Nonabelian dark matter: models and constraints

# Nonabelian dark matter: models and constraints

Fang Chen, James M. Cline, Andrew R. Frey Physics Department, McGill University, 3600 University Street, Montréal, Québec, Canada H3A 2T8
e-mail: fangchen, jcline, frey @physics.mcgill.ca
July 2009
###### Abstract

Numerous experimental anomalies hint at the existence of a dark matter (DM) multiplet with small mass splittings. We survey the simplest such models which arise from DM in the low representations of a new SU(2) gauge symmetry, whose gauge bosons have a small mass 1 GeV. We identify preferred parameters TeV, MeV, and the annihilation channel, for explaining PAMELA, Fermi, and INTEGRAL/SPI lepton excesses, while remaining consistent with constraints from relic density, diffuse gamma rays and the CMB. This consistency is strengthened if DM annihilations occur mainly in subhalos, while excitations (relevant to the excited DM proposal to explain the 511 keV excess) occur in the galactic center (GC), due to higher velocity dispersions in the GC, induced by baryons. We derive new constraints and predictions which are generic to these models. Notably, decays of excited DM states arise at one loop and could provide a new signal for INTEGRAL/SPI; big bang nucleosynthesis (BBN) constraints on the density of dark SU(2) gauge bosons imply a lower bound on the mixing parameter between the SU(2) gauge bosons and photon. These considerations rule out the possibility of the gauge bosons that decay into being long-lived. We study in detail models of doublet, triplet and quintuplet DM, showing that both normal and inverted mass hierarchies can occur, with mass splittings that can be parametrically smaller (e.g., keV) than the generic MeV scale of splittings. A systematic treatment of symmetry which insures the stability of the intermediate DM state is given for cases with inverted mass hierarchy, of interest for boosting the 511 keV signal from the excited dark matter mechanism.

###### pacs:
98.80.Cq, 98.70.Rc, 95.35.+d, 12.60Cn

## I Introduction

In the last year, it was intriguingly suggested that a variety of observed astrophysical anomalies might be tied together by a single theoretical framework, in which transitions between states in a dark matter (DM) multiplet, mediated by new GeV-scale gauge bosons, could lead to production of lepton pairs AH (). These could explain excess electron/positrons seen by the PAMELA pamela (), ATIC atic (), PPB-BETS ppb-bets (), HEAT heat () and INTEGRAL/SPI integral () experiments (the latter via the excited DM proposal (XDM) xdm ()). In addition, it has been proposed that such transitions could account for the DAMA/LIBRA annual modulation dama () via the inelastic DM mechanism (iDM) idm (). Synchrotron radiation from the leptons could explain the WMAP haze haze (). More recently Fermi/LAT fermi () and HESS hess () have made higher precision measurements of the spectrum at TeV energies, confirming an excess above the known background, although less pronounced than the ATIC data. The DM explanation for this excess has by now been studied by numerous authors Delahaye (); Cirelli:2008pk (); Ibarra:2008jk (); Yin:2008bs (); Cholis (); MPV (); Mardon:2009rc (); Rothstein (); Hamaguchi:2009jb (); Malyshev (); Berg (); Balazs:2009wm (); Meade (); Liu (), and a plethora of models has been proposed models (), including ones where the DM decays rather than annihilates decays (). Pulsars provide a more conventional astrophysical explanation111An even more conservative interpretation is that no new source is needed to fit the data; see for example ref. katz (), or concerning the 511 keV excess, ref. lingenfelter () for many of these anomalies, but the data do not yet clearly prefer them over the DM hypothesis pulsars (). However, constraints from secondary gamma rays produced by the charged leptons (or from primary neutrinos) are rapidly closing up the allowed DM parameter space Hisano (); Bergstrom:2008ag (); Borriello:2009fa (); Cirelli:2009vg (); CyrRacine:2009yn (); Regis:2009md (); Pinzke:2009cp (); Hisano:2009fb (); Spolyar:2009kx (); Profumo (); Slatyer (); Belikov:2009cx (); Huetsi (); Kuhlen (); Cirelli (); Cholis2 (); Kanzaki (). Anticipated new data from the Fermi telescope is expected to tighten these constraints in the near future.

The theoretical paradigm we focus on here assumes that the DM transforms nontrivially under a nonabelian gauge symmetry which is spontaneously broken below the 10 GeV scale. Radiative corrections from virtual gauge bosons induce mass splittings between the DM states of order , where is the fine structure constant of the new gauge symmetry and is a characteristic gauge boson mass after spontaneous symmetry breaking. Multiple exchanges of the light gauge (or Higgs) bosons gives a Sommerfeld enhancement AH (); Sommerfeld () which can explain the large annihilation cross section needed in the galaxy, compared to the smaller one in the early universe at the DM freeze-out temperature, expected from the relic density. In our previous paper us (), we presented an SU(2) model along these lines which was designed to more easily give a large enough 511 keV signal as observed by INTEGRAL while also accommodating the PAMELA/ATIC observations.

Our goal in the present paper is to give a more comprehensive survey of models based on SU(2) gauge symmetry, considering a few different possibilities for the means of coupling the DM to the standard model, for the representation of the DM multiplet, and that of the scalars which break the gauge symmetry. We also derive some new constraints on the gauge and Higgs couplings which are particular to this class of models. We start by discussing a number of general issues which transcend the individual models.

The paper is organized as follows. Section II details the mechanism of kinetic mixing of dark and standard model (SM) gauge bosons, including its possible UV origin, and we derive a new constraint on the gauge coupling from the induced DM transition magnetic moment in the nonabelian case. Section III discusses the alternative of communication between the dark and SM sectors by Higgs mixing. We derive new constraints on diagonal Yukawa couplings of the dark Higgs to DM, from direct detection and from antiproton production in the galaxy. In section IV we discuss the concept of an inverted DM mass hierarchy for boosting the predicted 511 keV INTEGRAL signal, and the symmetry and nonthermal DM history needed to make this idea work. Section V analyzes which regions of parameter space best fit the experimental anomalies (we do not insist on explaining DAMA, since the constraints on the iDM mechanism have become so severe dama-constraints (); dama-constraints2 (),BPR ()) and constraints from diffuse gamma rays, relic density, big bang nucleosynthesis, and laboratory constraints.

