4.1 Lighter pseudoscalar a

# Simplified dark matter models with two Higgs doublets: I. Pseudoscalar mediators

## Abstract

We study a new class of renormalisable simplified models for dark matter searches at the LHC that are based on two Higgs doublet models with an additional pseudoscalar mediator. In contrast to the spin-0 simplified models employed in analyses of Run I data these models are self-consistent, unitary and bounds from Higgs physics typically pose no constraints. Predictions for various missing transverse energy () searches are discussed and the reach of the 13 TeV LHC is explored. It is found that the proposed models provide a rich spectrum of complementary observables that lead to non-trivial constraints. We emphasise in this context the sensitivity of the , mono- and mono-Higgs channels, which yield stronger limits than mono-jet searches in large parts of the parameter space. Constraints from spin-0 resonance searches, electroweak precision measurements and flavour observables are also derived and shown to provide further important handles to constraint and to test the considered dark matter models.

1]Martin Bauer, 2,3]Ulrich Haisch 4]and Felix Kahlhoefer \affiliation[1]Institut für Theoretische Physik, Universität Heidelberg,
Philosophenweg 16, 69120 Heidelberg, Germany \affiliation[2]Rudolf Peierls Centre for Theoretical Physics, University of Oxford,
OX1 3NP Oxford, United Kingdom \affiliation[3]CERN, Theoretical Physics Department,
CH-1211 Geneva 23, Switzerland \affiliation[4]DESY, Notkestraße 85, D-22607 Hamburg, Germany \emailAddbauer@thphys.uni-heidelberg.de, ulrich.haisch@physics.ox.ac.uk, felix.kahlhoefer@desy.de \preprintCERN-TH-2017-011, DESY-17-010

## 1 Introduction

Simplified models of dark matter (DM) and a single mediator overcome many of the shortcomings of DM effective field theories, but remain general enough to represent a large class of popular theories of DM (see the reviews [1, 2, 3] for a complete list of references). In particular, including contributions from on-shell production of the mediators allows to capture the full kinematics of DM production at colliders, making meaningful comparisons with bounds from direct and indirect detection experiments possible.

In simplified DM models the interactions between the mediators and the Standard Model (SM) fermions are usually written as gauge or Yukawa couplings of mass dimension four. In many cases these interactions are however only apparently renormalisable, because in a full gauge-invariant theory they in fact arise from higher-dimensional operators or they signal the presence of additional particles or couplings that are needed to restore gauge invariance [4, 5, 6, 7, 8, 9]. These features can lead to parameter regions which are theoretically inaccessible or to misleading/unphysical predictions often related to unitarity violation. Models in which the mediators mix with the SM bosons avoid such inconsistencies. The existing LEP and LHC measurements of the -boson and Higgs-boson properties however severely restrict the corresponding mixing angles, and as a result classic searches like mono-jets are typically not the leading collider constraints in this class of simplified DM models [6, 10, 11].

In this article, we study a new class of simplified DM models for spin-0 mediators based on two Higgs doublet models (THDMs), which are an essential ingredient of many well-motivated theories beyond the SM. In contrast to inert THDMs, where the DM particle is the lightest neutral component of the second Higgs doublet and is stabilised by an ad-hoc  symmetry [12, 13, 14, 15], our focus is on the case where the DM candidate is a SM singlet fermion. To couple the DM particle to the SM, we introduce a new spin-0 mediator, which mixes dominantly with the scalar or pseudoscalar partners of the SM Higgs. In this way constraints from Higgs signal strength measurements [16] can be satisfied and one obtains a framework in which all operators are gauge invariant and renormalisable.

In what follows we will explore the phenomenology of pseudoscalar mediators, while scalar portals will be discussed in detail in an accompanying paper [17] (see also [18]). Pseudoscalar mediators have the obvious advantage of avoiding constraints from DM direct-detection experiments, so that the observed DM relic abundance can be reproduced in large regions of parameter space and LHC searches are particularly relevant to test these models. Similar investigations of THDM plus pseudoscalar simplified DM models have been presented in [19, 20, 21]. Whenever indicated we will highlight the similarities and differences between these and our work.

