Scalar Electroweak Multiplet Dark Matter
aa r X i v : . [ h e p - ph ] D ec ACFI-T18-17
Scalar Electroweak Multiplet Dark Matter
Wei Chao , ∗ Gui-Jun Ding , † Xiao-Gang He , , , ‡ and Michael Ramsey-Musolf , § Center for advanced quantum studies, Department of Physics,Beijing Normal University, 100875, Beijing, China Interdisciplinary Center for Theoretical Study and Department of Modern Physics,University of Science and Technology of China, Hefei, Anhui 230026, China Tsung-Dao Lee Institute, and School of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai 200240, China Department of Physics, National Taiwan University, Taipei 106, Taiwan National Center for Theoretical Sciences, Hsinchu 300, Taiwan Amherst Center for Fundamental Interactions, University of Massachusetts-Amherst,Department of Physics, Amherst, MA 01003, USA Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, CA 91125 USA
We revisit the theory and phenomenology of scalar electroweak multiplet thermal dark matter.We derive the most general, renormalizable scalar potential, assuming the presence of the StandardModel Higgs doublet, H , and an electroweak multiplet Φ of arbitrary SU(2) L rank and hypercharge, Y . We show that, in general, the Φ- H Higgs portal interactions depend on three, rather than twoindependent couplings as has been previously considered in the literature. For the phenomenologi-cally viable case of Y = 0 multiplets, we focus on the septuplet and quintuplet cases, and considerthe interplay of relic density and spin-independent direct detection cross section. We show thatboth the relic density and direct detection cross sections depend on a single linear combination ofHiggs portal couplings, λ eff . For λ eff ∼ O (1), present direct detection exclusion limits imply thatthe neutral component of a scalar electroweak multiplet would comprise a subdominant fraction ofthe observed DM relic density. I. INTRODUCTION
Determining the identity of the dark matter and thenature of its interactions is a forefront challenge for as-troparticle physics. A plethora of scenarios have beenproposed over the years, and it remains to be seenwhether any of these ideas is realized in nature. Onehopes that results from ongoing and future dark mat-ter direct and indirect detection experiments, in tandemwith searches for dark matter signatures at the LargeHadron Collider and possible future colliders, will even-tually reveal the identity of dark matter and the characterof its interactions.A widely studied possibility of continuing interest isthat dark matter consists of weakly interacting mas-sive particles (WIMPs). An array of realizations of theWIMP paradigm have been considered, ranging from ul-traviolet complete theories such as the Minimal Super-symmetric Standard Model to simplified models contain-ing a relatively small number of degrees of freedom andnew interactions. In the latter context, one may clas-sify WIMP dark matter candidates according to theirspin and electroweak gauge quantum numbers. The sim-plest possibility involves SU(2) L × U(1) Y gauge singlets.Null results from direct detection (DD) experiments andLHC searches place severe constraints on this possibil- ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ity, though some room remains depending on the specificmodel realization.An alternative possibility is that the dark matter con-sists of the neutral component of an electroweak multi-plet, χ . A classification of these possibilities is givenin[1]. Those favored by the absence of DD signalscarry zero hypercharge ( Y ), thereby preventing overly-large WIMP-nucleus cross sections mediated by Z ex-change. Stability of the χ requires imposition of a dis-crete Z symmetry unless the representation of the elec-troweak multiplet is of sufficiently high dimension: d=5for fermions and d=7 for scalars. These scenarios withsufficiently high dimension representation go under theheading “minimal dark matter”.In this work, we consider features of scalar electroweakmultiplet dark matter Φ, including but not restrictingour attention to minimal dark matter (as conventionallydefined). The phenomenology of scalar triplet dark mat-ter, involving a multiplet transforming as (1 , ,
0) underSU(3) C and electroweak symmetries, has been consideredpreviously in Refs. [2–8]. Extensive studies for other elec-troweak multiplets of dimension n have been reported inRefs. [9, 10]. The authors of Ref. [9] considered the InertDoublet model and the n = 3 , , σ SI , and indirect de-tection (ID) signals. Ref. [10] also considered the impactof Higgs portal interactions on the relic density and σ SI but did not analyze the implications for indirect detec-tion. The latter study also focused on a relatively lightmass for the dark matter candidate, for which it wouldappear to undersaturate the relic density.In what follows, we revisit the topic of these ear-lier studies, taking into account several new featuresthat may require modifying some of the conclusions inRefs. [9, 10]: • We find that the scalar potentials V ( H, Φ) given inRefs. [9, 10] are not the most general renormalizablepotentials and that, depending on the representa-tion of SU(2) L × U(1) Y there exist one or more ad-ditional interactions that should be included. Forthe Y = 0 representations, the Φ- H interaction rel-evant for both the relic density and DD cross sec-tion involves an effective coupling λ eff that is linearcombination of two of the three possible Higgs por-tal couplings. The specific linear combination isrepresentation dependent. • We update the computation of σ SI taking intoaccount the nucleon matrix elements of twist-twooperators generated by gauge boson-mediated boxgraph contributions as outlined in Refs. [11–14].We note that Ref. [10] considered only the Higgsportal contribution to σ SI and did not include theeffect of electroweak gauge bosons. We find that thegauge boson-mediated box graph contributions aresmaller in magnitude that given in Ref. [9], whichused the expressions given in Ref. [1]. In general,the Higgs portal contribution dominates the DDdetection cross section except for very small valuesof λ eff . • The presence of a non-vanishing λ eff can allow fora larger maximum dark matter mass, M , to beconsistent with the observed relic density than onewould infer when considering only gauge interac-tions. For the cases we consider below, this maxi-mum mass be as larger as O (20) TeV for perturba-tive values of λ eff . • For moderate values of the Higgs portal couplings,the spin-independent cross section, scaled by thefraction of the relic density comprised by Φ , is afunction λ eff and M . The present DD bounds on σ SI generally require M . λ eff – well below the maximum mass con-sistent with the