In the remainder of the paper we discuss several specific kinds of models, organized according to the SU(2) representation of the DM. Sections VI, VII and VIII respectively deal with DM in the doublet, triplet and quintuplet representations. In all of these models the gauge group is simply SU(2). For completeness and contrast, in section IX we consider one model with dark gauge group SU(2)U(1) and triplet DM, which illustrates the differences between the purely nonabelian models and ones where gauge kinetic mixing occurs between U(1) field strengths. We summarize our findings in X. Appendices A and B respectively give details of the transition magnetic moment and radiative mass computations, C computes the annihilation cross sections for freeze-out of DM in a general representation, and E treats the diagonalization of the gauge boson and DM mass matrices for the SU(2)U(1) model.

## Ii Kinetic mixing of gauge bosons

A simple way of generating couplings between one of the SU(2) gauge bosons and electrons is through nonrenormalizable couplings of the form

 ∑i1ΛiYμνBμνaΔai (1)

or

 ∑i1Λ2iYμνBμνah†iτahi (2)

where and are respectively triplet and doublet Higgs fields which are assumed to get a VEV. By having several triplet or doublet fields (labeled by index ) which get VEV’s in different directions, it is possible to get mixing with several colors of the gauge boson. In (1), note that only a single linear combination of vectors mixes with the SM. With a generic Higgs potential, we can always choose the linear combination of triplets in (1) to be . However, depending on the Higgs potential, the vector that mixes with the SM may be a linear combination of several mass eigenstates.

To understand the consequences of gauge boson mixing, it is useful to start with a simple example in which a massive abelian boson mixes with the photon. The kinetic term is

 −14(FμνFμν+BμνBμν−2ϵBμνFμν)+12μ2BαBα (3)

Since the U(1) gauge symmetry of the photon is unbroken, it must remain strictly massless. This restricts the form of the transformation which diagonalizes the kinetic term to

 Aμ=~Aμ+ϵ~Bμ (4)

Therefore all particles which couple to the photon acquire a coupling of strength to the massive gauge boson.

For the models we consider, the mixing takes the form , where for concreteness we take color 1 of the nonabelian gauge boson to mix with the standard model weak hypercharge, it is straightforward to show that eq. (4) generalizes to the similar form

 Aμ=~Aμ+ϵcosθW~Bμ1 (5)

where is the Weinberg angle. One must further transform and the gauge boson as

 Bμ1 = [~Bμ1−ϵsinθW(m2Zm2Z−μ2)~Zμ](1+O(ϵ2)) (6) Zμ = [~Zμ+ϵsinθW(μ2m2Z−μ2)~Bμ1](1+O(ϵ2)) (7)

where the tilded fields are those which diagonalize the kinetic term. Therefore the gauge boson acquires a coupling to the current of the boson, in addition to that of the photon. Figure 1(a) shows an example of a transition mediated by the .

The mass of the gets shifted by a fractional amount

 δmZmZ=ϵ2sin2θWμ22m2Z (8)

relative to its usual value. For the small values of and GeV which are of interest, this is a negligible shift.

With gauge boson mixing, the annihilation results in subsequent decays of with roughly equal branching ratios for all leptons with mass below . Due to the nondiagonal couplings of to the states, assuming they are Majorana, there is no -channel annihilation through a single virtual . Hence the annihilation into 4 leptons is guaranteed. For Dirac DM, such as in the doublet representation, this need not be the case, as we will discuss in section VI.

### ii.1 Microscopic origin of gauge kinetic mixing

The dimension-5 operator (1) can be induced at one loop by a heavy particle which carries both dark charge and weak hypercharge , if it also has a Yukawa coupling to the dark sector Higgs triplet. Suppose is a Dirac fermion which transforms as doublet of the SU(2), so the Yukawa interaction is

 hX¯¯¯¯¯Xi(τa)ijXjΔa (9)

The diagram is shown in figure 2(a). It generates the effective interaction which can be estimated as

 hXyXg16π2MXYμνBμνaΔa (10)

so that the mixing parameter is given by , where is the VEV of the triplet Higgs. For couplings of order unity and GeV, can be of order TeV to generate .

Similarly, the dimension-6 operator (2) can arise from a heavy doublet scalar field with a coupling to another dark higgs doublet (or perhaps the same one, ). If has weak hypercharge , the analogous diagram with replaced by gives rise to the operator (2) with .

### ii.2 Long-lived dark gauge bosons

It is noteworthy that pure SU(2) models generically predict small gauge mixing parameters , suppressed by powers of a heavy scale, whereas models with SU(2)U(1) gauge symmetry in the dark sector allow for renormalizable mixing of SM and dark hypercharge, in which case there is no reason to expect particularly small values of . A phenomenological advantage of small is that values on the order of give the gauge boson a lifetime of order s. Such a long lifetime lets ’s produced from DM annihilation propagate away from the galactic center before decaying. This delocalizes gamma rays produced by the leptonic decay products, allowing such models to evade HESS constraints Rothstein (); Meade (). However we will show in section V.5 that gauge bosons with a lifetime greater than s are ruled out by big bang nucleosynthesis for the models considered in this work.

### ii.3 Direct decay of excited DM to photon

Because there is no mixing of to in eq. (6), there is no tree level amplitude for the decay of excited DM directly to a photon. For example in the case of triplet DM, one would have the decay if such a mixing existed. Instead the dominant decay is mediated by the . However, in the class of models with kinetic mixing between SM hypercharge and one of the dark SU(2) gauge bosons, it is inevitable for the single photon final state to arise at the loop level, as we now show. Naively, one could draw the diagram where form a loop connecting to the photon, but this just renormalizes the kinetic mixing term, so it is not relevant. There is another process which occurs due to the nonabelian nature of the , illustrated in fig. 2(b).

