Higgs Portals for Thermal Dark Matter - EFT Perspectives and the NMSSM -

# Higgs Portals for Thermal Dark Matter - EFT Perspectives and the NMSSM -

Sebastian Baum, The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, Alba Nova, 10691 Stockholm, SwedenNordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, SwedenFermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USAEnrico Fermi Institute and Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USADepartment of Physics & Astronomy, Wayne State University, Detroit, MI 48201, USAHEP Division, Argonne National Laboratory, 9700 Cass Ave., Argonne, IL 60439, USA    Marcela Carena, The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, Alba Nova, 10691 Stockholm, SwedenNordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, SwedenFermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USAEnrico Fermi Institute and Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USADepartment of Physics & Astronomy, Wayne State University, Detroit, MI 48201, USAHEP Division, Argonne National Laboratory, 9700 Cass Ave., Argonne, IL 60439, USA    Nausheen R. Shah, The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, Alba Nova, 10691 Stockholm, SwedenNordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, SwedenFermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USAEnrico Fermi Institute and Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USADepartment of Physics & Astronomy, Wayne State University, Detroit, MI 48201, USAHEP Division, Argonne National Laboratory, 9700 Cass Ave., Argonne, IL 60439, USA    Carlos E. M. Wagner The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, Alba Nova, 10691 Stockholm, SwedenNordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, SwedenFermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USAEnrico Fermi Institute and Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USADepartment of Physics & Astronomy, Wayne State University, Detroit, MI 48201, USAHEP Division, Argonne National Laboratory, 9700 Cass Ave., Argonne, IL 60439, USA
###### Abstract

We analyze a low energy effective model of Dark Matter in which the thermal relic density is provided by a singlet Majorana fermion which interacts with the Higgs fields via higher dimensional operators. Direct detection signatures may be reduced if blind spot solutions exist, which naturally appear in models with extended Higgs sectors. Explicit mass terms for the Majorana fermion can be forbidden by a symmetry, which in addition leads to a reduction of the number of higher dimensional operators. Moreover, a weak scale mass for the Majorana fermion is naturally obtained from the vacuum expectation value of a scalar singlet field. The proper relic density may be obtained by the -channel interchange of Higgs and gauge bosons, with the longitudinal mode of the boson (the neutral Goldstone mode) playing a relevant role in the annihilation process. This model shares many properties with the Next-to-Minimal Supersymmetric extension of the Standard Model (NMSSM) with light singlinos and heavy scalar and gauge superpartners. In order to test the validity of the low energy effective field theory, we compare its predictions with those of the ultraviolet complete NMSSM. Extending our framework to include neutral Majorana fermions, analogous to the bino in the NMSSM, we find the appearance of a new bino-singlino well tempered Dark Matter region.

\preprint

NORDITA-2017-130
0 FERMILAB-PUB-17-611-T
0 EFI-17-25
0 WSU-HEP-1715

## 1 Introduction

While the Standard Model (SM) is extremely successful in describing the known particle interactions, it fails to explain the large scale structure of the Universe, since it does not provide a good Dark Matter (DM) candidate. The simplest addition to the SM particle content would be in the form of a SM gauge singlet and in this work we shall concentrate on the particular example of a fermion as our DM candidate. In recent years, DM-nucleon scattering experiments such as LUX, XENON1T and PandaX-II have set stringent bounds on the possible couplings of DM to SM particles Angloher:2015ewa (); Agnese:2017jvy (); Aprile:2017iyp (); Cui:2017nnn (); Akerib:2017kat (); Amole:2017dex (). In particular, the coupling of the 125 GeV Higgs boson to DM is significantly constrained. In addition, the vector coupling of DM to the Z gauge boson must be very small (see for instance Ref. Escudero:2016gzx ()), therefore we shall concentrate on singlet Majorana fermions, which couple only axially to the boson. Such a fermion has the same gauge quantum numbers as a right-handed neutrino. One can define a matter parity, based on the quantum numbers of particles, namely , and demand interactions to be invariant under such a parity. Assuming that the DM carries no baryon or lepton numbers, this forbids all renormalizable interactions of the DM with SM particles, while allowing all SM Yukawa terms and Majorana masses for the right-handed neutrinos.

