JHEP04(2018)069 3 d,f Z -channel Springer s boson (the , April 12, 2018 Z March 27, 2018 : January 8, 2018 : : Published Accepted Received symmetry, which in addi- 3 and Carlos E.M. Wagner Z e Published for SISSA by [email protected] https://doi.org/10.1007/JHEP04(2018)069 , Nausheen R. Shah c,d [email protected] , . 3 Marcela Carena, a,b 1712.09873 The Authors. Cosmology of Theories beyond the SM, Beyond Standard Model, Effective c

We analyze a low energy effective model of Dark Matter in which the thermal , [email protected] E-mail: [email protected] Chicago, IL 60637, U.S.A. Department of Physics &Detroit, Astronomy, MI Wayne 48201, State U.S.A. University, HEP Division, Argonne National9700 Laboratory, Cass Ave., Argonne, IL 60439, U.S.A. Alba Nova, 10691 Stockholm, Sweden Nordita, KTH Royal InstituteRoslagstullsbacken of 23, Technology and 10691 Stockholm Stockholm, University, Sweden Fermi National Accelerator Laboratory, P.O. Box 500, Batavia,Enrico IL Fermi 60510, Institute U.S.A. and Kavli Institute for Cosmological Physics, University of Chicago, The Oskar Klein CentreDepartment for of Cosmoparticle Physics, Physics, Stockholm University, b c e d a f ArXiv ePrint: Open Access Article funded by SCOAP neutral Majorana fermions, analogous toof the a bino new in bino-singlino the well NMSSM, temperedKeywords: we Dark find Matter the region. appearance Field Theories, Supersymmetric Effective Theories neutral Goldstone mode) playingshares a many relevant properties role with indard the the Model Next-to-Minimal annihilation Supersymmetric (NMSSM) process. with extensionorder light of to This singlinos the test model and Stan- the validity heavy ofwith scalar the those and low of energy gauge effective the superpartners. field ultraviolet theory, In we complete compare NMSSM. its predictions Extending our framework to include solutions exist, which naturallymass appear terms for in the modelstion Majorana with leads fermion to extended can a Higgs be reductionscale sectors. forbidden of mass the by number a Explicit for of the highervalue Majorana dimensional of operators. fermion a Moreover, scalar is ainterchange singlet naturally weak of field. obtained Higgs from The and proper the gauge relic vacuum bosons, expectation density may with be the obtained longitudinal by mode the of the Abstract: relic density is provided by avia singlet higher Majorana dimensional fermion operators. which interacts Direct with detection the signatures Higgs may fields be reduced if blind spot EFT perspectives and the NMSSM Sebastian Baum, Higgs portals for thermal Dark Matter. JHEP04(2018)069 , ) L − B 3( 1) − boson. Such a = ( Z P ]), therefore we shall 7 15 ]. In particular, the coupling of the 6 – 1 – 1 – 3 17 24 ) quantum numbers of particles, namely L 14 − 32 B 9 31 numbers, this forbids all renormalizable interactions of the DM 20 L 35 7 1 33 or lepton B Since the coupling of DM to SM mediators is strongly constrained, we shall consider ex- 3.1 Top-down EFT 4.1 Indirect detection 4.2 Collider constraints 2.1 Higgs sector 2.2 EFT: relic2.3 density EFT: direct2.4 detection Bounds on couplings and parameters of the EFT no baryon with SM particles,right-handed while neutrinos. allowing all SM Yukawa termstending and the SM Majorana by additional masses particles which for can mediate the interactions between DM and SM 125 GeV Higgs boson toof DM DM to is the significantly Z constrained.concentrate gauge on boson In singlet must Majorana addition, be fermions, the very whichfermion small vector couple has (see only coupling the for axially same instance to gauge ref. the quantummatter [ numbers parity, as based a on right-handed the neutrino. ( and One demand can interactions define to a be invariant under such a parity. Assuming that the DM carries provide a good Darkcontent Matter would be (DM) in candidate. theon form The the of particular simplest example a addition of SMscattering to a gauge experiments fermion the such singlet as as SM and our LUX, in particle on DM XENON1T this and candidate. the work PandaX-II possible In we have recent couplings set shall years, stringent of concentrate DM-nucleon bounds DM to SM particles [ 1 Introduction While the Standard Modelinteractions, (SM) it is fails extremely to successful explain in the describing large the scale known structure particle of the Universe, since it does not 5 Conclusions A EFT Lagrangian 4 Dark Matter phenomenology 3 NMSSM ultraviolet completion Contents 1 Introduction 2 An EFT for singlet Dark Matter JHEP04(2018)069 ], which consist boson exchange 8 Z ]. 19 ], and corrections to electroweak preci- – 25 13 – 23 – 2 – ]. 33 – 27 ]. As we shall see, the extended Higgs sector allows for the 12 – 9 ] which also allow for a dynamical explanation for the weak ], fulfills all of the above properties while additionally containing 20 where the interaction of the Higgs bosons with DM particles may 26 symmetry, which also reduces the allowed number of higher dimensional 3 Z blind spots ]. A particular virtue of the SUSY framework is that the stability of the 22 – 20 The light neutralino in the NMSSM is naturally mostly singlino-like, but with a non- Among the simplest SUSY extensions, the Next-to-Minimal Supersymmetric extension A possible realization of these class of models is provided by supersymmetric (SUSY) We shall require DM to be a weakly interacting massive particle (WIMP). In order the contributions from therelic non-standard density Higgs may bosons also becomeby also be either relevant. resonant obtained; contributions The thecontributions, of proper with thermal subdominant the contributions annihilation Higgs from the cross bosons,bosons, light CP-even the section or or latter non-resonant the is also CP-odd having dominated Higgs a large singlet component. In addition, a bino-like neutralino direct experimental constraints [ negligible Higgsino component. Hence,cross its section is spin mediated independent predominantlyon direct via the DM the SIDD detection SM-like cross Higgs (SIDD) sectionlike boson. lead Higgs The to current boson, relevant bounds constraints and on demand the couplings the of theory DM to to the be SM- in the proximity of blind spots, where of the SM (NMSSM)a [ rich Higgs and neutralinoobservables. spectrum. In This particular, may ifsinglets, have the an they lighter important would neutralinos impact be on andor predominantly low neutral from produced energy the Higgs in cascade bosons association decays are with of mainly heavier other SUSY Higgs particles, bosons and therefore can easily avoid current the SM-like Higgs bosonsion is and flavor naturally observables light tendAdditionally, to [ be low small, scale leading SUSY toprovides good a leads agreement natural to with DM the observations. candidate, unification namely the of lightest couplings neutralino. at high energies and operators and leads to a redefinition of the blindextensions spot of condition. thescale SM [ [ Higgs mass parameter under quantum corrections can be ensured. In minimal extensions, to obtain a weakproceed scale from mass the for vacuum thein expectation Majorana the value fermion process (vev) in of of a EWSB.of a natural The an singlet way, absence explicit scalar we of field, demand explicit which it masses develops to may be the result of the presence rana fermion bilinears and SMinvolve gauge interactions invariant operators. with the The lower Higgscalled dimensional fields Higgs operators portal and models constitute [ aexistence simple of generalization of thebe so- reduced, satisfying direct detectionobtaining constraints, the while observed still (thermal) allowing relic for the density possibility [ of gauge sector, while far lessof is the SM known about Higgs sector theof are SM adding so-called Higgs a two sector. second Higgs Higgsorigin A doublet doublet, of models well as the studied (2HDMs) commonly electroweak extension [ symmetry foundmay breaking in quite (EWSB) models generally mechanism. that be The provide described interactions a by of a dynamical DM set of non-renormalizable operators, including Majo- particles. Experimental precision tests of the SM strongly constrain extensions of the SM JHEP04(2018)069 , 2, 2 / 1 (2.2) (2.1) − , with u v = = Y i u is we discuss the H h d with 3 ., , H c d d . , v u H = + h H i d χχ H χ 2 and h 2 / m . − 0 d i ) = +1 H u 0 u H Y H † u H − − d + and used a dot notation for SU(2) products, with d H u DM region, where the DM candidate is mostly 5, the generic Lagrangian density describing u + H u H – 3 – † d H ≤ H H , ↔ ( d = Y γ d with the two Higgs doublets d H + χ H · d u H · H u well tempered δH h µ χχ boson, the neutral Goldstone mode. Much of the phenomenology in − Z = we use the language of Effective Field Theories (EFT) to outline the L 2 for our conclusions. we use the mature numerical tools available for the NMSSM to study the DM phe- 5 4 Including operators of dimension In section where we have imposed a symmetry Assuming, as usual, that both Higgs doublets acquire vevs, interactions of a Majorana fermion to the SM, wedoublets consider with a 2HDM opposite Higgswhich hypercharges sector, are more naturally specifically, responsiblerespectively, we as for shall in generating type take II masses two 2HDMs.the for Higgs scale Since the associated non-renormalizable with up interactions are the andthe suppressed masses dominant down-type by of interactions a quarks, to heavy be sector that associated was with integrated lower out, dimensional one operators. expects 2 An EFT for singletAs Dark motivated in Matter the introduction, weDM, will which consider has a model no of renormalizable SM interactions singlet with Majorana fermion SM particles. In order to couple DM dinal mode of the both regions can bealthough understood some from details are the only propertiessection found of in the complete models EFT such worked as out the in NMSSM. section We reserve projections for the future. WeDM identify phenomenology: two 1) viable a regions new ofbino-like parameter and thermal space production with proceeds different viathe resonant annihilation singlino-like or state, co-annihilation with andand 2) the the thermal region relic density where is the mainly DM achieved candidate via interactions is mediated mostly by singlino-like the longitu- necessary to simultaneously obtainaccommodate a a thermal phenomenologically relic density, consistentNMSSM satisfy Higgs as SIDD a sector. possible constraints, ultraviolet and Inping completion of section of our EFT EFT model, parameterssection and to demonstrate NMSSM the map- parameters utilizingnomenology, taking a into top-down account EFT current approach. collider and In astrophysical constraints, as well as large thermal annihilation crossby section co-annihilation yielding with the the proper next-to-lightest neutralino, relic generally density the can singlino. begeneric obtained requirements of aquired model extended with Higgs singlet sector. Majorana In fermion particular, DM we show and the identify correlations the of re- EFT parameters region with non-negligible Higgsino component may be present. In such a case, a sufficiently JHEP04(2018)069 0. are ∼ (2.8) (2.9) (2.3) (2.4) (2.5) (2.6) (2.7) λ (2.10) , hence χχh . Hence, 1 g
SM ] and S H 40 κ – iA appears to be 34 + h . S
; 2) large values of H γ , where resulting in i 2 NSM 1 S √ γ h and iA λ + may be obtained from the . δ , , , , + i ) = and h



S γ 0 0 0 0 . d d h d d δ 2 µ i 2 NSM = − H βH βH βH βH with H S β and β = χ 2 sin cos i i c u S + sin − sin 2 − + cos h + 0 0 δ u u 0 0 κ u u ( γ/δ . SM v are the non SM-like CP-even and CP-odd βH βH βH βH 2 = 2 µ H = 2 β √ χ 2 sin cos sin cos – 4 – , we can define the Higgs basis [ β solution, is given by s NSM ,H d m = ∗ A j 2 1 /v v SM u sin 2 2Re 2Im 2Re 2Im √ H v ij √ √ √ √ and  − = χχH g = = = = ], the interactions of β = β 2 to first order. Ignoring the charged Higgs fluctuations, we blind spot 0 s i ' d NSM 42 2 SM G is via the vev of a singlet , 2 H v SM NSM NSM H = 0. Since the observed 125 GeV Higgs state H µ 41 χχh A H H i g and tan ∼ → − 2 NSM h d . H H · h denote the neutral components of the respective Higgs doublets. These u NSM 0 u H A H and interaction eigenstate has the same couplings to SM particles as a SM Higgs and the scale v and 2 χ SM is the (neutral) Goldstone mode making up the longitudinal polarization of the and √ m 0 H 0 d ) = (174 GeV) NSM = G 2 u H ; and 3) a particular correlation of the two couplings H v i v It is interesting to consider a model in which there are no explicit mass terms or scales The + Note, that there are different conventions in the literature for the Higgs basis differing by an overall SM boson after EWSB, and 1 2  d H v the mass without loss of generalitydimensionless we parameters. can define sign of and hence the Lagrangian is scale invariant. In such a situation, a natural way to generate The interaction of a Majoranasuppressed fermion in with three the scenarios:µ SM-like 1) Higgs boson suppression listed of aboveThe the may last couplings be scenario, the so-called Hence, the SM-like Higgs coupling to DM becomes h close to SM-like in natureabove, approximating [ obtain, at linear order in the fields, boson, Z states, respectively. In particular, notethe that SM the vev is Higgs acquired basis by fields the are field defined corresponding such to that the all neutral component of where may be related to the usual type II 2HDM Higgs bosons by the relations ( JHEP04(2018)069 3 Z with (2.11) (2.12) (2.13) W θ δ , 3] Ψ (there- sin v . 2 ≡ = 6 operators c µ πi/ . √ d W i s + h with respect to the = exp[2 ! 5 terms are , u /µ → NSM  χ ≤ H ) † u d d m χχA H ) 2 | is not very small. One could u χH ., + µ ( c | . d  /µ κ . χH µ H ( χ 2 † d + h Z  m ) √ β , g µ H = 6 operators which will allow us to − 6 operators in such a case. We shall 2 W ξ Z ) d must be suppressed in order to satisfy = Qs δ cos 2 2 W κSχχ S 1 + d > δ ) originate from a theory where a heavier − −

v Qs is the hypercharge coupling. Note, that the 3 ) coupling to either the thermal annihilation 2 does not couple to SM particles beyond the χχA µ T d − ( W 2.11 √ ig has been integrated out, we can write all the S 3 H θ – 5 – 1 · T W = = g µ u ( κSχχ s χχS cos . Barring accidental cancellations between contri- 1 S W − 2 − g δH s δ ( e/ µ ! NSM − χχH ˆ S i∂ = µ µ g χχ µ  λ 1 χχH µ g − i∂ − ¯ σ  are the charge and weak isospin operators, † d = 1 µ 4 terms in eq. ( 3 H ¯ σ