The novel feature of the gauge mixing operator is that contains the term . There is thus a trilinear vertex coupling these gauge bosons to the weak hypercharge field strength, with strength . One consequence of this interaction is the generation of a transition magnetic moment for the DM. An example is shown in fig. 2(b) for the case of DM in the triplet representation. A magnetic moment interaction of the form arises, where is expected to be of order . A careful computation of the loop diagram given in appendix A gives

 μ23≅ϵg3cW128π2Mχ(lnMχμ−1) (11)

where is the scale of the nonabelian gauge boson masses and . It is straightforward to compute the rate for ,

 Γγ=μ2238π(δM23)3 (12)

where is the energy available for the decay. On the other hand, the rate for is approximately times the spin-averaged squared matrix element,

 ⟨|M|2⟩≅32g2e2ϵ2c2WM2χμ4(E+E−+→p+⋅→p−−m2e) (13)

(where and are the energy and 3-momenta of the electron and positron, respectively). This varies approximately linearly over the allowed phase space, so we estimate the integral as being

 Γ2e≅4ϵ2ααg(δM23−2me)3(δM23+2me)2/μ4 (14)

The branching ratio for the single photon versus the two lepton decay is thus

 BRγ=c2Wα2g/α8192π2μ4(δM23)3M2χ(δM23−)3(δM23+)2ln2Mχeμ (15)

where and . Taking but allowing for the possibility that , we can write

 BRγ ≅ 2.6×10−4α2gα(μ200 MeV)4(1 TeVMχ)2 (16) × (100 keVδM23−)3

The reference values chosen here are compatible with constraints which we will discuss in later sections, and small values of enhance the size of .

Even though the branching ratio for due to the magnetic moment is small, the observable signal due to this process, in the diffuse gamma ray background, is distinctive. If the dark matter was at rest, it would produce a monoenergetic photon with MeV. Since the central galactic DM has a velocity distribution with dispersion , the spectrum of the photon is Doppler broadened with a width of order keV for MeV. This is just below the keV resolution of SPI. The nonobservation of such a signal by INTEGRAL thus provides a new constraint on models with -parameter type mixing of the nonabelian gauge boson with weak hypercharge.

To determine the constraint, we can compare the new direct photon signal with that of the 511 keV line already observed by INTEGRAL. The latter is seen with a confidence level (c.l.) of 50 and a signal to background ratio () of a few percent. One can predict the c.l. of the new signal from that of the 511 keV line through the relation

 (c.l.)new=(c.l.)511BRγ(S/B)new(S/B)511(σ511σnew)1/2 (17)

where keV is the width of the 511 keV line and keV is the resolution of the detector (which is approximately the same as the intrinsic line width). To understand the dependence on width, notice that for fixed flux, increasing the width of a line reduces the signal proportionally (), but for fixed signal-to-background, it increases the counting statistics by since a wide line of a given intensity has more flux than a narrow one. These effects combine to give the dependence. The background for the 511 keV line is dominated by the positronium continuum and annihilations of positrons in the INTEGRAL telescope, effects which are both absent for the new signal. On the other hand, there is a broad instrumental line near 1.8 MeV which is the dominant background for the narrow galactic Al line al26 (), whose signal to background ratio is around 70/30. Putting these numbers together, and assuming that to avoid a detection, we find the limit

 αg\raisebox{-2.58pt}{~{}\lx@stackrel<∼~{}}0.08(200 MeVμ)2(Mχ1 TeV)(δM23−100 keV)3/2 (18)

(recall that ). It is interesting that such reasonable values of the dark gauge coupling could lead to an additional signal potentially detectable by INTEGRAL. However, it would require a nonthermal DM history, since we will show that smaller values of are needed for the correct relic density, eq. (V.2), or in the case of doublet dark matter, the bound (18) does not apply because the magnetic moment is suppressed by an additional factor of , as we will show in section VI.1.

## Iii Mixing through the Higgs sector

### iii.1 General features

An alternative way in which the dark matter might couple to the standard model is through renormalizable operators of the form

 λHS|H|2|S|2 (19)

where is the standard model Higgs doublet and is a Higgs field which is charged under the dark SU(2) gauge group. If gets a VEV and also has a Yukawa coupling to the DM, schematically of the form , then transitions such as can be mediated by the Higgs bosons as shown in figure 1(b). The Higgs sector has a mass matrix of the form

 (m2HλHSvHvSλHSvHvSm2S) (20)

where is the VEV of the SM Higgs . If the mixing is small, then the Lagrangian fields are related to the mass eigenstates by

 (HS)≅(1−θθ−1)(H′S′),θ=λHSvHvSm2H−m2S (21)

Therefore couples with strength to any SM model fermion whose Yukawa coupling to is . In addition, the couples to with strength . Thus the diagram involving exchange is of the same order in couplings as that with the , but at low momentum transfer it is suppressed by .

### iii.2 Constraints on diagonal couplings

#### iii.2.1 No antiproton production

An interesting qualitative difference between Higgs and gauge boson mixing is that in the former case, the Yukawa couplings are generally not off-diagonal. For example, triplet dark matter coupling to a quintuplet scalar as has diagonal couplings; similarly for doublet dark matter coupling to a triplet scalar via . In either case, the annihilation shown in fig. 3(a) occurs, resulting in quark or lepton pairs favoring the most strongly coupled fermions—the top quark. To avoid production of hadrons, since no antiproton excess is observed by PAMELA, one needs to have mixing with a scalar that has dominantly off-diagonal couplings so that annihilates primarily to a pair of bosons by virtual exchange. The bosons decay nearly on shell and hadron production can be suppressed if the is lighter than GeV. Note that it is impossible to keep the couplings strictly off-diagonal in the mass basis, once the relevant component of gets a VEV, since this contributes an off-diagonal mass term to the DM. Therefore the Higgs mixing scenario in its simplest form could be disfavored by the lack of any antiproton excess in the PAMELA data.