Since the coupling of DM to SM mediators is strongly constrained, we shall consider extending the SM by additional particles which can mediate interactions between DM and SM particles. Experimental precision tests of the SM strongly constrain extensions of the SM gauge sector, while far less is known about the SM Higgs sector. A well studied extension of the SM Higgs sector are so-called two Higgs doublet models (2HDMs) Branco:2011iw (), which consist of adding a second Higgs doublet, as commonly found in models that provide a dynamical origin of the electroweak symmetry breaking (EWSB) mechanism. The interactions of DM may quite generally be described by a set of non-renormalizable operators, including Majorana fermion bilinears and SM gauge invariant operators. The lower dimensional operators involve interactions with the Higgs fields and constitute a simple generalization of the so-called Higgs portal models Kanemura:2010sh (); Djouadi:2011aa (); Lebedev:2011iq (). As we shall see, the extended Higgs sector allows for the existence of blind spots where the interaction of the Higgs bosons with DM particles may be reduced, satisfying direct detection constraints, while still allowing for the possibility of obtaining the observed (thermal) relic density Perelstein:2012qg (); Cheung:2012qy (); Huang:2014xua (); Cheung:2014lqa (); Han:2016qtc (); Huang:2017kdh (); Badziak:2017uto ().

We shall require DM to be a weakly interacting massive particle (WIMP). In order to obtain a weak scale mass for the Majorana fermion in a natural way, we demand it to proceed from the vacuum expectation value (vev) of a singlet scalar field, which develops in the process of EWSB. The absence of explicit masses may be the result of the presence of an explicit symmetry, which also reduces the allowed number of higher dimensional operators and leads to a redefinition of the blind spot condition.

A possible realization of these class of models is provided by supersymmetric (SUSY) extensions of the SM Martin:1997ns () which also allow for a dynamical explanation for the weak scale Nilles:1983ge (); Haber:1984rc (); Martin:1997ns (). A particular virtue of the SUSY framework is that the stability of the Higgs mass parameter under quantum corrections can be ensured. In minimal extensions, the SM-like Higgs boson is naturally light Casas:1994us (); Haber:1996fp (); Degrassi:2002fi (), and corrections to electroweak precision and flavor observables tend to be small, leading to good agreement with observations. Additionally, low scale SUSY leads to the unification of couplings at high energies and provides a natural DM candidate, namely the lightest neutralino.

Among the simplest SUSY extensions, the Next-to-Minimal Supersymmetric extension of the SM (NMSSM) Ellwanger:2009dp (), fulfills all of the above properties while additionally containing a rich Higgs and neutralino spectrum. This may have an important impact on low energy observables. In particular, if the lighter neutralinos and neutral Higgs bosons are mainly singlets, they would be predominantly produced in association with heavier Higgs bosons or from the cascade decays of other SUSY particles, and therefore can easily avoid current direct experimental constraints Kang:2013rj (); King:2014xwa (); Carena:2015moc (); Ellwanger:2015uaz (); Costa:2015llh (); Baum:2017gbj (); Ellwanger:2017skc ().

The light neutralino in the NMSSM is naturally mostly singlino-like, but with a non-negligible Higgsino component. Hence, its spin independent direct DM detection (SIDD) cross section is mediated predominantly via the SM-like Higgs boson. The current bounds on the SIDD cross section lead to relevant constraints on the couplings of DM to the SM-like Higgs boson, and demand the theory to be in the proximity of blind spots, where the contributions from the non-standard Higgs bosons become also relevant. The proper relic density may also be obtained; the thermal annihilation cross section is dominated by either resonant contributions of the Higgs bosons, or non-resonant boson exchange contributions, with subdominant contributions from the light CP-even or the CP-odd Higgs bosons, the latter also having a large singlet component. In addition, a bino-like neutralino region with non-negligible Higgsino component may be present. In such a case, a sufficiently large thermal annihilation cross section yielding the proper relic density can be obtained by co-annihilation with the next-to-lightest neutralino, generally the singlino.

