Constraints on the Parameter Space in an Inert Doublet Model with two Active Doublets
PPrepared for submission to JHEP
Constraints on the Parameter Space in an InertDoublet Model with two Active Doublets
Marco Merchand a , Marc Sher a a High Energy Theory Group, William & Mary, Williamsburg, VA 23187, USA
E-mail: [email protected] , [email protected] Abstract:
We study a three Higgs doublet model where one doublet is inert and theother two doublets are active. Flavor changing neutral currents are avoided at tree-levelby imposing a softly broken Z (cid:48) symmetry and we consider type I and type II Yukawastructures. The lightest inert scalar is a viable Dark Matter (DM) candidate. A numericalscan of the free parameters is performed taking into account theoretical constraints such aspositivity of the scalar potential and unitarity of 2 → B physics lower limits on charged Higgsmasses, Electroweak Precision Observables, LEP II, LHC Higgs measurements, Planckmeasurement of the DM relic abundance and WIMP direct searches by the LUX andXENON1T experiments. The model predictions for mono-jet, mono Z and mono Higgsfinal states are studied and tested against current LHC data and we find the model tobe allowed. We also discuss the effects of abandoning the “dark democracy” assumptioncommon in studies of inert models. Projected sensitivities of direct detection experimentswill leave only a tiny window in the DM mass versus coupling plane that is compliant withrelic density bounds. a r X i v : . [ h e p - ph ] M a r ontents – 1 – Introduction
The nature of dark matter remains one of the biggest mysteries in physics. While evidencefor the existence of DM has been very well established over the last decades, the StandardModel (SM) lacks a good DM candidate. It is therefore necessary to consider theoriesbeyond the SM to address this issue.One of the most conspicuous examples of a DM candidate is the so called WIMPwhose mass is expected to be of order the electroweak scale m χ ≈ SU (2) L Higgs doublet. They called it the IDM2 and scanned itsparameter space imposing theoretical and experimental constraints to determine where DMabundance is acceptable and CP is violated. Although the issue of electroweak baryogenesiswas not addressed by the authors they used the difference between the average and themaximal values of the electron electric dipole moment and the basis-independent invariants,introduced by Gunion and Haber in [3], to provide a measure of the amount of CP violation.The same model was further studied by some of the same authors in Refs. [4, 5]. In[4] the authors refined the basis invariants used in [2] to include the effect from the extrainert doublet and include DM direct detection constraints in their study. In Ref. [5] thephenomenology of charged scalars at the LHC was studied.Another interesting scenario that allows for CP violation is that of a 2HDM plus aninert gauge singlet scalar [6]. This model has fewer parameters and the DM is more inert,i.e. it doesn’t have gauge interactions. In this model there are two independent portalcouplings that allow decoupling between DM annihilation and scattering off nucleons andthus one has to take into account isospin violation i.e. the effective couplings of DM to theproton and the neutron are different and one has to rescale the experimental cross sections.The CP conserved version of the IDM2 was studied in ref. [7] by Moretti and Yagyu,together with a model with 2 inert and one active doublet. They referred to these models asI(1+2)HDM and I(2+1)HDM respectively. They studied the constraints on the parameter– 2 –pace from perturbative unitarity by calculating all possible scalar boson 2 → H ± W ∓ Z vertex and study the parameter spacewhere the branching fraction H ± → W ∓ Z can be of order 10% when the charged scalar islighter than the top quark.The I(2+1)HDM [9–11] has two inert doublets and thus can alleviate the tension withdirect detection experiments in the low mass region. In the high mass region, it can bringthe model to testable territory by decreasing the mass or increasing the Higgs DM couplingwhile keeping the required amount of DM relic density. See Refs. [12–15] for further studieson this model.In the past five years or so there has been little investigation of the I(1+2)HDM. Anupdated revision of the parameter space confronted with the data from run 2 of the LHCwould be valuable. It is also important to do a detailed survey of the different mono objectsignals that arise as predictions of the model and to test them against LHC analyses.In this work we study the CP conserving I(1+2)HDM with both type I and type IIYukawa interactions. We take into account theoretical constraints such as positivity of thescalar potential and unitarity constraints on the quartic couplings. B physics bounds onthe charged Higgs mass m H ± as a function of tan β are utilized. The most recent LHCHiggs data is enforced as well as the DM relic density results by Planck. Direct detectionupper limits on the annihilation cross section as function of DM mass are used to findconsistent regions of parameter space. The model was implemented in FeynRules [16] andwe used micrOMEGAs [17–20] to calculate DM observables.The outline of the paper is as follows: the model notation and conventions are intro-duced in section 2 together with the relevant free parameters. In section 3 we present thetheoretical constraints that will be imposed. Section 4 contains the experimental restric-tions the model needs to satisfy to be consistent. The differences between this model anda simple superposition of the IDM and 2HDM are outlined in section 5. A numerical scanof the parameter space is performed in section 6. The predictions of the model for differentmono-object final states are examined in section 7. In Section 8, we discuss the effects ofabandoning the “dark democracy” assumption discussed in the next section; these effectscan be substantial in this model. In section 9 we discuss constraints from heavy Higgssearches at the LHC. Section 10 contains our conclusions. We devote two Appendices toinclude relevant formulas for the 2HDM parameters and for the oblique corrections. – 3 –he I(1+2)HDM has two active SU (2) L Higgs doublets that we parametrize as followsΦ = (cid:32) ϕ +1 ( v + ρ + iχ ) / √ (cid:33) , Φ = (cid:32) ϕ +2 ( v + ρ + iχ ) / √ (cid:33) , (2.1)while the inert doublet is written as η = (cid:32) χ + ( χ + iχ a ) / √ (cid:33) . (2.2)The model has a Z × Z (cid:48) symmetry, where the first factor is the inert-doublet Z : η → − η (all other fields are neutral) and a softly broken Z (cid:48) is introduced (Φ → Φ , Φ →− Φ , ) on the Higgs doublets to avoid tree level FCNC’s.In this work we follow the notation of Ref. [2] and we write the potential as V (Φ , Φ , η ) = V (Φ , Φ ) + V ( η ) + V (Φ , Φ , η ) , (2.3)where V is the regular 2HDM potential with softly broken Z , namely V (Φ , Φ ) = − (cid:110) m Φ † Φ + m Φ † Φ + (cid:104) m Φ † Φ + h.c. (cid:105)(cid:111) + λ † Φ ) + λ † Φ ) + λ (Φ † Φ )(Φ † Φ ) + λ (Φ † Φ )(Φ † Φ )+ 12 (cid:104) λ (Φ † Φ ) + h.c. (cid:105) , (2.4)while the inert sector potential