L T † † u , and ) ). Besides forbidding explicit fermion mass terms, imposing such χ d µ H β , g W d > † H θ symmetry removes the possibility of a blind spot as defined in + and 3] · = 5 operators must be suppressed, we will include χ u = 6 operators are suppressed by powers of 3  Q d sin 2 H Z πi/ d , 2 δ ( | ˆ S α v µ | 2 µ µ + χχ √ exp[2 δ + = − µ/λ → = 6 operators which would arise from integrating out such a field. Ignoring the µ = ). The contributions from the = d SM L S symmetry also forbids certain interactions. The remaining 2.10 χχH Assuming that the Imposing the The absence of explicit scale dependence could be understood as originating from a 3 g Z = 5 ones, and thus become most relevant if the ratio where the weak mixing angle charged gauge boson interactions, we get address this question later bythe considering qualitative an features ultraviolet found completion inquantitative of predictions the will the EFT indeed EFT. remain be Although affected valid to in some the degree complete by higher theory, dimensional theSU(2)-doublet terms. precise Dirac fermion withallowed mass contributions to the annihilation crossobtain section sufficiently from large annihilationThe cross most sections relevant tod avoid over-closure of theinquire Universe. about the impact of higher dimensional butions from different Higgs bosons,the the stringent coupling bounds from directthat detection the experiments. dimension Hence, sincein current the data following. implies Asappear, we shall enabling demonstrate, the this suppression will of again the allow SIDD for cross blind section. spot solutions In to addition, we find relevant eq. ( cross section relevant for theby relic singlet-doublet density mixing or since theHiggs the sector. SIDD singlet cross Hence, the section dominantcross is contributions further section to the suppressed will SIDD be and the proportional thermal to annihilation resulting in the following DM-Higgs sector interactions: symmetry, under which all scalar and fermionfore fields also transform like Ψ a JHEP04(2018)069 , ) ) 3 S 2.1 /µ | mass, (2.15) (2.16) (2.17) (2.18) (2.14) ˆ S ( χ ). O + A.1 2 fields, there is | to pairs of DM µ χ | 1. = 6 terms arising / ., , eq. ( SM c d . → H . Another important A = 1 S , β + h  S † χ S χχ m 2 ) Φ ∗ α χ /λ µ 0 for tan − im ξ ( → = -term arises because this originally ), which as we shall see turn out to β . , ) and therefore there is no blindspot δ − χ χ χ . ) NSM β µ m m 2.11 χ cos 2 → ) ∂ δ δ ∗ α m ) χχH + ∗ (or v µ sin 2 g − α 2 µ µ µ δ ) in the γ 2 ξ √  ∂ ← − S ( , which we have expanded around the vev of ( will lead to terms proportional to the – 6 – v ξ µ = ( 2 = ). On the other hand, all the µ ¯ σ χ 3 /λS i √ β = † ). Moreover, we have not included terms involving /µ NSM χ γ 3 = 2 ); instead are Φ S i sin 2 /µ ( χχH SM ). Observe, that if dealing with on-shell | − results from the application of the equations of motion. g ˆ 2.16 O S ( /µ NSM = χ + χ χχH ) leads to interactions with the CP-even Higgs bosons when O H g χ 2 -term when inserting the vev of the field m µ ) is that the presence of derivative terms allows for interactions κ and DM, absent in eq. ( ( αm + ' ¯ σ 2.13 2 0 i in eq. ( O ˆ S/µ † (the fluctuations of G 2.13 λ χ /µ χχh ˆ SM S g − (1 + Φ) H µ κ = 1 ), the coupling of the DM particles to the SM-like Higgs is given by in eq. ( ∂ /µ ) in terms of the Higgs basis states in the appendix ( 2 i → α , and the only modification from such terms would be a redefinition of the S 2.13 2 = 1 κ κ 2.13 /λ interactions with /λS ) by noting that χ 2.13 coupling The We see that the blind spot for the cancellation of the coupling of From eq. ( = 5 term was actually suppressed by 1 Note, that there are nosuch terms as proportional the to one for and we can further matchand the ( interactions dictated by the Lagrangians given in eqs. ( the CP-odd Higgs statesby when hermiticity. the derivative is acting on the Higgsnow doublets, occurs as for required derivatives, where Φ isproportional a to real scalarthe field. derivative is acting Note, on that the Majorana the fermion direct fields, expansion and to of derivative interactions the with derivative terms where the dependence on In general, we calculate the on-shell relationships by using the fact that, ignoring total point to note from eq.between ( the Goldstone be relevant for thewe thermal write annihilation eq. cross ( section. For the convenience of the reader, a Dirac doublet fermion.level coupling The DMSχχ interactions with singlets area redundancy dominated in by the theterms above tree proportional terms, to since the thewhich also application derivative appear of of from the the equation of motion on the d yielding 1 from integrating outinstead a of Dirac 1 fermionhigher are powers suppressed of the by singlet 1 field, since they are not expected to arise from integrating out term proportional to JHEP04(2018)069 , } S (2.25) (2.23) (2.19) (2.20) (2.21) (2.22) (2.24) extended h, H, h { interactions. for the CP-odd = ˆ S i } ), and the singlet h S χχ . . 2.6 necessary to retain a ,A  χ  )–( 2 2 v NSM v , ) 2.3 ) 2 . However, we are only con- . A 2 | α S | α { χ µ  . µ | 6 operators. In principle, this H − | are analogous to its CP-even − i β ]. These interaction eigenstates S ξ h    ξ ≤ . ( S S 29 ( β . 13 23 33 S d A sin 2 2 S, 2 S, 2 S, + A i 1 + α S 1 + a  cos 2 M M M χ  ∗ P κ α NSM µ κ m 12 22 23 + v + H 2 S, 2 S, 2 S, χ + + 2 | β β m δ µ M M M i | 2 NSM ) to obtain the interaction with the (neutral) NSM h  A – 7 – √ S 11 12 13 v sin 2 i sin 2 2 2 S, 2 S, 2 S, µ + 2.15 − i δλ √ , NSM δλ a M M M i 2 } = 2 2 SM P 2 µ S does not get rotated [ v    µ 0 = v 2 H 2
= =  S for the CP-even states and i i  A, a SM χχG h 2 a 2 S NSM { 2 g } iA are obtained from the diagonalization of the corresponding S √ S √ when including dimension j M = i i + , also has relevant interactions with DM, namely comes from a field renormalization of = − χχA i P i χ S g ,H a = with the CP-odd scalars are easy to read from the above as well. α = 6. The modification from the field renormalization is suppressed h H S NSM S χ ≤ and NSM A 2 d 1 j i χχA √ ’s one can use eq. ( χχH g ,H S g χ + SM H µ/λ { = , ˆ S , hence, this correction is only relevant for the renormalizable + 2 i − S | h The interactions of On the other hand, the interactions with the CP-even singlet state are given by µ | = mass matrices. We canbosons write in the the (symmetric) extended squared Higgs mass Basis matrix as of the CP-even Higgs and the CP-odd states as The mixing angles states, where the doubletS components are asmix defined into mass in eigenstates. eqs. We ( denote the CP-even mass eigenstates as 2.1 Higgs sector In the previous section we haveHiggs motivated doublets a and structure one for singlet, the allHiggs scalar three sector Basis of consisting which of acquire two a vev. We can define the Finally, the interactionscounterpart, of the CP-odd singlet state Goldstone mode The orthogonal state, sidering operators of by For instance, althoughfields, the for Goldstone on-shell interactions involve derivatives of the Goldstone Here, the dependence on canonical kinetic term for field renormalization introduces corrections to all couplings of JHEP04(2018)069 , 11 22 2 S, 2 S, (2.28) (2.29) (2.30) (2.31) (2.27) (2.32) (2.33) (2.26) M M 11 . 2 S, 2  M . The matrix 23 v q 2 S, , , , , ≡ M   , ,    12 22 22   33 33   2 S,   2 S, 2 S,   2 S, 2 S, 33 33 +4 22 11 33 11 only, the eigenvalues are 33 22 2 S, 2 S, 2 M 2 S, 2 S, 2 S, 2 S,  S, 2 S, 2 , η − M − M 33 − M − M 13 S S 2 S, − M − M 2 S, 2 H 2 H 2 h 2 h and − M − M − M − M − M − M 2 h 2 h m m ε S S m m M h 2 h 2 2 H 2 H . 2 h 2 h m m η −M v m m m m     m m     2   22 ≡         11 11 11 11 2 S, 22 22 13 12 12 23 13 2 S, 2 S, 2 S, 2 S, 13 23 2 S, 2 S, 2 S, 2 S, 2 S, 2 S, 2 S, M 2 S, 2 S,  M M M M M   η − M − M − M − M r η − M − M − M mixing as − M − S S − − 2 H 2 H ∓ − 2 h 2 h 2 h 2 h S 13 1, and we have defined – 8 – 13 m m 23 33 23 23 , m m m m 23 2 S,   2 S, H S, 2 2 S, 2 S, 2 S,     2 S, –  12 − − M M − − are only linear in − − M M 2 S, . Barring the possibility of very degenerate diagonal M M 2 2 M SM 12 2 2 2 2 12 13 + 33 , η   13 12 M 23 12   2 S,   2 S, H 2 S,  2 S, 2 S, 2 S, 22 13 13 2 S, 2 S, 23 23 12 12 M 2 S, M 2 S, 2 S, ≡ M 2 S, 2 S, M M M 2 S, 2 S, M M η  η η and η  M M 11 M 12 M M − − M M − −   − − 2 S, 2 S,     ' 2 2 and 2 2 η η NSM M   M = = ,H 13 S H = = q = = 2 h 2 S, – S become proportional to the square of the Higgs vev h SM h ≡ SM h S S S NSM h M S S SM S S H h 12 SM S h SM S H S h η S NSM NSM 13 H H S S NSM h 2 S, S H S S S S 2 S, S = M , m are small parameters M ε 13 11 η 2 S, = 2 S, M 12 and 2 S, matrix element. Observe, that after imposing the minimization conditions  'M 2 h M 11 Keeping terms to linear order in the small parameters m 2 S, for the singlet-like mass eigenstate. for the doublet-like eigenstate, and for the SM-like mass eigenstate, The eigenvectors are elements mass terms, these relations ensurenon-standard that states the and SM-like that stateM its has only mass small squared mixings canand with be the approximately identified with the corresponding to where and similarly, Since the observed Higgs boson is predominantly SM-like, we can parametrize the elements JHEP04(2018)069 . χ S SM SM (2.40) (2.35) (2.37) (2.38) (2.34) (2.36) (2.39) χχA g i S arise via a P 1 , . + 2 ,  b F 3  x 23 NSM 2 ) and the mixing 1 2 S, x + a  0 χχA M NSM 2.22 with SM particles. The g  O ≡ χ ,G ,H )–( (b) F + i x NSM + 4 a i SM b NSM ’s with the doublet-like Higgs 2 x , P 6 2.14 χ  H A ) σv = h + 33 i , η a 2 S, ( χχa O , ) = i 23 = − M 4  2 S, v s 22 h S h respectively. The right diagram (b) appears / SM h ( χ χ 2 S, , g M 3 S S 2 O S , M cm NSM F +  . with SM particles depicted in figure x 26 A i χχH i SM h 2 2 r − g NSM – 9 – h χ / i v S 1 S S SM SM h σv h + 10 and h b − S can be expanded as   × 33 + , which mix (indicated by the cross) with the Higgs basis 2 + T 2 S, 3 , the thermal relic density is obtained S . a NSM ! , F A 2 1 = NSM S  ,H /x S S h i − M χ ≈ NSM  h v and S SM S m χχH F 22 S H 25 g x SM h