In Section 2 we use the language of Effective Field Theories (EFT) to outline the generic requirements of a model with singlet Majorana fermion DM and identify the required extended Higgs sector. In particular, we show the correlations of EFT parameters necessary to simultaneously obtain a thermal relic density, satisfy SIDD constraints, and accommodate a phenomenologically consistent Higgs sector. In Section 3 we discuss the NMSSM as a possible ultraviolet completion of our EFT model, and demonstrate the mapping of EFT parameters to NMSSM parameters utilizing a top-down EFT approach. In Section 4 we use the mature numerical tools available for the NMSSM to study the DM phenomenology, taking into account current collider and astrophysical constraints, as well as projections for the future. We identify two viable regions of parameter space with different DM phenomenology: 1) a new well tempered DM region, where the DM candidate is mostly bino-like and thermal production proceeds via resonant annihilation or co-annihilation with the singlino-like state, and 2) the region where the DM candidate is mostly singlino-like and the thermal relic density is mainly achieved via interactions mediated by the longitudinal mode of the boson, the neutral Goldstone mode. Much of the phenomenology in both regions can be understood from the properties of the EFT worked out in Section 2, although some details are only found in complete models such as the NMSSM. We reserve Section 5 for our conclusions.

## 2 An EFT for Singlet Dark Matter

As motivated in the introduction, we will consider a model of SM singlet Majorana fermion DM, which has no renormalizable interactions with SM particles. In order to couple DM to the SM, we consider a 2HDM Higgs sector, more specifically, we shall take two Higgs doublets with opposite hypercharges , with and with , which are naturally responsible for generating masses for the up and down-type quarks, respectively, as in type II 2HDMs. Since non-renormalizable interactions are suppressed by the scale associated with the masses of a heavy sector that was integrated out, one expects the dominant interactions to be associated with lower dimensional operators.

Including operators of dimension , the generic Lagrangian density describing interactions of a Majorana fermion with the two Higgs doublets , is

 L=−χχμ[δHu⋅Hd+γ(H†dHd+H†uHu)]−mχ2χχ+h.c. , (1)

where we have imposed a symmetry and used a dot notation for products,

 Hu⋅Hd=H+uH−d−H0uH0d . (2)

Assuming, as usual, that both Higgs doublets acquire vevs, , , with and , we can define the Higgs basis Georgi:1978ri (); Donoghue:1978cj (); gunion2008higgs (); Lavoura:1994fv (); Botella:1994cs (); Branco99 (); Gunion:2002zf ()111Note, that there are different conventions in the literature for the Higgs basis differing by an overall sign of and .

 HSM =√2Re(sinβH0u+cosβH0d), (3) G0 =√2Im(sinβH0u−cosβH0d), (4) HNSM =√2Re(cosβH0u−sinβH0d), (5) ANSM =√2Im(cosβH0u+sinβH0d), (6)

where and denote the neutral components of the respective Higgs doublets. These may be related to the usual type II 2HDM Higgs bosons by the relations

 Hid=ϵijHj∗1,Hiu=Hi2. (7)

The interaction eigenstate has the same couplings to SM particles as a SM Higgs boson, is the (neutral) Goldstone mode making up the longitudinal polarization of the boson after EWSB, and and are the non SM-like CP-even and CP-odd states, respectively. In particular, note that the Higgs basis fields are defined such that all the SM vev is acquired by the field corresponding to the neutral component of , hence and . Since the observed 125 GeV Higgs state appears to be close to SM-like in nature Aad:2015zhl (); Khachatryan:2016vau (), the interactions of with may be obtained from the above, approximating to first order. Ignoring the charged Higgs fluctuations, we obtain, at linear order in the fields,

 Hu⋅Hd→−v22s2β−v√2(s2βHSM+c2βHNSM+iANSM). (8)

Hence, the SM-like Higgs coupling to DM becomes

 gχχh≃gχχHSM=√2vμ(δsin2β−2γ). (9)

The interaction of a Majorana fermion with the SM-like Higgs boson listed above may be suppressed in three scenarios: 1) suppression of the couplings and ; 2) large values of ; and 3) a particular correlation of the two couplings and resulting in . The last scenario, the so-called blind spot solution, is given by

 sin2β=2γ/δ. (10)

It is interesting to consider a model in which there are no explicit mass terms or scales and hence the Lagrangian is scale invariant. In such a situation, a natural way to generate the mass and the scale is via the vev of a singlet . Hence, without loss of generality we can define and , where and are dimensionless parameters.