observed relic density.In what follows, we provide the detailed analysisleading to these conclusions. For the structure of V ( H, Φ) we consider Φ to be a general representationof SU(2) L × U(1) Y . Previous studies have considered indetail electroweak singlets ( n = 1), doublets ( n = 2), andtriplets ( n = 3). In all three cases, stability of the DMparticle requires that one impose a discrete symmetryon the Lagrangian. Going to higher dimension repre-sentations, it has been shown in Ref. [1] that for n = 4,stability of the neutral component also requires imposi-tion of a discrete symmetry, while for n = 5, the neutralcomponent can only decay through a non-renormalizable dimension five operator with coefficient suppressed byone power of a heavy mass scale Λ. In the latter case, itis possible to ensure DM stability on cosmological timescales by either imposing a discrete symmetry or bychoosing Λ to be well above the Planck scale. At n = 7,the first non-renormalizable, decay-inducing operatorappears at higher dimension, and DM stability may beensured even without imposition of a discrete symmetryby choosing Λ below the Planck scale.With the foregoing considerations in mind, we focuson the n = 5 and 7 cases for purposes of illustratingthe dark matter phenomenology. Since the group theoryrelevant to construction of V ( H, Φ) is rather involved, weprovide a detailed discussion in Appendices A and B. InSection II, we start with a general formulation, followedby treatment of specific model cases. Section III givesthe calculation of the relic density, including the effectsof coannihilation and the Sommerfeld enhancement. Wecompute σ SI in Section IV. We summarize in Section IV.Along the way, we point out where we find differenceswith earlier studies. II. MODELS
We consider the renormalizable Higgs portal interac-tions involving H and Φ for two illustrative cases. Werestrict our attention to Φ being a complex scalar with Y = 0. The form of the potential for Φ being a realrepresentation of SU(2) L with Y = 0 is relatively simple.The corresponding features have been illustrated in pre-vious studies wherein Φ is either an SU(2) L singlet or realtriplet. Consequently, we focus on complex representa-tions, using the n = 5 and n = 7 examples, to illustratethe new features not considered in earlier work.To proceed, we first introduce some notation. It is con-venient to consider both Φ and the associated conjugate¯Φ, whose components are related to those of Φ asΦ j,m = ( − j − m Φ ∗ j, − m . (1)As we discuss in Appendix A, Φ and Φ transform inthe same way under SU(2) L . One may then proceed tobuild SU(2) L invariants by first coupling Φ, Φ, H , and H pairwise into irreducible representations and finally intoSU(2) L invariants. For example, (cid:0) ΦΦ (cid:1) = ( − j √ j + 1 Φ † Φ , (2)which in general is a distinct invariant from (ΦΦ) exceptin special cases when Φ is a real scalar multiplet satisfyingΦ = Φ. Note that for j = 1 / vanishes, so thatthere is only one quadratic invariant in this case as well.Quartic interactions can be constructed in a variety ofways, such as (cid:2) (ΦΦ) J (cid:0) Φ Φ (cid:1) J (cid:3) (3)for Φ self-interactions or (cid:2)(cid:0) HH (cid:1) L (cid:0) ΦΦ (cid:1) L (cid:3) (4)with L = 0 , (cid:0) HH (cid:1) (ΦΦ) (5)that is distinct from the L = 0 operator in Eq. (4) forΦ being a complex integer representation. We note thatprevious studies have not in generally included all three ofthe possible Higgs portal interactions. The classificationof the Φ self-interactions is more involved, and it is mostilluminating to consider them on a case-by-case basis. A. Setptuplet
The interactions can be written as V =+ M A (Φ † Φ) + (cid:8) M B (ΦΦ) + h . c . (cid:9) − µ H † H + λ ( H † H ) + λ ( H † H )(Φ † Φ) (6)+ λ [( HH ) (ΦΦ) ] + [ λ ( HH ) (ΦΦ) + h . c . ] , where H is the Higgs doublet and Φ is a complex elec-troweak septuplet with(ΦΦ) = 1 √ X m = − ( − − m φ ,m φ , − m = 1 √ (cid:0) φ , φ , − − φ , φ , − +2Φ , φ , − − φ , Φ , (cid:1) (7)( HH ) = 1 √ (cid:2) ( H + ) ∗ H + + ( H ) ∗ H (cid:3) (8)and ( HH ) = ( H ) ∗ H +1 √ (cid:2) ( H ) ∗ H − ( H + ) ∗ H + (cid:3) − ( H + ) ∗ H (9)(ΦΦ) = A− √ P m = − mφ ∗ ,m φ ,m B (10)with A =+ √ φ ∗ , − φ , − + √ φ ∗ , − φ , − + √ φ ∗ , − φ , + √ φ ∗ , φ , + √ φ ∗ , φ , + √ φ ∗ , φ , (11) B = −√ φ ∗ , − φ , − − √ φ ∗ , − φ , − − √ φ ∗ , φ , − −√ φ ∗ , φ , − √ φ ∗ , φ , − √ φ ∗ , φ , (12)After electroweak symmetry breaking, whereinRe H → ( v + h ) / √ L mass = (cid:0) φ ,k φ ∗ , − k (cid:1) M A + λ v + √ kλ v √ ( − k +1 n M B + √ λ v o √ ( − k +1 n M ∗ B + √ λ ∗ v o M A + λ v − √ kλ v (cid:18) φ ∗ ,k φ , − k (cid:19) (14)By setting φ , = ( φ , +) + iφ , − ) ) / √
2, the neutral scalar mass matrix can be written as M A + λ v − √ Re( M B ) − √ Re( λ ) v √ Im( M B ) + √ Im( λ ) v √ Im( M B ) + √ Im( λ ) v M A + λ v + √ Re( M B ) + √ Re( λ ) v ! (15)in the basis ( φ , +) , φ , − ) ) T . Then we have the mass eigenvalues M φ ± k = M A + 12 λ v ± s(cid:12)(cid:12)(cid:12)(cid:12) √ M B + 1 √ λ v (cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) √ kλ v (cid:19) (16) M φ , ± ) = M A + 12 λ v ± (cid:12)(cid:12)(cid:12)(cid:12) √ M B + 1 √ λ v (cid:12)(cid:12)(cid:12)(cid:12) (17)where for each isospin projection k , the “ ± denotes theupper or lower sign in Eqs. (16,17) and where the nota-tion ˆ φ , ± k indicates the mass eigenstate.From these expressions we conclude that • If λ is nonzero, there will be no dark matter sinceone may have M φ k, − ) < M φ , − ) for k = 0. Oneneeds λ ∼
0, otherwise there may exist long-livedcharged scalars. • For λ = 0, we have two real septuplets S A = 1 √ ˆ φ ∗ , − i ˆ φ ∗ , − ˆ φ ∗ , − i ˆ φ , − ) ˆ φ , − i ˆΦ , − ˆ φ , − S B = 1 √ ˆ φ , i ˆ φ , ˆ φ , i ˆ φ , +) ˆ φ ∗ , i ˆ φ ∗ , ˆ φ ∗ , (18)The corresponding mass eigenvalues eigenvalues are M S A ,S B = M A + 12 λ v (19) ± (cid:12)(cid:12)(cid:12)(cid:12) √ M B + 1 √ λ v (cid:12)(cid:12)(cid:12)(cid:12) , where the lower (upper) sign corresponds to S A ( S B ). • In general, the neutral component of S A – denotedhere as the real scalar χ – will be the DM par-ticle. Radiative corrections will give rise to themass splitting between the neutral and chargedcomponents. In the limit M A ≫ M W,Z , one has M Q − M ≈ Q ∆ M , with ∆ M = (166 ±
1) MeV [1]being the mass splitting between the Q=1 and 0components.From the full scalar potential, one may obtain darkmatter self interactions L self χ = − ˜ λ self χ , (20)which may be important in solving the core-cusp prob-lem [15, 16]. The relevant terms are J X J =0 κ k [(ΦΦ) k (Φ Φ) k ] + J X k =0 { κ ′ k [(ΦΦ) k (ΦΦ) k ] + κ ′′ k [(Φ Φ) k (ΦΦ) k ] + h . c . (cid:9) (21)Note that each component of (ΦΦ) j ( j = 0 , . . . ,