Moreover, diagonal couplings are constrained by direct dark matter searches, by the process shown in fig. 3(b). Translating the limit quoted in eq. (11) of ref. AH () to the present case (and assuming 200 MeV), a diagonal Yukawa coupling is bounded by

 θhsyN<16π×10−8αem ⇒ θhs<4×10−6 (22)

Here is the Higgs-nucleon coupling Cheng (). Assuming that the SM Higgs mass is much heavier than , this implies

 hsλHS<2.7×10−4(mS200 MeV)2(mH130 GeV)2(1 GeVvS) (23)

To illustrate how severe (or mild) this constraint might be, consider the case of triplet DM coupled to a quintuplet (traceless symmetric tensor) scalar , via , and the cross-coupling to the SM Higgs . Suppose for example that gets a VEV GeV to induce mixing with , with mixing angle . In addition to radiatively generated mass splittings of the DM (as we will discuss below), there is a tree level contribution so that the mass eigenstates become linear combinations, . The fluctuations of thus couple to the mass eigenstates as . Therefore the ground state can annihilate directly into a single fermion pair through a single intermediate scalar. The latter is always far off shell, so this annihilation channel is dominated by production of top quarks which hadronize and produce antiprotons, contrary to the observations. However notice that the two diagrams in fig. 3(a) interfere destructively. We can estimate the effect of these diagrams by integrating out the intermediate scalar, and using the fact that , to get the effective dimension-6 operator

 θythSm2HM4χχ¯tt (24)

On the other hand, the annihilation by exchange can be estimated from the dimension-5 operator

 h2SMχχχS′2 (25)

Assuming that the initial ’s are nonrelativistic, the ratio of the corresponding cross sections is of order

 σ(χχ→¯tt)σ(χχ→S′S′)∼θ2y2th2Sm4HM4χ (26)

The top quarks decay to quarks before hadronization, and each quark produces antiprotons (using MicrOMEGAs micro ()), so the number of antiprotons per positron is of the same order. The observed flux of antiprotons to electrons is approximately , and given that no antiprotons in excess of standard expectations are observed, we should demand that the ratio (26) not exceed this limit. For definiteness, if TeV we obtain the rather weak constraint

 λHS\raisebox{-2.58pt}{~{}\lx@stackrel<∼~{}}85hS(Mχ1 TeV)(1 GeVvS) (27)

Both (27) and the direct detection constraint (23) can be satisfied using reasonable values of the couplings.

Furthermore, if there are additional contributions to the DM mass splittings, it is possible to parametrically suppress the diagonal couplings. For example, consider a second quintuplet Higgs with coupling , and a VEV which splits the masses diagonally, . In this case, the mass eigenstates are not maximal mixtures of the flavor states; rather , , with (assuming is small). If the coupling is negligible, then the overall effect is to reduce the diagonal couplings by the factor , while leaving the off-diagonal couplings unsuppressed.

The constraint due to the assumed lack of production of antiprotons would be weakened even further if the recent claim of ref. Kane () is verified. This work questions the assumption that the observed antiproton background is actually understood in terms of physics other than dark matter annihilation.

#### iii.2.2 No two-lepton final states

In section V.4 we will discuss the fact that recent constraints on DM annihilation from the diffuse gamma ray background are more severe for models in which than final states (where is a charged lepton) due to the harder spectrum in two-body decays. This is not an issue when the intermediate particle is an SU(2) gauge boson, since its couplings are automatically off-diagonal and thus two bosons must be emitted in the annihilation, but it might be an issue for intermediate Higgs bosons with diagonal couplings. However, the result (26) can be directly adapted to the case of decays to a lepton pair instead of a top pair by substituting the lepton Yukawa coupling for that of the top. Even for the heaviest lepton, , the result is suppressed by . These annihilations are thus much more rare than those with the final states, and do not provide a stronger constraint than the one derived above, even if we only demand that the ratio be rather than .

### iii.3 Long-lived dark Higgs boson

In order to realize the long-lived intermediate state proposal of ref. Rothstein (), it is interesting to know how small a mixing angle is required to get the Higgs lifetime to be s. In section V it will be aruged that Higgs masses in the range mass MeV are the most promising for fitting PAMELA/Fermi observations, such that only the final state is available. Using the decay rate , we find that

 θ=6×10−12(100 MeVmS)1/2 (28)

is the required value. We will show in section V.5.3 that such small values are strongly excluded by constraints on the density of dark gauge bosons, which must decay before BBN.

## Iv Inverted mass hierarchy and Z2 symmetry

In the following models, a recurring theme will be whether it is possible to have a stable excited DM state which is slightly lighter than the highest excited state (the one that decays into leptons plus ground state). This “inverted hierarchy” is shown in figure 4(a), in contrast to the “normal hierarchy,” fig. 4(b). We proposed the inverted hierarchy in ref. us () as a means of boosting the galactic 511 keV signal from excited dark matter, since the transition requires less energy than and therefore benefits from a larger proportion of the DM velocity distribution.

Let us first review the mechanism of radiative mass splitting of a DM multiplet by virtual massive gauge bosons, through diagrams like that shown in fig. 5(a). Although the correction to the mass is logarithmically divergent, mass differences between members of the multiplet are finite. By choosing a suitable counterterm, the finite part which contributes to the mass splitting can be defined as

 δMi≅−12αg∑jμjTjiaTjai (29)

where and the sum runs over all the gauge bosons, with mass , which contribute in the intermediate state. The approximation (29) is valid when . Details of the derivation are given in appendix B.

If Higgs mixing rather than gauge boson kinetic mixing is the dominant portal between the dark and SM sectors, it is likely that the dominant souce of mass splittings is the tree level contributions from the Higgs VEVs. It is possible however that the analogous radiative corrections with the intermediate Higgses, fig. 5(b), have an important effect. In appendix B it is shown that the analogous formula to (29) in this case is

 δMi≅+14αy∑jmj (30)

where is the relevant Yukawa coupling for the Higgs multiplet in the loop, and is the mass individual components of that multiplet.