is simply written as V ( η ) = m η η † η + λ η η † η ) , (2.5)and the most general mixing terms between active and inert doublets is given by V (Φ , Φ , η ) = λ (Φ † Φ )( η † η ) + λ (Φ † Φ )( η † η )+ λ (Φ † η )( η † Φ ) + λ (Φ † η )( η † Φ )+ 12 (cid:104) λ (Φ † η ) + h.c. (cid:105) + 12 (cid:104) λ (Φ † η ) + h.c. (cid:105) . (2.6)As we are not interested in investigating electroweak baryogenesis in this scenario wewill assume CP conservation for simplicity and take all the parameters in the scalar sectorto be real. We also note that CP non-conservation introduces three mixing angles in theactive scalar sector and requires some parameters to be complex. A full account of CPviolation can be found in Refs. [2, 4].We adopt the ”dark democracy” of the quartic couplings λ a ≡ λ = λ ,λ b ≡ λ = λ ,λ c ≡ λ = λ , (2.7)– 4 –his simplification reduces the number of parameters significantly. As we will see in Section8, relaxing this assumption, which is common in IDM studies, can have a substantial effectin this particular model.The softly broken Z (cid:48) gives rise to four different Yukawa interactions. For a review seeRef. [21]. In this work we focus on type I model in which all fermions have charge − Z (cid:48) and couple to Φ and the type II model which has down quarks and leptons to beneutral under Z (cid:48) thus coupling to Φ and the up quarks are odd which couples them to Φ .The other two types of models called lepton-specific and flipped have identical couplingsto quarks as the type I and type II model respectively and are not considered here. The diagonalization of the CP odd fields as well as the charged scalars is carried out bythe orthogonal transformation (See Appendix A for details) (cid:32) χ χ (cid:33) = (cid:32) c β − s β s β c β (cid:33) (cid:32) G A (cid:33) , (cid:32) ϕ +1 ϕ +2 (cid:33) = (cid:32) c β − s β s β c β (cid:33) (cid:32) G + H + (cid:33) , (2.8)where c β = cos β , s β = sin β and tan β ≡ v /v . G is the neutral Goldstone boson and A is the physical pseudoscalar while G ± is the charged Goldstone boson and H + is thecharged Higgs.The physical CP even scalars are obtained by the rotation (cid:32) ρ ρ (cid:33) = (cid:32) c α − s α s α c α (cid:33) (cid:32) Hh (cid:33) , (2.9)where h and H correspond to the lighter and heavier CP even scalar states respectively.Notice that there are 8 real parameters in the scalar potential V and tan β = v /v giving a total of 9 parameters. However the minimization conditions of the potential reducethe number of free parameters down to 7. Here we choose as free parameters the following S = (cid:8) m h , m H , m A , m H ± , m , α, β (cid:9) , (2.10)which we will call the ”active” set as it corresponds to the active Higgs doublets. We write m explicitly in this set as it can have positive or negative values.Since the inert doublet is endowed with a discrete Z symmetry its field componentsdo not mix with the Higgs eigenstates and the mass matrices are trivially diagonal in thissector. One can thus solve for the quartic couplings in favor of the squared masses and thequadratic mass term m η as follows λ a = 2( m χ ± − m η ) v , (2.11) λ b = m χ a − m χ ± + m χ v , (2.12) λ c = − m χ a + m χ v . (2.13)– 5 –he inert sector V + V is thus characterized by 5 parameters S = (cid:8) m χ , m χ a , m χ ± , m η , λ η (cid:9) , (2.14)which we call the ”inert” set of parameters. The full set of free parameters is thus givenby S = S + S . The SM Higgs boson is fixed to m h = 125 GeV therefore the effectivenumber of free parameters is reduced to 11. Notice that we write m η explicitly in S as itcan have positive or negative values.When the active doublets get a vev the quadratic mass term for the neutral componentof the inert doublet is given by V ⊇ m η η † η + v (cid:16) λ a η † η + λ b | η | + λ c Re [( η ) ] (cid:17) , (2.15)therefore in order for η not to develop a vev one has to require m η + v λ a + λ b + λ c ) = m χ > , (2.16)which is automatically satisfied and where we used the expressions for the quartic couplingsgiven above. In this section we provide the theoretical constraints that will be imposed on our numericalscan later in section (6). These formulas have been derived before, see Refs. [2, 7], but weinclude them here for completeness.
The following inequalities involving the quartic couplings provide the sufficient conditionsfor positivity of the scalar potential [2] λ > , λ > , λ η > , (3.1) λ x > − (cid:112) λ λ , λ y > − (cid:112) λ λ η , λ y > − (cid:112) λ λ η , (3.2) λ y ≥ ∨ (cid:16) λ η λ x − λ y > − (cid:113) ( λ η λ − λ y )( λ η λ − λ y ) (cid:17) . (3.3)where λ x = λ + min (0 , λ − | λ | ) , (3.4) λ y = λ a + min (0 , λ b − | λ c | ) . (3.5)In Ref. [2] these conditions were presented as necessary and sufficient. However it has beenshown in Ref. [22] that these conditions are only sufficient but not necessary. Thus therecan be regions of parameter space which violate the conditions above but still have a scalarpotential that is bounded from below. In this work we will implement these constraints toguarantee positivity. – 6 – .2 Unitarity The magnitude of the quartic couplings can also be constrained by requiring unitarity ofthe S-matrix. The calculation of the s-wave amplitude matrix for all possible 2 → | x i | < π, ( i = 1 , ... , (3.6) | y ± j | < π, ( j = 1 , ... , (3.7)where x i are the eigenvalues of the following matrices X = λ η λ a + λ b λ a + λ b λ a + λ b λ λ + λ λ a + λ b λ + λ λ , X = λ η λ b λ b λ b λ λ λ b λ λ , X = λ η λ c λ c λ c λ λ λ c λ λ , (3.8)and y ± = λ + 2 λ ± λ , (3.9) y ± = λ a + 2 λ b ± λ c , (3.10) y ± = λ ± λ , (3.11) y ± = λ a ± λ c , (3.12) y ± = λ ± λ , (3.13) y ± = λ a ± λ b , (3.14)notice that in the most general case there are 18 eigenvalues while in our scenario with thedark democracy assumption the are only 15. The formulas for the most general case aregiven in Ref. [7].As shown in [23] several vacuum solutions of the 2HDM may coexist at tree-leveland one has to check that the vacuum chosen corresponds to the global minimum of thepotential. However in Refs.[4, 5] it has been mentioned that checking of this restriction iscomputationally very expensive and that it only eliminates about order 10 % of the pointsin parameter space that satisfy all other restrictions. Even if one imposes the tree levelglobal minimum conditions, running effects could drive some quartic couplings negative athigh energy scales rendering the minimum metastable. A dedicated study of these runningeffects is beyond the scope of this paper. Flavor observables, e.g. the b-meson decay B → X s γ receive corrections from charged Higgsboson loops and therefore its branching fraction impose constraints on the charged Higgsmass m H ± . The mass of the inert charged Higgs field χ ± is innocuous to flavor observablesas it doesn’t couple to fermions. The most recent fits of the 2HDM to experimental data– 7 –n flavor physics constraints have been presented in Refs. [24–26], for all four types ofYukawa interactions.For the type I model we use the most conservative bounds of Ref. [25] while for thetype II model we use the tan β independent bound m H ± >