6) is de-termined by(ΦΦ) j,m = X m ,m C j,m ,m ;3 ,m φ ,m φ ,m . (22)From the property of Clebsch-Gordan coefficients: C j,mj ,m ; j ,m = ( − j − j − j C j,mj ,m ; j ,m . (23) If j − j − j is an odd (even) integer, the correspondingcontraction of two Φ fields is antisymmetric (symmetric).Consequently, (ΦΦ) , (ΦΦ) and (ΦΦ) vanish. For themost general case leading to the mass-squared matrix inEq. (15), the expression for the DM quartic self inter-action is rather involved and not particularly enlighten-ing. For completeness, in Appendix C we give an expres-sion for the quartic interactions in terms of φ , ± ) , fromwhich one can determine the DM self interaction by ex-pressing the φ , ± ) in terms of the mass eigenstates. Toillustrate, we give here the result for the special case ofreal M B and λ with 2 √ M B + λ v < λ self =+ 17 [ κ + 2Re( κ ′ ) + 2Re( κ ′′ )]+ 421 √ κ + 2Re( κ ′ ) + 2Re( κ ′′ )]+ 677 [ κ + 2Re( κ ′ ) + 2Re( κ ′′ )]+ 100231 √
13 [ κ + 2Re( κ ′ ) + 2Re( κ ′′ )] , (24)where the factor 4 comes from the fact that φ , =( φ , +) + iφ , − ) ) / √
2. In general, ˜ λ self depends on12 free parameters in Eq. (21). We defer an explorationof the possible additional physical consequences of theseindependent interactions to future work. B. Quintuplet
The analysis for the electroweak scalar quintuplet darkmatter is similar to the septuplet case. For purposes ofcompleteness, we include some of the important featuresbelow. The complex quintuplet scalar field with j = 2and Y = 0 is denoted byΦ = φ , φ , φ , φ , − φ , − . (25)The mass term and interactions of quintuplet are thesame as those of the septuplet given in Eq. (6), wherewe set λ = 0 to ensure the presence of a stable neutralcomponent. To derive the mass eigenvalues we considerthe contractions of the two scalar multiplets ΦΦ. Ac-cording to general decomposition rule, one has(ΦΦ) = φ , − φ , − φ , + 2 φ , − φ , √ . (26)By setting φ , = ( α ′ + iβ ′ ) / √
2, the mass matrix of theneutral scalars can be written as12 ( α ′ β ′ ) M A + λ v + √ Re M B − √ Re( λ ) v − √ Im( M B ) + √ Im( λ ) v − √ Im( M B ) + √ Im( λ ) v M A + λ v − √ Re M B + √ Re( λ ) v ! (cid:18) α ′ β ′ (cid:19) (27)The mass eigenvalues are M α ′ ,β ′ = M A + 12 λ v ± (cid:12)(cid:12)(cid:12)(cid:12) √ M B − √ λ v (cid:12)(cid:12)(cid:12)(cid:12) , (28)which are also mass eigenvalues of the two real quintu-plet.The self-coupling can be derived following the samestrategy of the septuplet case, and we give the results inAppendix C. III. RELIC DENSITY
In this work, we assume that dark matter in the earlyUniverse was in the local thermodynamic equilibrium.Decoupling ocurred when its interaction rate drops be-low the expansion rate of the Universe. The correspond-ing evolution of the dark matter number density n , isgoverned by the Boltzmann equation:˙ n + 3 Hn = −h σv M /o ller i ( n − n ) , (29)where H is the Hubble constant, σv M /o ller is the total an-nihilation cross section multiplied by the M / oller velocity, v M /o ller = ( | v − v | − | v × v | ) / , brackets denote ther-mal average and n EQ is the number density at thermalequilibrium. It has been shown that h σv M /o ller i = h σv lab i = 12 [1+ K ( x ) /K ( x )] h σv cm i , (30)where x = m/T , K i are the modified Bessel functions oforder i .In a general framework that includes co-annihilation,the dynamics depend on a set of species { χ i } with masses { m i } and number densities { n i } . It has been shown thatthe total number density of all species taking part in the co-annihilation process, n ≡ P i n i , obeys Eq. (29). Inthis case h σv M /o ller i can be written as [17, 18] h σv M /o ller i = R ∞ m χ dss / K (cid:16) √ sT (cid:17) P Nij β ij g i g j g χ σ ij ( s )8 m χ T hP Ni g i g χ m i m χ K (cid:0) m i T (cid:1)i (31)where g i is the number of degrees of freedom, s is theMandelstam variable, σ ij = σ ( χ i χ j → all), and the kine-matic factor β f ( s, m i , m j ) is given by β ij = s(cid:20) − ( m i + m j ) s (cid:21) (cid:20) − ( m i − m j ) s (cid:21) . (32)The number density of the dark matter at the end willbe n χ = n . The relic density of the dark matter todaycan be written asΩ χ h = 1 . T γ √ g ∗ M pl ρ crit (cid:18) T χ T γ (cid:19) (cid:20)Z x f dx h σv M /o ller i ( x ) (cid:21) − (33)where ρ crit ≡ . × − ( h ) GeV /cm is the criticaldensity, M pl denotes the Planck mass, T γ and T χ are thepresent temperatures of photon and dark matter, respec-tively. According to entropy conservation in a comovingvolume, the suppression factor ( T χ /T γ ) ≈ / A. The single species case
We first calculate the dark matter relic density as-suming only a single species, i.e. , including no co-annihilation. To show the interplay between the Higgsportal and gauge interactions in the annihilation dynam-ics, we compute the relic density analytically. For com-pleteness, we show the thermal average of various anni-hilation cross sections: h σv i hh = λ ( p m − m h πm + − m + 5 m h πm p m − m h h v i ) (34) h σv i ¯ tt = λ (cid:26) m t ( m − m t ) / πm (4 m − m h ) + ∆ ( t ) h v i (cid:27) (35) h σv i ZZ = λ ( p m − m z (4 m − m m z + 3 m z )8 πm (4 m − m h ) + ∆ ( Z ) h v i ) (36) h σv i W W = λ ( p m − m w (4 m − m m w + 3 m w )8 πm (4 m − m h ) + ∆ ( W ) h v i ) + λ eff g v πm (4 m − m h ) + c n g πm (37)where λ eff is an effective coupling given by a linear com-bination of the independent Higgs portal couplings. As-suming real M B and λ one has λ eff = λ ± q λ , septuplet λ ∓ q λ , quintuplet , (38)where we have set λ = 0 as above; where the upper (lower) signs correspond to 2 √ M B + λ v being negative(positive); where the parameter c n = ( n −
64 (39)accounts for the effective couplings of the dark matterwith the W boson; and where∆ ( t )= m t p m − m t ( − m − m m t + 2 m ( m h + 18 m t ))32 πm (4 m − m h ) (40)∆ ( v )= − m + 176 m m v − m h m v − m (3 m h m v + 52 m v ) + 12 m (2 m h m v + 9 m v )64 πm (4 m − m h ) p m − m v (41)The present relic density of the DM is simply given by ρ χ = M n χ . The relic density can finally be expressed interms of the critical densityΩ h ≈ . × GeV − x F M pl √ g ∗ ( a + 3 b/x F ) , (42)where a and b , which are given in Eqs. (34-37), areexpressed in GeV − and g ∗ is the effective degrees offreedom at the freeze-out temperature T F , x F = M/T F ,which can be estimated through the iterative solution ofthe equation x F = ln " c ( c + 2) r g π M M pl ( a + 6 b/x F ) √ g ∗ x F , (43)where c is a constant of order one determined by match-ing the late-time and early-time solutions. It is conven-tional to write the relic density in terms of the Hub-ble parameter, h = H / − Mpc − . Observa-tionally, the DM relic abundance is determined to beΩ h = 0 . ± . λ eff = 0 ,
2, and 5. Thetop (bottom) panel gives the septuplet (quintuplet) case.To obtain the correct relic density, one has M = 9 .