2 S, NSM = H  S SM) i 2 F M , ,H NSM / h 1 + T 1 → 0 S NSM ≈   SM  A + ≈ ∗ ≈ χχ H (a) g S 80 ( S S h . After integrating over the thermal history of the Universe until the S S SM NSM  h σ S SM h h S S S S /T 12 χ . χχH S S ≈ ≈ ≈ g i ≡ h m H A v i = 0 SM h SM H S H = χχ 2 . Exemplary diagrams illustrating the interactions of singlet DM NSM S H S NSM H S σ h 5 interactions (indicated by the solid black disc) of pairs of x S S h = Ω ≥ − i The interactions of the singlet fermion If we use the approximate eigenmasses, we find for the SM-like mass eigenstate d χχh χ χ g the couplings to the extendedof Higgs extended basis Higgs states basis given states in into eqs.( mass eigenstates, where freeze-out temperature 2.2 EFT: relicIn density the absence ofpair co-annihilation, of the DM thermally particles averaged at annihilation temperature cross section for a and for the other mass eigenstates via states arising when integrating out a heavy Dirac fermion SU(2)-doublet. Figure 1 left diagram (a) depicts interactionswith arising the via scalar the singlet tree-levelstates states interaction from of the a Higgs pair doublets of DM singlets JHEP04(2018)069 - ) Z s 2.45 (2.48) (2.45) (2.47) (2.41) (2.42) (2.43) (2.44) (2.46) ) since ) from the 2.45 ¯ q . , q . Note, that i χ .  , → = 0 2 χ m v d χ is suppressed by i i 2
m , m 2 χ ) and ( m 4 χχ qh √ F i g v x m u ZZa β − i 2 i 4 qa g i m 2.44 g √ 2 h σv i − = χχh h tan i m g i β 2 a a  χχa − i − NSM g m , a i NSM tan a W P − = v d i + 2 i ¯ q iP q h . = m W √ = , ¯ i = q
A i χ i q a χ 2 χ i − ua
m A m m da 2 χ = 4 0 qZ , g -wave suppressed while the annihilation , m 0 g − p qG 4 2 boson i g

dG , -wave. For typical freeze-out temperatures SM 2 Z h i ¯ q − , g 0 s , g Z q h 2 S χχZ m

v 2 Z g u , ¯ i 2 Z q v A 2 d 0 v q a χχG 2 m q χ m m g ) annihilations can also be mediated by the – 10 – √ i m G A √ , g ¯ q m m X − q

= !
v i u − 2 i 2 β X = β /

m 3 → √ = i 0 2 ¯ q i NSM h / ! ¯ G q q the width of the Higgs mass eigenstate Φ ZZh tan 1 tan q Z S 2 q 2 χ i A = χχ from annihilation into pairs of quarks ( do not couple to SM particles, thus, assuming a type II ! A Φ m m 0 + i 2 q 2 χ F S NSM h x m i − uG m S A i SM , g h g 1 S − − σv i

with Γ h SM h

1 i bosons. This is accounted for by extending the sum in eq. ( and χ S i SM h

T = Φ m S S Z 2 W i Γ π 3 v 3 4 H 2 m uh = 2 g π 3 i = 2 -wave amplitudes mediated by both the longitudinal polarization of the |  30 compared to the contribution from CP-odd Higgs bosons, as long as = s / χ dh = i F ¯ q,s 1 g q x h m i 2 − 25, the contribution from CP-even Higgs bosons to 1 such that the kinematic correction from the quark mass is irrelevant. F ¯ q,p ∼ q x W σv − i ' χ h +  i σv Φ m W χ F h 4 g m Besides via Higgs bosons, ( The contribution to / | /T /m F q χ T -channel exchange of the CP-even Higgs bosons is given by The couplings of the Goldstone mode to up-type and down-type quarks, respectively, are boson, i.e. the (neutral) Goldstone mode as well as the transversal polarizations of the 3 m channel exchange of to include the for there is nothe interference scalar between Higgs the bosons exchange contributionscross contribution section listed is via in pseudoscalar eqs. Higgsm bosons ( is and from the exchange of CP-odd Higgs boson by s For completeness, we record the couplings to pairs of vector bosons 2HDM Yukawa structure, thegiven by couplings of the mass eigenstates to up-type quarks are and to down-type quarks by The singlet states JHEP04(2018)069 - ) ) ¯ t s . t 2 may . In − 2.13 χ → (2.52) (2.51) (2.49) (2.50) Z m , boson to χχ 2

Z j ) or CP-odd ). Regardless a j i 300 GeV) ) annihilations h h k / i ¯ q χ A h q ), approximately 2.13 m ( k denotes a scalar or . → → X 2 2.48

i ) annihilations are × − j , such that ( 2 t χχ / a  χχ i 1 023 h . 0 #) χ > m ) and ( 2 → m
χ j ∼ − a m 300 GeV 2.46 χχ ) 2 χ m β .  2 χ ) are proportional to the Yukawa -channel exchange of the Majorana 2 . m − ), ( m 4 ! 4 i W 1

h cos 2 / t/u 2.47 0 s g 2 Z 2.45 4 m 1 W 1 . )–( m s g 0 χχG ( = ] is obtained from ( 2 g − −

α 43 1 2.44 dZ 2 ) annihilations, where Φ

| g 2 = j , eqs. ( – 11 – v µ 3 0 -wave contribution listed above it is not chirality #" − 2 χ 173 GeV and bottom quarks for lighter Φ | 12 [ i . s s contact interaction terms in eq. ( 2 G m 0 −
= cm Φ j ∼ A j a / Φ ∼ t = 26 i  uZ Z → 2 χ m -channel, or via 2 − g Φ ) annihilation discussed above, there may be relevant s 2 t A h ¯ q m + 10 χχZ q 4 m > m χχ i χχ -wave suppressed, while ( g h × p χ ) annihilations will be dominated by the heaviest accessible → 2 m ¯ q m q -wave contribution to the thermally averaged annihilation cross ∼ ) is s 1. j − from ( χχ . F ¯ q boson to the Majorana fermion can be read off from eqs. ( a → 0 1 q x i F i a x Z -wave contribution from the transversal polarization of the boson in the boson is suppressed with respect to that of its longitudinal polarization i (" p χχ | ∼ σv h 0 → Z 2 χ Z σv h -wave contribution to the annihilation cross section from the transversal 1 s χχG χχ πm g ) annihilation. For example, if we assume | ¯ q 64 q or the (heavy) SU(2)-doublet Dirac fermion we integrated out. In our EFT, the = → χ F ) ha x i χχ In addition to the ( It is interesting to consider the size of the thermally averaged cross section obtainable All the amplitudes appearing in eqs. ( A.1 σv h wave. The corresponding section is given by with a Higgs orfermion a last possibility proceeds via the of the type ofHiggs diagram, bosons the ( annihilation into a pair of CP-even ( Hence, the correct relicfor density couplings Ω contributions to pseudoscalar Higgs mass eigenstate. Such processes can be mediated either via diagrams (neutral) Goldstone mode,lead for to become relevant, since in contrastsuppressed to (hence, its not proportional to the Yukawa couplings). via ( is kinematically allowed, and assume the dominant annihilation channel to be via the couplings. Due to thecross hierarchy section of the from Yukawa ( couplings,quarks, the i.e. contribution to top-quarks the for the thermal latter case the Note, that the polarization of the (the neutral Goldstone mode) by quarks are The coupling of the or ( and the relevant axial-vector coupling of the transversal polarizations of the JHEP04(2018)069 1 k } ). = 0 Φ − h j h S Φ i ) and a m (2.53) (2.54) (2.55) (2.56) (2.57) in the Φ h ,A,G g S -channel S → S a a a { t/u → χχ , = χ ) annihilation. 1 k ¯ t χχ − t → . GeV -channel 4   s χχ −  , j 10 2 a ) taking into account the mediated by a j 2 χ 2 a j m , respectively], the × m a i m 4 A.1 4 + h Ψ ’s to the Higgs mass eigenstates j , i χχZ ) processes are ( − ∼ − 2 a A -channel proceeding via contact 2 h j χ g i k respectively, which as mentioned

2 h 2 Z m Φ Φ m , i S j  2 ) couplings of dimension (mass) ). ) or ( t/u  m m a a k χ Φ bosons in the i ∗ 2 χ S h κξ Φ − a µ m g j 2 m Z S j boson 2.24 → i 2.13 2 χ h a 4 k NSM h Φ i g + Φ m S Z − ), ( χχ 4 χχ β χχ k χχh g g j 2 Φ and ) which is relevant only very close to threshold NSM a 2 χ 2.23 1 + m P S sin 2 − – 12 – m   a 1 2 S j 2.52 W µ ) for the coupling to the neutral Goldstone mode g = λδ s − ha j are the (  g χχa a i = g j 2.50 v h Ψ µ χχZ i Φ j i i g a A k i Φ − − h Φ

-channel exchange of the Majorana fermion χχh g χχ are the couplings of pairs of A = ≈ g 2 i ) and ( t/u j

can be achieved for Φ -wave amplitudes mediated by CP-odd scalars Φ − a S s i 1 χχ A h Z = − g 2.20 s j A SM a 3 i h χ cm χχH A ), the ), canonical values of the thermally averaged annihilation cross section g

