band, the correct density is obtained. We compare to constraints from Fermi-LAT gamma ray searches in dwarf galaxies (cyan) [89,90], in the Virgo cluster (dark blue) [91], and in the inner Milky Way (dark red) [92], to exclusion limits from AMS-02 positron data (orange), and to a combination of x-ray and gamma-ray bounds from a compilation by Essig et al. (magenta) [93]. We plot the CMB constraint for a narrow window m k ∈ [m W,SM , m Z,SM ] where ∆ > 0 and use ε = 10 −3 for this constraint.
annihilation rate and delaying freeze-out unless g d is increased or m k is lowered. In this regime, "cannibal processes" such as K 1 K 1 K 2 → K 1 K 3 need to be taken into account [16]. We also note the small dip in the g d = const contours in figure 3 (a) around m k = m Z . In this region, the mixing between K 3 and the Z is large even for very small ε.

Direct detection
Direct detection of Impeded DM in the SU(2) d model is complementary to indirect searches as it is sensitive to the K 1 K 2 ff coupling, which does not contribute significantly to DM annihilation, being velocity and ε 2 suppressed (see table 1). The relevant processes for direct detection are the spin-independent t-channel reactions K 1,2 q → K 2,1 q, mediated by γ, Z, and K 3 . Since the typical momentum transferred in DM-nucleus scattering is m k , m Z , the photon mediated diagram dominates over the Z and K 3 mediated

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diagrams, so we neglect the latter. In principle, the dark Higgs boson φ can also mediate DM-nucleus scattering via mixing with the SM Higgs through the Higgs portal, eq. (2.4). However, the corresponding amplitudes are suppressed by the mass of φ, the small Yukawa couplings of the Higgs, and by our assumption that Higgs mixing is tiny. Therefore, we neglect φ-mediated scattering here. The spin-independent (SI) DM-nucleus scattering cross-section in the SU(2) d model is where E r is the nuclear recoil energy, Z is the nuclear charge, α em ≡ e 2 /(4π) and α d ≡ g 2 d /(4π) are the electromagnetic and SU(2) d fine structure constants, respectively, E in = m k v 2 in /2 is the kinetic energy of the incoming DM particle, and F SI (E r ) is the nuclear form factor [94]. To obtain the spin-independent result in eq. (2.17), we have computed the scattering of K 1 , K 2 on a scalar particle of charge Z, thus neglecting the nuclear spin. Spin-dependent scattering exists as well, but as usual constraints are much weaker as the Z 2 enhancement is absent. Note that eq. (2.17) has some similarity with the scattering cross section for dipolar dark matter [95,96]. This is not surprising as we argued in section 2.1 that the K 1 K 2 γ coupling is in fact a magnetic dipole coupling.
To calculate direct detection constraints, we use data from the LUX experiment corresponding to 332 live days [97], (see also PandaX-II results [52] from first 98.7-day data, which has comparable limit to LUX). The LUX constraint is presented in ref. [97] as a mass-dependent limit on the total DM-nucleon scattering cross-section σ n , assuming the latter to be independent of the DM velocity. This assumption is violated for the photonmediated scattering processes relevant in our SU(2) d model. Therefore, we first compute σ n in a contact operator model with a fermionic DM candidate χ, for instance L ⊃χγ µ χqγ µ q and choose the coupling such that the LUX limit is saturated. We then compute the differential event rate dR/dE r for this operator, taking into account the Maxwell-Boltzmannlike DM velocity distribution, and multiply by the efficiency for nuclear recoil events in LUX [51]. We integrate dR/dE r over the energy range 1.1 keV < E r < 100 keV to obtain the maximum total number of events N max consistent with LUX data. We then compute dE r dR/dE r also in our model. By requiring the result to match N max determined for the contact operator, we obtain a constraint on the coupling g d . This constraint is shown in figure 3 (a) in brown. We see that it is stronger than indirect bounds and collider bounds for DM masses between 10 GeV and 10 TeV.