### iv.2 Z2 symmetry

The idea of exciting the intermediate state depends on it being significantly populated and stable on cosmological time scales. One possibility is for it to be absolutely stable, which should be guaranteed by some symmetry. Another, which has been explored in ref. FSWY (), is that the state is only metastable. In section V.6 we will discuss that this scenario is strongly constrained by direct detection considerations. In this paper we will highlight models that admit a discrete parity, which not only ensures the stability of the intermediate state, but also forbids transitions between it and the neighboring states, that could be coupled to currents of SM particles.222Such transitions, if they exist, can always mediate decays , as in fig. 17(a). The absence of these transitions makes the models safe from the direct detection constraints. (For other references discussing symmetries which stabilize DM, see walker ().)

The simplest example is triplet DM in which only one gauge boson color, say , mixes with the SM hypercharge. In this case we can assign conserved charges to the fields

 χ1, χ3,B1,B3 (31)

and to no others. Suppose that is the ground state and the heaviest state. Because of the symmetry, can never decay into plus SM particles. It could in principle decay into , but this is kinematically blocked by the mass of the . From the point of view of the symmetry, there is no light particle that can appear in the final state to compensate the charge of the .

Alternatively, we can state the condition that would make it impossible to keep the intermediate state stable. From the above argument we see that a necessary requirement is to be able to assign charge to the ground state. Therefore the highest excited state of interest must also be charged. If any gauge boson which mediates transitions between the intermediate state and either of the charged states mixes with SM hypercharge, then charges cannot be consistently assigned.

### iv.3 Z2×Z′2 symmetry for quintuplet DM

The issue of having a stable intermediate state does not arise for DM in the doublet representation, but it can be applied to higher representations, such as the symmetric tensor (quintuplet). We can label the canonically normalized states of by

 ⟨χ⟩=⎛⎜ ⎜⎝A−B/√3CDC2B/√3FDF−A−B/√3⎞⎟ ⎟⎠ (32)

The transitions mediated between these states by the three gauge bosons are shown in figure 6. Let us consider how to assign charges to the states in a systematic way. First, suppose that one of the gauge bosons, say , mixes with SM hypercharge. Then must not carry charge, while the other ’s do; call these . This implies that some subset of ’s which appear only linearly and not bilinearly in the gauge interactions of the ’s should also be charged. The states in must also have the property that they only appearly bilinearly and not linearly in the interactions of .

Using this logic, we can make an exhaustive list of the possible charge assignments for a given choice of the that mixes with hypercharge, which we denote by . In the process, we discover that actually the global symmetry is larger than just ; for a given subset of states, its complement could also have been chosen. This means that we can assign charge to states in , and a separate to states in . Meanwhile, the two gauge bosons other than transform under both and . The result is

 B1↔e+e−: X={C,D}, Y={A,B,F} B2↔e+e−: X={C,F}, Y={A,B,D} B3↔e+e−: X={D,F}, Y={A,B,C} (33)

It turns out that the and states always mix to form the heaviest () and lightest () mass eigenstates. We therefore take the heaviest state relevant for the INTEGRAL transition to be one in , and this dictates that the intermediate state whose stability is to be guaranteed is the lightest one in . The symmetry then insures that the -charged intermediate state cannot decay into the -charged lowest state, since both symmetries would be violated. We will give explicit examples in section VIII.2.1.

In order for this to work, at least one of the ’s must be left unbroken by the VEVs of the Higgs fields. If there is only one triplet Higgs which gets a VEV to accomplish kinetic mixing, this presents no difficulty since then the Higgs components can transform in just the same way as the corresponding gauge fields, preserving both ’s. Moreover components of a quintuplet Higgs can be given the same charges as the corresponding DM components, so one can be preserved as long as VEVs appear only in the or subsets, but not both.

VEVs of additional doublets break all of the discrete symmetries, but multiple triplet VEVs can be consistent with the symmetries if they are orthogonal. Consider two triplets with VEVs and in the 1 and 2 directions, respectively, and suppose that is used to generate kinetic mixing between and the SM. Then a single is preserved, under which the fields , , , and change sign, while , and do not. Adding a third triplet with VEV in the 3 direction is also consistent with the , if transforms under it.

### iv.4 Nonthermal history

Even though symmetry guarantees the stability of the intermediate state , it cannot prevent depletion of its density in the early universe, through exactly the same process needed for the INTEGRAL signal, namely followed by decay. Even more simply, the depletion could occur directly by . In ref. us (), we noted that this depletion could be prevented if the ’s were produced out of thermal equilibrium rather than through the standard freeze-out. If the ’s are decay products of a supermassive scalar , their initially high energies suppress the annihilation cross section sufficiently long to keep the excitation or the relaxation out of thermal equilibrium in the early universe.

In more detail, suppose that the gauge coupling is too large to yield the right relic density from freeze-out. It was envisioned that could decay at a low temperature MeV, resulting in mildly relativistic DM with momenta . The Sommerfeld enhancement is initially absent for DM with such large velocity, and in fact the rate of annihilations remains always less than the Hubble rate before cosmological structure begins to form, because and both scale like . Only when DM begins to concentrate in halos does the rate of annihilations become significant.

## V Fitting PAMELA/Fermi/HESS versus INTEGRAL/SPI, cosmology and laboratory bounds

### v.1 Fits to PAMELA/Fermi/HESS

Ref. Meade () has identified regions in the parameter space of and for the process (followed by ) which are compatible with the PAMELA/Fermi/HESS observations, as well as the HESS constraints on inverse Compton gamma rays produced by the electrons and positrons coming from DM annihilation.333A recent analysis of preliminary Fermi observations of gamma rays from the inner galaxy is also consistent with this annihilation channel Cholis2 (). As we will discuss in further detail below, additional constraints from extragalactic diffuse gamma ray production favor the models in which the ’s from annihilation decay only to and no heavier leptons. This implies that the mass of the ’s, , must be less than twice the mass of the muon, MeV. The allowed region for this scenario is reproduced in fig. 7. It should be emphasized that the two-body decay mediated by a single exchange is excluded because its electron spectrum ends too abruptly due to its near monoenergeticity Meade (); this channel also provides a very poor fit to the PAMELA data MPV (). Moreover in the class of models considered here, it would be impossible to forbid the channels where is any SM fermion, if is unsuppressed.