The absence of explicit scale dependence could be understood as originating from a symmetry, under which all scalar and fermion fields transform like  (therefore also ). Besides forbidding explicit fermion mass terms, imposing such a symmetry also forbids certain interactions. The remaining terms are

 L=−χχμ(δHu⋅Hd)−κSχχ+h.c., (11)

resulting in the following DM-Higgs sector interactions:

 gχχHSM=√2vμδsin2β ,gχχHNSM=√2vμδcos2β ,gχχANSM=i√2vμδ ,gχχHS=igχχAS=−√2κ. (12)

Imposing the symmetry removes the possibility of a blind spot as defined in Eq. (10). The contributions from the coupling to either the thermal annihilation cross section relevant for the relic density or the SIDD cross section is further suppressed by singlet-doublet mixing since the singlet does not couple to SM particles beyond the Higgs sector. Hence, the dominant contributions to the SIDD and the thermal annihilation cross section will be proportional to . Barring accidental cancellations between contributions from different Higgs bosons, the coupling must be suppressed in order to satisfy the stringent bounds from direct detection experiments. Hence, since current data implies that the dimension operators must be suppressed, we will include operators in the following. As we shall demonstrate, this will again allow for blind spot solutions to appear, enabling the suppression of the SIDD cross section. In addition, we find relevant contributions to the annihilation cross section from operators which will allow us to obtain sufficiently large annihilation cross sections to avoid over-closure of the Universe. The most relevant operators are suppressed by powers of with respect to the ones, and thus become most relevant if the ratio is not very small. One could inquire about the impact of higher dimensional operators in such a case. We shall address this question later by considering an ultraviolet completion of the EFT. Although the qualitative features found in the EFT remain valid in the complete theory, the precise quantitative predictions will indeed be affected to some degree by higher dimensional terms.

Assuming that the terms in Eq. (11) originate from a theory where a heavier -doublet Dirac fermion with mass has been integrated out, we can write all the allowed operators which would arise from integrating out such a field. Ignoring the charged gauge boson interactions, we get

 L= −δχχμ(Hu⋅Hd)(1−λ^Sμ)−κSχχ⎛⎝1+ξH†dHd+H†uHu|μ|2⎞⎠+h.c.+α|μ|2{χ†H†u¯σμ[i∂μ−g1sW(T3−Qs2W)Zμ](χHu)+χ†H†d¯σμ[i∂μ−g1sW(T3−Qs2W)Zμ](χHd)}, (13)

where , and are the charge and weak isospin operators, with the weak mixing angle , and is the hypercharge coupling. Note, that the term proportional to (the fluctuations of ) in the -term arises because this originally term was actually suppressed by , which we have expanded around the vev of , yielding . On the other hand, all the terms arising from integrating out a Dirac fermion are suppressed by instead of . Moreover, we have not included terms involving higher powers of the singlet field, since they are not expected to arise from integrating out a Dirac doublet fermion. The DM interactions with singlets are dominated by the tree level coupling , and the only modification from such terms would be a redefinition of the coupling . Observe, that if dealing with on-shell fields, there is a redundancy in the above terms, since the application of the equation of motion on the terms proportional to the derivative of will lead to terms proportional to the mass, which also appear from the -term when inserting the vev of the field . Another important point to note from Eq. (13) is that the presence of derivative terms allows for interactions between the Goldstone and DM, absent in Eq. (11), which as we shall see turn out to be relevant for the thermal annihilation cross section. For the convenience of the reader, we write Eq. (13) in terms of the Higgs basis states in the Appendix A, Eq. (108).