600 GeV.
The 1-loop corrections to the gauge bosons two-point functions can be encoded by thePeskin-Takeuchi S , T and U parameters, also known as oblique parameters. The differencebetween the value of this parameters in this model and the SM is written as∆ S [I(1+2)HDM] = ∆ S A + ∆ S I , ∆ T [I(1+2)HDM] = ∆ T A + ∆ T I , ∆ U [I(1+2)HDM] = ∆ U A + ∆ U I . (4.1)where the subscript A stands for the contribution of loop effects due to the active Higgsdoublets Φ and Φ while the subscript I stands for the inert doublet η contribution, i.e.,at 1-loop the effects coming from the active and inert doublets are simply additive.The oblique parameters in the general 2HDM were calculated in [27] and in the IDMin [28]. The formulas for the I(1+2)HDM were presented in Ref. [7] and we include themin Appendix B.With U = 0 fixed, the current measured values of the S and T parameters assuming m h = 125 GeV are given by [29]∆ S = 0 . ± . , ∆ T = 0 . ± . , (4.2)with correlation coefficient 91%. For every point of parameter space we calculate the S and T parameters and perform a chi-square test taking into account the correlation coefficientand exclude all points that lie outside the 2 σ confidence level contour. The widths of the gauge Z and W bosons have been measured very precisely at LEPexperiments [30–32] . Thus in order to ensure that the decay of the gauge bosons to inertand active scalar sectors are kinematically forbidden we impose the following constraintson the scalar masses 2 m H ± > m Z , m χ ± > m Z , (4.3)for Z → H + H − , χ + χ − , m A + m H > m Z , m χ + m χ a > m Z , (4.4)for Z → HA , XX a and m H + m H ± > m W , m A + m H ± > m W , (4.5) m χ + m χ ± > m W , m χ a + m χ ± > m W , (4.6)– 8 –or W ± → HH ± , AH ± , χχ ± , χ a χ ± .We also take into account the LEPII MSSM limits applied to the IDM as derived in[33]. These results exclude the intersection of conditions m X <
80 GeV , m X a <
100 GeV , m X a − m X > . (4.7)Furthermore, LEP collaborations have searched for charged Higgs bosons [34] and havefound no significant excess relative to SM backgrounds. In this work we take the conser-vative lower bound m H ± , m X ± >
70 GeV , (4.8)as was found in Ref. [35], see also Ref. [25]. As discussed in section (2), the I(1+2)HDM is determined by 12 free parameters. Howeverwe fix the value of m h = 125 GeV such that the field h corresponds to the SM Higgsboson. We also notice that the Higgs boson couplings to vector bosons normalized to theSM value are g hV V = sin ( β − α ) and measurements of this coupling at the LHC are veryconsistent with the SM value of g SM hV V = 1 at the level of 10% and is expected to improveat the HL-LHC [36] to about 2% accuracy.In this work we focus on the case where χ is the lightest particle in the inert sector,i.e. M χ < M χ a and M χ < M χ ± . For light enough dark matter, M χ < m h / h → χχ ) = ( λ a + λ b + λ c ) π v m h sin ( β − α ) (cid:115) − M χ m h . (4.9)Constraints on the Higgs boson branching ratio to invisible final states have beenreported by the ATLAS collaboration to be BR ( h → invisible) <
28% at the 95% C.L.The most recent constraint has been reported by CMS group and is given by [37] BR ( h → invisible) < , (4.10)at the 95% C.L. We will use this bound in our numerical scan.It is well known that in the 2HDM the effective coupling of the Higgs bosons to pairsof photons receive contributions from charged scalar loops. In the I(1+2)HDM the Higgscouplings to photons receive extra contributions from loops of inert charged Higgs χ ± . Wehave implemented the effective gg and γγ couplings of the Higgs bosons in micrOMEGAs.To take into account the experimental constraints from all different Higgs signalstrengths measured at the LHC and the Tevatron we have used the Lilith library [38].In addition, exclusion limits from heavy Higgs searches are taken into account by use ofthe HiggsBounds [39] code. – 9 – .5 Relic Density The latest results from the Planck collaboration [40] give the following value for the DMrelic density Ω DM h = 0 . ± . . (4.11)In the numerical scan we allow the model to predict the dark matter under-abundanceas other field components could contribute to the relic density and impose the upperbound as an experimental constraint. The relic density Ω PlanckDM h was evaluated withthe micrOMEGAs package [17]. The annihilation into three body final state with a virtual Z and W bosons were included in the calculation of the relic density.We assume 10% theoretical uncertainty as the calculation of the relic density is per-formed at tree-level. Thus we inflated the experimental error to 10% of the central valueand use two standard deviations to set limitsΩ limitDM h = 0 . ± × . . (4.12) Experiments such as LUX [41], PANDAX-II [42] and the XENON1T [43] place constraintson the spin-independent cross section of Weakly interacting Massive Particles (WIMP) offnucleons as a function of the WIMP mass.We have used the micrOMEGAs package [17] to evaluate the spin-independent crosssection for DM scattering off a proton. To constraint the model parameters we rescale thecross section by a factor Ω DM / Ω PlanckDM that takes into account that χ represents only a partof the total DM.To calculate the spin independent (SI) amplitude micrOMEGAs calculates the effectivecoupling of the DM candidate with quarks and automatically takes into account loopcontributions from box diagrams following the model independent calculation of Ref. [44].The DM-quark amplitudes are related to the DM-nucleon amplitudes by form factors thatare stored as global parameters. We have checked that the effect of isospin violation isnegligible in this scenario in contrast to the 2HDM plus a gauge singlet scalar [6] wherethis effect was found to be negligible in the type-I model allowing the direct use of theexperimental upper limits, while it was more significant for the type-II.We note that there are two loop triangle diagrams with either h or H being exchanged.In the alignment limit sin ( β − α ) = 1 only h exchange is supported and the couplingstrength scales as λ abc ≡ λ a + λ b + λ c = 2 m χ − m η v , (4.13)therefore the DM-nucleon interaction is determined by the difference between the DMmass and the quadratic mass term of the inert doublet potential. This is the same scalingbehavior of the invisible decay width of the Higgs boson, see equation (4.9). Contrary toa naive expectation, the potential mass term m η is phenomenologically relevant as for agiven DM mass value it moderates several DM observables.In the IDM the bounds from indirect detection experiments, e.g. AMS-02 or Fermi-LAT are much weaker than the LHC and direct detection constraints above, see Refs.