17 TeVfor the septuplet and M = 4 .
60 TeV for the quintupletby taking λ eff = 0. B. Co-annihilation
The mass splittings between the neutral and chargedcomponents of the septuplet is about 166 MeV [1], sothe effect of co-annihilation should be considered. Therelevant processes are listed in Table. I.
Process Mediator s − channel t − channel u-channel 4P s + Q s − Q → W + W − h, Z, γ s Q − s − Q +1 s + Q s − Q +1 → W + Z ( γ ) W + s Q − s − Q s + Q s − Q → ZZ ( γγ, Zγ ) h s Q s − Q s + Q s − Q → ¯ f f h, W, Z, γs + Q s − Q → hh h s + Q s − Q s + Q s − Q → hZ ( γ ) Z S + Q s − Q s Q s − Q +1 → hW + W s Q s − Q +1 TABLE I: A complete set of process relevant to the co-annihilation of the scalar multiplet dark matter, 4P representsfour point interactions.
The eq. (31) can be simplified as [17] h σv M /o ller i = X ij A ij n (44)where n eq in the denominator is n eq = T π X i g i m i K (cid:16) m i T (cid:17) , (45) × - M Ω h n = M Ω h n = FIG. 1: Dark matter relic density as a function of the darkmatter mass. The solid (red), dashed (blue), and dot-dashed(green) curves correspond to λ eff = 0 , ,
5, respectively. Thehorizontal line is the observed relic density. and A ij in the numerator can be written as A ij = T π Z ds √ sβ ij g i g j W ij k (cid:18) √ sT (cid:19) , (46)with W ij being a dimensionless Lorentz invariant, definedas W ij = 4 E i E j σ ij v ij .To illustrate the impact of including co-annihilationprocesses, we plot in Fig. 2 the value of λ eff neededto reproduce the observed relic density as a functionof the DM mass. The upper (lower) panel corresponds v ij is defined by v ij = q ( p i · p j ) − m i m j /E i E j [21], where E i and p i are the Energy of four-momentum of particle i . M λ e(cid:0)(cid:1) = c(cid:2) - annihilationSommerfeld M λ (cid:3)(cid:4)(cid:5) n = Co - annihilationSommerfeld FIG. 2: Contours of the dark matter relic density (Ω h =0 . M − λ eff plane. The solid line (red) correspondsto the co-annihilation case, the long-dashed line (blue) rep-resents the one-species scenario. The short-dashed (black)line includes the Sommerfeld enhancement effect (see SectionIII C below). The top (bottom) panel describes the septuplet(quintuplet) dark matter case. to the septuplet (quintuplet) case. The dashed blueline gives the result for single species annihilation case,while the solid red curve indicates the result includ-ing co-annihilation. We observe that the presence ofmore species initially in equilibrium with the DM re-quires a larger effective interaction strength to avoid over-saturating the observed relic density. The reason can beseen from Eq. (44), for which the denominator can be ap-proximated as n eq ≈ (2 j +1) n ,s with j and n eq ,s being,respectively, the total isospin of the multiplet and thenumber density of a single component in equilibrium. As j increases, so does n eq . On the other hand, the numer-ator factor, P ij A ij only accounts for the combinationsof multiplet components that are able to annihilate, andit does not grow as fast as n eq with increasing j . Con-sequently, one must (a) increase λ eff (for fixed M ); (b)decrease M (for fixed λ eff ); or (c) introduce some combi-nation of both in order to maintain the total cross sectionas compared to the single species scenario. We refer thereader to Ref. [22] for a similar discussion regarding the n = 6 , C. Sommerfeld enhancement
Now we investigate the effect of the non-perturbativeelectroweak Sommerfeld enhancement [23–25], where thegauge bosons mediate an long-range effective force be-tween the annihilating DM particles. To that end, wefirst observe that in the SM, there is no true phase tran-sition between the electroweak symmetric phase and thebroken phase, but the cross over is located at T c = 159 ± M c ≈ . x F ∼ .
05 with T F ∼ T c ) , electroweak symmetry is restored; W and Z bosons can be taken as massless particles; and triplescalar couplings go to zero as they are proportional to thevaccum expectation value of neutral component of theHiggs doublet. According to the calculation performedin the last subsection, both the septuplet and the quin-tuplet DM are heavier than M c for a sizable λ eff , so wetake the massless gauge boson limit and vanishing triplescalar coupling to evaluate the Sommerfeld enhancement.Note that we do not consider here the impact of DM-DM bound states, which can lead to an additional en-hancement of the annihilation cross section for certainvalues of M . The impact of a bound state on DM an-nihilation dynamics is most pronounced when the tem-perature is . E B , where E B is the binding energy. Asanalyzed in detail in Ref. [27], however, the Sommerfeldenhancement is plays the most significant role in settingthe relic density at temperatures well above E B . Thus,one would expect the presence of the bound states tohave a subdominant effect on the overall relic density.Consequently, neglect of the bound state effects appearsto be reasonable in the present context.To proceed, we consider the Coulomb potential associ-ated with the electroweak gauge bosons is given by [28] V ≡ ar = g πr [(2 N + 1) + 1 − n ] (47)where N is the total isospin of the initial state contain-ing two annihilating DM particles and n is the dimen-sion of the SU(2) L irreducible representation of the DM.Since DM only annihilates into SM final states, one has N = 0 , ,
2, depending on the specific process. Of thesepossibilities, which there exist more N = 0 final SM finalstates that those with N = 0 ,
1, so we concentrate onthe N = 0 case. Note that for n >
1, the correspondingpotential is attractive.