The best fit is in the vicinity of cm/s and TeV. This cross section exceeds that needed for the correct thermal relic density ( cm PDG ()) by a factor of , which is thus the required boost factor, assuming a thermal origin for the DM. Even if a nonthermal origin is assumed, the thermal component should be suppressed by having an even larger cross section, and thus 330 should be regarded as an upper bound on the required boost factor.

The example shown assumes an isothermal radial density profile, which eases the constraints from HESS on the inverse Compton photons by lowering the DM density near the galactic ridge. For preferred profiles such as Einasto, the fit to PAMELA/Fermi is nearly ruled out. The isothermal profile is considered to be unrealistically flat near the center compared to the results of the best N-body simulations, but it was noted in Meade () that long-lived intermediate bosons (the gauge bosons in our case) could justify such an effective profile, due to the ’s traveling away from the galactic center before decaying Rothstein (). We will show that in section V.5.1 below that big bang nucleosynthesis constraints rule out such a long-lived in the present class of models, hence the mechanism of long-lived intermediate states cannot work here.

Another way of decentralizing the region of DM annihilation has been proposed in AH (), however, which could have a quantitatively similar effect to the softer halo profile; namely DM subhalos which populate the halo could dominate as annihilation sites, due to their lower velocity dispersion and hence larger Sommerfeld enhancement. The small-velocity subhalo scenario has recently been studied in detail in ref. Kuhlen () (see also Robertson (); Bovy:2009zs (); vialactea ()), with reference to models favored by the pre-Fermi analysis of MPV (), in particular with TeV, MeV and . This happens to be close to the preferred values mentioned above; we will show in the next section that this value of is just slightly larger than the one needed to get the right relic density for triplet DM.

Still, to avoid the stronger inverse-Compton constraints on the preferred Einasto profile, it may be necessary to reduce the annihilation rate near , in addition to providing alternative subhalo regions for the annihilation. Recent work on halo formation including the effects of baryons indicates that the velocity profile steepens considerably (diverging like ) for kpc instead of leveling off to smaller values RD () as in pure DM simulations. (This reference also finds that the DM density profiles are softened near the center, a result not corroborated by other simulations which include baryons Abadi (), but the latter work does qualitatively confirm the steepening of the velocity profile julio ().) Moreover the overall magnitude of the velocity is somewhat increased for kpc. Because the Sommerfeld enhancement of the annihilation cross section scales like , this should have a similar effect to erasing the cusp of the density profile, making it more similar to the isothermal profile.

### v.2 Relic density

We have computed the early-universe annihilation cross section of DM in any SU(2) representation into dark gauge bosons. For the three representations we focus on in this paper, the result is

 ⟨σannvrel⟩≅πα2gM2χ×⎧⎪⎨⎪⎩0.14,doublet0.88,triplet5.18,quintuplet⎫⎪⎬⎪⎭ (34)

Details are given in appendix C. Using the standard value cm needed for thermal relic abundance PDG (), comparison with the cross section (34) indicates that the values of required are

 αg = ⎧⎪⎨⎪⎩0.077,doublet0.031,triplet0.013,quintuplet⎫⎪⎬⎪⎭×(Mχ1 TeV) (relic density value)

As mentioned above, there are motivations to question the assumption that DM has a thermal origin, such as our inverted mass hierarchy proposal us () (see also FSWY () and brand ()). It is important to notice that to justify a nonthermal origin, the thermal contribution must be smaller than usual so that it is subdominant to the nonthermal contribution; thus the annihilation cross section would be larger. The values (V.2) should then be regarded as lower bounds.

These bounds can be evaded if the DM has stronger Yukawa couplings to dark Higgs fields; for example triplet DM can have the coupling to a quintuplet Higgs . If then the freeze-out density is determined by and the gauge coupling can be smaller than in (V.2). In such a case, it should be kept in mind that the annihilation in the galaxy will probably also be dominated by Higgs boson exchange; notice that the mass scale of the Higgs bosons cannot naturally exceed that of the gauge bosons by a large factor, since the scale of spontaneous breaking of the dark SU(2) gauge symmetry is dictated by the mass scales in the Higgs sector. Thus late-time annihilations would likely be dominated by Sommerfeld-enhanced Higgs exchange diagrams. The expected boost factor would thus still be even in cases where is much smaller than indicated in (V.2).

### v.3 Mass splittings and the XDM (iDM) mechanism

In contrast to the above values of , the paradigm of ref. AH () would at first seem to suggest smaller values , because the radiative mass splittings of the DM multiplets go like (where is the scale of the gauge bosons masses) and it was presumed that GeV as the largest value compatible with no production of antiprotons by the decays of the gauge bosons after annihilation in the galaxy. Since the XDM hypothesis requires mass splittings of order MeV, MeV/GeV would be indicated.

However we have argued above that lighter gauge boson masses MeV are in better agreement with gamma ray constraints. The generic estimate gives MeV for such masses and the preferred coupling from section V.1. This is in just the right range for having excited DM states which can decay to and the ground state.

For other applications, like the iDM mechanism for DAMA, or our inverted mass hierarchy variant of XDM us (), it is desirable to have splittings which are perhaps smaller than the MeV scale. In section VII.2.2 we will show that with sufficiently complicated Higgs sectors (three triplets in this example) it is possible to reduce the mass splittings below the generic level of . It is also possible to design the gauge symmetry breaking (by appropriate choices of VEV’s or the DM representation) so that no mass splittings are induced by gauge boson radiative corrections; for doublet dark matter this is true regardless of the Higgs respresentations. In that case, the splittings must come from Yukawa couplings and then it is possible to decouple the scale of the gauge boson masses from that of the splitting.