– 10 –45, 46]. Hence we do not consider constraints coming from direct detection experimentsin this work. Although some constraints e.g. the change in the Peskin-Takeuchi parameters at leadingorder, are a simple sum of the active and inert sector contributions, it must be emphasizedthat in general the I(1+2)HDM is not just a simple superposition of the regular IDM and2HDM.The addition of an extra active doublet to the regular IDM gives rise to notable dif-ferences from just the regular IDM or 2HDM. These differences are encoded in the quarticcouplings which parametrize the interaction between the two sectors and are given in eqs.(2.7). The most important ones in this paper are the following: • Positivity and Unitarity. At the theoretical level, the quartic couplings lead to non-trivial positivity as well as unitarity conditions which are not just the sum of condi-tions given in the IDM and 2HDM, see sections 3.1 and 3.2. • Effective coupling to photons. The Higgs coupling to a pair of photons receives loopcontributions from both active and inert charged Higgs states. Depending on thesign and size of the quartic and trilinear Higgs couplings there can be cancellationsor enhancements between the two diagrams, an effect that is not present in eitherthe IDM or 2HDM. • Invisible decays and DM annihilation. The invisible decay channel of Higgs statesas well as the decay mode into charged inert states h, H → χ + χ − arise naturally inthis scenario while they are absent in just the 2HDM. These decay modes are furthercontrolled by the mixing angles of the active sector so that g hχχ ∝ λ abc sin ( β − α )and g Hχχ ∝ λ abc cos ( β − α ) hence in the alignment limit only the SM Higgs is allowedto decay invisibly. This also affects the amplitude of DM annihilation via the Higgsmediated diagrams which is important mostly in the low mass region. The are alsoquartic couplings which are independent of the mixing angles, namely g χχhh = g χχHH = λ abc , g χχAA = λ a + λ b − λ c , (5.1)which are mainly relevant in the high mass region. • Mono object production. The cross sections for mono-jet, mono Z and mono Higgsfinal states pick up contributions from the active doublet states which are not presentin the IDM alone. Although for mono-jet and mono Z these effects turn out to benegligible for the total cross section, they can be substantial for mono Higgs finalstates which are also controlled by trilinear Higgs couplings coming from the 2HDMexclusively. – 11 –
Numerical Scan of Parameter Space
We perform a random scan of the free parameters of the model, according to the followingranges 2 ≤ tan β ≤ , (6.1)0 ≤ β − α ≤ π, (6.2)10 GeV ≤ m ≤ , with m = m H , m A , m χ , m χ a , (6.3) m Z / ≤ m ≤ , with m = m H ± , m χ ± , (6.4) − ≤ m ≤ , (6.5) − ≤ m η ≤ , (6.6)0 ≤ λ η ≤ π. (6.7)The scan is done in a succession of cuts as follows:1. Cut 1. We first random scan the parameters imposing the positivity, unitarity andB physics constraints of sections (3.1), (3.2) and (4.1) respectively. The conditions m χ < m χ a and m χ < m χ ± are implemented to make sure χ is the lightest inertscalar. We also avoid the degenerate regime where the CP even active scalar hasmass in the range m H ∈ [123 , σ confidence regionwhere EWPO are satisfied and impose LEP constraints given in sections (4.2) and(4.3) respectively.3. Cut 3. We apply the constraints coming from the Higgs boson signal strengths andheavy Higgs searches given in section (4.4). We discard all points that give DMoverabundance see equation (4.12).4. Cut 4. Finally we throw away all points in the parameter space that produce spin-independent cross section above the quoted limits by LUX and XENON1T experi-ments. The bounds from PANDAX-II experiment are less severe and thus are notimposed. Data from XENON1T was taken from [47].There is almost no correlation between the active and inert parameter sets, S and S ,that we introduced in section (2.2) thus we choose to present the results of the parameterscan in figure 1 in terms of the most relevant parameters that affect the relic density, namelythe DM mass m χ and the quadratic mass term of the inert sector m η . From the plots itis evident that as the various parameter scan cuts are applied the m χ and m η parametersare forced to become more degenerate. The reason can be seen from Eq. (4.13), where onecan see that the degeneracy is a result of λ abc not being too large.– 12 – igure 1 : Allowed points of parameter space that survived the succession of cuts. InCut 1 (upper left) we applied unitarity, positivity and B physics constraints, in Cut 2(upper right) shows points that survive after imposing EWPO and LEP constraints, inCut 3 (lower left) the LHC constraints on the Higgs boson signal strengths and heavyHiggs searches limits were imposed as well as the upper bound on DM relic abundance areapplied and finally in Cut 4 (lower right) the LUX and XENON1T experimental resultson the spin independent cross section were implemented. In the vertical axis we took thesquare root of the absolute value of inert sector mass parameter m η so that negative allowedvalues correspond to − m η in the scalar potential.During the scan we only imposed the Planck upper limit (4.12) on the relic density asthere can be other components or dark sectors that contribute to DM production. The mostsalient result from these figures is the identification of two regions where the experimentallower bound on the relic density is also satisfied. There is a low mass region with DM massin the range [57 ,
73] GeV and a high mass region with m χ in the range [500 , β − α )-tan β parameter space consistent with the parameter scan in figure 2. Superimposed and colored– 13 –reen are the points that also comply with Planck lower limits on the relic density. Onecan notice that DM constraints have a moderate impact on the parameter space of themixing angles. The upper panel plots agree with the overall shape from the latest 2HDMfit presented in Ref. [26]. Also shown is the cos ( β − α )- λ abc plane which shows that relicdensity constraints are not affected by the allowed range of cos ( β − α ) but only by theabsolute value of the portal interaction λ abc . Figure 2 : Parameter space points in the cos ( β − α )-tan β (upper) and cos ( β − α )- λ abc (lower) planes. All points survived the cut 4 of the parameter scan while the pink dots arealso compliant with Planck lower limit.When the projected upper bounds from the LZ collaboration [48] are applied to the”LDM+HDM” data set we find that the allowed parameter space can be dramaticallydiminished sending the model to a fine tuned region where | λ abc | ≤ × − (correspondingto | m η − m χ | ≤ . | λ abc | ≤ .