60 80 1001 <> == FIG. 3: Sommerfeld enhancement factor for the quintupletand septuplet as the function of x ( M/T ). The Sommerfeld enhancement factor S = σ/σ perturbative for the Coulomb potential can bewritten as S = − π aβ − exp( πaβ ) (48)where β is the relative velocity between the annihilat-ing particles (note that a < N = 0 and n > s -wave annihilation, one can use the Sommerfeldenhancement averaged over the thermal distribution, de-fined as [29] h S i = x / √ π Z Sβ exp (cid:0) − xβ / (cid:1) dβ (49)where x = M/T with T the temperature.In Fig. 3 we show the thermal average of the Som-merfeld enhancement as the function of x . A numericalcalculation gives h S i ∼ . . x = x F , which will be used in the calculation of thedark matter relic density. As can be seen from Eq. (47), ahigher dimensional representation for the multiplet givesrise to a larger enhancement factor. The resulting impactof the Sommerfeld enhancement is shown in Fig. 2, wherethe dotted black line corresponds to the case of includ-ing both co-annihilation and Sommerfeld enhancementeffects. As expected, the presence of this enhancementcounteracts the effect of coannihilation, allowing for asmaller value of λ eff (for fixed M ) or larger value of M (for fixed λ eff ). IV. DIRECT DETECTION
For conventional Higgs portal dark matter models,constraints from dark matter direct detection are quitesevere. The parameter space of these models is stronglyconstrained by the limits obtained by the LUX [30],PandaX-II [31], and Xenon1T [32] experiments. In whatfollows, we consider how the presence of the Higgs portalinteractions affects the interpretation of these experimen-tal results. To that end, we consider all the terms in theeffective Lagrangian for low-energy DM interactions withSM particles relevant to the scalar DM scenario consid-ered in this paper. In the limit M DM ≫ M W ≫ M q , onehas[11–14] L eff = 12 λ eff m h Φ n, ¯ qm q q + f T M Φ n, ( i∂ µ )( i∂ ν )Φ n, O qµν where O qµν = 12 ¯ qi (cid:18) D µ γ ν + D ν γ µ − g µν /D (cid:19) q (50)is the twist-two quark bilinear with coefficientfunction[33] f T = α m W n − (4 Y + 1)4 (cid:26) ω ln ω + 4 + (4 − ω )(2 + ω ) arctan 2 b ω / √ ωb ω √ ω (cid:27) (51)and with ω = m W /m , b ω = p − ω/ M if = 2 m N (cid:18) f N λ eff m h + 34 f T f PDF N (cid:19) , (52) where f N ≈ . . f PDF N = 0 .
526 [35] is the second moment of the nucleon(proton or neutron) parton distribution function (PDF)evaluated at µ = M Z ; and where we have taken a normal-ization appropriate to non-relativistic nuclear states. Wenote that the expression (51) for f T is µ -independent. In-clusion of NLO QCD corrections in the DM-parton scat-tering amplitude will generate a µ -dependence in f T thatmust compensate for the scale dependence of the PDF.We defer a detailed discussion of this feature to Ref. [33].The spin-independent cross section then can be writtenas σ SI = | M fi | π ( m N + m Φ ) = µ π m N m (cid:18) f N λ eff m h + 34 f T f PDF N (cid:19) (53)where µ = m N M/ ( m N + M ).In Fig. 4, we plot as a function of M the cross sectionof the DM-proton cross section, scaled by the fractionof the relic density corresponding to the value of M asobtained in our computation of Section III. Taking theseptuplet for illustration, the dashed dashed (red), solid(blue), and dotted (green) lines correspond to λ eff = 2, 1,0 respectively. The gray, cyan and black dotted lines give the exclusion limits of LUX, PandaX-II and XENON1Trespectively. We observe that the Higgs portal interac-tions dominate the scaled spin-independent cross sectionfor a sizable λ eff . The contribution of twist-2 effectiveoperator, indicated by the λ eff = 0 curve, becomes rela-tively sizable only for heavy DM, though its impact stilllies well below the sensitivity of the present direct de-tection experiments. The situation is different in the0 λ eff = λ eff = λ eff = - - - - - M ( GeV ) σ S I Ω h / ( Ω h ) E x p LUX PandaX - IIXENON1T
FIG. 4: DM-nucleus scattering cross section as a function ofthe dark matter mass for n = 7. The dashed dashed (red),solid (blue), and dotted (green) lines correspond to λ eff =2, 1, 0 respectively. The gray, cyan and black dotted linesgive the exclusion limits of LUX, PandaX-II and XENON1Trespectively. evaluation of the relic abundance, where the gauge in-teractions dominate the annihilation. As a result, onecan easily find the parameter space that may give rise toan observed relic abundance and a small direct detectioncross section. Conversely, including the effects of bothco-annihilation and the Sommerfeld enhancement, we ob-serve that saturating the observed relic density and evad-ing the present direct detection limits require a rathersmall value of | λ eff | . To illustrate, consider the septupletcase. From Fig. 1 we see that obtaining the relic densityrequires M in the vicinity of 9 TeV for vanishing λ eff .On the other hand, for λ eff = 1, the present direct de-tection results constrain M to be no larger than aboutone TeV – a value for which the fraction of the relic den-sity would lie well below the observed value. Lookingahead to next generation direct detection experimentsand assuming that the only thermal WIMP is the neu-tral component of the septuplet, we conclude that theobservation of a non-zero signal would likely require thepresence of a significant, non-zero λ eff . In this case, theseptuplet would comprise at most only a modest fractionof the relic density, with the remaining corresponding toa non-thermal and/or non-WIMP species. V. CONCLUSIONS
In this paper we have revisited earlier analyses of scalarelectroweak multiplet dark matter. After presenting themost general, renormalizable potential for a electroweakmultiplet Φ that interacts with the SM Higgs doublet, we show that in general the Higgs portal coupling dependson three independent parameters in the potential. In or-der to ensure that the neutral component of Φ yieldsthe lowest mass state, ensuring its viability as a DMcandidate, one of these couplings must be vanishinglysmall. The resulting dynamics of DM annihilation andDM-nucleus scattering then depend on a single effectivecoupling, λ eff . After evaluating the DM relic abundanceby considering effects of both co-annihilation and Som-merfeld enhancement, we calculated for the first time thespin-independent direct detection cross section by takinginto account the contribution of the twist-2 effective op-erators, which turns to be important for a heavy scalarDM. Focusing on the electroweak quintuplet and septu-plet for illustration, we find that for λ eff ∼ O (1) presentDM direct detection limits imply that the electroweakmultiplet mass scale M most be . Acknowledgments
WC was supported in part by the Natural ScienceFoundation of China under Grant No. 11775025 andthe Fundamental Research Funds for the Central Uni-versities. GJD was supported in part by the support ofthe National Natural Science Foundation of China underGrant No 11522546. XGH was supported in part by theMOST (Grant No. MOST 106-2112-M-002-003-MY3 ),and in part by Key Laboratory for Particle Physics, As-trophysics and Cosmology, Ministry of Education, andShanghai Key Laboratory for Particle Physics and Cos-mology (Grant No. 15DZ2272100), and in part by theNSFC (Grant Nos. 11575111 and 11735010). MJRMwas supported in part under U.S. Department of Energycontract de-sc0011095.