χ ) [eq. ( 26 − m 2.56 | ≈ 10 2.40 S ). If kinematically accessible, they can compete with ( = 2 )–( × S i h ] for details). The 2 a S χχha ) are the dimensionful trilinear couplings between different Higgs mass eigenstates 2.53 -channel 29 m g a 0 ∼ | and the transversal polarizations of the s G F + j 0 → x Ignoring such Higgs exchange diagrams, the amplitude After accounting for suppression of couplings arising from the requirement of an Φ i i i h Φ χχG -channel may play an important role. However, their relevance to the total cross section χχ g m σv Ignoring the kinematic correction in eq.( ( h is dictated by the couplingabove strengths are not relatedthese to couplings our are EFT small, parameters. rendering these Hence processes for irrelevant for simplicity,Dirac we the fermion, relic will which density. assume we that integrated out, is most relevant for the final state ( 125 GeV SM-like Higgs masscondition, the eigenstate most and relevant final from states( for approximately the satisfying ( the blindspot For both these channels,s the amplitudes mediated by the singlet-like CP-odd related to the parametersof of our ref. EFT, [ butwhich arise can from the be Higgs readmixing potential off of (see the from appendices Higgs the mass Lagrangian eigenstates, eqs. eqs. ( ( In eqs. ( given in eq. ( g ( (between the neutral Goldstone mode and two Higgs mass eigenstates), which are not and the amplitude from the the amplitudes mediated byinteraction the terms Dirac after fermion integrating in out the the Dirac fermion Ψ in the the amplitude mediated by transversally polarized where the sum includes the JHEP04(2018)069 χ 1, . 0 (2.58) (2.59) (2.60) ∼ , anni- t 0 -channel ), although χχG t/u > m g ), and correc- χ . Final states χ 2.59 t can be achieved m m 1 2 ), ( . The annihilation . − t s ≈ > m κ (1) suppression), the 3 2 S )] do not play an im- 2.58 χ -wave suppressed since O a √ p − > m cm m m ∼ χ . W 26 is achieved for + S 2 + − m F / S 1 x . h -wave suppressed. 10 W i  χχh p ( 2 m g , ×  σv h 2 2 ZZ / χ 1 ∼ m  µ → F 300 GeV x i  χ coupling. 700 GeV χχ such that boson are also suppressed: the coupling of 2  m 0 σv . h κ 0 Z ∼ 300 GeV χχG  g | ∼ – 13 – κξ S 15 . + 2 bosons, respectively). Annihilations into a pair of χχh 0 ) contact interaction terms in our EFT which would g β − ) the processes associated with a Majorana fermion W | ∼ W S κ | ≈ | + h | sin 2 7 and hence are strongly suppressed. For S S and a δλ ≥ -wave. However, such annihilations proceeding via χχa leading to the observed relic density can be easily obtained when -channel Higgs or s d χχW g Z → for the estimate on | ( s 1 − µ s χχ 3  v cm χχZZ 26 − 10 ). Annihilation into pairs of vector bosons is suppressed because it proceeds × -channel can be relevant. Assuming again for simplicity that these processes 2 ) annihilations are kinematically allowed, i.e. when ¯ t is small), neglecting the threshold corrections for ( 2.51 t ∼ t/u S F a -channel mediator to one transversally and one longitudinally (a pair of transversally) → x To summarize, the proper value of the thermally averaged annihilation cross section Annihilations into pairs of vector bosons [ For the channel ( S s i a S h σv χχ cf. eq. ( through smaller couplings.pairs of If Higgs kinematically mass allowed, eigenstatesthe may the annihilation become into annihilation large pairs cross enough,EFT of cf. parameters section top eqs. conspire into quarks ( to tends suppress to the be competitive or dominant unless the coupling. The Higgs mediated channel is furthermore also h ( cross section will then typically bethe neutral dominated Goldstone by mode annihilations for into which top the quarks proper value mediated of by only appear at dimension hilations mediated by an the polarized vector bosons isof proportional the Higgs to and the Goldstone gauge bosons couplings to quarks, (squared). instead, The are proportional couplings to the top Yukawa Goldstone modes fortransversally the polarized vector bosons orpolarized one vector transversally boson polarized are andexchange one of longitudinally the neutral (charged)out components correspond of to the SU(2)-doublet fermion we integrated where we assumed portant role forconsisting obtaining of the two vector thermal bosonsthey relic with correspond longitudinal density to polarizations annihilations as are into long a pair as of CP-odd scalars (i.e. the neutral and charged This implies g tions from singlet-doublet mixing (thethermally latter averaged annihilation potentially leading cross to section for a DM coupling to the singlets sponds to couplings in the dominate compared to the one associated to the interchange of a singlet pseudoscalar (i.e. where we have assumed the mixing of the CP-odd Higgs bosons to be small. This corre- JHEP04(2018)069 2 1, 13. . ≈ 0 (2.61) (2.62) ≈ SM h avoiding -channel S  t t p T q f , ) annihilation 2 < m b ¯ b    χ u,d,s  i = m → q ]. While we are mainly  d d P 47 χχ a – m -channel is boosted via − 44 s  [ 1 d }  F 2 27 is furthermore suppressed by + , 0447 . 0 + i i ). 0 , G  p T s u χχh f u 125 GeV and composition g 2.41 a , but interpreting them as m + i 0191 q . ≈ 1 0  p qh T d , m boson) in the h g u f ] we focus on SIDD in the remainder of i F m 2 h Z =  1 ) and ( 6 0153 , . m d S ). 0 5 ] are much stronger than those on the spin- F { Z 2 4 1 with 2.40 – √ m = – 14 – 1 X , achieving a sufficiently large annihilation cross h,H,h h t } -wave contribution, and couplings are usually not − = ≈ i p p T s χ = 15 and    , f . i < m m is more difficult. In this case, ( 2 ) while simultaneously satisfying collider constraints. 0 p T d  χ  χ 1 (2 q q ≈ , f χ − m m a Φ s m χ 2 p  T u , the observation of the 125 GeV Higgs boson at the Large m 3 f m  m p { T q < + p f cm ≈ 2.1 p i m a χ 26 are given in eqs. ( m − m i m u,d,s 4.3.5  boson mediated = qh 10 2 p q + g Z i π m × P h 2 2 is the momentum transfer. In contrast, the exchange of CP-even Higgs m − and ) scattering relevant for direct detection proceeds via the same diagrams as = ∼ 1 q i parametrize the contribution to the scattering amplitude from one Higgs mass  SI p χq F micrOMEGAs i x σ 4 χχh ) 27 i − q g is the mass of the proton, and the form factors (at zero momentum transfer) are + σv p χq h /m p T u , where q m f -proton SIDD cross section can be written as a 2 Z As discussed in section Summing coherently over the contribution from all CP-even Higgs mass eigenstates, For lighter DM candidates, χ Note, that there are considerable uncertainties on the form factors. In this work, we use the default = 2 /m u 2 Hadron Collider (LHC) with couplingsmust close contain to a CP-even that Higgs of eigenstate a SM Higgs implies thatvalues our used model by interested in the DM-Higgs couplingsfor which the are values not of directly the affected form by factors the can form be factors, compensated a by different (small) choice redefinitions of other parameters. eigenstate, where the where F The ( this section. the scattering, and the contributionq of the Goldstone mode bosons leads to spin-independentnucleon scattering. scattering Since (SIDD) bounds crossdependent on section scattering the [ (SDDD) spin-independent cross DM- section [ 2.3 EFT: directElastic detection ( annihilation in the earlyexchanges Universe of depicted Higgs in bosons. figure The exchange of CP-odd Higgs bosons leads to spin-dependent mass spectrum ( Annihilation into pairs ofsections vector either since bosons they usually areover-closure does controlled of by not the gauge achieve Universe couplings.hilation sufficiently requires Hence, cross large large for sections cross couplings via inresonant a annihilation, the 2 Higgs Higgs sector boson unless Φ the (a anni- section is dominated by the sufficiently large to obtainof the Higgs proper bosons annihilation requiresorder cross large to section. couplings be between sufficiently Annihilation effective; the into in different pairs addition Higgs it mass is eigenstates not in easy to obtain a light enough Higgs JHEP04(2018)069 4 h S ∝ h 2 may /m ) 1 q (2.63) , since to the and 0 )  χχh χ /m g q H χχG 4 H a 2.61 g ). It is also /m . We therefore . The coupling 1 ]. For values of 2 S | 2 2.16 ∝ µ 15  H | , and therefore at ]. Hence, values of 2 h . 4 β ) v/  q χ d d χχh pb [ a m g /m m -mediated diagram, such q 10 h  a , − d 4 F 10 − +  . Hence, the current stringent × h h 3 .  3 h u u m a . m , or both mechanisms may be at work  125 GeV -channel. We find from eq. ( ). In general, the couplings of S t ), is approximately fulfilled. u  h F 2 interferes destructively with those mediated  2.51  2.16 h 2 or dominantly composed of and 1  . 1 necessary to obtain the observed relic density in the . – 15 – χχh h 0 χ 0 g H h χ m = 300 GeV) m  ∼ m χ +  p 0 m pb p m ( H 9 m χχG SI p − m . Under such conditions, the SIDD cross section will be g σ  2 ) 10 contributions to the SIDD cross section [ = 5 operators with the leading contribution suppressed by 2 p i = 6 operators and is suppressed by S h d × π m d S 2 5 may be estimated by considering the SIDD cross section, taking NSM − = ∼ H (1 the suppression induced by its large mass may be compensated by χχh h g
β and necessary to fulfill the constraints on the SIDD cross section may only SI p arises via } ≤ σ 2 0 ) 1. In addition, to avoid bounds on additional Higgs bosons from the LHC SM χχh Higgs boson to down-type quarks is enhanced by tan g H is already suppressed by proximity to the blind spot condition. In order to i χχG NSM h g (1), as we shall use in our work, the contribution of the non-standard Higgs h must be either heavy }  S 025 are necessary to fulfill these constraints. This range of values of S NSM ( h . O S , 0 h 2 H ,S ) = arises from dimension . i The bounds on , while β SM h and NSM h S ( S χχh χχh are in general suppressedthat with destructive respect to interference the willlike amplitude Higgs only of be the effectivedemonstrate these when properties, the we shall contributionfields briefly from are consider heavy the the and simple SM- play case no in role which in the the singlet low energy effective theory. In the previous sectionexperimental we argued limits, that the inpossible order model that to the must suppress amplitude the sit mediatedby SIDD by close the the cross to section other below CP-even thesimultaneously. mass blind As eigenstates argued spot, above, eq. the amplitudes ( of the diagrams mediated by v/µ conclude that under the requirementthe of coupling obtaining an acceptablebe relic density, obtained the if values the of blind spot condition,2.4 eq. ( Bounds on couplings and parameters of the EFT be compared with the valuesas of discussed in theHiggs previous mass eigenstates section, are cf. expected tothey eq. be arise ( of at similar the magnitude, org same larger, order, than or lower, in our EFT expansion. In particular, the coupling while the experimental limitg is bounds on the SIDD cross section lead to relevantinto constraints account on only the diagrams with an between the tan bosons to the SIDDor, cross section in will the either case{ be of suppressed by mostly ( singletdominated states, by by the their contribution small from doublet the components SM-like ( state and the Large Electron-Positron ColliderH (LEP), the remaining CP-evenof mass the eigenstates large values of tan an enhancement of the coupling. This case allows for effective destructive interference { JHEP04(2018)069 12 . 0 Left: = 2. ∼ β 2 h 025. As can . 0 . χχh g = 300 GeV, and the non-standard χ m values of the EFT parameters consistent with = 500 GeV, and decoupled singlet states. 500 GeV and perfect alignment is assumed in NSM – 16 – A = 500 GeV, and moderate values of tan Right: }  m S A = NSM A , m m S NSM H H = m m { NSM H m , this bound may be slightly relaxed in the presence of destructive = 300 GeV, 2 pb (dashed and solid lines). The couplings of the CP-odd Higgs bosons are χ m 10 shows a representative example, illustrating the bounds on the parameters of − 2 10 . EFT parameters and couplings of DM to the CP-even and CP-odd Higgs bosons re- = 2, × β Figure = 3 SI p be seen ininterference figure with other Higgs bosons.couplings of The orange DM shadedthe to region boundaries the denotes denoting the CP-even the values Higgsσ values of of the bosons these consistent couplings saturating with the the current SIDD SIDD bound bounds, with The singlets are decoupled the doublet sectorthe such left that panel the wefunction Higgs show of basis the the states DM couplings couplingignoring coincide of to the with the the other SM-like mass CP-even DM Higgs Higgsboson eigenstates. to boson. bosons the needs contributions, As In non-standard the explained to coupling in Higgs be to the bosons the last suppressed as SM-like section, to Higgs a satisfy the SIDD constraints, our EFT and the couplingssatisfy the of SIDD DM experimental to constraints the andWe CP-even obtain used the and proper characteristic CP-odd relic Higgs valuesHiggs density bosons concurrently. of boson necessary the to masses DM mass, the couplings shown in thethis left panel panel. are in Dashed one-to-one and correspondence solid lines, with as similar well lines as and the areas shaded shown areas in shown the in left panel. for tan the orange shaded-region boundedcouplings by consistent solid with and the dashed SIDDof lines bounds, DM represents while to the the the CP-evenfor blue Higgs (neutral) and CP-even bosons Goldstone black Higgs mode ellipses couplings denotethin and denoted the lines the couplings by denoting heavy the CP-odd two different corresponding Higgs solutions. boson solid yielding or Ω dashed lines, with thick and Figure 2 quired to obtain the correct thermal relic density while concurrently satisfying SIDD constraints, JHEP04(2018)069 . ) 12, . NSM 0 2.20 A ∼ ), ( 2 and h 0 2.18 G ), ( 2.14 The solid and dashed 3 ) annihilation are kinematically ]. The LHC phenomenol- hA → 48 , χχ 26 (blue) needed to obtain Ω 0 χχG g . For couplings of DM to the CP-even Higgs