It should be emphasized that getting the excited dark matter (XDM) mechanism to produce a large enough signal to explain the INTEGRAL/SPI observations is not as easy as just having the right DM mass splitting; one must generically saturate partial wave unitarity bounds for the excitation cross section to get a large enough rate PR (). We leave the details of reanalyzing this problem to work in progress new (). The same can be said (even more so) of the iDM mechanism for DAMA. The region of parameter space consistent with the DAMA annual modulation as well as other direct detection experiments is essentially excluded dama-constraints2 (), BPR (). We give less emphasis to trying to implement the iDM mechanism.

### v.4 Overcoming diffuse gamma ray and CMB constraints

We have already seen that constraints from gamma rays originating as brehmsstrahlung or inverse Compton scattering of the emitted leptons can often rule out models which would have provided good fits to the PAMELA and Fermi observations Hisano ()-Kuhlen (). Not only annihilations within our own galaxy provide such constraints, but the accumulated effect from early redshifts and other halos on the CMB and diffuse gamma ray background can be severe. For example, ref. Slatyer () obtains the 95% c.l. CMB bound

 ⟨σvrel⟩<4−8×10−24cm3/s (36)

for the model with and TeV (where their efficiency factor for transfering energy to the intergalactic medium is approximately ). This is barely compatible with the fit to PAMELA/Fermi/HESS for the same model in ref. Meade (), reproduced in fig. 7.

Many papers which place gamma ray constraints on annihilating DM assume that only two leptons are produced, instead of the four which are predicted by the class of models we are considering. Given that the preferred models are near the borderline of being excluded, subject to large astrophysical uncertainties, the distinction between the relatively hard, monoenergetic input spectrum for two-lepton annihilations versus the softer four-body final states is important. In particular, ref. MPV () (see section 4.1.3) has quantitatively shown this to be the case.

Furthermore, in excluding a given model, one should keep in mind the correlation between the best fit model parameters (the DM mass, annihilation cross section, and gauge boson decay branching ratios) with the assumed DM galactic density profile, since varying the latter can cause significant changes in the former. For example some papers refer to best-fit models as determined by ref. Meade (), but use different DM profiles to compute the constraints than those used to fit the PAMELA/Fermi data, making it unclear which models are really ruled out.

### v.5 Relic dark gauge (or Higgs) bosons and big bang nucleosythesis

In this section we consider cosmological constraints on the lightest stable or metastable particle in the dark sector. Since we have identified the mass scale MeV for the portal boson as being favored by fits to the PAMELA/Fermi/HESS data, we will take this to be the lightest particle, be it the gauge boson in the case of gauge kinetic mixing, or a Higgs boson in the case of Higgs mixing. By this assumption we avoid the introduction of any scales which are even lower than 100 MeV.

Some of the bounds we derive implicitly assume that the dark gauge bosons were in equilibrium with the rest of the plasma at a high temperature, so that their abundance is known around the time when they are becoming nonrelativistic. Even if the mixing parameter is too small for interactions with electrons to achieve thermal equilibrium with the dark sector, one should remember that kinetic mixing arises from some higher scale physics, such as a heavy particle which transforms under both the dark and the SM gauge symmetries; recall eq. (9). Even for small values of , such an origin for the kinetic mixing can insure equilibrium between the dark and SM sectors at the TeV scale.

#### v.5.1 Long-lived gauge bosons

In previous sections, it was noted that dark gauge bosons with long s lifetimes could have provided an escape from gamma ray constraints on annihilating DM through the mechanism of ref. Rothstein (), but we now argue these would also dominate the energy density of the universe at the time of BBN, assuming the DM was produced thermally. Let us consider the least dangerous case of MeV gauge bosons. Further, suppose that the SM becomes supersymmetric above the weak scale, so that the number of degrees of freedom is doubled; if instead there is a desert of no new states, this will only make the BBN constraint stronger. When the DM particles freeze out between and , they transfer their entropy to the dark gauge bosons. This increases the energy density of the latter by at most a factor of two, since there are more gauge degrees of freedom than DM ones. In the meantime, between temperatures of 1 TeV and MeV, the SM degrees of freedom are differentially heated relative to the dark gauge bosons by a factor of approximately , due to the change in the number of degrees of freedom from 214 to 11, and the fact that the gauge bosons had been heated by a factor of by the DM annihilations. (The precise value depends on the dimension of the DM representation, but for the small- models we consider, this has no effect on the ensuing bound.) Thus at MeV, the energy density in dark gauge bosons is suppressed by a factor of per degree of freedom. By MeV this suppression has gone down to a factor of 2.1 due to the gauge bosons being nonrelativistic. However there are 3 colors and 3 polarizations, so this counts as approximately 4.5 extra species, and is ruled out. We conclude that the gauge boson lifetime should be less than 1 s (the time corresponding to MeV), requiring that

 ϵ>4×10−11(100 MeVμ)1/2 (37)

We used the decay rate for .

Even if the thermal relic DM density is highly depleted by having a large annihilation cross section, the above arguments hold, since most of the energy of the original thermal DM population is deposited in the gauge bosons, regardless of how much DM is left. The only obvious way to avoid the above constraint on is to somehow dilute the original DM even more relative to the SM, e.g., by having even more extra degrees of freedom present at a TeV than in the minimal supersymmetric standard model. We note that the bosons will not equilibrate with the SM for values of lower than (37), so equilibration cannot serve to dilute the dark gauge bosons.

#### v.5.2 Stable gauge bosons

Typically only one color of the dark gauge bosons mixes with the SM, say , while transitions between and can be mediated by the nonabelian mixing interaction which we referred to previously in section II.3, leaving the lighter of these two states stable against decay. We must verify that its relic density is not too large.

For definiteness, suppose the stable gauge boson is . The most efficient process for depleting is the scattering , shown in fig. 8, followed by the decays . We will show that this is true even if is heavier than .

The cross section for can be estimated as

 ⟨σv⟩∼e−ΔE/Tα2gμ2(δμμ)1/2 (38)

where is the energy barrier: if , and if . The factor of arises from the velocity of the final state particles, which is in the more familiar case of annihilation to light final states. The freeze-out temperature for this reaction is determined as usual by setting equal to the Hubble rate, using the equilibrium density of a massive particle for ; one finds that

 xf=μTf = ln(0.04α2g√g∗mPμ)−2lnxf1+ΔE/μ (39) ≅ 35.4−2lnxf1+ΔE/μ

for MeV and . This implicit equation quickly converges to a solution by iteration. Values of as a function of are shown in figure 9.