Appendix A: SU (2) group theory The Lie algebra of the SU (2) group is specified by[ J i , J j ] = iǫ ijk J k , i, j, k = 1 , , . (A1) SU (2) has only one Casimir operator J = J + J + J . (A2)We can define familiar raising and lowering operators: J ± = J ± iJ . (A3)They satisfy the following commutation relation[ J , J ± ] = ± J ± . (A4)1The eigenstate | j, m i can be labelled by the eigenvaluesof J and J : J | j, m i = j ( j + 1) | j, m i ,J | j, m i = m | j, m i , (A5)where j can be any half integer, and m = − j, − j +1 , . . . , j − , j . The different states within a multipletcan be generated by acting with the raising and loweringoperators, J ± | j, m i = p ( j ∓ m )( j ± m + 1) | j, m ± i (A6)Consequently we have h j, m ′ | J + | j, m i = p ( j − m )( j + m + 1) δ m ′ ,m +1 , h j, m ′ | J − | j, m i = p ( j + m )( j − m + 1) δ m ′ ,m − (A7)We can form a 2 j + 1 representation by choosing thefollowing 2 j + 1 orthogonal states as base vectors: ... ≡ | j, j i , ... ≡ | j, j − i , · · · ... ≡ | j, − j i . (A8) The representation matrices for the generators J + , J − and J are J + = √ j . . . p j −
1) 0 . . . p j − . . . . . . . . . . . . . . . . . . . . . . . . . . . p j −
1) 00 0 0 0 . . . √ j . . . , (A9) J − = . . . √ j . . . p j −
1) 0 . . . p j − . . . . . . . . . . . . . . . . . . . . . . . . . . . p j −
1) 0 00 0 0 . . . √ j , (A10) J = j . . . j − . . . j − . . . . . . . . . . . . . . . . . . . . . . . . − j + 1 00 0 0 . . . − j (A11)The representation matrix for each group element of SU (2) can be expressed asexp i X k =1 α k J k ! , (A12)where α k ( k = 1 , ,
3) are real parameters and J = 12 ( J + + J − ) , J = − i J + − J − ) . (A13)It is well-known that SU (2) has a unique irreducible rep-resentation for each spin j . Hence each representationshould be equivalent to its complex conjugate represen-tation. We find the unitary transformation relating rep- resentation and its complex conjugate is V = . . . . . . − . . . . . . . . . . . . . . . − j − . . . − j . . . , (A14)which fulfills V ik = ( − i +1 δ i + k, j +2 . Note that the uni-tary transformation V reduces to the familiar form for j = , V = (cid:18) − (cid:19) , for j = 12 . (A15)2One can easily check that − V J ∗± V − = J ∓ , − V J ∗ V − = J , (A16)which leads to − V J ∗ k V − = J k , k = 1 , , . (A17)Consequently we have V exp i X k =1 α k J k !! ∗ V − = exp i X k =1 α k J k ! , (A18)which implies each representation and its complex con-jugate are really equivalent, and the similarity transfor-mation is indeed given by V . As a result, for a SU (2)multiplet Φ in the representation j withΦ = φ j,j φ j,j − φ j,j − ...φ j, − j +1 φ j, − j , (A19)where the subscript denotes the eigenvalues of J and J .The state Φ would transform in the same way as Φ withΦ = V Φ ∗ = φ ∗ j, − j − φ ∗ j, − j +1 φ ∗ j, − j +2 ... ( − j − φ ∗ j,j − ( − j φ ∗ j,j . (A20)Note that it is very convenient to construct SU (2) invari-ant from Φ instead of Φ ∗ . Appendix B: The renormalizable scalar potential ofHiggs and a scalar multiplet
If we extend the standard model by introducing ascalar electroweak multiplet Φ of isospin j , the one-loopbeta function of SU (2) gauge coupling for the StandardModel would be modified into β ( g ) = g π (cid:2) −
196 + 19 j ( j + 1)(2 j + 1) (cid:3) . (B1)We can see that β ( g ) remains negative only for j ≤ .For j ≥
2, it becomes positive and hits the Landau pole.For instance adding a scalar multiplet with isospin j ≥ SU (2) gauge coupling atΛ ≤
10 TeV and it is even smaller Λ ≤
180 GeV for j ≥
10. Therefore, perturbativity of gauge coupling atthe TeV scale constraints the isospin of the multiplet tobe j ≤ → SU (2) multipletwould have isospin j ≤ / j ≤
4. In the following, we shall report the mostgeneral renormalizable scalar potential V (Φ) for Φ andthe interaction potential V (Φ , H ) between Φ and H . Thehypercharge of Φ is denoted by Y .