) annihilation mediated dominantly by ¯ t 2 χ t – 17 – m → 4 symmetry is defined so that all chiral superfields 3 − χχ Z 2 Z (black) and m / shows the three independent combinations of the param- NSM 1 2 ∼ χχA g
2 χ m 4 − , while gauge superfields transform trivially. 3 NSM πi/ 2 2 A e symmetry, however, this region can nonetheless quite simply be mapped onto m 3 Z / ). 1 × 2.21 Reviews of the NMSSM can be found in refs. [ The right panel of figure Note, that the masses of the Higgs bosons are chosen such that ( : 3 3 Z transform by ogy of the NMSSM Higgsforbidden, and hence, the neutralino relic density is sectors set by has ( recently been investigated in NMSSM which allows us to computethe particle spectra DM and and couplings andcontains collider subsequently a study phenomenology second of regionunder of the the parameter model. space whereour the As EFT. DM we candidate In shall does the see, not SUSY transform the case such NMSSM a also an additional SU(2)-doublet Diracout, fermion yielding which interactions is of assumedcontent the is to Majorana very similar be fermion to the heavy with Higgsthe and the and gluinos neutralino Higgs integrated as sector doublets. well of the as Thiscan NMSSM. the particle serve SUSY Hence, decoupling partners as of an thediscussed SM ultraviolet above. fermions completion from for the In theory, the the addition, EFT, NMSSM preserving a all range the of essential mature features numerical tools are available for the 3 NMSSM ultraviolet completion The EFT model discussedadditional in SU(2)-doublet the previous Higgs section field,fermion extends both as the well of SM as particle which content transform a by as complex an singlets scalar under field the and SM a gauge Majorana group. There is also detection constraints. The shadedcorrespondence with regions, those solid in the lines, leftstood and panel, and from dashed the the shown lines dependence correlations are may ofand be in the easily ( couplings one-to-one under- on these parameters, eqs. ( bosons in the shadedexperimental area, limit, for the whichwould couplings the take of SIDD intermediate values the cross between section CP-odd the would ones Higgs represented be boson by smallereters the and than of two the ellipses. the the Goldstone EFT that mode play a role in the determination of the relic density and the direct one-to-one correspondence with similarbetween points the in couplings the to other thebe ellipses. Goldstone understood mode The from and clear toassociated the correlation the with fact heavy the CP-odd that Higgs propagatorfactor for boson contributing 3 the may to weaker given for the Higgsm the annihilation and heavy cross CP-odd DM section Higgs masses, is boson the about than suppression for a the Goldstone mode of mass ellipses show the values of consistent with the SM-likeeven Higgs Higgs couplings couplings denoted andlines by the the represent boundary solid two values and differentof of dashed solutions values lines, the for of respectively. heavy the the Thick CP- and CP-odd/Goldstone CP-even thin Higgs couplings bosons for couplings, each with set points in each of the ellipses in constrained only by the demand of obtaining a proper relic density. JHEP04(2018)069 ) e ]. ), b Y , L ( 57 3.1 – (3.3) (3.4) (3.5) (3.1) (3.2)      b i , with Q
 52 and the , and , λ κµ d H 4 , is a chiral κ Y 19 3 + , , b . S b i S κ u β 16 ) as vectors in 3 2 i Y κ A   s ¯ e b κ λ i ( λ κ + µ 2 λ + κµ ¯ κ d b i λ β − j + 2 κA ) β s 3 d  2 2 2 2 A s i κ λ b µ H 2 2 M − 2 A ( bosons after electroweak µ i λ − κµ  4 M  3 b β L β ]. κ λ Z 2 2 ¯ e = 246 GeV convention, while we b − 3 2 s − e 1 26 2 v 2 A Y β µ  + 2 M 2 λvµc and , − β s to be positive, while  . } − 2 , and one charged Higgs 2 j A β λvµ h µ κ s
)  2 β 2 2 W 0 2 M d 2 s d A h 2 µ and the λ ,A b  4 H M v H κµ λ 2 2 ( β  i v λ λv + 2 b 1 2 β are the dimensionful trilinear soft SUSY Q 2 λ 2 h ¯ λ 1 d b s √ κ and tan d  − A 2  ). The remaining components of the Higgs h + sin β A Y β, µ, A v  β d 2 Z 2 2 λ 2 0 2 u s c − m H β λ
β ( ). Including the usual soft SUSY breaking and 2 j – 18 – 2 1 2 tan µ and ≡ ) v βH s 2 u 2 u
can have both signs [ 2.6 λ Z λ sin 2 2 b H H v κ A cos i h M ( λ, κ, − 2 , two CP-odd neutral Higgs bosons, the imaginary )–( i ≡ A { λ 2 Z S b λ κµ Q 2 A 3 − m = 2.3 ¯ b u 2 Z i u M 2 Im , and and − p + Y m 4 S √ β λ 2 A -invariant NMSSM contains no dimensionful parameters and is 2 − 2 , and A , the symmetric tree-level squared mass matrix for the CP-odd A 3 , and we have introduced − s = d M } j } are the usual Higgs doublet supermultiplets, and , β  2 Z A ) M µ S S ) the vev of H µ d 2 and down-type right-handed quarks (leptons) d   d M b β , b v sin H ,A ¯ 2 2 H  b u NSM ( ,H u ( s ], and studies of the DM phenomenology include refs. [ i 2 A ≡ ) H u ], and ignoring radiative corrections for presentational purposes, see v λv u 53 v 2 the vev of β NSM ] uses the parameter 2 NSM – and b λ 1 s 26 H i √ A 29 ( and , u + { S h b are the dimensionless couplings appearing in the superpotential eq. ( ,H 49 S β h b and β , S H ] 2 2   λ κ c d h SM 33 26 2 Z cos = – ], the symmetric squared mass matrix for the neutral CP-even Higgs bosons in ij /v H m  -terms [ with u 2 P {  ≡ and 29 v 27 i D      = , β M S λ c = h = 16 = 174 GeV. W λ In the basis The Higgs sector is analogous to that of our EFT model, and can be rotated to the The Higgs sector of the NMSSM consists of three neutral CP-even Higgs bosons, the 2 S β Note, that ref. [ v 4 ≡ M - and use where neutral Higgs bosons is the basis dimensionful parameters extended Higgs basis usingF eqs. ( e.g. ref. [ where µ breaking couplings. Assuming CPall conservation, parameters one can to without be loss real, of generality and choose furthermore components of tan doublets make up thesymmetry longitudinal breaking. components The of Higgs the sector is controlled by the parameters should be understood toas be well matrices, as and the thefamily up-type left-handed space. quark (lepton) doublets supermultiplet which transforms as a singlet under the SMreal gauge components group. of The superpotential of the is given by [ where we have written the SU(2) indices explicitly. The Yukawa couplings refs. [ JHEP04(2018)069 , ) λ and 3.8 (3.8) (3.9) (3.6) (3.7) 0 d e H )] is also 3.6 which allows 2 h , m  β limit, or by setting to , 2 ) ]. Perfect alignment is sin 2        β e S, 2 β , β 5 λ i κ ) 32 , the fermionic component . 2 , ˜ t h 0 0 e N S 2 λvs λvc λvs κµ/λ − 29 m + 2 − − 1 decoupling , the neutral Higgsinos 0 u  β 3 β e s H s β + ∆( 4 f W i W limit [ 2 β µ W c , the 2 4 0 2 2 N s Z − s 11 Z 2 + sin m S, v m 0 d 2 − . = e λ ]. H to accommodate the SM-like Higgs boson µ β 3 2 β c i A mixing, is satisfied for the same values of 2 + 58 c [  M µ N β W SM M 2 2 W and s 0 c 7. Intriguingly, the first alignment condition in 33 c + – 19 – H . Z NSM 2 Z A Z S, 0 3 m H m m M and neutral wino f , W ∼ M − , the symmetric tree level neutralino mass matrix is , − ) = } 12 e 2 λ B e β 22 S 0 N 11 , M S, SM 0 u NMSSMTools 2 S, β + 1 3 if H e H M 2 e cos(2 , M B M . are obtained from the diagonalization of the neutralino mass 5 0 ]. This can be achieved either by making the remaining CP- 1 d 2 Z i sin        e = β ij H 2 N m , 42 2 h alignment-without-decoupling v = N , 3 2 − = 0 m heavy, 41 χ i f 2 h W χ S m 7. Alignment with the singlet [the second condition in eq. ( M e . B, h 0, the are the bino and wino soft SUSY breaking masses. The neutralino mass 0 { = 2 ≈ 2 . ] ). Decoupling some of the neutralinos allows us to write down simple M and λ λ 13 29 3.8 ) are radiative corrections that are common to the MSSM. Note, that with 2 S, H ˜ t h . and 6 m . 1 ), which suppresses the ≈ M M 3.6 12 The neutralino sector of the NMSSM consists of the superpartners of the neutral As in the case of our EFT model, the NMSSM must contain a CP-even 125 GeV . In the basis belonging to the respective Higgs doublets, and the singlino Note, that we use the same notation for the Higgs states as in section b S, 2 5 S 0 u e where the mixing angles matrix eq. ( approximations for various mixing anglesthe of particular singlino interest. and For wino example, if contributions, we diagonalizing neglect the neutralino mass matrix eq. ( where eigenstates are given in terms of the interaction eigenstates by electroweak gauge bosons, the bino H of for a 125 GeV SM-likefor Higgs moderate boson values mass ofeq. tan without ( the need fornamely large 0 radiative corrections easily achieved by judicious choices of and in the alignment limit the mass of the SM-like Higgs mass eigenstate is given by where ∆( respect to the MSSM, one obtains an additional contribution ( M achieved for [ numerical results obtained with mass eigenstate mostly composedobserved out at of the LHCeven [ states Radiative corrections may be relevant and are incorporated at the two-loop level in the JHEP04(2018)069 ) 2.40 (3.13) (3.12) (3.14) (3.15) (3.16) (3.10) (3.11) , . i i ) ) , , β β c ) ) s β β 14 14 c s N N 14 14 N − N + β β + − s c β β 13 13 ] c s N N 13 13 16 , )( )( N N # , , , , we can construct a top-down ), this also results in different 12 12 )( )( . β . # # # c 2 N N s 2 . 12 12 β 1 2 β β 2 2 ) , ) c 2 c s g g 2 2 N N − ) 1 2 ) ) ) 2 2 /µ /µ + h  − − g g − − − 1 1 , /µ /µ /µ /µ χ χ β e 2 2 ] β  14 15 1 B 1 1 1 − − 11 11 s c χ χ χ χ m e 15 m 2 2 N N 14 11 B ) ) ( N N 11 11 1 m 1 1 m m m N N N ( ( ( + g g − /µ /µ N N + ( M ( 15 1 1 1 1 1 − + 2 1 − − β + ( χ χ g g 2 2 13 15 c ( − 1 1 1 β κN m m 2 2 N N 13 11 ) ) + ( " s – 20 – ( ( − β β − N N " v " " µ c ) + ( ) s symmetry since it stems from a gauge superfield. β β 2 v µ 1 + v v 14 1 µ µ 14 14 c s 3 g 2 √ λ λ  1 N N N 1 + Z 14 14 g √ −  13 = = = N − N + β β = = = + − 15 14 15 13 15 c s λN β β N 11 N N N N 13 13 11 13 11 14 s c 2 [ N N N N N N N 13 13 √ ( ( N N = ( ( 15 15 S 15 15 A λN λN j 2 2 χ λN λN i √ √ 2 2 χ h h i i √ √ ig = = = = = 0 S G SM H 1 NSM NSM 1 ] H χ χ A 1 H 1 1 1 16 χ 1 χ χ χ 1 χ g 1 g χ 1 The singlino will play the role of the Majorana singlet in our EFT model. However, In terms of the mixing angles, the couplings of the lightest neutralino to the Higgs χ g χ g g 3.1 Top-down EFT In order to connect theEFT NMSSM for to the our NMSSM EFTapproach Higgs from is and valid section as neutralino long sector as by the integrating Higgsino mass out is the large Higgsinos. compared to This the mass of the lightest bino does not transformBesides under allowing the for ancouplings explicit to mass the term Higgs ( region where doublets the and bino the isstraightforward way, singlet the as DM than we candidate will those can see found nonetheless in be for the connected the following to section. singlino. our EFT The in a From these, the couplingsand to the the mixing Higgs angles mass of eigenstates the can Higgs be mass obtained eigenstates. the using NMSSM eq. contains ( a second SU(2)-singlet neutralino, the bino. Unlike the singlino, the basis states are while if we neglect the bino and the wino component, we find [ yields [ JHEP04(2018)069   . e e B B  (3.22) (3.20) (3.21) (3.17) (3.18) (3.19) † † . . ) ) c c . 0 0 . d u H H ( ( as the left- +h + h 2 2 1 1 . 0 0 d d  0 d  g g ,  † √ √ e e e † † e B H H † ) H e e e − + B e e  B B . Neglecting the B B )  † 0 e e ( 0 u e } S S 6 (the same order u 0 B d ) µ 0 d 0 0 u d † 0 u H H ) Z , µ H ≤ H 0 2 ( 1 2 d 1 H 1 2 λH λH W ( 1 d g 2 g 1 √ H √ † θ M g g ( √ 1 ) √ 0 g d 2   − − 1 + µ µ g † + † H √  { acquires a vev, and because † e 2 sin ( Z Z e † S S e i † 2 1 S † e 2 + S 2 ) S ) † W W g † 0 ) 0 d − θ θ d e ) S M 1 1 0 u 0 + u 0 u µ g g H H e ( ( H ∂ S H ( λ ( λ e 2sin 2sin  B when λH λ λ i i  i µ . + 0 d 0 . This allows us to use the equation of d   + ¯ − + σ  † e i + µ as the right-handed and µ H µ µ   ) † H † e ) 0 B 0 0 u µ d ∂ ∂ u Z † ) 0 u † 0 u ) 0 e d , e e ) Z   H is H H 0 e d e W H 0 e d ( H  µ µ µ H θ  0 ( † W u H 1 e ¯ ¯ σ σ † H B θ e ( e g i i B ( 1 † H → † + ( – 21 – λS ) g   − 2 ) 1 λS 0 † † u 0 0 0 u d 2 sin 0 u g u √ + e e i e e B B 2 sin H + H H H H 0 0 0 u ( i d u ), we find † 0 ( u 0 + , d + ) e 2  1 H 2 e H H 2 e 0 1 − u e H µ µ  H g S √ . g 2 2 and vice versa. Keeping only terms leading to dimension 1 1 ∂ 0 √  u µ Z H c  g g . µ 0 ) and keeping terms of dimension ( ∂ d √ √ λS  − µ h − W Z e  µ H λH − + e θ Z 1 S e 2 +h 1 ¯ σ  S † † µ 0 W d i 3.17 g 0 d e e e ¯ W σ θ S S √ B λg i 1 θ † † S 1 e 1 ) ) g † λS B 2 sin λH + S g 0 0 λH d u 1 1 i † S  e  S 2 2 − 1 2 sin S H H e M + 2 sin S λ i ( ( 2 1 + 0 i u λS λ λ λ µ + 0 d + −   ∂ H − e − S e − 0 µ d H  S S e µ S 0 ∂ u † † = = µ H ∂ e 1 1  2 S S 0 ¯ H σ u to integrate out 0 d  κS 2 2 i λ µ e 0 † e u H µ λ λ  H ¯ σ ) − λS e ¯ i σ 0 u H i + + −   0 d e H e ( H + 1 0 = λS 0 = 6 operators when substituting the equations of motion into the Lagrangian, we obtain L ⊃ L ⊃ ≤ Substituting these into eq.used ( for our generic EFT in section and Due to the Higgsinoof mass their term identical couplings,handed we component can of interpret amotion Dirac for fermion with mass d and for The corresponding equation of motion for Yukawa terms and ignoring theinvolving charged the current neutral interactions, components the of terms the in Higgsinos the are Lagrangian neutralino. We will also assume the winos to be heavy, JHEP04(2018)069 , 3 ), Z 2 2.14 (3.25) (3.26) (3.27) (3.23) (3.24) instead 6. e . B . Keeping 0 0 e B in section † G ∼ is allowed. ) χ 0 u . In particular, λ 1 1 H , M and ( M † ! ) . 06 to Higgs doublet 2 0 3 . = d 0 ˆ ), or from eqs. ( 0 , while assuming the µ S χ H NSM 3 ) to those in the EFT,