The mono- phenomenology of the considered simplified pseudoscalar models turns out to be surprisingly rich. We examine the constraints from searches for  [22, 23],  [24, 25, 26, 27],  [28, 29, 30],  [31, 32, 33, 34] and  [35, 36] and present projections for the 13 TeV LHC. In particular, we provide benchmark scenarios that are consistent with bounds from electroweak (EW) precision, flavour and Higgs observables including invisible decays [37, 38]. For the simplified pseudoscalar model recommended by the ATLAS/CMS DM Forum (DMF) [3] constraints from mono-jet searches dominate throughout the parameter space [39], whereas for the model considered here , mono- and mono-Higgs searches yield competitive bounds and often provide the leading constraints. See Figure 1 for an illustration of the various processes that are of most interest in our simplified model. This complementarity of different searches is the result of the consistent treatment of the scalar sector, inducing gauge and trilinear scalar couplings of the mediator beyond the ones present in the DMF pseudoscalar model.

It is particularly appealing that the and signatures are strongest in the theoretically best motivated region of parameter space, where the couplings of the light Higgs are SM-like. In this region of parameter space, couplings of the new scalar states to SM gauge bosons are strongly suppressed and play no role in the phenomenology, leading to gluon-fusion dominated production and a very predictive pattern of branching ratios. In consequence, a complementary search strategy can be advised, with the exciting possibility to observe DM simultaneously in a number of different channels, some of which are not limited by systematic errors and can be improved by statistics even beyond  of luminosity. The importance of di-top resonance searches [40, 41] to probe neutral spin-0 states with masses above the threshold is also stressed, and it is pointed out that for model realisations with a light scalar partner of the SM Higgs, di-tau resonance searches should provide relevant constraints in the near future. We finally comment on the impact of bottom-quark ( initiated production.

This paper is structured as follows. In Section 2 we describe the class of simplified DM models that we will study throughout our work, while Section 3 contains a comprehensive review of the non- constraints that have to be satisfied in order to make a given model realisation phenomenologically viable. The partial decay widths and the branching ratios of the spin-0 particles arising in the considered simplified DM models are studied in Section 4. The most important features of the resulting phenomenology are described in Section 5. In Section 6 we finally present the numerical results of our analyses providing summary plots of the mono- constraints for several benchmark scenarios. The result-oriented reader might want to skip directly to this section. Our conclusions and a brief outlook are given in Section 7.

## 2 THDM plus pseudoscalar extensions

In this section we describe the structure of the simplified DM model with a pseudoscalar mediator. We start with the scalar potential and then consider the Yukawa sector. In both cases we will point out which are the new parameters corresponding to the interactions in question.

### 2.1 Scalar potential

The tree-level THDM scalar potential that we will consider throughout this paper is given by the following expression (see for example [42, 43])

 Missing or unrecognized delimiter for \big (1)

Here we have imposed a symmetry under which and to suppress flavour-changing neutral currents (FCNCs), but allowed for this discrete symmetry to be softly broken by the term The vacuum expectation values (VEVs) of the Higgs doublets are given by with and we define . To avoid possible issues with electric dipole moments, we assume that the mass-squared terms , the quartic couplings  and the VEVs are all real and as a result the scalar potential as given in (1) is CP conserving. The three physical neutral Higgses that emerge from are in such a case both mass and CP eigenstates.

The most economic way to couple fermionic DM to the SM through pseudoscalar exchange is by mixing a CP-odd mediator with the CP-odd Higgs that arises from (1). This can be achieved by considering the following interaction terms

 VP=12m2PP2+P(ibPH†1H2+h.c.)+P2(λP1H†1H1+λP2H†2H2), (2)

where and are parameters with dimensions of mass. We assume that does not break CP and thus take to be real in the following. In this case does not develop a VEV and remains a pure CP eigenstate. Nevertheless, this term does lead to a soft breaking of the symmetry. Notice that compared to [19, 20, 21] which include only the trilinear portal coupling , we also allow for quartic portal interactions proportional to  and . A quartic self-coupling of the form has instead not been included in (2), as it does not lead to any relevant effect in the observables studied in our paper.