Constraints from the cosmic microwave background
Another important constraint on any model in which DM can annihilate arises from observations of the cosmic microwave background (CMB). In particular, the extra energy injected into the primordial plasma due to DM annihilation would delay recombination and thus leave observable imprints in the CMB [98][99][100][101]. The impact of DM on the CMB is characterized by the "energy deposition yield" [102,103] p ann = f eff σv m DM . (2.18)

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Here, f eff gives the efficiency with which the energy released in DM annihilation is absorbed by the primordial plasma.
For the specific case of the SU(2) d model, we need to consider the annihilation processes shown in table 1. As in the previous sections, we neglect DM annihilation to φγ, assuming that φ is sufficiently heavy for this channel to be closed. We also note that annihilation via K 1 K 2 → ff , W + W − is subdominant at the CMB epoch because of v 2 rel and ε 2 suppression, as is K 1 K 1 , K 2 K 2 → γγ because of ε 4 suppression.
The annihilation cross-section for K 1 K 1 , K 2 K 2 → K 3 K 3 is phase space suppressed by the factor v 2 rel /4 − 2∆/m k , therefore we need to estimate the DM velocity at the time of CMB decoupling. To do so, we need to determine the temperature at which DM kinetically decouples from the SM, i.e. the temperature at which K 1,2 f → K 2,1 f scattering freezes out. (Scattering of K 1,2 on photons via t-channel K 1,2 exchange is negligible as the cross-section is proportional to ε 4 .) It turns out that, in most of the parameter space considered here, this happens no later than at T 1 MeV, when e + e − annihilation reduces the density of SM fermions by ∼ 10 orders of magnitude. Afterwards, the kinetic energy of DM drops quickly as a −2 , where a is the scale factor of the Universe. Therefore, by the time of recombination, the dark sector temperature has dropped to 10 −6 eV. We conclude that, at the CMB epoch the DM temperature is typically too low to overcome the mass splitting |∆| ∼ m k ε 2 in the process K 1 K 1 , K 2 K 2 → K 3 K 3 , except at very small ε and in a small mass window with m W,SM < m k < m Z,SM where ∆ > 0 (see eq. (2.9)). We plot the CMB constraint from K 1 K 1 , K 2 K 2 → K 3 K 3 in this narrow window, for ε = 10 −3 in figure 3 (b). We see that the resulting limit is g d 0.2(10 −3 /ε) 1/4 . Finally, we need to consider the annihilation process K 1 K 1 , K 2 K 2 → K 3 γ. For this annihilation channel, f eff can be written as where E K 3 ≈ 5 4 m k and E γ ≈ 3 4 m k . In eq. (2.19), the contributions to f eff from K 3 (f K 3 eff ) and from photons (f γ eff ) is weighted by respective energy fraction because the CMB is sensitive to energy injection into the primordial plasma. f K 3 eff is given by Here, the sum runs over all SM final states into which K 3 can decay, and f SM i SM i eff are the corresponding efficiency factors for each final state. We take these, as well as f γ eff from ref. [104]. We make the approximation here that the energy of each SM i particle is E K 3 /2 in the laboratory frame. Their actual energy is distributed around E K 3 /2, but since the energy of K 3 is very close to its mass m k , the distribution is very close to a delta function. Moreover, f SM i SM i eff changes only mildly with E SM i , therefore our assumption is reasonable. For m k smaller than the QCD scale, the calculation of f K 3 eff follows the procedure from ref. [105]. Demanding p ann < 4.1 × 10 −28 cm 3 s −1 GeV −1 [53], we obtain the constraints shown in green in figure 3 (a). We see that CMB constraints from K 1 K 1 , K 2 K 2 → K 3 γ are particularly strong at low DM mass, where the annihilation cross-section is large. In

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conventional WIMP models, they exclude thermal relic DM lighter than ∼ 10 GeV, while in our SU(2) d model, they can always be avoided by choosing ε 10 −2 , a condition that is imposed anyway by dark photon searches (gray region in figure 3 (a)).