→ ≈ A Z m . 3.22 2 arises because scalar and O 0 / 3.22 2 1 + α g → 2 − | ξ 1 µ | κ , ξ = = δ → → S † /µ . /µ . 1 S χ acquires a vev. In contrast, the bino does χ 2 and correspondingly λ , κ m S λ m )] ˆ − S λ , κ → couplings = , = + ) directly to those in the NMSSM via – 22 – → β 2.16 α ! β 2 3 ˆ µ 2.13 µ/λ S sin 2 and , λ sin 2

→ 2 1 δ 2 , λ O g S 2 ) we see that the bino couples to ( + λ → ˆ 2 S α and hence a soft SUSY breaking mass term µ 2.13 λ → − 3 = − Z α δ 1 µ − ) and ( = transforms trivially and has a Majorana mass → δ χ around its vev 1 3.22 λS S , which can be compensated for by changing the sign of the coupling of the χχ ) 0 u ), via H 4, recalling that the presence of the 125 GeV SM-like Higgs implies )( . 0 0 d 2.13 Note, that the bino couples with characteristic strength The couplings of pairs of (on-shell) singlinos to the extended Higgs basis states and In contrast to the singlino, the bino couples to different combinations of the Higgs Expanding H ∼ 2 states, whereas the singlinoλ couples to Higgs doublet states withthe characteristic (neutral) strength Goldstone mode can be directly read off from eq. ( The blind spot condition for the bino region is then of ( binos to the CP-oddthis states in coming mind, from the weeq. Higgs ( can doublets, map i.e. the couplings of the bino in eq. ( doublets and the singlet.for Such the interactions Higgs would beMajorana doublets fermion obtained and by the writingcomparing down singlet eqs. the transforming ( EFT under the of gauge supermultiplets) share theleads same to the couplings blind in spot SUSY models. condition [cf. The eq. mapping ( above The simple relationfermion between components the of chiral supermultiplets (vector and fermion components in the case we can appreciate the similarities withsinglino the has Lagrangian of the our same EFT model.and structure we for In can particular, the the map couplings the as couplings the in Majorana eq. fermion ( singlino-singlino interaction term. Note, thatsymmetry, because it only the gets singlino a transforms mass undernot when the transform the under singlet the where in the last line we have included the standard bino mass term and the singlet- JHEP04(2018)069 . 1 M (3.36) (3.37) (3.38) (3.39) (3.40) (3.41) (3.42) (3.28) (3.30) (3.31) (3.32) (3.33) (3.34) (3.35) (3.29) = χ m , ,  1 2 λ β , λg 2 , | 2 v 2 v 1 , µ β ,
| g ), sin 2 e 2 S 2 2 λ 1 | − β , 2 m β . v g | v µ sin 2 1 χ | 2 3.24 χ µ | − 2 2 | 2 v ), keeping in mind the switch in |  m µ β , λ m e χ | B µ cos 2 sin 2 √ κ | 2 v 2 β , 1 1 2 m m χ 2 √ + 3.26 β . √ | √ sin 2 λg λg i m β , µ β 2 + 1 | 2 v v β , + − cos 2 β , − β ,

β √ 2 1 cos 2 β λg β 2 e e 1 S S i g 2 2 g sin 2 cos 2 2 2 2 λ m m cos 2 µ 2 2 | 1 1 + µ | | cos 2 2 v 2 sin 2 cos 2 1 g g v 2 µ sin 2 2 sin 2 v µ µ v 1 χ | + + | | χ 2 2 µ µ 2 1 √ λ g 3 2 √ | λg e e λ λ m 2 2 2 2 B B v v i – 23 – λ v m µ v √ v v λ λg | 2 2 2 µ √ √ µ m m − i − 2 2 2 2 2 v µ µ µ 2 µ √ √ )] give to good approximation the same numerical v 2 µ 2 − i i √ √ v i i = = = = = ) couplings and the additional mass term 0 √ 0 − − − − S = = = = = ). The differences between the two may be understood SM 3.37 0 S e = = = = = NSM NSM e BG BA SM e e e 0 B e χχG S B BH )–( e SG NSM NSM SA e 3.16 e g BA e BH SM e . For on-shell binos and singlinos, these couplings are B e e B B SH ig e e e e SG NSM NSM B e SA g B g SA e e SH e B g e S )–( g S ig e e SH e 3.33 B = g B g e SA e e SH S g g ig e S e = S g S κµ/λ ) and ( g g 3.10 2 = S e ) also induces bino-singlino-Higgs couplings which might be relevant BH S ≈ e e B SH NSM g 1 e B ) [eqs. ( e SH 3.22 g e S M g χχA 3.32 )–( ), using the mapping of the parameters in eq. ( 3.28 2.22 )–( For mostly singlino-like (bino-like) DM with small Higgsino admixture, the couplings Note that eq. ( 2.18 in eqs. ( results as those in eqs. ( for thermal production viamass co-annihilation degenerate, when the bino and singlino are approximately in agreement with the mappingthe of sign parameters of in the eq. ( ( Similarly, the couplings of pairs of (on-shell) binos are given by ( JHEP04(2018)069 ] to (4.1) 62 – because = 5 and 58 ) in pow- κ [ d A 3.11 and 5.1.2 5, a 125 GeV SM- µ . For points passing scan. ) and ( 125 GeV Higgs boson ≤ ] for details). In partic- ≈ β 3.10 58 h m NMSSMTools NMSSMTools NMSSMTools in eqs. ( . . The choice of parameter ranges ), (see ref. [ 3] i  1 . 2 3.6 7] κ λ ) 75] TeV 5] TeV 75] TeV . . . . − , when replacing these expressions into /µ 5; 5] . 2 5; 0 3; +0 χ . . β [1 0 m 5; +1 /µ [0 ) and ( 1 . [0; 1] TeV ( 75; +0 75; +0 − . 1 . [ sin 2 0 0 − 3.4 − NMSSMTools [ 1 – 24 – − − parameter points, drawing the parameters from  [ [ h 9 µ = 2 λ β ). ]. is chosen to be larger than those for A 1 λ κ λ 64 tan λ κ µ A A M , 3.37 A 63 )–( , . NMSSM parameter ranges used in 47 – 3.28 44 [ ) is sufficient to reproduce the couplings obtained from the Table 1 . Keeping only the first term, 1 4.3.5 3.16 2 ). The range for /µ )–( 2 χ 3.7 m 3.12 We perform a random scan over 10 = 6 operators, eqs. ( like Higgs boson iscf. obtained eq. without ( the needapproximate alignment for implies, large from radiative eqs. corrections ( to its mass, these constraints we compute themicrOMEGAs relic density and the direct detection cross sectionlinear-flat with distributions over theis ranges motivated listed by in table the phenomenology of a SM-like Higgs: for tan much larger than theparticle. SUSY scale, Furthermore, we and require ifwith the the couplings spectrum lightest to to neutralino SM contain particles isLHC. an compatible not We with also the those require lightest of compatibility SUSY additional the with SM-like Higgs constraints Higgs from bosons LEP, observed and Tevatron, at and sparticles the the as LHC implemented on in properties that are sharedbetween with the the EFT EFT approach, andcompute while the NMSSM also spectra full determining and theory the couplingsa predictions. differences and subset to of subsequently We the test use parameter constraintsular, implemented points points in against are excluded if they feature an unphysical global minimum, soft Higgs masses 4 Dark Matter phenomenology Besides serving as an ultraviolet completionnient of computational our basis EFT, since the mature NMSSM numericalcollider also tools and provides are a DM available for conve- phenomenology. the In analysisof this of section, the both we NMSSM, study going the beyond phenomenological the properties EFT validity conditions. Doing so, we can identify those ers of eqs. ( d as originating from corrections duewith to the higher dimensional expansion operators of in the the denominator EFT associated JHEP04(2018)069 5 ]. ), ≡ χ are t 65 µ X 3.16 appears )–( /µ 1 3.12 in the range χ , the annihila- will practically m 3 U 2 | and to the elements = 2 TeV, and, in ). In addition to M λ 1 3 A = M ) and the case where κµ/λ M } 2 3 | Q  M ) , while annihilations into F κµ/λ , the heaviest neutralino x 2 i |} . This can be quiet generically µ, κµ/λ , while we chose the range of σv | F { 2 h x µ i | µ, < κµ/λ 2 1 σv | h , M | µ | , 1 allowing for moderate radiative corrections for the mixing angles also captures effects , such that we omit the phenomenologically 6 ) for the CP-even (CP-odd) Higgs bosons as M 1 ij χ (20 %) of P ( – 25 – 12, annihilations into top quarks typically make  { O . ) = 0, 0 2 ij β S M ∼ 3, the singlino mass parameter 2 . tan h 0 NMSSMTools is chosen such that we obtain both the case where the µ , we scan over the stop mass 1 − 1 | ≤ b such that the bino can be both lighter and heavier than the M κ A | 1 ( is negligible (i.e. when min( M ≡ 1 . Computing the couplings from mixing angles, eqs. ( χ b is mostly bino-like (i.e. when X 1 χ for the neutralinos and . Including non-minimal third generation squark mixing would not affect our results, since NMSSMTools ij ) = 0 and β N (80 %) of the thermally averaged cross section 5] TeV and, for definiteness, we set the stop and sbottom mixing parameters = 10 TeV. Note, that due to this choice of parameters we also satisfy all LHC O cot . µ 2 NMSSMTools In the NMSSM, apart from the processes discussed in detail in section In all figures presented in this section, we compute the couplings of the neutralinos to Due to our choice of parameters − This choice minimizes radiative corrections to soft breaking parameters such as M 6 t 75; 2 . A of the Higgs massfrom matrices, such that we canwe are easily only match interested analytical in the treeradiative phenomenology level corrections of results can the to Higgs be and numerical compensated neutralino results for sectors for by which shifts the of corresponding the tree level parameters. tion cross section can be enhancedFor either by the resonant non-resonant annihilation or annihilation bywith co-annihilation cross [ an sections, acceptable we relicup checked density numerically Ω for that forpairs points of Higgs bosons usually account for components. These would onlyor appear for in wino the effects EFTfermion through from in higher including addition dimensional operators to operators, arisingthermore, the from operators using integrating from the out integrating outputinduced out an from by an SU(2)-triplet the SU(2)-doublet running fermion. of Fur- the parameters from the SUSY scale to the electroweak scale. Higgs mass eigenstates and ofing the SM angles particles to theoutput Higgs from mass eigenstates from thetakes mix- into account sub-leading effects from the (small) admixture of additional neutralino always heavier than the lightestdisfavored Higgsino neutralino DM region and we canas map an results expansion onto parameter. our EFT, Hence, where mostly our DM singlino-like candidate, the or lightest mostly neutralino,mediated will bino-like processes. be with either small Higgsino admixture facilitating Higgs will be wino like, while theFurthermore, wino since component we of chose the lighter massalways eigenstates will be be negligible. smallerthe than bino the mass Higgsino parameter singlino mass and the parameter Higgsinos. This also ensures that the Higgsinos with mass parameter to the Higgs masses.sfermion We mass decouple parameters the remaining toorder SUSY 3 to TeV, particles minimize the the by wino gluino settingto component all mass of remaining parameter the lightest to neutralino,bounds the on wino sfermions mass and parameter gluinos. lightest neutralino the bino component of the parameters listed[0 in table ( The range of the bino mass JHEP04(2018)069 95 . 0 boson ≥ Z i 2 1 N 95, or singlino . P 0 ≥ 2 14 N + . However, in this case 2 13 β N if sin 2 e H 95 (blue). Similarly, points are . 0 ). In the left panel, we show points 06 for binos while for singlinos the →  ≥ . 0 2 15 2.46 /µ ≈ N χ 2 if 50 % and satisfy bounds from direct detection m / 95, Higgsino . 2 1  e S 0 g 12 ≥ . 2 11 – 26 – = 0 N 2 if h 4. Secondly, for all the different DM compositions, shows the SIDD cross section vs. the contribution . e B 0 3 ∼ vs. the contribution of the longitudinal mode of the 2 λ SI p ) when taking into account only the Goldstone amplitude in 95 (red), singlino σ . ), figure 0 ¯ t t 95 but none of the individual contributions is sufficiently large to put 2.45 . ≥ 0 → 2 11 ≥ : first of all, we note that both the SIDD and the annihilation cross N i 2 ) with the amplitude given in eq. ( 1 χχ if N 3.1 e B 2.45 P and 95. Similarly, points are denoted as mixed if the sum of the square of corresponding . 2 0 . SIDD cross section ). The left panel shows all points from our scan passing the Higgs and LHC con- ≥ 2 , which can be suppressed independently from each other. The SIDD cross section 15 2.46 G N i if σv e couplings to Higgs bosons arecouplings proportional are to proportional to a striking feature ofh this plot is theis relative suppressed independence close of to the the SIDD blind cross spot section conditions and denoted as bino denoted as mixed if thebut sum none of of the the individual square contributionsous of is categories. corresponding sufficiently large The mixing to behavior angles putin shown is them sections can in one be of understood thesection previ- quite for intuitively from mostly our bino findings DM are smaller than those for mostly singlino DM because the to the thermal annihilationmode cross obtained from section eq. fromeq. ( the ( diagrams mediatedstraints by described the above but Goldstone beforedirect requiring detection the limits. correct relic The density dominant or composition compatibility of with the DM candidate is color coded and understood to be due tofor the the difficulty in second obtaining finalConcentrating a on state, Higgs ( spectrum while light evading enough to Higgs allow phenomenology and collider constraints. required to have the correct relicexperiments. density Ω The color codinglegend. indicates We the denote compositions points as ofS purely the bino DM candidate asmixing indicated angles in is the them in one of the previous categories. Figure 3 (i.e. the Goldstone mode ofas the Higgs defined doublets) in to the eq.( from thermally our averaged parameter annihilation scan cross passing section collider constraints, while points shown in the right panel are also JHEP04(2018)069 In 7 ) for (4.2) ]. β 43 sin 2 12 [ . µ ( 0 50 %. The width / χ  ∼ uncertainties are of m 2 12 boson is generally . h Z = 0 2 h boson or one of the Higgs , NMSSMTools Z 2 remains sizable and thus the is always large in this region). ! 2 | 1 2 χ 2 χ µ -wave suppressed. As argued in | m M m p 4 4 ]. Typical β/ − − 66 i 2 Z 2 a cos 2 m m χ which set the relic density. We have thus × m is mostly bino-like with very small couplings, 2 χ β ∝ 1 χ i – 27 – 0 m NSM a m tan P χχG