The interactions in the scalar potential (1) mix the neutral CP-even weak eigenstates and we denote the corresponding mixing angle by . The portal coupling appearing in (2) instead mixes the two neutral CP-odd weak eigenstates with representing the associated mixing angle. The resulting CP-even mass eigenstates will be denoted by and , while in the CP-odd sector the states will be called and , where denotes the extra degree of freedom not present in THDMs. The scalar spectrum also contains two charged mass eigenstates of identical mass.

Diagonalising the mass-squared matrices of the scalar states leads to relations between the fundamental parameters entering and . These relations allow to trade the parameters , , , , , , , , for sines and cosines of mixing angles, VEVs and the masses of the physical Higgses. This procedure ensures in addition that the scalar potential is positive definite and that the vacuum solution is an absolute minimum. In the broken EW phase the physics of (1) and (2) is hence fully captured by the angles , , , the EW VEV , the quartic couplings , , and the masses , , , , . We will use these parameters as input in our analysis.

### 2.2 Yukawa sector

The couplings between the scalars and the SM fermions are restricted by the stringent experimental limits on flavour observables. A necessary and sufficient condition to avoid FCNCs associated to neutral Higgs tree-level exchange is that not more than one of the Higgs doublets couples to fermions of a given charge [44, 45]. This so-called natural flavour conservation hypothesis is automatically enforced by the aforementioned symmetry acting on the doublets, if the right-handed fermion singlets transform accordingly. The Yukawa couplings are explicitly given by

 LY=−∑i=1,2(¯QYiu~HiuR+¯QYidHidR+¯LYiℓHiℓR+h.c.). (3)

Here are Yukawa matrices acting on the three fermion generations and we have suppressed flavour indices, and are left-handed quark and lepton doublets, while , and are right-handed up-type quark, down-type quark and charged lepton singlets, respectively. Finally, with denoting the two-dimensional antisymmetric tensor. The natural flavour conservation hypothesis can be satisfied by four discrete assignments, where by convention up-type quarks are always taken to couple to :

 Y1u=Y1d=Y1ℓ=0,(type I),Y1u=Y2d=Y2ℓ=0,(% type II),Y1u=Y1d=Y2ℓ=0,(% type III),Y1u=Y2d=Y1ℓ=0,(% type IV). (4)

The dependence of our results on the choice of the Yukawa sector will be discussed in some detail in the next section.

Taking DM to be a Dirac fermion a separate symmetry under which can be used to forbid a coupling of the form At the level of renormalisable operators this leaves

 Missing dimension or its units for \hskip (5)

as the only possibility to couple the pseudoscalar mediator to DM. In order to not violate CP we require the dark sector Yukawa coupling to be real. The parameter and the DM mass are further input parameters in our analysis.

## 3 Anatomy of the parameter space

In this section we examine the anatomy of the parameter space of the model introduced above and discuss a number of important simplifications. We briefly explain the alignment/decoupling limit and describe the dependence of the predictions on the choice of Yukawa sector. The constraints on the mixing angles, quartic couplings and Higgs masses from spin-0 resonance searches, flavour physics, EW precision measurements, perturbativity and unitarity are also elucidated.

### 3.1 Alignment/decoupling limit

After EW symmetry breaking the kinetic terms of the Higgs fields lead to interactions between the CP-even mass eigenstates and the massive EW gauge bosons. These interactions take the form

 L⊃(sin(β−α)h+cos(β−α)H)(2M2WvW+μW−μ+M2ZvZμZμ). (6)

In order to simplify the further analysis, we concentrate on the well-motivated alignment/decoupling limit of the THDM where . In this case meaning that the field  has SM-like EW gauge boson couplings. It can therefore be identified with the boson of mass  discovered at the LHC and the constraints from the Run I combination of the ATLAS and CMS measurements of the Higgs boson production and decay rates to SM final states [16] are readily fulfilled. Notice that in the alignment/decoupling limit the scalar  does not interact with -boson or -boson pairs at tree level because in this limit one has .