As long as the interactions of fig. 8 are in equilibrium, the abundance tracks that of , whose principal connection with the SM is through the decays and inverse decays . The decay rate is suppressed by , and for the small values of we obtain in the ensuing bound, it is consistent to neglect scattering processes whose rate goes like . In appendix D we show that the processes are able to keep the gauge bosons in kinetic equilibrium with themselves down to a temperature given by , so would maintain the equilibrium abundance of a nonrelativistic particle until this temperature. At lower temperatures, it disappears due to its decays:

 YB1=Yeq(xk)e−Γt=Yeq(xk)e−(Γ/H(μ))x2 (40)

where is the decay rate, , and is the Hubble rate at . Since , we find that . The analysis of ref. KT () shows that a good estimate of the relic abundance of is obtained by evaluating (which is the source for in the Boltzmann equation) at : . On the other hand, the present abundance of stable bosons must not exceed the observed DM abundance. Using baryons as a reference,

 YB2(∞)<ΩDMΩbmNμηb≅3×10−8 (41)

where , , is the mass of the nucleon, and we took MeV. Putting these results together, we obtain the bound

 ϵ > 1xf⎛⎝3(ln(13×108)−xk+32lnxk))1.67√g∗αμMp⎞⎠1/2 (42) = 9×10−9xf

Since (the value when ), this is approximately an order of magnitude stronger than the bound (37) from nucleosynthesis.444If , then the bound is slightly modified since maintains equilibrium density until :

#### v.5.3 Long-lived Higgs bosons

We now consider the case where the Higgs boson that mixes with the SM is the lightest metastable state of the dark sector. The gauge bosons provide no more constraint in this case since they are presumed to be heavier, and although they are stable, they efficiently annihilate into dark sector Higgses with a negligible relic density, smaller than the closure density.

If the coupling of the Higgs to the SM is too strongly suppressed by the small mixing angle , there will be similar problem as the one involving metastable gauge bosons, discussed above. The Higgs should decay before nucleosynthesis to avoid dominating the energy density of the universe. We can directly adapt the result (37) by replacing , , (the electron Yukawa coupling, ):

 θ>4×10−6(100 MeVmS)1/2 (43)

Of course, this also forbids the possibility of a long-lived intermediate state Rothstein () for transporting them outside the galactic center before decaying into .

### v.6 Long-lived intermediate DM states and direct detection constraint

In section IV we discussed the implications of an absolutely stable intermediate DM state, protected by a discrete symmetry. This symmetry also made the models safe from downward transitions mediated by nuclear recoil in direct detection experiments, since the gauge boson was forbidden from mixing with the SM. However, if the symmetry is not present and does mix with hypercharge, interesting constraints can arise, since the state generically has a lifetime longer than the age of the universe, has a significant relic density, and can undergo in the detector BPR (). The latter process is not kinematically suppressed since it is exothermic, and it leads to strong constraints on the mixing parameter . Ref. BPR () finds the 90% c.l. limit from CDMS for TeV, keV for the small splitting which would be relevant for the iDM explanation of DAMA, and GeV. As explained above, we prefer MeV, which makes the constraint even more severe,

 ϵ<2×10−8(μ100 MeV)2 (44)

since the -nucleon cross section scales like .

Notice that the window between (44) and our BBN or relic density bounds (37,42) is only a few orders of magnitude. This region of parameter space is also below those which could be probed by complementary experiments, as illustrated in fig. 10, taken from ref. Bjorken () (see also ref. posp (),batt ().) In the models we consider, the bound (44) can be evaded if we insist upon the symmetry which forbids the transitions leading to direct detection. This makes it possible to have models which could also be probed by laboratory experiments such as beam dumps. Another way to evade (44) can arise if the mass splitting between the intermediate and ground state is too large batt (), since direct detection experiments do not look for very large recoil energies. The inverted mass hierarchy could thus be useful for this purpose even if there is no symmetry and the intermediate state is only metastable.

## Vi Doublet dark matter

We now begin our investigation of more specific classes of models, organized according to the SU(2) representation under which the DM transforms. If the DM is in the doublet representation, it must be vector-like (Dirac) in order to have a bare mass term,

 M¯χiχi (45)

In this case, DM number becomes conserved. Its abundance could be due to its chemical potential rather than freeze-out, similar to the baryon asymmetry, and so a nonthermal origin could be considered more natural than for Majorana DM.

There is no way to split the masses of the doublet through radiative corrections from the gauge bosons, because each member of the doublet has equal-strength interactions with all three gauge bosons. For example suppose only were to get a mass ; the contribution to the mass matrix is . But we can get a splitting through the VEV of a triplet via the Yukawa interaction

 hχ†τaχΔa3 (46)

The suffix on is a mnemonic for the fact that (for convenience) we take its VEV to be in the direction, since this gives the mass splitting between the Dirac states and .

### vi.1 Gauge kinetic mixing

Let us first consider the case of gauge kinetic mixing as the portal to the SM. With the above mass splitting, either or must mix with the SM hypercharge so that transitions between and can occur, with the production of . The triplet VEV which generates the mass splitting is not suitable for generating the kinetic mixing of the gauge boson via . In fact, such mixing is dangerous from the standpoint of constraints from direct DM searches, since it would induce diagonal couplings via of the DM to nuclei. One possibility is to have an additional triplet, , coupling as in eq. (1), which gets a VEV along the 1 (or 2) direction. The extra triplet VEV serves another purpose, by completely breaking the SU(2) gauge symmetry, whereas a single triplet would break SU(2)U(1). Assuming that gets its VEV along the 1 direction, the spectrum of the gauge bosons is

 μ1=gΔ3,μ2=g√Δ23+Δ21,μ3=gΔ1 (47)

With this spectrum and the couplings described above, is stable, but can decay into via the nonabelian gauge mixing interaction