1. Integer isospin j The electroweak multiplet Φ has 2 j + 1 componentfields, and the coupling of each component of Φ to the Z boson is proportional to T − Q sin θ W with the elec-tric charge Q = T + Y /
2. If the hypercharge is nonzero Y = 0, the neutral component of Φ has unsuppressedvector interaction with Z such that it can not be darkmatter candidate because of the constraints from directdetection. On the other hand, for Y = 0, the neutralcomponent of Φ could be potential dark matter candi-date. The scalar multiplet Φ can be real or complex. IfΦ is a real multiplet, there is a redundancy Φ = Φ suchthat the constraint φ j,m = ( − j − m φ ∗ j, − m should be ful-filled. For complex multiplet, each component representsa unique field, and it can be decomposed into two realmultiplets as follows A = 1 √ (cid:0) Φ + Φ (cid:1) , B = i √ (cid:0) Φ − Φ (cid:1) . (B2)It is easy to check that both A and B fulfill the real-ness condition A = A and B = B . Therefore a gen-eral model with a complex multiplet Φ is equivalent toa model of two interacting real multiplets A and B . Weshall present the concrete form of the scalar potentials V (Φ) and V ( H, Φ) for different cases of Y = 0 and Y = 0. • Complex Φ with Y = 0 and Y = ± V (Φ)= M Φ † Φ + j X J =0 λ J [(ΦΦ) J (Φ Φ) J ] ,V ( H, Φ)= α ( H † H )(Φ † Φ) + β [( HH ) (ΦΦ) ] , (B3)where only the terms of even J lead to nonzero con-tribution, H = iσ H ∗ with σ being Pauli matrix.The contraction [(ΦΦ) J (Φ Φ) J ] is given by [(ΦΦ) J (Φ Φ) J ] = X m C , J ,m ; J , − m (ΦΦ) J ,m (Φ Φ) J , − m , (B4) J ,m = X m ,m C J ,mj,m ; j,m φ j,m φ j,m , (Φ Φ) J ,m = X m ,m ( − j − m − m × C J ,mj,m ; j,m φ ∗ j, − m φ ∗ j, − m . (B5)Consequently the contraction (ΦΦ) J vanishes for odd J . Notice that all the independent self interactionsof Φ are included here while only two terms are con-sidered in [9, 10]. • Complex Φ with Y = 2 V (Φ)= M Φ † Φ + j X J =0 λ J [(ΦΦ) J (Φ Φ) J ] ,V ( H, Φ)= α ( H † H )(Φ † Φ) + β [( HH ) (ΦΦ) ] +[ µ (( HH ) Φ) δ j, + h.c.] , (B6)We see that an additional term [( HH ) Φ] and itshermitian conjugate are allowed if Φ is a isospintriplet with j = 1 and Y = 2. This term woulddisappear if one adopts a Z symmetry under whichall SM particles are Z even and extra scalar Φ is Z odd. • Complex Φ with Y = − V (Φ)= M Φ † Φ + j X J =0 λ J [(ΦΦ) J (Φ Φ) J ] ,V ( H, Φ)= α ( H † H )(Φ † Φ) + β [( HH ) (ΦΦ) ] + (cid:2) µ (( HH ) Φ) δ j, + h.c. (cid:3) , (B7) • Complex Φ with Y = 0 V (Φ) = M Φ † Φ + h M ′ (ΦΦ) +h.c. i + δ ,j (mod 2) h µ (Φ(ΦΦ) j ) + µ (Φ(ΦΦ) j ) + h.c. i + j X J =0 λ J [(ΦΦ) J (Φ Φ) J ] + j X K =0 (cid:26) λ ′K [(ΦΦ) K (ΦΦ) K ] + λ ′′K [(ΦΦ) K (ΦΦ) K ] + h.c. (cid:27) , (B8) V ( H, Φ) = h µ ( HH ) Φ δ j, + µ (( HH ) Φ) δ j, + h.c. i + α ( H † H )(Φ † Φ) + β [( HH ) (ΦΦ) ] + (cid:2) γ ( HH ) (ΦΦ) + h.c. (cid:3) . (B9) Notice that not all the interaction terms[(ΦΦ) K ](ΦΦ) K ] for K = 0 , , . . . , j are indepen-dent from each other. For j = 0 , ,
2, there isonly one independent interaction (ΦΦ) (ΦΦ) . Wefind two independent contractions (ΦΦ) (ΦΦ) and[(ΦΦ) (ΦΦ) ] for j = 3 , ,
5. However, thereare four independent interaction terms (ΦΦ) (ΦΦ) , [(ΦΦ) (ΦΦ) ] , [(ΦΦ) (ΦΦ) ] and [(ΦΦ) (ΦΦ) ] in the case of j = 10. For any given isospin j , we can straightforwardly find all the indepen-dent contractions among [(ΦΦ) K (ΦΦ) K ] with K =0 , , . . . , j . The same holds true for the contractions[(ΦΦ) K (ΦΦ) K ] . • Real Φ with Y = 0 V (Φ)= 12 M Φ † Φ + µ [Φ(ΦΦ) j ] δ ,j (mod 2) + j X K =0 λ K [(ΦΦ) K (ΦΦ) K ] V ( H, Φ)= µ ( HH ) Φ δ j, + µ [( HH ) Φ] δ j, + α ( HH ) (ΦΦ) . (B10)As regards the quartic self interaction terms[(ΦΦ) K (ΦΦ) K ] with K = 0 , , . . . , j , there isonly one independent contraction (ΦΦ) (ΦΦ) for j = 0 , ,
2. We find two independent contractions(ΦΦ) (ΦΦ) and [(ΦΦ) (ΦΦ) ] for the case of j =3 , ,
2. Half integer isospin j A scalar multiplet Φ of half integer isospin is alwayscomplex for any value of hypercharge Y . In other words,the realness condition Φ = Φ can not be fulfilled any-more. • Generic Φ with Y = 0, Y = ± Y = ± V (Φ)= M Φ † Φ + j X J =1 λ J [(ΦΦ) J (Φ Φ) J ] ,V ( H, Φ)= α ( H † H )(Φ † Φ) + β [( HH ) (ΦΦ) ] , (B11)where J should be odd otherwise the contraction(ΦΦ) J is vanishing. • Φ with Y = 0 V (Φ) = M Φ † Φ + j X J =1 λ J [(ΦΦ) J (Φ Φ) J ] + j X K =1 n λ ′K [(ΦΦ) K (ΦΦ) K ] + λ ′′K [(ΦΦ) K (ΦΦ) K ] + h . c . o ,V ( H, Φ) = α ( H † H )(Φ † Φ) + β [( HH ) (ΦΦ) ] + (cid:8) γ [( HH ) (ΦΦ) ] + h . c . (cid:9) , (B12)Both interaction terms [(ΦΦ) K (ΦΦ) K ] and[(ΦΦ) K (ΦΦ) K ] are vanishing for j = 1 /
2. Thereis only one independent contraction [(ΦΦ) (ΦΦ) ] for j = 3 / , / , /