From Eq. (13), the coupling of the DM particles to the SM-like Higgs is given by

 gχχh≃gχχHSM=√2vμ[δsin2β−(ξ−α)mχμ∗], (14)

where the dependence on results from the application of the equations of motion. In general, we calculate the on-shell relationships by using the fact that, ignoring total derivatives,

 i(∂μΦ)χ†i¯σμχ=−iΦχ†i¯σμ(\lx@stackrel←∂μ+\lx@stackrel→∂μ)χ=imχΦχχ+h.c. , (15)

where is a real scalar field. Note, that the direct expansion of the derivative terms proportional to in Eq. (13) leads to interactions with the CP-even Higgs bosons when the derivative is acting on the Majorana fermion fields, and to derivative interactions with the CP-odd Higgs states when the derivative is acting on the Higgs doublets, as required by hermiticity.

We see that the blind spot for the cancellation of the coupling of to pairs of DM now occurs for

 sin2β=(ξ−α)mχμ∗δ, (16)

and we can further match the interactions dictated by the Lagrangians given in Eqs. (1) and (13) by noting that

 γ=(ξ−α)mχ2μ∗. (17)

The interactions with are

 gχχHNSM=√2vμδcos2β. (18)

Note, that there are no terms proportional to (or ) and therefore there is no blindspot such as the one for in Eq. (16); instead for .

On the other hand, the interactions with the CP-even singlet state are given by

 gχχHS=−√2{v22μ2δλsin2β+κ[1+(ξ−α)v2|μ|2]}. (19)

Here, the dependence on comes from a field renormalization of necessary to retain a canonical kinetic term for when including dimension operators. In principle, this field renormalization introduces corrections to all couplings of . However, we are only considering operators of . The modification from the field renormalization is suppressed by , hence, this correction is only relevant for the renormalizable interactions.

The interactions of with the CP-odd scalars are easy to read from the above as well. For instance, although the Goldstone interactions involve derivatives of the Goldstone fields, for on-shell ’s one can use Eq. (15) to obtain the interaction with the (neutral) Goldstone mode

 gχχG0=−i√2mχv|μ|2αcos2β. (20)

The orthogonal state, , also has relevant interactions with DM, namely

 gχχANSM=i√2vμ(δ+mχμ∗αsin2β). (21)

Finally, the interactions of the CP-odd singlet state are analogous to its CP-even counterpart,

 gχχAS=i√2{v22μ2δλsin2β+κ[1+(ξ−α)v2|μ|2]}. (22)

### 2.1 Higgs Sector

In the previous section we have motivated a structure for the scalar sector consisting of two Higgs doublets and one singlet, all three of which acquire a vev. We can define the extended Higgs Basis, for the CP-even states and for the CP-odd states, where the doublet components are as defined in Eqs. (3)–(6), and the singlet does not get rotated Carena:2015moc (). These interaction eigenstates mix into mass eigenstates. We denote the CP-even mass eigenstates as ,

 hi=SSMhiHSM+SNSMhiHNSM+SShiHS, (23)

and the CP-odd states as ,

 ai=PNSMaiANSM+PSaiAS. (24)

The mixing angles and are obtained from the diagonalization of the corresponding mass matrices. We can write the (symmetric) squared mass matrix of the CP-even Higgs bosons in the extended Higgs Basis as

 (25)

Since the observed Higgs boson is predominantly SM-like, we can parametrize the elements corresponding to and mixing as

 M2S,12≡ϵ¯¯¯M2S,12,M2S,13≡η¯¯¯M2S,13, (26)

where and are small parameters , and we have defined , and similarly, . Barring the possibility of very degenerate diagonal mass terms, these relations ensure that the SM-like state has only small mixings with the non-standard states and that its mass squared can be approximately identified with the matrix element. Observe, that after imposing the minimization conditions and become proportional to the square of the Higgs vev . The matrix elements and are only linear in .