### 3.2 Yukawa assignments

Working in the alignment/decoupling limit the fermion-scalar interactions most relevant for the further discussion are given by

 Missing dimension or its units for \hskip (7)

where denote the SM Yukawa couplings and are the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. The couplings encode the dependence on the choice of Yukawa sector (4). In terms of one has

 ξIt=ξIb=ξIτ=−cotβ,(type I),ξIIt=−cotβ,ξIIb=ξIIτ=tanβ,(type II),ξIIIt=ξIIIb=−cotβ,ξIIIτ=tanβ,(type III),ξIVt=ξIVτ=−cotβ,ξIVb=tanβ,(type IV). (8)

Since the production of the pseudoscalar mediator as well as is driven by top-quark loops that enter the gluon-fusion () channel at the LHC (see for instance [46] for a discussion in the context of searches) we will in the following focus on the region of small . In this limit the couplings of to down-type quarks and charged leptons in (7) are strongly Yukawa suppressed irrespectively of the chosen Yukawa assignment (8). As a result existing bounds on the neutral scalar masses from flavour observables such as that are known to receive enhanced corrections [49] are within experimental limits [50] even for a light scalar spectrum.

### 3.3 Di-tau searches

In order to understand whether the existing LHC searches for heavy neutral Higgses in fermionic final states such as pose constraints on the low region of our simplified model, it is important to realise that while the pseudoscalars  and couple both to DM, the heavy scalar  does not, as can be seen from (7). If the channels are open, the discovery potential for is therefore generically larger than that for the corresponding pseudoscalar modes. In fact, the constraints from are most stringent for model realisations with and , so that the decays , and are kinematically forbidden and in consequence is forced to decay to light SM fermions (see Section 4.3).

The typical restrictions that result from LHC searches for heavy scalars can be illustrated by considering and employing the 95% confidence level (CL) limit  [47, 48] that is based on of 13 TeV data. Using the next-to-next-to-next-to-leading order results [51] for inclusive  production in gluon fusion, we then find that the current di-tau searches only exclude a narrow sliver of parameters in the plane with and in the case of a Yukawa sector of type II. A reduction of the quoted upper limit on the production cross section times branching ratio to would however improve the range of excluded  values to . As we will see in Section 6.4, such a constraint would be very valuable because probing models with and turns out to be difficult by other means.

### 3.4 Di-top searches

Heavy scalar and pseudoscalar bosons decaying dominantly into top-quark pairs can be searched for by studying the resulting invariant mass spectra . In contrast to di-top searches for spin-1 or spin-2 states, a peak in the distribution that one generically expects in the narrow-width approximation (NWA) is however not the only signature of a spin-0 resonance in this case. Indeed, the signal will interfere with the QCD background which at the LHC is mainly generated by the gluon-fusion channel . The signal-background interference will depend on the CP nature of the intermediate spin-0 boson, its mass and its total decay width. The observed interference pattern can be either constructive or destructive, leading to a rather complex signature with a peak-dip structure in the spectrum [52, 53]. The channel provides hence an interesting but challenging opportunity for hadron colliders to search for additional spin-0 bosons (see for instance [54, 55] for recent phenomenological discussions).

The first LHC analysis that takes into account interference effects between the signal process and the SM background is the ATLAS search [40]. It is based on of 8 TeV LHC data and considers the spectrum in final states with a single charged lepton (electron or muon), large and at least four jets. The search results are interpreted in the context of a pure THDM of type II for two different mass points and employ the alignment/decoupling limit, i.e. . For a neutral scalar  (pseudoscalar ) with a mass of , the ATLAS analysis excludes the parameter values () at the 95% CL, while for the mass point no meaningful constraint on can be set. Recasting these limits into bounds on the parameter space of spin-0 simplified DM models is straightforward [41] and we will analyse the resulting restrictions on our model in Section 6.4.

### 3.5 Flavour constraints

Indirect constraints on the charged Higgs-boson mass arise from  [56, 57, 58],  [59, 60, 61] and mixing [62, 63, 64, 65] since the latter processes receive corrections from the and terms in (7). We find that provides the strongest indirect constraint on for small values in models of type I and III at present, while oscillations represent the leading indirect constraint in the other two cases. For we obtain the bound from a combination of -meson physics observables irrespective of the choice of the Yukawa sector. A model-independent lower limit of can also be obtained from the requirement that the top-quark Yukawa coupling remains perturbative [43]. The latest LHC search limits on the charged Higgs mass in the channel [66, 67] are satisfied for  if is assumed, and therefore provide no relevant constraint.