02 (corresponding to | m η − m χ | ≤ . m χ - λ abc parameterspace points for LDM+HDM and LDM. – 14 – igure 3 : LDM+HDM (left) and LDM (right) regions where points colored brown wouldbe excluded by LZ projected upper limits while points colored green would still be allowed.One can also observe that for the LDM region there would be a tiny window thatsurvives with 70 ≤ m χ ≤ . m χ we fixed m η such that λ abc remains fixed. We chose three benchmarkpoints: BP1 with the maximum absolute value for λ abc and BP2 and BP3 which wheredrawn from the LDM+HDM region that would not be ruled out by the projected LZbounds, corresponding to green colored points in figure 3. The rest of the free parametersremained fixed and the benchmark points are displayed in table 1. Since for each benchmarkvalue chosen we are varying the DM mass, we indicate with color green on each curve thesegment that remains consistent with all the constraints.tan β c β − α m m H m A m H ± m χa m χ ± λ η λ abc BP1 2 . − .
02 49 261 307 222 919 920 6 − . . − .
38 91 295 129 281 924 920 0 . − . . − . −
225 82 100 203 363 142 3 . − × − Table 1 : Benchmark points as drawn originally from the numerical scan which correspondto the curves of figure 4 .We took the square root of the absolute value of active sectormass parameter m so that a negative value corresponds to − m in the scalar potential.All mass parameters are in GeV units. – 15 – igure 4 : DM spin independent scattering cross section with nucleons as function of m χ for benchmark points of table 1. The upper limits from XENON1T (dashed) and projectedfrom LZ (dashed-dotted) experiments are shown for comparison. For each curve, the greensegments are allowed by all theoretical and experimental constraints. We devote this section to studying mono object signals plus missing transverse energy andthe prospects for the LHC to probe them. Of particular interest are processes with a jet,a Z boson or a Higgs boson in the final state. For the rest of this section, unless otherwisenoted, whenever we refer to the allowed parameters in our scan we imply all points thatfall into the whole LDM + HDM region where relic density is fully accounted by the inertsector of this model. The study of signals at lepton colliders in this model are left for futurework. In the IDM this has been investigated in Refs. [49–51].In the calculation of the production rates we implemented the following configurationsand cuts:1 For the proton initial states we used the PDF NNPDF23 lo as 0130 qed [52, 53] thatis implemented within CalcHEP.2 The QCD renormalization and factorization scales were set equal to the missingtransverse momentum of the final states for all processes.3 A minimum cut of 100 GeV is placed on the missing transverse momentum for allprocesses.4 For all processes, the proton and jets have been defined as composite states of a gluonand all quarks except the top.The calculation of the cross sections is performed by CalcHEP at leading order inperturbation theory. In this work, we are not including higher order corrections. In adetailed study of the inert doublet model [54], these corrections were included, includinga discussion of the choice of renormalization and factorization scales and the K-factorsfor the production cross sections. This model, however, has many more parameters thanthe simple inert doublet model, some of which (such as heavy scalar masses) will not be– 16 –easured in the near future, and thus it would seem premature to include these effects.Including them would not have a substantial qualitative effect on our results.What are the experimental bounds and prospects? A very detailed analysis is beyondthe scope of this paper, and would also depend on several extra parameters. However onecan get a rough idea from the IDM analyses in recent works of Refs. [54] and [55]. Thelatter paper gives approximate bounds from Run 2 (their Figure 13) and show that theupper bounds are one to two orders of magnitude above the expected value in the IDM.Our results are the same order of magnitude for light m χ (which is the expected regionif χ is the dark matter). After 3000 fb − at the HL-LHC, the expected reach becomescomparable to the IDM results for low masses, and the same is true in our case. The sameis true for the mono Z signature in the next section. Of course, should an additional activedoublet be detected, a much more detailed analysis would be in order. In addition to Ref. [54], mono jet signals have been studied in the context of the inerttwo Higgs doublet model, see e.g. Refs. [1, 46, 55], see also Ref. [56] for related studiesin other Higgs portal scenarios. In a similar way, the I(1+2)HDM predicts two differentkinds of jet plus missing transverse energy final states. The two possibilities are pp → jχχ and pp → jχχ a respectively.The process pp → jχχ is determined by the couplings of the Higgs eigenstates tofermions which control the production cross section by gluon fusion. The most importantcontribution comes from top quark loops and the couplings to quarks scale as g htt , g Htt ∝ / tan β for both type-I and -II models.The other parameter affecting the mono-jet production is given by the interaction ofthe Higgs eigenstates with the DM particle which goes as λ abc sin ( β − α ) for h and as λ abc cos ( β − α ) for H where λ abc ≡ λ a + λ b + λ c was defined in eq. (4.13). Therefore inthe alignment limit only the SM Higgs will mediate this process.Thus we expect that the biggest effects will come from the maximum absolute valueallowed for the Higgs DM coupling and for small values of tan β . We have chosen threebenchmark points (see table 2) with maximum absolute value of λ abc ≈ . β − α ). Cross sections for the process pp → jχχ are presented infigure 5 below. For BP4 we chose the minimum value of cos ( β − α ) for comparison. Sincein type-II model the deviations from alignment limit are more stringent, we present crosssections for the type-I only. – 17 –an β c β − α m m H m A m H ± m χa m χ ± λ η λ abc BP4 4 .
26 6 × − −
80 108 407 88 894 895 3 .
48 0 . .
65 0 . . . . − .