2. We find only two independentterms [(ΦΦ) (ΦΦ) ] and [(ΦΦ) (ΦΦ) ] in case of j = 9 / • Φ with Y = 1 V (Φ)= M Φ † Φ + j X J =1 λ J [(ΦΦ) J (Φ Φ) J ] ,V ( H, Φ)= α ( H † H )(Φ † Φ) + β [( HH ) (ΦΦ) ] + n γ [( H Φ) j + (Φ Φ) j + ] δ ,j − (mod 2) + γ [( H Φ) j − (Φ Φ) j − ] δ ,j + (mod 2) + h.c. o + n κ [( HH ) (Φ Φ) ] + κ [( HH ) ( H Φ) ] δ j, + κ [( HH ) ( H Φ) ] δ j, + h.c. o . (B13) • Φ with Y = − V (Φ)= M Φ † Φ + j X J =1 λ J [(ΦΦ) J (Φ Φ) J ] ,V ( H, Φ)= α ( H † H )(Φ † Φ) + β [( HH ) (ΦΦ) ] + n γ [( H Φ) j + (Φ Φ) j + ] δ ,j − (mod 2) + γ [( H Φ) j − (Φ Φ) j − ] δ ,j + (mod 2) + h . c . o + n κ [( HH ) (Φ Φ) ] + κ [( HH ) ( H Φ) ] δ j, + κ [( HH ) ( H Φ) ] δ j, + h . c . o (B14) • Φ with Y = 3 V (Φ)= M Φ † Φ + j X J =1 λ J [(ΦΦ) J (Φ Φ) J ] ,V ( H, Φ)= α ( H † H )(Φ † Φ) + β [( HH ) (ΦΦ) ] + δ j, (cid:8) γ [( HH ) ( H Φ) ] + h . c . (cid:9) . (B15) • Φ with Y = − V (Φ)= M Φ † Φ + j X J =1 λ J [(ΦΦ) J (Φ Φ) J ] ,V ( H, Φ)= α ( H † H )(Φ † Φ) + β [( HH ) (ΦΦ) ] + δ j, (cid:8) γ [( HH ) ( H Φ) ] + h . c . (cid:9) . (B16) Appendix C: Self-Interactions
Starting from Eq. (21) a direct calculation yields theself interactions among the neutral fields φ , +) and φ , − )
14 [˜ κ + 2Re(˜ κ ) + 2Re(˜ κ )] φ , +) − [2Im(˜ κ ) + Im(˜ κ )] φ , +) φ , − ) (C1)+ 12 [˜ κ − κ )] φ , +) φ , − ) + [2Im(˜ κ ) − Im(˜ κ )] φ , +) φ , − ) + 14 [˜ κ + 2Re(˜ κ ) − κ )] φ , − ) , (C2)with ˜ κ ≡ κ + 421 √ κ + 677 κ + 100231 √ κ , ˜ κ ≡ κ ′ + 421 √ κ ′ + 677 κ ′ + 100231 √ κ ′ , ˜ κ ≡ − κ ′′ − √ κ ′′ − κ ′′ − √ κ ′′ . (C3)In the most general case, the dark matter is linear combi-nation of φ , +) and φ , − ) , since the mass matrix for φ , +) and φ , − ) shown in Eq. (15) is not diagonal,the dark matter self interactions can be easily extractedfrom Eq. (C5). In the limit of both M B and λ are real,the lightest one of φ , +) and φ , − ) is the DM can-didate, accordingly the self interaction can be read outstraightforwardly.For the electroweak scalar quintuplet φ , = ( φ , +) + iφ , − ) ) / √
2, the the self interactions among the neutralfields φ , +) and φ , − ) read as14 [˜ κ + 2Re(˜ κ ) + 2Re(˜ κ )] φ , +) − [2Im(˜ κ ) + Im(˜ κ )] φ , +) φ , − ) (C4)+ 12 [˜ κ − κ )] φ , +) φ , − ) + [2Im(˜ κ ) − Im(˜ κ )] φ , +) φ , − ) + 14 [˜ κ + 2Re(˜ κ ) − κ )] φ , − ) , (C5)with ˜ κ ≡ κ + 27 √ κ + 635 κ , ˜ κ ≡ κ ′ + 27 √ κ ′ + 635 κ ′ , ˜ κ ≡ κ ′′ + 27 √ κ ′′ + 635 κ ′′ . (C6) [1] M. Cirelli, N. Fornengo, and A. Strumia, Nucl. Phys. B753 , 178 (2006), hep-ph/0512090.[2] P. Fileviez Perez, H. H. Patel, M. Ramsey-Musolf, and K. Wang, Phys. Rev.
D79 , 055024 (2009), 0811.3957.[3] W.-B. Lu and P.-H. Gu, Nucl. Phys.
B924 , 279 (2017),1611.02106. [4] O. Fischer and J. J. van der Bij, JCAP , 032 (2014),1311.1077.[5] F.-X. Josse-Michaux and E. Molinaro, Phys. Rev. D87 ,036007 (2013), 1210.7202.[6] T. Basak and S. Mohanty, Phys. Rev.
D86 , 075031(2012), 1204.6592.[7] T. Araki, C. Q. Geng, and K. I. Nagao, Phys. Rev.
D83 ,075014 (2011), 1102.4906.[8] P. B. Pal, Phys. Lett.
B205 , 65 (1988).[9] T. Hambye, F. S. Ling, L. Lopez Honorez, andJ. Rocher, JHEP , 090 (2009), 0903.4010, [Erratum:JHEP05,066(2010)].[10] S. S. AbdusSalam and T. A. Chowdhury, JCAP ,026 (2014), 1310.8152.[11] J. Hisano, K. Ishiwata, N. Nagata, and T. Takesako,JHEP , 005 (2011), 1104.0228.[12] J. Hisano, D. Kobayashi, N. Mori, and E. Senaha, Phys.Lett. B742 , 80 (2015), 1410.3569.[13] R. J. Hill and M. P. Solon, Phys. Rev.
D91 , 043504(2015), 1401.3339.[14] R. J. Hill and M. P. Solon, Phys. Rev.
D91 , 043505(2015), 1409.8290.[15] W. J. G. de Blok, Adv. Astron. , 789293 (2010),0910.3538.[16] S. Tulin and H.-B. Yu, Phys. Rept. , 1 (2018),1705.02358.[17] J. Edsjo and P. Gondolo, Phys. Rev.
D56 , 1879 (1997),hep-ph/9704361.[18] T. Nihei, L. Roszkowski, and R. Ruiz de Austri, JHEP , 024 (2002), hep-ph/0206266.[19] K. A. Olive, D. N. Schramm, and G. Steigman, Nucl.Phys. B180 , 497 (1981).[20] Planck, P. A. R. Ade et al., Astron. Astrophys. , A13 (2016), 1502.01589.[21] M. Beneke, C. Hellmann, and P. Ruiz-Femenia, JHEP , 115 (2015), 1411.6924.[22] H. E. Logan and T. Pilkington, Phys. Rev. D96 , 015030(2017), 1610.08835.[23] J. Hisano, S. Matsumoto, and M. M. Nojiri, Phys. Rev.Lett. , 031303 (2004), hep-ph/0307216.[24] N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer, andN. Weiner, Phys. Rev. D79 , 015014 (2009), 0810.0713.[25] T. R. Slatyer, JCAP , 028 (2010), 0910.5713.[26] M. D’Onofrio, K. Rummukainen, and A. Tranberg, Phys.Rev. Lett. , 141602 (2014), 1404.3565.[27] M. Cirelli, A. Strumia, and M. Tamburini, Nucl. Phys.
B787 , 152 (2007), 0706.4071.[28] A. Strumia, Nucl. Phys.
B809 , 308 (2009), 0806.1630.[29] J. L. Feng, M. Kaplinghat, and H.-B. Yu, Phys. Rev.
D82 , 083525 (2010), 1005.4678.[30] LUX, D. S. Akerib et al., Phys. Rev. Lett. , 021303(2017), 1608.07648.[31] PandaX-II, X. Cui et al., Phys. Rev. Lett. , 181302(2017), 1708.06917.[32] XENON, E. Aprile et al., Phys. Rev. Lett. , 111302(2018), 1805.12562.[33] M. Ramsey-Musolf et al., In preparation.[34] G. Belanger, F. Boudjema, A. Pukhov, and A. Semenov,Comput. Phys. Commun. , 960 (2014), 1305.0237.[35] J. Hisano, K. Ishiwata, and N. Nagata, JHEP , 097(2015), 1504.00915.[36] K. Hally, H. E. Logan, and T. Pilkington, Phys. Rev. D85 , 095017 (2012), 1202.5073.[37] K. Earl, K. Hartling, H. E. Logan, and T. Pilkington,Phys. Rev.