### 3.6 EW precision constraints

A scalar potential with two doublets such as the one introduced in (1) leads to additional Higgs interactions compared to the SM, which can violate the custodial symmetry present in the SM Higgs sector. It can be shown [68, 69, 70, 71, 72] that the tree-level potential is custodially invariant for or . Only in these two cases can or have a sizeable mass splitting from the rest of the Higgses without being in conflict with EW precision measurements, most importantly . Since the potential (2) mixes the pseudoscalar degree of freedom in  with , in the theory described by there are however additional sources of custodial symmetry breaking compared to the case of the pure THDM. In the alignment/decoupling limit and taking , we find that the extended scalar sector gives rise to the following one-loop correction

 Δρ=1(4π)2M2H±v2[1+f(MH,Ma,MH±)+f(Ma,MH,MH±)]sin2θ, (9)

with

 Missing dimension or its units for \hskip (10)

Notice that in the limit in which the two CP-odd weak eigenstates are also mass eigenstates or if the scalar mass spectrum is fully degenerate. In the alignment/decoupling limit with , the custodial symmetry is instead not broken by  and as a result one has at the one-loop level.

From the above discussion it follows that only cases with are subject to the constraints from the EW precision measurements, while scenarios with are not. In order to derive the resulting constraints in the former case, we employ the 95% CL bound

 Δρ∈[−1.2,2.4]⋅10−3, (11)

which corresponds to the value extracted in [73] from a simultaneous determination of the Peskin-Takeuchi parameters , and . The fact that (9) is proportional to the product of mass differences and as well as implies that the existing EW precision data allow to set stringent bounds on if the relevant mass splittings in the scalar sector are sizeable. Taking for instance and , we find that for () the inequality () has to be satisfied in order to be compatible with (11). We will see in Section 4.3 that the restrictions on can have a visible impact on the decay pattern of the scalar , which in turn affects the mono- phenomenology discussed in Section 6.4.

### 3.7 Perturbativity and unitarity

Perturbativity [74, 75] and unitarity [76, 77, 78, 79] also put restrictions on the scalar masses and the magnitudes and signs of the quartic couplings. In our numerical analysis we will restrict our attention to the parameter space that satisfies and always keep , and of or below. For such input parameter choices all constraints discussed in this section are satisfied if is not too far below 1. We also only consider parameters for which the total decay widths of and are sufficiently small so that the NWA applies, i.e.  for . This requirement sets an upper limit on the mass of the charged Higgs boson that is often stronger than bounds from perturbativity.

## 4 Partial decay widths and branching ratios

This section is devoted to the discussion of the partial decay widths and the branching ratios of the spin-0 particles arising in the simplified DM model introduced in Section 2. For concreteness we will focus on the alignment/decoupling limit of the theory. We will furthermore pay special attention to the parameter space with a light DM particle, small values of and scalar spectra where the new pseudoscalar and the scalar are the lightest degrees of freedom.

### 4.1 Lighter pseudoscalar a

As a result of CP conservation the field has no couplings of the form , and . In contrast the vertex is allowed by CP symmetry but vanishes in the alignment/decoupling limit. At tree level the pseudoscalar can thus only decay into DM particles and SM fermions. The corresponding partial decay widths are given by

 Γ(a→χ¯χ)=y2χ8πMaβχ/acos2θ,Γ(a→f¯f)=Nfc(ξMf)28πm2fv2Maβf/asin2θ, (12)

where is the velocity of the particle in the rest frame of the final-state pair and we have defined . Furthermore denotes the relevant colour factor for quarks (leptons) and the explicit expressions for the couplings can be found in (8). At the loop level the pseudoscalar  can also decay to gauge bosons. The largest partial decay width is the one to gluon pairs. It takes the form

 Γ(a→gg)=α2s32π3v2M3a∣∣∑q=t,b,cξMqf(τq/a)∣∣2sin2θ, (13)

with

 f(τ)=τarctan2(1√τ−1). (14)

For small and non-zero values of the couplings of to DM and top quarks dominate over all other couplings. As a result, the decay pattern of is in general very simple. This is illustrated in the panels of Figure 2 for two different choices of parameter sets. The left panel shows the branching ratio of for a very light DM particle with . One observes that below the threshold one has while for both decays to DM and top-quarks pairs are relevant. In fact, sufficiently far above the threshold one obtains independent of the specific realisation of the Yukawa sector. In the right panel we present our results for a DM state of . In this case we see that below the threshold the pseudoscalar  decays dominantly into bottom-quark pairs but that also the branching ratios to taus and gluons exceed the percent level. Compared to the left plot one also observes that in the right plot the ratio is significantly larger for due to the different choice of .