Indirect detection
In this section, we will investigate indirect astrophysical constraints on Impeded DM in the SU(2) d model, in particular from searches for anomalous signals in continuum gamma rays, charged cosmic rays, and gamma ray lines.
The differential flux of continuum photons from a solid angle interval dΩ is where σv X is the thermally averaged annihilation cross-section for a process X, dN X γ /dE γ is the differential photon spectrum for a single annihilation reaction, and the sum runs over all accessible final states. The factor c is a symmetry factor, which is c = 4 for vector DM. It would be c = 1 (c = 2) if DM was a Majorana (Dirac) fermion. 3 The factor J(θ, φ) in eq. (2.21) is the integral over the squared DM density along the line of sight (l.o.s.) oriented in direction (θ, φ). It is given by We describe ρ DM as an NFW profile with a local DM density ∼ 0.3 GeV/cm 3 [106][107][108], and a scale radius of 20 kpc. The cosmic ray e + and e − spectra are obtained from an expression analogous to eq. (2.21), replacing dN X γ /dE γ by the corresponding spectra dN X e ± /dE e ± . The dominant contribution to continuum gamma ray and charged cosmic ray signals in the SU(2) d model comes from the annihilation channel K 1 K 1 , K 2 K 2 → K 3 K 3 . Even though we have seen above that this process is kinematically forbidden at the CMB epoch, it opens up again later, when DM particles are reaccelerated as they fall into the gravitational potential wells of newly forming galaxies and clusters. Observable signals arise from K 1 K 1 , K 2 K 2 → K 3 K 3 when K 3 decays to SM particles through its kinetic mixing with the photon and the Z. These decays contribute to cosmic e + and e − fluxes through K 3 → e + e − , and to e + , e − , and to gamma ray fluxes through final state radiation and K 3 → mesons, followed by meson decays. For m k 3 GeV, we compute the spectra dN e ± ,γ /dE e ± ,γ from e + e − → hadrons data following ref. [105]. At larger m k , we compute the K 3 decay rates to quark and lepton pairs and then use ref. [109] to obtain the resulting cosmic ray spectra.
The high-energy e + and e − can also upscatter ambient photons to gamma-ray energies via inverse Compton scattering (ICS), providing an additional secondary contribution to the gamma-ray flux. This contribution depends on the propagation of the charged particles, and so has additional uncertainties relative to the prompt photon emission from

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annihilation. For the constraints we discuss below, only those from Fermi observations of the Virgo cluster include the ICS component.
We also consider DM annihilation to final states containing mono-energetic photons, where the dominant signal channel is K 1 K 1 , K 2 K 2 → K 3 γ. In this case dN X γ /dE γ is just a δ function. As in the previous sections, we do not consider K 1 K 2 → φγ, assuming this channel to be kinematically forbidden.
We compare the predicted cosmic ray spectra to the following data sets • Fermi-LAT observations of dwarf galaxies. We use the bin-by-bin likelihood provided by the Fermi-LAT collaboration [89], based on observations of 15 nonoverlapping dwarf galaxies. Using eq. (2.14), we can translate this likelihood into limits on the annihilation cross-section σv rel 11→33 and hence g d . In computing σv rel 11→33 , we account for the different root mean square (rms) velocity v 0 in each dwarf galaxy, and we use v rel = (2/ √ π)v 0 . This approach is valid in the regime where the cross-section is linearly dependent on velocity; in the forbidden regime where ε > v 0 , it may mis-estimate σv rel (since in this case the cross-section will be sensitive to the high-velocity tail of the velocity distribution), but in this regime the cross-section will in any case be very small.
• Fermi-LAT observations of the Virgo cluster. This constraint is based on three years of Fermi-LAT data, presented in ref. [91] as upper limits on σv rel SM i SM i (m DM ), the thermally averaged DM annihilation cross-section into different final states consisting of pairs of SM particles SM i . We impose that should be below the limiting value of σv rel SM i SM i (m k /2). Here, c is the same symmetry factor as in eq. (2.21), and the last term describes the average number of K 3 decays to SM i SM i . In computing σv rel 11→33 and σv rel 22→33 , we use the rms velocity of the Virgo cluster, v 0 = 525 km/sec, and set again v rel = (2/ √ π)v 0 . We find that the most constraining K 3 decay modes are τ + τ − at m k 40GeV, bb at intermediate m k ∈ [40,200] GeV, and e + e − at m k 200GeV. The strong constraint on annihilation to e + e − at high masses arises from inverse Compton scattering of the electrons on the CMB, which produces photons in the Fermi-LAT energy range. Note that the authors of [91] multiply the DM annihilation cross-section by a boost factor to account for enhanced annihilation in overdense DM subhalos. We do not include boost factors here because (a) the size of this boost factor is highly uncertain, so constraints assuming a large boost factor are difficult to make robust, and (b) since the rms velocity in DM subhalos is much lower than in the host halo, σv rel 11→33 and σv rel 22→33 will be lower for DM particles bound in subhalos, especially for the very small subhalos that typically contribute much of the boost.