34 86 256 67 238 818 815 5 . − . Table 2 : Benchmark points as drawn originally from the numerical scan which correspondto the curves of figure 5 .We took the square root of the absolute value of active sectormass parameter m so that a negative value corresponds to − m in the scalar potential.All mass parameters are in GeV units. Figure 5 : Cross sections for mono jet production jχχ as a function of DM mass for thebenchmark points presented in table 2. The mass squared parameter m η was also variedso that the portal coupling λ abc remained fixed as displayed on the table.The other possibility for monojet final state is given by the process pp → jχχ a . Thisis even simpler to describe as the only relevant parameters are the DM masses m χ and m χa . A Z boson is mediated in this process and the Zχχ a vertex is fixed by electroweakparameters. The effect is more significant for small mass separation. This effect can beappreciated in figure 6 where we plot the cross section as a function of m χ for differentmass separations. Figure 6 : Cross sections for the process pp → jχχ a for different mass separations.– 18 – .2 Mono Z Another interesting signal that arises within this model is the mono Z final state. Therelevant diagrams for this process are displayed in figure 7. There are Higgs mediateddiagrams which are determined by the λ abc coupling and are displayed in the first row offigure 7. Figure 7 : Feynman diagrams contributing to the mono Z process.The interaction between the active heavy states and a Z boson scales as g AHZ ∝ sin ( β − α ) while the heavy Higgs coupling to DM is proportional to cos ( β − α ), thereforeaway from the alignment limit the pseudoscalar and heavy Higgs contribute to the crosssection as shown in the third diagram of the first row. This effect is however insignificantand the total cross section is dominated by the diagrams of the second row. This typeof diagrams (second row) do not involve Higgs bosons and are completely determined bygauge interactions with m χa as the only relevant parameter. In figure 8 we show the crosssections as a function of DM mass with other parameters fixed to the values correspondingto BP5 which has the highest deviation from alignment limit and maximum value of λ abc .– 19 – igure 8 : Cross sections for the mono-Z object final state for BP5 of table 2. Theparameter m χ a was varied according to different mass separations as shown in the legends. There is also the possibility of mono Higgs final states. Similar to the mono-jet productionthere are two possible final states, namely hχχ or hχχ a . Example diagrams for the formerare displayed in figure 9. In this case the possibilities are more diverse having diagrams thatscale with g hhXX = λ abc as shown on the first diagram of the first row and also diagramsthat scale with the trilinear Higgs couplings g hhh , g hhH and g hHH as manifested in thesecond diagram of the first row. The Higgs trilinear couplings are given by g hhh = − vc β s β (cid:2) − m c β − α c β + α + 2 m h ( c β − α + 3 c β + α ) s β (cid:3) , (7.1) g hhH = c β − α vs β c β (cid:20) m (cid:18) − c α s α c β s β (cid:19) − (cid:0) m h + m H (cid:1) s α (cid:21) , (7.2) g hHH = 8 s β − α vs β (cid:20) − m + s α (cid:18) m h + 2 m H − m s β (cid:19)(cid:21) , (7.3)where we used the shorthand notation c α ≡ cos α etc. We thus see from these expressionsthat m and m H determine the strength of the trilinear interactions for fixed α and β .From the parameter scan we chose two benchmarks where this values are maximal andgiven by g hhh = − . v , g hhH = 0 . v and g hHH = 3 . v for BP7 and g hhh = − . v , g hhH = 1 . v and g hHH = − . v for BP8. These benchmark points are presented in table3. – 20 – igure 9 : Feynman diagrams contributing to the mono-Higgs production via the process pp → χχh . tan β c β − α m m H m A m H ± m χa m χ ± λ η λ abc BP7 4 . − .
36 20 88 224 255 847 842 8 . − . . − .
20 102 211 386 399 733 728 7 . − . Table 3 : Benchmark points as drawn originally from the numerical scan which correspondto the curves of figure 10 .We took the square root of the absolute value of active sectormass parameter m so that a negative value corresponds to − m in the scalar potential.All mass parameters are in GeV units.The results for the production cross sections are presented in figure 10 where forcomparison we show the cross sections for the benchmarks BP4, BP5 and BP6. It can beseen that BP5, which corresponds to cos ( β − α ) = 0 . | λ abc | and is enhanced in the low mass regionup to about the Higgs threshold when it starts to decrease. Figure 10 : Cross sections as a function of m χ for mono-Higgs production via the χχh final state. The benchmark points are presented in tables 2 and 3.– 21 –inally, the Feynman diagrams for the process pp → hχχ a are shown in figure 11. Thefirst two diagrams are mediated by the pseudoscalar and only contribute when cos ( β − α ) (cid:54) =0. The biggest cross sections correspond to small mass separation between m χ a and m χ .We present plots for the cross sections in figure 12. We can see that for cos ( β − α ) (cid:54) = 0BP5 and BP6 yield cross sections reduced relative to the alignment limit corresponding toBP4. Figure 11 : Feynman diagrams contributing to the mono-Higgs production via the process pp → χχ a h . Figure 12 : Cross sections for χχ a h final state as a function of DM mass for benchmarkpoints BP4, BP5 and BP6. The mass parameter m χ a was varied such that the massdifference between χ a and χ was kept fixed as shown in the legends.– 22 – .4 Discussion From the cross sections presented above we can conclude that the strongest effect comesfrom the mono-jet final state jχχ with cross sections of about O (1) pb for a DM mass inthe range 50 ≤ m χ ≤
70 GeV.To test these predictions against the LHC data we used the CheckMATE 2 [57–63]software package. We implemented a grid search for different benchmark points in theplane of λ abc and m χ in the range 1 ≤ λ abc ≤ ≤ m χ ≤
100 GeV. For each point inthe grid we generated partonic events using CalcHEP [64] with the same cuts and scalesas specified at the beginning of this section.Parton showering and hadronization using Pythia 8 [63] was performed within Check-MATE 2. The program then performs a fast detector simulation using DELPHES 3 [58].We found all points in the grid to be allowed at 95% CL with the most sensitive analysisgiven in Ref. [65] which corresponds to an ATLAS search for DM using 36 . f b − of dataat 13 TeV center of mass energy.This implies that, using mono-jet final states, current LHC data or at least the analysescurrently implemented within CheckMATE 2 cannot rule out the I(1+2)HDM even formaximal values of λ abc allowed by theoretical and experimental constraints. We checkedthat the LDM and HDM regions are trivially allowed as they lead to very small crosssections either due to very small λ abc in LDM or very high mass in HDM. This effect canbe better appreciated in figure 3 (left) where we show the region in the λ abc , m χ plane. Onthe right of that figure we show the zoomed LDM region where the points colored pink areexcluded by the projected bounds from LZ collaboration.Our results agree with those of the IDM, see Ref. [55]. In that reference the authorsobtained upper limits for current and projected luminosities by doing a shape analysis of themissing transverse momentum distribution. Only by combining the final states jχχ + jχχ a for small mass separation m χ a = m χ + 1 GeV, they found that DM masses very close tothe Higgs threshold m h / − of integrated luminosity. Furtherwith 3000 fb − all the region with m χ < m h / In this section we explore the consequences of slightly relaxing the assumption of darkdemocracy of quartic couplings by taking λ (cid:54) = λ and calling λ c ≡ λ , (8.1) λ d ≡ λ − λ c , (8.2)with λ a and λ b as defined in (2.7). In this case, once again we can solve for the quarticcouplings in favor of the scalar mass parameters and one obtains λ a = 2( m χ ± − m η ) v , (8.3)– 23 – b = m χ a − m χ ± + m χ v , (8.4) λ c = m χ − m χ a v − λ d cos β, (8.5)with λ a and λ b given by the same expressions as before. For this scenario the active set ofparameters is modified to include the extra quartic coupling λ d S (cid:48) = (cid:8) m χ , m χ a , m χ ± , m η , λ η , λ d (cid:9) , (8.6)in this case the pseudoscalar would have a non vanishing coupling to pairs of inert statesas g Aχχ a = λ d v cos β sin β. (8.7)As a byproduct of this non-vanishing coupling, the process pp → j χ χ a will also bemediated by a virtual pseudoscalar A that decays into χ χ a modifying the cross sections thatwe calculated for the Dark democracy case in the preceding section. The extra diagramswith a pseudoscalar mediator thus depend on m A , tan β and the new quartic λ d . We presentthe cross sections in figure 13 for λ d = 0 . λ d = 1 with tan β = 2 and m A = 200 GeVfixed. We notice that there is a significant enhancement relative to the democratic case ofabout one order of magnitude if λ d = 1. Figure 13 : Cross sections for the process pp → j χχ a as a function of DM mass fortan β = 2 and m A = 200 GeV. The mass parameter m χ a was varied such that the massdifference was fixed to m χ a − m χ = 1 GeV.As noted in Section 7.1, the current experimental limits on the process are approxi-mately one or two orders of magnitude above the expected value in this model, so a largevalue of λ d will bring them closer. Certainly, the HL-LHC will cover a substantial part ofthe region. Constraints on the parameter space of 2HDMs with softly broken Z symmetry due toexperimental heavy Higgs searches at the LHC have been studied before, see e.g. Ref. [24].– 24 –ere, we present predictions for production cross sections times branching fractions usinga benchmark value of the parameters and we will compare with experimental searches toindicate if the model can be probed or not.We focus on the decay mode H → hh for concreteness. The production cross sectiondepends on the heavy Higgs couplings to quarks and on the trilinear coupling g hhH givenin eq. (7.2). From the parameter scan we chose a benchmark value where the decay widthhas its maximum value. This benchmark is presented in table 4 below.tan β c β − α m m H m A m H ± m η m χ m χa m χ ± λ η BP9 3 .