Keeping terms to linear order in the small parameters and only, the eigenvalues are

 m2h≃M2S,11 ,m2hS,H≃M2S,22+M2S,33∓√(M2S,22−M2S,33)2+4(M2S,23)22 . (27)

The eigenvectors are

 SNSMhSSMh =−η¯¯¯M2S,13M2S,23−ϵ¯¯¯M2S,12(m2h−M2S,33)(M2S,23)2−(m2h−M2S,22)(m2h−M2S,33), (28) SShSSMh =−ϵ¯¯¯M2S,12M2S,23−η¯¯¯M2S,13(m2h−M2S,22)(M2S,23)2−(m2h−M2S,22)(m2h−M2S,33), (29)

for the SM-like mass eigenstate,

 SSMHSNSMH =−η¯¯¯M2S,13M2S,23−ϵ¯¯¯M2S,12(m2H−M2S,33)η2(¯¯¯M2S,13)2−(m2H−M2S,11)(m2H−M2S,33), (30) SSHSNSMH =−ϵη¯¯¯M2S,12¯¯¯M2S,13−M2S,23(m2H−M2S,11)η2(¯¯¯M2S,13)2−(m2H−M2S,11)(m2H−M2S,33), (31)

for the doublet-like eigenstate, and

 SSMhSSShS =−ϵ¯¯¯M2S,12M2S,23−η¯¯¯M2S,13(m2hS−M2S,22)ϵ2(¯¯¯M2S,12)2−(m2hS−M2S,11)(m2hS−M2S,22), (32) SNSMhSSShS =−ϵη¯¯¯M2S,12¯¯¯M2S,13−M2S,23(m2hS−M2S,11)ϵ2(¯¯¯M2S,12)2−(m2hS−M2S,11)(m2hS−M2S,22), (33)

for the singlet-like mass eigenstate.

If we use the approximate eigenmasses, we find for the SM-like mass eigenstate

 SSMh≈1,SNSMhSSMh,SShSSMh=O(ϵ,η), (34)

and for the other mass eigenstates

 SSMH ≈SSMhS≈0 , (35) −SSHSNSMH (36) SNSMH ≈SShS≈⎡⎢⎣1+⎛⎝SNSMhSSShS⎞⎠2⎤⎥⎦−1/2 . (37)

### 2.2 EFT: Relic Density

In the absence of co-annihilation, the thermally averaged annihilation cross section for a pair of DM particles at temperature can be expanded as

 ⟨σχχv⟩≡⟨σ(χχ→SM)v⟩=a+b⟨v2⟩+O(⟨v4⟩)=a+6bx+O(1x2), (38)

where . After integrating over the thermal history of the Universe until the freeze-out temperature , the thermal relic density is obtained

 Ωh2=0.12(80g∗)1/2(xF25)(2.3×10−26cm3/s⟨σv⟩xF),⟨σv⟩xF≡a+3bxF. (39)

The interactions of the singlet fermion with SM particles depicted in Fig. 1 arise via the couplings to the extended Higgs basis states given in Eqs.(14)–(22) and the mixing of extended Higgs basis states into mass eigenstates,

 gχχhi=SSMhigχχHSM+SNSMhigχχHNSM+SShigχχHS,gχχai=PNSMaigχχANSM+PSaigχχAS. (40)

The singlet states and do not couple to SM particles, thus, assuming a type II 2HDM Yukawa structure, the couplings of the mass eigenstates to up-type quarks are given by

 guhi=⎛⎝SSMhi+SNSMhitanβ⎞⎠mu√2v,guai=iPNSMaitanβmu√2v, (41)

and to down-type quarks by

 gdhi=(SSMhi−SNSMhitanβ)md√2v,gdai=iPNSMaitanβmd√2v. (42)

For completeness, we record the couplings to pairs of vector bosons

 gW+W−hi=2m2WvSSMhi,gZZhi=m2ZvSSMhi,gW+W−ai=gZZai=0 . (43)

The contribution to from annihilation into pairs of quarks () from the -channel exchange of the CP-even Higgs bosons is given by