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• Gamma ray constraints from the inner Milky Way. These limits are derived in analogy to the Virgo limits, but based on the results of ref. [92], assuming an NFW profile for the Milky Way. We assume an rms velocity v 0 = 220 km/sec for the Milky Way, but we remind the reader that the velocity dispersion in the Galactic Center region is highly uncertain, see for instance [110].
• Combined x-ray, gamma ray, and e + e − limits for light DM. For low mass DM (1 MeV m k 10 GeV), Essig et al. [93] have compiled x-ray and gamma ray constraints for the annihilation channel DM DM → e + e − . They use data from the HEAO-1 [111], INTEGRAL [112], COMPTEL [113], EGRET [114], and Fermi [115] satellites. We translate these limits into bounds on g d in the same way as for Fermi-LAT limits from the Virgo cluster and the Milky Way.
• AMS-02 data on e + and e − fluxes. Monoenergetic e + e − pairs produced in K 3 decays can generate bump-like features in the cosmic electron and positron fluxes observed by AMS-02. We use in particular the AMS-02 measurement of the positron flux [116] and follow the approach of ref. [90] to derive a bound on σv rel 33 from it. In computing σv rel 33 , we assume v 0 = 220 km/sec. Note that our bound is more conservative than the one from ref. [117] since we assume larger magnetic fields in simulating e + e − propagation [118,119].
• Gamma ray line searches in Fermi-LAT and H.E.S.S. Even though the crosssection for K 1 K 1 , K 2 K 2 → K 3 γ is suppressed by ε 2 , we expect competitive limits from these channels thanks to the cleanliness of gamma ray line signatures. We derive these limits using the data from ref. [54,55], 4 (see also [121]).
The above results are summarized in figure 3 (b). We see that AMS-02 provides the most stringent constraints for 3 GeV m k 400 GeV, followed by gamma ray constraints from the inner galaxy. The constraints from Fermi-LAT gamma ray searches in dwarf galaxies provide the strongest bound for DM masses around a few GeV, where the annihilation products mostly lie below the energy threshold of AMS-02. The dwarf bounds are only a factor of few weaker than those from AMS-02 and Fermi observations of the inner Galaxy over the remainder of the mass range, and have smaller systematic uncertainties. At DM masses below 1 GeV, the dominant decay channel of K 3 is e + e − . In this case, the strongest constraint arises from limits on x-ray and gamma ray photons produced as final state radiation from DM annihilation in the Milky Way. Galactic observations offer on the one hand large statistics, and on the other hand large DM velocities, which is important for the v rel -suppressed annihilation channel K 1 K 1 , K 2 K 2 → K 3 K 3 . Note that when ε 10 −3 (above the range assumed in figure 3 (b), and in tension with other limits according to figure 3 (a)), |∆| is large and shuts off the K 1 K 1 , K 2 K 2 → K 3 K 3 channel, see eq. (2.14). This happens first in dwarf galaxies, where v 0 is lowest. Table 2. Field content and quantum numbers of the dark pion model, where stands for the fundamental representation of the dark SU(N ). We show here only the field content necessary for the Impeded DM phenomenology, but it is important to keep in mind that additional particles like heavy dark leptons are necessary for anomaly cancellation.