35 0 .
51 512 693 603 488 366 372 379 375 5 . Table 4 : Benchmark point as drawn originally from the numerical scan which correspondto the curve of figure 14 . All mass parameters are in GeV.
Figure 14 : Production cross section times branching ratios for the process pp → H → hh ( bbbb ) as a function of m H for the benchmark BP9 given in table 4. We also show the95% CL observed upper limit from Ref. [66].The magnitude of the observed experimental upper bounds depends on the decay modeof the final state Higgs bosons of the respective search analysis. In figure 11 of Ref. [67]upper bounds for the production cross section times branching fraction are presented cor-responding to different final states of each SM Higgs boson. One can see that the strongestconstraints are given by searches focused on the bbbb final state. As no signal excess wasobserved in the experiment we plot the 95 % confidence level upper limit (observed) on thecross section in figure 14 together with the cross section for benchmark BP9. We noticethat heavy Higgs searches on this decay mode can not test the cross section.One clear distinction of the I(1+2)HDM is the possibility of heavy Higgs decays intoinert scalars, a possibility that exists in neither the usual 2HDM nor the usual IDM. Thisis an invisible decay that would have the effect of suppressing the branching ratio to otherstates. Note that many effects of heavy Higgs bosons are included in the previous section,including the effects of invisible decays in mono-jet, mono-Z and mono-Higgs searches. Wenow focus on the suppression of the branching ratios of heavy scalars.– 25 –ince the invisible decay of the H depends on the Hχχ vertex, which is λ abc v cos( β − α ),and since the HZZ vertex also depends on cos( β − α ), the ratio of the two decays will beindependent of the mixing angles. For typical values of λ abc given in table 2, the ratio ofthe decays to ZZ vs. into χχ varies from roughly 0 . τ + τ − may become critical (the decays into ¯ bb and ¯ cc are much more difficult tosee due to backgrounds). The ratio of H → χχ to H → τ τ is (neglecting phase space)Γ( H → χχ )Γ( H → τ τ ) = λ abc v m H m τ cos ( β − α ) sin β sin α (9.1)For typical values of λ abc in Table 2, and for an H mass of 200 GeV, this is approximately3 cos ( β − α ) sin β/ sin α . This could be reduced by phase space, of course. In thealignment limit, this vanishes, but if one is at the larger values of cos( β − α ) allowed intype I models, it can be substantial. Note that one could make the ratio dominate in thelimit of α very small, but that is a region of parameter-space in which the H → τ τ decayis unmeasurable. Nonetheless, this invisible decay could suppress a promising signature ofthe H .For the pseudoscalar, A , the A → τ τ signature is more promising, even for largermasses, since the decay into W ’s and Z ’s is absent. However, as noted in the last section,the decay A → χχ a vanishes in the ”dark democracy” limit, thus invisible decays wouldbe absent. If one does not adopt dark democracy, then there will be a signal. Defining λ d as in the last section, the ratio of invisible to tau decays isΓ( A → χχ a )Γ( A → τ τ ) = λ d sin β m τ /v ) (9.2)times the usual phase space factors. Unless one is very close to the dark democracy limit,this expression will be quite large, and if kinematically allowed, the invisible decay of the A will dominate, swamping the visible decay.If the χ is the dark matter particle, then χ a will be heavier, and the decay is onlykinematically allowed for m A greater than about 145 GeV. Above a mass of 210 GeV, thedecay into h + Z can be important (away from the alignment limit), and thus the rangeof masses in which the invisible decay is dominant may be small. Nonetheless, a study ofinvisible decays in this case could be interesting.
10 Conclusions
In this paper we have investigated the constraints on the parameter space of the I(1+2)HDM.A random scan of the free parameters was performed taking into account positivity of thepotential and unitarity of the quartic couplings as theoretical constraints. Experimentalconstraints such as B physics, EWPO, LEP bounds on gauge bosons decays, LHC data onthe SM Higgs boson, heavy Higgs searches, upper limit of the DM relic density and currentdirect detection constraints on DM-nucleon scattering have been imposed.– 26 –wo regions that satisfy also the lower limit on the relic density and thus are nonunder-abundant have been identified. There is a low mass region we called LDM with m χ ∈ [57 ,
73] GeV and a high mass region HDM with m χ ∈ [500 , m χ , the Higgscoupling to DM is controlled by the inert sector mass squared parameter as shown inequation 4.13.The projected sensitivity of the LZ experiment has the potential to exclude a significantamount of parameter space leaving a maximum separation of | m η − m χ | ≤ . | m η − m χ | ≤ .