 ⟨σv⟩q¯q,pxF=32π34Tmχ(1−m2qm2χ)3/2∣∣ ∣∣∑iAq¯qhi∣∣ ∣∣2,Aq¯qhi=−gχχhigqhimχ(m2hi−4m2χ), (44)

and from the exchange of CP-odd Higgs boson by

 ⟨σv⟩q¯q,sxF=32π(1−m2qm2χ)1/2∣∣ ∣∣∑iAq¯qai∣∣ ∣∣2,Aq¯qai=−gχχaigqaimχ(m2ai−4m2χ), (45)

for with the width of the Higgs mass eigenstate . Note, that there is no interference between the contributions listed in Eqs. (44) and (45) since the scalar Higgs bosons exchange contribution is -wave suppressed while the annihilation cross section via pseudoscalar Higgs bosons is -wave. For typical freeze-out temperatures , the contribution from CP-even Higgs bosons to is suppressed by compared to the contribution from CP-odd Higgs bosons, as long as such that the kinematic correction from the quark mass is irrelevant.

Besides via Higgs bosons, () annihilations can also be mediated by the -channel exchange of bosons. This is accounted for by extending the sum in Eq. (45) to include the -wave amplitudes mediated by both the longitudinal polarization of the boson, i.e. the (neutral) Goldstone mode ,

 Aq¯qG0=−gχχG0gqG0mχ(m2Z−4m2χ), (46)

as well as the transversal polarizations of the boson

 Aq¯qZ=−mqmχgχχZgqZmχ(m2Z−4m2χ). (47)

The couplings of the Goldstone mode to up-type and down-type quarks, respectively, are

 guG0=imu√2v ,gdG0=−imd√2v , (48)

and the relevant axial-vector coupling of the transversal polarizations of the boson to quarks are

 guZ=−gdZ=g14sW . (49)

The coupling of the boson to the Majorana fermion can be read off from Eqs. (13) or (108)

 gχχZ=−v2|μ|2αg12sWcos2β . (50)

Note, that the -wave contribution to the annihilation cross section from the transversal polarization of the boson is suppressed with respect to that of its longitudinal polarization (the neutral Goldstone mode) by .

All the amplitudes appearing in Eqs. (44)-(47) are proportional to the Yukawa couplings. Due to the hierarchy of the Yukawa couplings, the contribution to the thermal cross section from () annihilations will be dominated by the heaviest accessible quarks, i.e. top-quarks for GeV and bottom quarks for lighter . In the latter case the -wave contribution from the transversal polarization of the may become relevant, since in contrast to its -wave contribution listed above it is not chirality suppressed (hence, not proportional to the Yukawa couplings).

It is interesting to consider the size of the thermally averaged cross section obtainable via () annihilation. For example, if we assume , such that is kinematically allowed, and assume the dominant annihilation channel to be via the (neutral) Goldstone mode, for , Eqs. (45), (46) and (48), approximately lead to

 ⟨σv⟩q¯qxF∼2×10−26cm3s(∣∣gχχG0∣∣0.1)2(mχ300GeV)−2. (51)

Hence, the correct relic density  Ade:2015xua () is obtained from () annihilations for couplings .

In addition to the () annihilation discussed above, there may be relevant contributions to from () annihilations, where denotes a scalar or pseudoscalar Higgs mass eigenstate. Such processes can be mediated either via diagrams with a Higgs or a boson in the -channel, or via -channel exchange of the Majorana fermion or the (heavy) -doublet Dirac fermion we integrated out. In our EFT, the last possibility proceeds via the contact interaction terms in Eq. (13). Regardless of the type of diagram, the annihilation into a pair of CP-even () or CP-odd Higgs bosons () is -wave suppressed, while () annihilations are -wave. The corresponding -wave contribution to the thermally averaged annihilation cross section is given by

 ⟨σv⟩haxF=164πm2χ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩⎡⎢ ⎢ ⎢⎣1−(mhi+maj)24m2χ⎤⎥ ⎥ ⎥⎦⎡⎢ ⎢ ⎢⎣1−(mhi−maj)24m2χ⎤⎥ ⎥ ⎥⎦⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭1/2∣∣ ∣∣∑kAhiajk∣∣ ∣∣2, (52)

where the sum includes the -wave amplitudes mediated by CP-odd scalars in the -channel