Model
We now switch gears and discuss a second realization of Impeded DM in a concrete model. In particular, we introduce a composite hidden sector based on an SU(N ) × U(1) gauge symmetry, analogous to the strong and electromagnetic interactions of the SM (see refs. [3,4,6,[122][123][124][125][126][127][128][129][130][131][132][133][134][135] for similar models). We assume the existence of two species of light "dark quarks" u d and d d with the charge assignments listed in table 2. We also introduce a dark scalar field φ that breaks U(1) by two units, giving mass to the dark photon. In analogy to QCD, the global chiral symmetry of the dark quarks is broken at energies below the strong coupling scale Λ N . The associated Nambu-Goldstone bosons (dark pions), π + d , π − d constitute excellent DM candidates, stabilized by a Z 2 symmetry, a residual of the broken dark U(1) symmetry. (Note that the superscripts here refer to the U(1) charge of the dark pions, not an electromagnetic charge.) Their neutral partner, π 0 d , is unstable and can decay through the chiral anomaly to dark photons.
In the broken phase of chiral symmetry, the effective Lagrangian of the model is [126] L = 1 4 where we will refer to f π as the dark pion decay constant (even though the dark π ± d are stable). The matrix U is defined as U ≡ exp(iπ a d σ a /f π ) with the Pauli matrices σ a , M is the 2 × 2 mass matrix for u d and d d , and Note that the mass matrix M is diagonal; since φ carries two units of U(1) charge, it cannot induce mixing between u d and d d even after breaking U(1) . Dark pion DM can behave as Impeded DM if there is a mass splitting between π ± d and π 0 d . Such a mass splitting could have two different origins: different u d and d d masses, m u d = m d d , and U(1) radiative corrections. We assume for simplicity that u d and d d are degenerate in mass, i.e. that dark isospin is unbroken. This means in particular that we neglect the η 0 d meson, which could mix with π 0 d if m u d = m d d . Such mixing would be

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proportional to m u d − m d d and would give a mass splitting between π ± d and π 0 d of order [126].
The mass splitting between π ± d and π 0 d is then obtained from the self-energy diagrams of π ± d through A . For light A , the mass splitting is estimated to be [136][137][138][139], In the following, we use the value Λ N = 4πf π for the dark sector confinement scale. Note that ∆ in the dark pion model is always positive, i.e. DM is always heavier than its annihilation product π 0 . Thus, annihilation is never kinematically forbidden.
In the following, we will also need the rate of the anomaly-mediated decay π 0 d → A A , which we calculate to be (3.5)

Constraints from relic abundance, direct and indirect detection
Annihilation of the DM particles π ± d in the dark pion scenario is dominated by the process is the dark pion mass. The cross-section is then given by at freeze-out, (3.6) where σ 0 = 9/(64π)m 2 π /f 4 π . The estimate in the second line of eq. (3.6) is for the time of DM freeze-out, where v rel ∼ 0.47, and assuming ∆/m π 1. From the requirement of obtaining the correct thermal relic cross-section, we then obtain (3.7) In the following, we will use this condition to determine f π as a function of m π . DM can annihilate also via π + d π − d → A A , with cross-section In the following, we will neglect this second annihilation channel on the grounds that the U(1) gauge coupling g should be much smaller than the m π /f π to keep the model QCDlike. Requiring that (σv rel ) A A < 0.1(σv rel ) 00 leads to the requirement g 0.01 m π /GeV.