### 4.2 Lighter scalar h

For sufficiently heavy pseudoscalars the decay pattern of resembles that of the SM Higgs boson in the alignment/decoupling limit. For on the other hand decays to two on-shell mediators are possible. The corresponding partial decay width reads

 Missing dimension or its units for \hskip (15)

with

 Missing dimension or its units for \hskip (16)

Notice that the coupling contains terms proportional to both and . These contributions result from the trilinear and quartic couplings in the scalar potential (2), respectively. In our THDM plus pseudoscalar extension, decays are even possible in the limit , which is not the case in the simplified model considered in [19, 20, 21].

Since the total decay width of the SM Higgs is only about , three-body decays of  into final states with a single can also be relevant in the mass range . Phenomenologically the most important three-body decay is the one where is accompanied by a pair of DM particles but decays to an and SM fermions are also possible. The corresponding partial decay widths are given by

 Missing dimension or its units for \hskip (17)

with [80]

 Missing or unrecognized delimiter for \right (18)

In Figure 3 we show the branching ratios of the SM Higgs for two different values of the DM mass. We observe that for a light pseudoscalar mediator one has in both cases . In fact, the total decay width of the lighter scalar exceeds for masses . Such large values of are in conflict with the model-independent upper limits on the total decay width of the Higgs as measured by both ATLAS and CMS in LHC Run I [81, 82]. Notice that since the pseudoscalar decays with 100% to DM pairs for the considered values of one has . This implies that for light DM the simplified model presented in Section 2 is subject to the constraints arising from invisible decays of the Higgs boson [37, 38]. We will analyse the resulting restrictions on the parameter space in Section 6.4. The right panel finally illustrates that in cases where  is close to a quarter of the SM Higgs mass also decays such as with can have branching ratios of a few percent (or more) for a narrow range of  values. Notice that for the choice used in the figure the result for does not depend on the particular Yukawa assignment.

### 4.3 Heavier scalar H

In the alignment/decoupling limit of the pseudoscalar extensions of the THDM model the heavier scalar does not couple to and pairs. In addition the vertex vanishes. Under the assumption that and taking to be sufficiently heavy, the scalar can hence decay only to SM fermions or the and final state at tree level. The corresponding partial decay widths are

 Γ(H→f¯f)=Nfc(ξMf)28πm2fv2MHβ3f/H,Γ(H→aa)=132πg2HaaMHβa/H,Γ(H→aZ)=116πλ3/2(MH,Ma,MZ)M3Hv2sin2θ, (19)

with

 Missing dimension or its units for \hskip (20)

denoting the coupling. We have furthermore introduced

 λ(m1,m2,m3)=(m21−m22−m23)2−4m22m23, (21)

which characterises the two-body phase space for three massive particles. Notice that the appearance of and in the partial decay width indicates again a qualitative difference between the scalar interactions considered in [19, 20, 21] and the more general potential (2). At the one-loop level the heavier scalar can in addition decay to gluons and other gauge bosons, but the associated branching ratios are very suppressed and thus have no impact on our numerical results.

The dominant branching ratios of as a function of are displayed in Figure 4 for two parameter sets. In the left panel the case of a scalar with and is shown. One observes that for the decay mode has the largest branching ratio, while for heavier the channel represents the leading decay. Notice that for model realisations where the decay channel dominates, interesting mono- signatures can be expected [20, 21]. We will come back to this point in Section 5.3. The decay pattern of is however strongly dependent on the mass of  since for the mixing angle is constrained to be small by EW precision measurements (see Section 3.6). This behaviour is easy to understand from (19) which in the limit of small , and large  imply that , and . For the decay mode  can hence dominate over the whole range of interest. This feature is illustrated on the right-hand side of the figure for and . One also sees from this panel that the branching ratio of can be relevant as it does not tend to zero in the limit if the combination of quartic couplings is non-zero. For and , can even be the largest branching ratio for . This happens because the terms proportional to and in (20) both give a sizeable contribution to the  coupling, while the coupling is suppressed by .