g we show points with mostly singlino DM candidate in ∼ 4 50 % and satisfying bounds from SIDD and SDDD exper- we show the SIDD cross section vs. s yielding the observed relic density Ω  / 0  i shows the points from our scan having approximately the 5 2 a G (which does not play a role in direct detection) which has 3 12 A A . we show bino-like points in the (bino mass)-(singlino mass) cm 2 bino region: χ  4 26 = 0 m − 2 10 h × parameter, and consequently the Higgsinos, tends to be about the same 50 %, with the color coding indicating the possibility of resonant annihi- 2 µ  ]. we show points from our numerical scan which have an acceptable relic den- i ∼ 12 well tempered . 4 68 , ]. For mostly singlino DM, the contribution from the Goldstone mode mediated σv , the contribution from the transverse polarizations of the h 6 67 = 0 – 3 2 2.2 h In the left panel of figure In figure The right panel of figure Here and in the following we consider the relic density to be acceptable if Ω 7 requiring the correct relic density or compatibility withof direct this detection band is limits. motivatedand by by For the the bino strong uncertainties sensitivity inthe of the order the calculation of relic of a density particlegenerators few calculation masses [ GeV, to [ the as particle can spectrum, for example be seen by comparing the results obtained by other spectrum significantly larger couplings, an observationallyThe consistent value relic of density the is easilyorder obtained. as shown for the singlino-like DM in thepoints right from panel. our scan passing the Higgs and collider constraints described above but before that co-annihilation yields theannihilations correct of relic the density. mostlyfound a singlino In new like theevading latter direct case, detection constraints it easily.degenerate is However due singlino-like in to the fact presence the of an almost mass the (singlino mass)-(Higgsino mass) plane,resonant demonstrating annihilation that is neither relevant for co-annihilationIn the nor the singlino region left ( panelplane. of We find figure that pointsmass either eigenstates, resonantly annihilate or, via feature the binos approximately mass degenerate with the singlino such contrast, we find that for mostlylarge bino DM amplitudes the to Goldstone mode avoidboth does over-closure not resonant mediate of annihilation sufficiently the and co-annihilation Universe. play a Hence very in importantsity, the Ω role. bino DM scenario, lation. In the right panel of figure correct relic density Ω iments [ amplitudes to the thermally averaged annihilationical cross value section is of the order of the canon- and the contributions fromsection CP-even Higgs bosonsalso are much smaller than the contribution from the Goldstone mode. corresponding contribution to thethe annihilation contributions cross to section the is thermalare not annihilation usually cross suppressed. section smaller Note, from than the that thoseof Higgs from the mass the eigenstates same Goldstone order, mode: the while contributions the from couplings the are CP-odd typically eigenstates are suppressed by the coupling to the Goldstone mode JHEP04(2018)069 ) = β sin 2 µ points with ( / χ m ) plane. For both µ Right: ) plane. κµ/λ the spin independent cross section ) — Higgsino mass ( Right: κµ/λ as indicated in the legend. – 28 – χ m for the points passing the required experimental collider ) — singlino mass (2 2 . 1 SI p 3 ≈ M σ ), where the blind-spot conditions are satisfied for β sin 2 µ ( / χ points from our parameter scan with bino-like lightest neutralino and relic density the SIDD cross section m 50 % in the bino mass (  Left: Left: -channel mediator with mass 12 . . s . 1) for the singlino-Higgsino (bino-Higgsino) case. = 0 − 2 h constraints vs. +1( for the same pointsThe vs. color coding the is coupling the of same as the if DM figure candidate to the SM-like Higgs mass eigenstate. Figure 5 singlino-like lightest neutralino in thepanels, singlino the mass color (2 codewith indicates the points where the lightest neutralino can pair-annihilate resonantly Figure 4 Ω JHEP04(2018)069 β 0, ≈ sin − /µ = χ lies on the ) ignoring Higgs basis m i /µ h ). The dashed χ 2.61 SM . Note that for m β 2.61 H sin ≈ interaction eigenstates S /µ χ H m and assuming only one CP-even Higgs i
SI p NSM σ H – 29 – coupling. In the case of mostly singlino DM we find . Hence, if the contribution from one of the e Bh SI p e we show the SIDD cross section vs. the DM coupling to vs. the contribution B for which we do not find any parameter points with an σ g 5 SI p µ multiplied by the sign of its amplitude, cf. eq. ( = σ }