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In figure 4, where we plot the parameter space of the dark pion model, this condition is satisfied below the diagonal black line.
To keep π 0 d in thermal equilibrium with the SM sector throughout DM freeze-out, the dark sector should have appreciable interactions with SM particles. This can be achieved for instance through a kinetic mixing term of the A , where F µν and F µν are the field strength tensors of the photon and the A , respectively. Requiring the scattering rate for A + f → γ + f to be larger than the Hubble rate at freeze-out gives the constraint Note that A decay to ff and its inverse are less efficient than A + f → γ + f in keeping A in thermal equilibrium at π ± freeze out if m A m π . If the A mass is similar to m π , then A decay and scattering will have similar efficiency in keeping A in equilibrium. Demanding also that π 0 d and A are in equilibrium through π 0 d ↔ A A leads to the additional requirement This condition is satisfied above the horizontal gray line in figure 4.
In direct detection experiments, dark pion DM can scatter on protons via t-channel A exchange. The scattering cross-section is (3.12) Based on this expression, we derive constraints on the model parameters from LUX data [50,51]. The result is shown in figure 4 (brown contours) for different values of ε(1 GeV/m A ), as indicated in the plot. Dark pion DM is also constrained by indirect astrophysical observations, where annihilation via π + d π − d → π 0 d π 0 d , followed by π 0 d → A A and A → SM SM leaves an imprint. We show the resulting constraints in figure 4. In this plot, we have taken m A ∼ m π 0 d /2, so that A particles decay nearly at rest. Changing the mass of A will not dramatically change our result. Constraints are obtained in the same way as for the SU(2) d model, see sections 2.4 and 2.5.
As in the SU(2) d model, the DM velocity relevant for CMB bounds is much smaller than ∆/m π , so that eq. (3.6) reduces to (σv rel ) 00 10 −23 cm 3 s −1 g m π /GeV , (3.13) where we have again determined f π from eq. (3.7), and ∆ from eq. (3.4). In fact, for the dark pion model, v 2 rel < ∆/m π holds even in galaxy clusters as long as g is not tiny. It holds in particular for g large enough to keep π 0 d in equilibrium in the   [50,51] and indirect searches. The indirect detection constraints are similar to those shown in figure 3 (b). We focus on the annihilation process π + d π − d → π 0 d π 0 d . For each combination of m π and g , the dark pion decay constant f π is determined from the relic density requirement eq. (3.7). In the large-g region above the diagonal black line, this condition is not strictly valid as annihilation via π + d π − d → A A becomes relevant. In the region below the horizontal gray line, the relic density is modified by a small π 0 d width, preventing π 0 d from maintaining equilibrium with the SM.
early Universe, i.e. above the horizontal gray line in figure 4. Therefore, we can always compute the annihilation cross-section using the v rel -independent expression in eq. (3.13). However, we now include substructure enhancement in the computation of limits from the Virgo cluster. In the plot we have used a substructure boost factor of 3 000 for the Virgo cluster. Such large boost factors may exist if there is sufficient small-scale substructure (as discussed in [91]), although assuming them could lead to overly stringent constraints. We see, however, that even for such large boost factors, the limits from Virgo are superseded by other bounds.
We see from figure 4 that constraints from dwarf galaxies and from AMS-02 are strongest for m π above few GeV, just as they were for the SU(2) d model in figure 3. The constraints on the e + e − final state by Essig et al. [93] also provide interesting limits on the dark pion model, but they are weaker than bounds from the CMB, which give the strongest constraints at m π GeV. The reason why CMB bounds are so powerful in the dark pion model, while being subdominant in the SU(2) d model from section 2 is of course