### 4.4 Heavier pseudoscalar A

For and assuming that decays to are kinematically inaccessible, the pseudoscalar can only decay to DM, SM fermions and the final state at tree level. In the alignment/decoupling limit the corresponding partial decay widths take the form

 Γ(A→χ¯χ)=y2χ8πMAβχ/Asin2θ,Γ(A→f¯f)=Nfc(ξMf)28πm2fv2MAβf/Acos2θ,Γ(A→ah)=116πλ1/2(MA,Ma,Mh)MAg2Aah, (22)

with

 gAah=1MAv[M2h−2M2H−M2A+4M2H±−M2a−2λ3v2+2(λP1cos2β+λP2sin2β)v2]sinθcosθ, (23)

denoting the coupling, and the analytic expression for the two-body phase-space function can be found in (21). Like in the case of , loop-induced decays of the heavier pseudoscalar  can be neglected for all practical purposes.

In Figure 5 we present our results for the branching ratios of the pseudoscalar as a function of for two different parameter choices. The left panel illustrates the case and one sees that for such an the branching ratios are all above 10% and the hierarchy is observed for . As shown on the right-hand side of the figure, this hierarchy not only remains intact but is even more pronounced for a moderately heavy until the threshold is reached. For larger values only decays to and final states matter and the ratio of their branching ratios is approximately given by irrespective of the particular Yukawa assignment. Notice that a sizeable branching ratio is a generic prediction in the THDM plus pseudoscalar extensions with small , since the charged Higgs has to be quite heavy in this case in order to avoid the bounds from and/or -meson mixing. Since is typically the dominant decay mode of the lighter pseudoscalar , appreciable mono-Higgs signals are hence a firm prediction in a certain region of parameter space of our simplified model. This point will be further explained in Section 5.4.

### 4.5 Charged scalar H±

Since in the alignment/decoupling limit the vertex vanishes, the partial decay widths of the charged scalar that are relevant in the small regime read

 Γ(H+→t¯b)=Ntc|Vtb|2(ξMt)28πm2tv2MH±(1−m2tM2H±)2,Γ(H+→HW+)=116πλ3/2(MH±,MH,MW)M3H±v2,Γ(H+→AW+)=116πλ3/2(MH±,MA,MW)M3H±v2cos2θ,Γ(H+→aW+)=116πλ3/2(MH±,Ma,MW)M3H±v2sin2θ, (24)

where in the case of we have neglected terms of in the expression for the partial decay width. Notice that in THDMs of type II and III also the decay can be important if . The result for can be obtained from the expression given above for by obvious replacements.

The main branching ratios of the charged Higgs are displayed in Figure 6. On the left-hand side of the figure the case of and is displayed and one observes that for the shown values of . Notice that for scenarios with the hierarchy is a rather model-independent prediction since in such cases EW precision measurements require to be small and . The same is not true for the hierarchy between and which depends sensitively on the choice of since . It follows that for values of the channel can also be the dominant decay mode. In model realisations with there are no constraints from  on and in turn the branching ratio can dominate for sufficiently large mixing in the pseudoscalar sector. This feature is illustrated by the right panel in the figure using and . For this choice of input parameters we find that for masses . Since the pseudoscalar  predominantly decays via it follows that THDM plus pseudoscalar extensions with can lead to a resonant mono- signal. We will discuss the LHC prospects for the detection of such a signature in Section 5.5.

## 5 Anatomy of mono-X signatures

In this section we will discuss the most important features of the mono- phenomenology of the pseudoscalar extensions of the THDM. We examine the mono-jet, the , the mono- and the mono-Higgs signature. The and mono- channel are also briefly considered. Our numerical analysis of the mono- signals is postponed to Section 6.

### 5.1 Mono-jet channel

A first possibility to search for pseudoscalar interactions of the form (7) consists in looking for a mono-jet signal, where the mediators that pair produce DM are radiated from heavy-quark loops [39, 46, 83, 84, 85, 86, 87, 88, 89]. Representative examples of the possible one-loop Feynman diagrams are shown in Figure 7.