i  S
1 SI p σ M

h, H, h { = i we show the contributions to the SIDD cross section when taking into h 6 50 %.  . SIDD cross section 12 . In figure In the right panel of figure = 0 2 h account only one Higgs mass eigenstate at a time as obtained from eq. ( eigenstate, and SIDDachieved cross by sections suppression satisfying of the this the correlation current to experimental beby bounds looser, additional can indicating mechanisms. that be the SIDD cross sections must be suppressed the SM-like 125 GeV Higgsstate, mass which eigenstate. vanishes at the Besides respective theto blind coupling the spots, to this coupling takes the from intoto account the the (small) the contributions admixtures 125 of GeVsection the mass is eigenstate. very tightly We correlated find with that the for coupling mostly to bino the 125 points GeV the SM-like SIDD Higgs cross mass acceptable relic density in ourwith the scan. singlino This nor is resonantwould annihilation because be with in required the this to Higgs regionthe boost bosons neither Universe. is the co-annihilation possible, thermal one annihilation of cross which section to avoid over-closure of DM, the SIDD crossis section approximately is satisfied, suppressed while whensinglino the for dominated, blind all the spot other SIDD condition compositionsbino cross section of DM is the candidates suppressed DM wewhich for candidate, also corresponds usually find to suppression of the SIDD cross section for the SIDD cross sectioncontributions is lie dominantly outside mediated theSIDD by dashed cross that diagonal section. mass lines, eigenstate.is The they bino left interfere On (singlino) (right) destructively theΩ panel to contrary, like. if yield shows the parameter the For points total both where cases, the we lightest show neutralino points from our parameter scan which satisfy Figure 6 mass eigenstate diagonal lines indicate diagonal lines and the contributions from the remaining mass eigenstates lie within the triangle, JHEP04(2018)069 ]. The XENON1T , destructive inter- 5 6 and the SM-like mass S h 5 kg yr [ . = 129  due to the presence of the tree-level S 1, while binos couple with characteristic 173 GeV are ruled out by current SIDD H . 0 ≈ ∼ t 4 λ – 30 – < m ∝ . χ e Sh m e 2 S g ). Hence we see the necessity of destructive interference ∝ 004. In addition, compared to bino DM, singlino DM has . SI p 3.35 0 σ ] for the same points. As noted above, current experimental ∼ 6 neutrino floor 4 – / 3 4 1 ) and ( g also shows that improvements in SDDD experiments will probe ∝ 7 3.30 e Bh = 20 t yr. e 2 B pb can be obtained by suppression of the coupling to the SM-like Higgs  g we show the constraints from direct detection experiments for points from = 1 t yr, while XENONnT could probe most of the singlino region assuming 13 pb, out of reach of direct detection experiments for the foreseeable future. ∝ to suppress the SIDD cross section below the experimental limits. 7  − h 13 SI , cf. eqs. ( p 10 − κ σ , parameter points satisfying SIDD limits almost always satisfy current constraints ∼ 10 7 SI p Finally, let us mention that out of the points passing the main phenomenological In figure Although blindspot cancellation or destructive interference arguably require some fine For mostly singlino DM, as shown in the right panel of figure . σ SI p experiment will haveexposure a of sensitivity roughlyan one exposure order of of magnitude better assumingconstraints an described at thelike beginning Higgs of boson this compatible section, with which in LHC particular measurements lead and to evade a collider SM- constraints as improvement in the bounds fromscattering SIDD. by In approximately particular, two orders improvingprobe of the most sensitivity magnitude of of with SDDD by the respect rescaling to singlino current current region. bounds boundscross will from A section LUX which estimate na¨ıve were on of obtained the with the spin-dependent an sensitivity WIMP-neutron exposure can of scattering be obtained SIDD limits are much morefigure constraining for our model thanon SDDD the limits: SDDD as scattering we cross canis section see not from in direct true. detectionthe experiments, remaining while Figure parameter the space reverse of our model more efficiently than a correspondingly large our scan with an acceptableshows relic the density and SIDD satisfying cross colliderwith section constraints. DM vs. The masses left below the panel constraints. the DM top The mass. right mass panel Note, showsand a that comparison SDDD of almost experiments the all constraining [ parameter power of points current SIDD tuning, we stress that weσ readily find points in ourSuch dataset small with cross SIDD sections cross aredue sections below challenging to to the probe presence of with the current direct detection strategies a much larger coupling tocoupling the scalar singlet state between the contributions fromeigenstate the singlet like mass eigenstate needed only for even smaller cross sections. ference between different Higgsexperimental mass bounds eigenstates on is the almoststrength SIDD of cross always singlino section. required couplings to This canstrength satisfy be the understood from the typical against the full SIDD crossfrom section in our the dataset left satisfying (right) allacceptable panel Higgs/collider for relic constraints mostly density. described bino (singlino) above Foras points and mostly featuring bino an DM, wemass eigenstate find alone. that Destructive SIDD interference cross between different sections Higgs as mass small eigenstates is the sum over the CP-even Higgs mass eigenstates. We show these contributions plotted JHEP04(2018)069 ]. To 50 %, Right: 6  12 . 100 GeV if the = 0 consistent with 1 2 . for points from our h − s ]. For DM masses χ ], most relevant for χ ], respectively. 3 4 m m 71 70 , cm 69 26 − 10 × 2 bosons, current bounds from DM leptons [ ∼ τ F ] and PandaX-II [ for DM masses W x 3 i 1 − σv s 35 % satisfy SDDD and SIDD constraints. h 3 ∼ cm quarks or 26 – 31 – b − ], or SDDD scattering of protons from PICO-60 [ 10 5 × 2 15 % have an acceptable relic density Ω i ∼ ∼ , σv h NMSSMTools the SIDD cross section vs. the mass of the DM candidate we have focused on direct detection constraints on our model. Any DM 50 %. The solid black and dashed red lines indicate the most constraining experimental . indirect detection bounds are much weaker, constraining thermal annihilation Left:  3 4 t . 12 . m = 0 & 2 χ h cross sections much larger than the typicalan values acceptable relic density. Relevantof constraints DM might matter further particles arise captureda via from flux elastic the of scattering annihilation in neutrinos the observableour Sun, at case potentially Earth. if giving Under rise DM certain to annihilates assumptions dominantly [ into pairs of production, we focus onindirect constraints detection from constraints on photon WIMPof emission DM here. Milky come from Way The Fermi-LATannihilation satellite and currently cross MAGIC strongest galaxies, analyses sections rulingannihilation out is canonical dominantly values intom of pairs of the thermally averaged 4.1 Indirect detection In section candidate is also subjectastrophysical to uncertainties constraints on from indirect indirect detection detection experiments. constraints arising Due from to charged large particle as in figure implemented in and of the points with the correct relic density SIDD vs. SDDD cross sectioncross in units section, of at the respective each observedand limit respective PandaX-II. for DM For the SDDD same mass scattering points. weSDDD we For use use scattering SIDD the of the more neutrons stronger constrainingguide from of of the LUX the the eye current [ two bounds we limits forquadrant indicate either from satisfy the XENON1T all current bounds current direct with detection thin dashed bounds. lines; The points color lying coding in in the both lower left panels is the same Figure 7 parameter scan passing the requiredΩ experimental collider constraints withupper acceptable limits relic on density the SIDD cross section from XENON1T [ JHEP04(2018)069 ) − W + bosons. is below or Higgs W Z couples to µ Z Higgs basis → SM SM H χχ H ] requires the presence of an 42 , such that their production cross 41 S A 2 TeV such that they are not directly 350 GeV, or, be dominantly composed ∼ ≥ or 2 in our model. The non-observation of t , the observation of a 125 GeV Higgs boson / S m Z 3 boson, such that the additional decay width decaying dominantly into pairs of top quarks H t Z ]. – 32 – and > m 76 2 > m – 2.1 / χ h 72 m m  χ ) will only have small increases in sensitivity at future LHC m ]. Note, that most of the conventional Higgs searches such as ) are suppressed for our preferred region of parameter space. − 32 Furthermore, DM is coupled to the SM via the Dirac fermion (the 8 W /γγ/Zh + − τ ]. Note, that in our NMSSM ultraviolet completion we set the masses /W + τ − 53 τ 173 GeV. Furthermore, SIDD constraints require the coupling of DM to + → ]. ≈ bosons. Thus, current indirect detection limits do not pose relevant constraints 80 t boson or the SM-like Higgs into DM particles must be small, even before taking bb/τ ¯ – W m H/A 125 GeV Higgs mass eigenstate approximately aligned with the Z 77 → & As discussed previously in sections The parameter region in our model yielding an acceptable relic density and compatible ∼ → Note, that this choice is made to allow for a straightforward connection of the NMSSM to our EFT 8 h χ gg χχ in the appendix of( ref. [ model and allowing forfrom lighter masses the will LHC generallyLHC allow not runs. effect for our much analysis. smaller Current masses experimental of bounds the SUSY particles, probeable at ongoing and future signals from additional Higgs bosonsbosons at must the either LHC furthermore have masses impliesof that heavier than additional the Higgs 2 additionalsection singlet is interaction suppressed. states for A additional comparison Higgs bosons of decaying the into constraining pairs power of of SM particles conventional searches can for example be found m interaction eigenstate. Direct detectionof limits furthermore DM strongly to constrain both theof the coupling the SM-like Higgs andinto the account the fact that Higgsinos in the case ofThus, the NMSSM) this which model in can turnbosons, be couples to or, tested the via at Higgs production the and of the LHC the either additional by Higgs DM bosons.with pair couplings production compatible via with those of a SM Higgs [ The Dirac fermion we integratedthe out TeV may scale be [ accessibleof all at additional the particles LHCsuperpartners such if as its of the mass the SUSY SMaccessible partners gauge at of bosons the the LHC. to SM values fermions and the charged LAT [ 4.2 Collider constraints Our EFT DM modeleigenstates, extends two the of neutral them particle CP-even content and of two the CP-odd, SM and by a four SM Higgs singlet mass Majorana fermion. annihilation is suppressed. This ispairs because of of all the Higgs basison states, our only model since the( annihilation modes most constraining forNote indirect however, detection that bounds DM withmay mass explain the excess of photons emitted from the Galactic Center observed by Fermi- DM-proton scattering cross section [ with current SIDD constraintsm typically features DM massesthe heavier SM-like than Higgs the boson top to quark be suppressed which in turn implies that ( capture and annihilation yield the most constraining upper limits on the spin dependent JHEP04(2018)069 Z ) can however be S /ha states. If kinematically S S Zh H → ) or into a SM-like Higgs boson and A hh ] have set stringent limits on WIMP 6 NSM → – 1 H H ) and ( boson. Due to the presence of the SM-like S Z /Za S – 33 – hh → H symmetry. In such a case the new fermions acquire mass 3 symmetry reduces the number of allowed operators involv- Z 3 Z Higgs basis state with the bosons only via axial couplings. In this article, we explore an EFT SM Z ) are suppressed since the corresponding couplings are proportional to H ]. Zh 33 , → 32 A , ]. The latter constraint is naturally satisfied for Majorana DM, since Majorana ( 7 29 Z In order to test the validity of the results derived in our EFT model, we compare them can be reduced tocomputational our basis. EFT model The atthe Majorana low NMSSM, singlet energies, and in such the our thatthe EFT EFT we qualitative scale is can features with use identified obtainedthe with it the in as precise the mass the an singlino values of EFT explicit of in are the preserved the heavier in Higgsinos. couplings, the relic We complete density, show theory, and that while cross sections are modified by the direct detection cross section.spin We independent find direct DM blind detection spot constraints,These while conditions blind yielding that spot the allow observed conditions relic for can density. the be evasion easily of characterized into those terms obtained of in the the supersymmetric EFTWe NMSSM, which parameters. demonstrate features an that analogous in Higgs sector. the case of heavy gauge and fermion superpartners the NMSSM candidate. In addition, the ing the singlet Majorana fermioneasily and the extend doublet the and EFT singletenergy scalar to fields, Higgs dimension such spectrum that 6, we including and can in derivative the this operators. couplings model, of We derive and DM the use to low the these neutral results Higgs to and compute gauge the bosons relic density and spin independent scale is identified withfrom the the theory. mass Furthermore, we ofmay assume a be that realized all heavy by explicit SU(2)-doublet imposing massthrough Dirac a terms the are fermion forbidden, vev integrated which of out induces the electroweak singlet symmetry field. breaking, we If naturally obtain this a vev weak is scale mass generated for by the the DM same mechanism that bosons [ fermions couple to describing the interactions ofcomprised Majorana of fermion a type WIMPsan II with extension 2HDM an of and Higgs extended portal a Higgs models. SM sector, gauge In particular, singlet. we This study the model case can where be the interpreted EFT as mass 5 Conclusions Current bounds from direct detectionDM experiments scenarios [ where the relicand density proceeds gauge from bosons. thermal production Inpling mediated particular, of by direct the Higgs detection SM-like experiments Higgs boson strongly to constrain DM the particles cou- and the vector-like coupling of DM to the mixing of the allowed, the branching ratiossizeable. of The ( collider signatures arisingmodel through [ such decays can be effective probes of our models with extended Higgs sectorsmass and potentially eigenstates large such couplings between as differentinto ours Higgs lighter Higgs can bosons be or a probedHiggs, light effectively decays Higgs and into by a pairs decays ofand of SM-like a heavy Higgs Higgs bosons bosons ( runs because they are increasingly dominated by systematic errors. On the other hand, JHEP04(2018)069 well tempered 6 operators in boson, plays a Z d > 1 by direct detection . 0 . χχh g invariant Majorana fermion DM 3 Z boson or a Higgs mass eigenstate. Z – 34 – , the correct relic density for bino-like DM may also be is obtained via co-annihilation of the bino with the singlino. well tempered region well tempered region Finally, collider searches for the Majorana DM particles are mostly restricted to the In both the cases of bino-like and singlino-like DM, we find that the current constraints In the NMSSM, when considering the case of light binos, we find a new Both in the EFT and in the NMSSM we show that the coupling of (singlino-like) Office of High Energyby Physics. accepting The the United article States forretains Government publication, a retains non-exclusive, acknowledges paid-up, and that irrevocable, the the world-widepublished publisher, United license form to States of publish Government or this reproduce manuscript,purposes. the or allow others Work to at do so, University for of United Chicago States Government is supported in part by U.S. Department of We would like to thankSB Bibhushan acknowledges Shakya for support interesting from discussionsthe the related Oskar Swedish Klein to Research this Centre Council (Contract work. University. No. (Vetenskapsr˚adet)through This 638-2013-8993). manuscript NRS has isContract No. supported been DE-AC02-07CH11359 by authored with Wayne by the State U.S. Fermi Department Research of Energy, Alliance, Office LLC of Science, under Higgs bosons. Going beyondmissing the energy effective searches theory, these in searches thethe may production NMSSM, be and by complemented related decay by searches of for the heavy Dirac superpartners. doublet fermion and,Acknowledgments in coming from SDDD are subdominantalso compared find to that those coming ancross from improvement sections SIDD. of could However, efficiently two we probe orders most of models magnitude consistent with inusual channels a the of blind single bounds SM-particles spot plus on in missing energy, SIDD. the in particular SDDD those associated with the blind spot conditions forin the DM the coupling to theBeyond SM-like Higgs the state. The relicobtained density via resonant annihilation with either the singlet and/or non SM-likethe CP-even cross doublet section. Higgs boson is required toDM further region. suppress For bino-likefor DM singlino-like the one couplings and to direct the detection bounds Higgs are bosons hence tend evaded to by be a smaller mild than proximity to experiments, while simultaneously consistency withthrough thermal the annihilation DM via couplings relic ofThe density DM to can neutral the be remaining Goldstone Higgs obtained and mode,prominent gauge comprising bosons. role the in obtaining longitudinalevade the mode direct thermal of annihilation detection the cross bounds,be section. reduced, not Moreover, but only in in order the addition to coupling destructive to interference of the the SM-like SM-like Higgs Higgs boson boson must with the the EFT description.candidate, We e.g. also the discuss bino in thein the case a NMSSM, straightforward of and manner show a to that include our this EFT scenario. model canDM be to generalized the SM-like Higgs is constrained to values below presence of small corrections which could be included by dimension JHEP04(2018)069 (A.2) (A.1) . i i . boson can be 2 2 c i i ) ) . 2 2 0 Z ) ) 0 G 0 +h (  NSM G G 0 i − A 2 G ) 2 +( +( ) µ 2 2 +( ) ) i∂ 2 ) NSM NSM A A NSM NSM NSM NSM A A +( i ) reads in terms of the Higgs H ( ( +( 2 . . H , ) 2 c c + − − ) . . 2 2 +( NSM 2.13 ) ) ... 2 A NSM +h +h io ) i i j NSM . NSM µ 0 0 0 0 c H Φ A . NSM NSM io io G H G G G i∂ µ ∂ to Higgs basis states written explicitely 0 0 L i H H +( ∂ i∂ ( ( G G +h +( +( 2 Φ ... ) 2 2 NSM j NSM NSM NSM − ∂ − ) ) 0 SM Φ 2 χχ 2 A A A 2 i H ) ) G NSM NSM i Φ H SM SM + + +
− ∂χ H H + SM 0 SM S – 35 – χχ +( H H in order to retain a canonical kinetic term and ( g 2 − − G H ≡ H ) NSM iA NSM NSM µ χ − +( χχ NSM ... + H i H H +( +( SM i∂ j NSM NSM
A S SM SM Φ µ S H i SM A A SM SM SM SM SM H Φ i∂ vH vH H
iA H H H +( SM SM 2 2 2 2 χχ S vH v vH 1 + + ), the EFT Lagrangian ( + + g + √ 2 √ H H 2 √ √ 2 SM S 0 iA  2 2 χ − √ + + √ + h G µ β H ), + µ − NSM NSM NSM 2 µ λ µ vH ¯ ( σ χ 2 S † +2 2 +2 h 2 µ is v i∂ O 2.6 2 1 NSM NSM 2 χ ∂ H h 2 vH vH vH 2 √ √  v v µ µ v 2 2 2 2 χ 2 − √ 2 2 )–( ¯ Z vA vA + σ | µ h h h √ √ √ i +2  2 2 µ λ µ χχ ¯ † h σ h h | β W 2 1 β β καv β † 2.3 2 2 2 2 2 2 2 h s g √ √ χ v β β β 2 c s s 2 µ 2 χ h h 2 2 2 2 2 − | | 2 s ic i c i c 2 2 h n n n λδ | | µ µ κ α κξ | | √ α α µ µ − + + + + + × × × 2 2 | | 2  χχ + − + + − − µ δ 2 = L The couplings of the Majoranaread fermions off to from the here. Higgsin Note, basis this that states work couplings and are the normalized as Rescaling the Majorana fermionkeeping terms field to order basis states, eqs. ( U.S. Department of Energy underNS Contract and No. CW was DE-AC02-06CH11357. partiallyby The performed National work at Science of the Foundation MC, grant Aspen PHY-1607611. Center for Physics, which is supported A EFT Lagrangian Energy grant number DE-FG02-13ER41958. Work at ANL is supported in part by the JHEP04(2018)069 , ]. ]. ]. 8 ] ] ]. F (A.3) 3 ] ] SPIRE SPIRE SPIRE ]. ]. SPIRE IN IN IN IN ][ ][ ][ ][ Theory and SPIRE SPIRE IN ]. IN ][ ][ (2017) 251302 arXiv:1208.0833 arXiv:1106.0034 [ [ SPIRE arXiv:1005.5651 [ 118 IN arXiv:1509.01515 ]. ][ [ ]. (2012) 1 arXiv:1705.06655 arXiv:1702.02924 arXiv:1708.06917 [ Can WIMP Dark Matter arXiv:1609.09079 [ SPIRE [ . [ IN 516 Implications of LHC searches for (2013) 075003 SPIRE ][ (2016) 25 arXiv:1112.3299 ... arXiv:1702.07666 [ IN i Prospects and Blind Spots for (2010) 055026 [ ][ Φ Toward (Finally!) Ruling Out Z and Phys. Rev. Lett. L D 88 C 76 (2017) 107 , ∂ (2016) 029 arXiv:1211.4873 D 82 ∂Z ∂ Low-mass dark matter search with CDMSlite (2017) 181301 [ 04 (2017) 181302 2 (2012) 65 Phys. Rept. 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