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Model SU(2) d dark gauge boson dark pion mass splitting ∆ − 1 2 ε 2 m DM , eq. (2.9) ∆ g 2 f 2 π /(2m π ), eq. (3.4) 10 −7 ε 10 −3 ε 10 −3 g 0.05 Table 3. Mass splittings ∆ and annihilation cross-sections σv rel for the two Impeded DM models discussed in this paper. In the SU(2) d dark gauge boson model, ∆ depends on the kinetic mixing parameter ε, while in the dark pion model it depends on the U(1) (dark electromagnetic) gauge coupling g . Note that the annihilation cross-section in galaxies clusters receives a boost factor (BF ) from halo substructure in the dark pion model, while a similar boost is absent in the SU(2) d model as σv rel drops at small v rel .
the different sign of ∆: for ∆ > 0, as in the dark pion model, DM annihilation can be significant even at very small DM velocity. Figure 4 shows that the dark pion model is not constrained by indirect searches for m π O(1)GeV and g ∈ 10 −3 , 1 , assuming annihilation to π 0 d π 0 d dominates. The available parameter space is thus larger than for conventional WIMP dark matter, which CMB constraints [53] and Fermi dwarf galaxy observations [89] force to be heavier than O(10 ∼ 100) GeV. This is the main success of the Impeded Dark Matter paradigm.

Conclusions
In summary, we have studied a class of dark matter models dubbed "Impeded DM", which are characterized by a very small mass splitting ∆ between the DM and its annihilation products. ∆ can be either positive or negative. For negative ∆, Impeded DM is characterized by an annihilation cross-section σv rel that grows linearly with v rel . This behavior allows for a regular thermal freeze-out, while constraints from low-v rel environments (CMB, dwarf galaxies) are suppressed. For positive ∆, the annihilation cross-section can be suppressed by the small ratio ∆/m DM .
We presented two specific models that realize the Impeded DM phenomenology (see table 3 for a summary of σv rel for the two models under different conditions). In the first one, DM comes in the form of massive gauge bosons associated with a dark sector SU(2) d group. When SU(2) d is broken, the mass of one of the three gauge bosons is changed by a small amount, typically upwards (∆ < 0). The lighter gauge bosons constitute the DM, while the slightly heavier gauge boson interacts with the SM sector through a non-Abelian kinetic mixing term induced by a dimension six operator.

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In the second Impeded DM model, the dark matter is composite. It features a confining gauge group SU(N ) and two species of dark quarks, which form dark pions. An additional U(1) (dark electromagnetism) splits the pion triplet in such a way that the DM particles π ± d are typically heavier than the neutral π 0 d , into which they annihilate. The dark and visible sectors are coupled through kinetic mixing of the dark and visible photons.
For both models, we have presented detailed investigations of the phenomenology and have constrained the parameter space using all available data from cosmology, direct and indirect detection.
We conclude that Impeded DM populates a new niche in DM model space, but a niche that is becoming more and more interesting as CMB and dwarf galaxy constraints on DM annihilation put conventional thermal relic models under severe pressure.

Acknowledgments
JK would like to thank Fermilab and the Aspen Center for Physics for hospitality and support during the final stages of this project, as well as Lufthansa for discounted WiFi access and a power outlet at a particularly critical time. WX is grateful to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and its partial support during the completion of this work. JL and XPW would like to thank Yang Bai for helpful discussion.  where v = v rel /2. We see that all of these cross-sections are proportional to v 2 rel , i.e. they are p-wave suppressed. Proportionality to ε 2 leads to an additional suppression.
The p-wave nature of K 1 K 2 annihilation to SM particles can be understood by considering that all of the above processes involve the coupling (K µ 1 ∂ µ K ν 2 − K µ 2 ∂ µ K ν 1 ). In the non-relativistic limit, only contributions involving derivatives with respect to time could in principle be unsuppressed by v 2 rel . These contributions have the form m k (ξ 0 1 ξ i 2 − ξ 0 2 ξ i 1 ), where ξ µ are the polarization vectors of the DM particles, and i = 1, 2, 3. This is a p-wave state [141], therefore the overall annihilation cross-section must be p-wave suppressed.
B K 3 decay to SM particles in the SU(2) d model The partial decay widths of K 3 for decays to SM particles are obtained from the non-Abelian kinetic mixing term eq. (2.5), after removing the mixing and rotating to mass eigenstates according to eq. (2.7). We find Γ(K 3 → uū) =