25 GeV in HDM. For the LDM there is a tiny window that would survivewith 70 ≤ m χ ≤
73 GeV and quartic coupling values of about λ abc ∼ O (10 − ). Of course,in any parameter scan, one can miss “funnel” regions where different processes cancel; wehave not considered the possibility of this fine-tuning.The maximum deviation from the alignment limit in LDM and HDM regions wasfound to be in agreement with the overall shape of the parameter space of mixing angles oftype-I and type-II models [26]. This demonstrates that relic density constraints are onlydependent on the quartic λ as is evident from figure 2.Predictions of the model for mono-jet, mono Z and mono Higgs final states have beenstudied. For well motivated benchmarks it has been shown that the most competitivesignal is given by pp → jχχ with cross sections of about O (1) pb for DM mass in therange 50 < m χ <
70 GeV and a Higgs DM interaction of about λ abc ≈ .
2. The model hasbeen tested using CheckMATE 2 and we found that it is allowed at 95 % CL by the LHCanalyses implemented in this package.We then considered the “dark democracy” assumption common in IDM studies. Thisassumption is often made to simplify the parameter-space. However in this model (unlikeeither the IDM or 2HDM models), the process A → χχ a can occur once the dark democracyassumption is lifted. This decay would be a convincing signature of the model, and canlead to a significant enhancement of the mono-jet cross section.We have shown that searches for DM at the LHC in final states with a jet offer adifficult way to test this model however future direct detection experiments will be able tochallenge this scenario as a model that can account for all the DM in the universe. As isthe case in the IDM, the current LHC bounds on mono-particle processes are not sufficientto test the model, but the HL-LHC will, after 3000 fb − , be able to probe a substantialpart of the parameter space. Acknowledgments
After the completion of this work we became aware that the I(1+2)HDM was being studiedin the context of the minimal extended seesaw framework in Ref. [68]. We thank Najimud-din Khan for pointing us to this reference. MM thanks Alexander Belyaev and Igor Ivanovfor helpful correspondence. The authors would also like to thank Eloy Romero Alcalde forhelping with the installation of CheckMATE 2. This work was supported by the NationalScience Foundation under Grant PHY-1819575.– 27 – ppendix A 2HDM parameters
The minimization conditions on the potential V allows us to solve for the quadratic massterms as m = v ( λ cos β + λ sin β ) − m tan β, (A.1) m = − m cot β + v ( λ cos β + λ sin β ) , (A.2)where we define λ ≡ λ + λ + λ . (A.3)The mass squared matrix for the CP-odd and charged Higgs bosons are given by M = (cid:32) / m − v λ sin 2 β ) tan β − m / v λ cos β sin β − m / v λ cos β sin β / β ( m − v λ sin 2 β ) (cid:33) , (A.4) M = (cid:32) − v ( λ + λ ) sin β + m tan β − m + v ( λ + λ ) cos β sin β − m + v ( λ + λ ) cos β sin β − v ( λ + λ ) cos β + m cot β (cid:33) , (A.5)respectively. Diagonalization of this matrices, by eq. 2.8, yield a zero eigenvalue corre-sponding to the Goldstone bosons that gets eaten to become the Z and W bosons longitu-dinal polarizations. We choose to solve for the quartic couplings in favor of the pseudoscalarand charged scalar masses squared m A , m H ± as λ = − m A + m csc 2 βv , (A.6) λ = m A − m H ± + m csc 2 βv . (A.7)The CP-even mass squared matrix is given by M = (cid:32) v λ cos β + m tan β/ − m / v λ cos β sin β − m / v λ cos β sin β m cot β/ v λ sin β (cid:33) , (A.8)where the angle that diagonalizes this matrix is given by the formulatan 2 α = 2 M , M − M . (A.9)We use the condition R ( − α ) M R ( α ) = (cid:32) m H m h (cid:33) , (A.10)where R ( α ) is the rotation matrix of equation (2.9) to solve to solve for λ and λ and theoff diagonal element to solve for λ . They are given by λ = m H cos α + m h sin α − / m tan βv cos β , (A.11) λ = m H sin α + m h cos α − / m cot βv sin β , (A.12) λ = m − ( m h − m H ) sin 2 α v cos β sin β . (A.13)– 28 – ppendix B EWPO Formulas ∆ S A = 14 π (cid:40) s β − α F (cid:48) ( m Z ; m H , m A ) − F (cid:48) ( m Z ; m H ± , m H ± )+ c β − α (cid:104) F (cid:48) ( m Z ; m h , m A ) + F (cid:48) ( m Z ; m H , m Z ) − F (cid:48) ( m Z ; m h , m Z ) (cid:105) + 4 m Z c β − α (cid:104) G (cid:48) ( m Z ; m H , m Z ) − G (cid:48) ( m Z ; m h , m Z ) (cid:105)(cid:41) , (B.1)∆ T A = 116 π α em v (cid:40) F (0; m H ± , m A ) + s β − α [ F (0; m H ± , m H ) − F (0; m A , m H )]+ c β − α (cid:104) F (0; m H ± , m h ) + F (0; m H , m W ) + F (0; m h , m Z ) − F (0; m h , m W ) − F (0; m A , m h ) − F (0; m H , m Z )+ 4 G (0; m H , m W ) + 4 G (0; m h , m Z ) − G (0; m h , m W ) − G (0; m H , m Z ) (cid:105)(cid:41) , (B.2)∆ U A = 14 π (cid:40) F (cid:48) ( m W ; m H ± , m A ) − F (cid:48) ( m Z ; m H ± , m H ± )+ s β − α [ F (cid:48) ( m W ; m H ± , m H ) − F (cid:48) ( m Z ; m A , m H )]+ c β − α (cid:104) F (cid:48) ( m W ; m H ± , m h ) + F (cid:48) ( m W ; m W , m H ) − F (cid:48) ( m W ; m W , m h ) (cid:105) − c β − α (cid:104) F (cid:48) ( m Z ; m A , m h ) + F (cid:48) ( m Z ; m Z , m H ) − F (cid:48) ( m Z ; m Z , m h ) (cid:105) + 4 m W c β − α (cid:104) G (cid:48) ( m W ; m H , m W ) − G (cid:48) ( m W ; m h , m W ) (cid:105) − m Z c β − α (cid:104) G (cid:48) ( m Z ; m H , m Z ) − G (cid:48) ( m Z ; m h , m Z ) (cid:105)(cid:41) , (B.3)∆ S I = 14 π (cid:104) F (cid:48) ( m Z ; m η H , m η A ) − F (cid:48) ( m Z ; m η ± , m η ± ) (cid:105) , (B.4)∆ T I = 116 π α em v (cid:104) F (0; m η ± , m η A ) + F (0; m η ± , m η H ) − F (0; m η A , m η H ) (cid:105) , (B.5)∆ U I = 14 π (cid:104) F (cid:48) ( m W ; m η ± , m η H ) + F (cid:48) ( m W ; m η ± , m η A ) − F (cid:48) ( m Z ; m η ± , m η ± ) − F (cid:48) ( m Z ; m η H , m η A ) (cid:105) , (B.6)where F (cid:48) ( m V ; m , m ) = [ F ( m V ; m , m ) − F (0; m , m )] /m V and G (cid:48) ( m V ; m , m ) =– 29 – G ( m V ; m , m ) − G (0; m , m )] /m V . 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