For and only graphs involving the exchange of the light pseudoscalar will contribute to the signal. As a result the normalised kinematic distributions of the mono-jet signal in the pseudoscalar extensions of the THDM are identical to those of the DMF pseudoscalar model. Working in the NWA and assuming that is small, the ratio of the fiducial cross sections in the two models is thus approximately given by the simple expression

 Missing dimension or its units for \hskip (25)

Here () denotes the DM-mediator (universal quark-mediator) coupling in the corresponding DMF spin-0 simplified model. Notice that the above relation is largely independent of the choice of Yukawa sector as long as since bottom-quark loops have only an effect of a few percent on the distributions (see for instance [90] for a related discussion in the context of Higgs physics). Using the approximation (25) it is straightforward to recast existing mono-jet results on the DMF pseudoscalar model such as those given in [23] into the THDM plus pseudoscalar model space. The numerical results presented in the next section however do not employ any approximation since they are based on a calculation of the cross sections including both top-quark and bottom-quark loops as well as the exchange of both and mediators.

### 5.2 t¯t/b¯b+ET,miss channels

A second channel that is known to be a sensitive probe of top-philic pseudoscalars with large invisible decay widths is associated production of DM and pairs [39, 86, 89, 91, 92, 93, 94]. Figure 8 displays examples of tree-level diagrams that give rise to a signature in the pseudoscalar extensions of the THDM model.

In the case that is again much heavier than , the signal strength for in our simplified model can be obtained from the prediction in the DMF pseudoscalar scenario from a rescaling relation analogous to the one shown in (25). Using such a simple recasting procedure we find that the most recent ATLAS [24] and CMS searches for  [25] that are based on and of 13 TeV LHC data, respectively, only allow to set very weak bounds on . For instance for , and a lower limit of is obtained. The constraints on the parameter space of the pseudoscalar extensions of the THDM are however expected to improve notably at forthcoming LHC runs. The numerical results that will be presented in Section 6.4 are based on the search strategy developed recently in [94] which employs a shape fit to the difference in pseudorapidity of the two charged leptons in the di-leptonic channel of .

Besides also production [91, 92] has been advocated as a sensitive probe of spin-0 portal couplings to heavy quarks. Recasting the most recent 13 TeV LHC searches [26, 27] by means of a simple rescaling similar to (25) we find that no relevant bound on the parameter space of our simplified model can be derived unless the  coupling is significantly enhanced. From (8) we see that such an enhancement can only arise in THDMs of type II and IV, while it is not possible for the other Yukawa assignments. Since in the limit of large also direct searches for the light pseudoscalar  in final states containing bottom quarks or charged leptons are relevant (and naively even provide the leading constraints) we do not consider the channel in what follows, restricting our numerical analysis to the parameter space with small .

### 5.3 Mono-Z channel

A mono- signal that is strongly suppressed in the case of the spin-0 DMF models [88] but will turn out to be relevant in our simplified DM scenario is the mono- channel [21]. A sample of one-loop diagrams that lead to such a signature are displayed in Figure 9. Notice that the left diagram in the figure allows for resonant production through a  vertex for a sufficiently heavy scalar . Unlike the graph on the right-hand side it has no counterpart in the spin-0 DMF simplified models.

As first emphasised in [20] the appearance of the contribution with virtual and exchange not only enhances the mono- cross section compared to the spin-0 DMF models, but also leads to quite different kinematics in production. In fact, for masses the predicted spectrum turns out to be peaked at

 EmaxT,miss≃λ1/2(MH,Ma,MZ)2MH, (26)

where the two-body phase-space function has been defined in (21). Denoting the lower experimental requirement on in a given mono- search by the latter result can be used to derive a simple bound on for which a significant fraction of the total cross section will pass the cut. We obtain the inequality

 MH≳Ma+√M2Z+(EcutT,miss)2. (27)

Given that in the latest mono- analyses [28, 29, 30] selection cuts of are imposed it follows that the scalar has to have a mass of if one wants to be sensitive to pseudoscalars with masses up to the threshold .

Our detailed Monte Carlo (MC) simulations of the signal in Section 6.4 however reveals that the above kinematical argument alone is insufficient to understand the shape of the mono- exclusion in the plane in all instances. The reason for this is twofold. First, in cases where is small is often not the dominant  decay mode and as a result the measurements lose already sensitivity for masses