 AhiajΦk=−2mχgχχΦkghiajΦkm2Φk−4m2χ, (53)

the amplitude mediated by transversally polarized bosons in the -channel

 AhiajZ=−gχχZg1sWPNSMajSNSMhim2hi−m2ajm2Z−4m2χ, (54)

the amplitudes mediated by the Dirac fermion in the -channel proceeding via contact interaction terms after integrating out the Dirac fermion

 AhiajΨ=−2gχχhiajmχ, (55)

and the amplitude from the -channel exchange of the Majorana fermion

 Ahiajχ=−2gχχhigχχaj⎡⎢ ⎢⎣1+2m2aj4m2χ−(m2hi+m2aj)⎤⎥ ⎥⎦. (56)

In Eqs. (53)–(56), the are the couplings of pairs of ’s to the Higgs mass eigenstates given in Eq. (40) [Eq. (20) and (50) for the coupling to the neutral Goldstone mode and the transversal polarizations of the boson , respectively], the () are the dimensionful trilinear couplings between different Higgs mass eigenstates (between the neutral Goldstone mode and two Higgs mass eigenstates), which are not related to the parameters of our EFT, but arise from the Higgs potential (see Appendices of Ref. Carena:2015moc () for details). The are the () couplings of dimension (mass) which can be read off from the Lagrangian Eqs. (13) or (108) taking into account the mixing of the Higgs mass eigenstates, Eqs. (23), (24).

After accounting for suppression of couplings arising from the requirement of an GeV SM-like Higgs mass eigenstate and from approximately satisfying the blindspot condition, the most relevant final states for the () processes are () and (). If kinematically accessible, they can compete with () annihilation. For both these channels, the amplitudes mediated by the singlet-like CP-odd in the -channel may play an important role. However, their relevance to the total cross section is dictated by the coupling strengths and respectively, which as mentioned above are not related to our EFT parameters. Hence for simplicity, we will assume that these couplings are small, rendering these processes irrelevant for the relic density.

Ignoring such Higgs exchange diagrams, the amplitude mediated by a -channel Dirac fermion, which we integrated out, is most relevant for the final state (). Ignoring the kinematic correction in Eq. (52) which is relevant only very close to threshold (), canonical values of the thermally averaged annihilation cross section can be achieved for

 ∣∣gχχhaS∣∣≈∣∣gχχHSMAS∣∣≈∣∣∣−ivμ(λδμsin2β+2κξμ∗)∣∣∣∼4×10−4GeV−1, (57)

where we have assumed the mixing of the CP-odd Higgs bosons to be small. This corresponds to couplings

 δλsin2β+2κξ∼(μ700GeV)2 . (58)

For the channel () the processes associated with a Majorana fermion in the -channel can be relevant. Assuming again for simplicity that these processes dominate compared to the one associated to the interchange of a singlet pseudoscalar (i.e. is small), neglecting the threshold corrections for (), and corrections from singlet-doublet mixing (the latter potentially leading to suppression), the thermally averaged annihilation cross section can be achieved for a DM coupling to the singlets

 ∣∣gχχaS∣∣≈∣∣gχχhS∣∣∼0.2(mχ300GeV)1/2. (59)

This implies

 |κ|∼0.15 (mχ300GeV)1/2, (60)

where we assumed for the estimate on such that .

Annihilations into pairs of vector bosons [] do not play an important role for obtaining the thermal relic density as long as . Final states consisting of two vector bosons with longitudinal polarizations are -wave suppressed since they correspond to annihilations into a pair of CP-odd scalars (i.e. the neutral and charged Goldstone modes for the and bosons, respectively). Annihilations into a pair of transversally polarized vector bosons or one transversally polarized and one longitudinally polarized vector boson are -wave. However, such annihilations proceeding via -channel exchange of the neutral (charged) components of the -doublet fermion we integrated out correspond to (