Simplified dark matter models with two Higgs doublets: I. Pseudoscalar mediators
Martin Bauer Ulrich Haisch Felix Kahlhoefer

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    #11]Martin Bauer,2,3]Ulrich Haisch4]and Felix Kahlhoefer
        [1]Institut für Theoretische Physik, Universität Heidelberg, 
         
        Philosophenweg 16, 69120 Heidelberg, Germany[2]Rudolf Peierls Centre for Theoretical Physics,
           University of Oxford, 
         OX1 3NP Oxford, United Kingdom[3]CERN, Theoretical Physics Department, 
         CH-1211 Geneva 23, Switzerland[4]DESY, Notkestraß e 85, D-22607 Hamburg, Germanybauer@thphys.uni-heidelberg.de, ulrich.haisch@physics.ox.ac.uk,  felix.kahlhoefer@desy.deWe 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 (E_T,  miss) 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 tt̅ + E_T,  miss,  mono-Z 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.CERN-TH-2017-011, DESY-17-010Simplified dark matter models with two Higgs doublets:  I. Pseudoscalar mediators
            [
            A. Mahmoodzadeha.mahmoodzadeh@iau-boukan.ac.ir, B. MalekolkalamiB.Malakolkalami@uok.ac.ir 
        ==============================================================================================
        
        
        
        § 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 <cit.> 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 SU(2)_L × U(1)_Y 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 <cit.>. 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 Z-boson and Higgs-boson properties however  severely restrict the corresponding mixing angles, and as a result classic E_T,  miss searches like mono-jets are typically not the leading collider constraints in  this class of simplified DM models <cit.>.
        
        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 Z_2 symmetry <cit.>, 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 <cit.> 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 <cit.> (see also <cit.>). 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 <cit.>. Whenever indicated we will highlight the similarities and differences between these and our work.
        
        The mono-X phenomenology of the considered simplified pseudoscalar models turns out to be surprisingly rich.  We examine the constraints from searches for j+ E_T,  miss <cit.>, tt̅/b b̅ + E_T,  miss <cit.>,  Z+E_T,  miss <cit.>, h+E_T,  miss <cit.>  and W+E_T,  miss <cit.> 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 <cit.>. For the simplified pseudoscalar model recommended by the ATLAS/CMS DM Forum (DMF) <cit.> constraints from mono-jet searches dominate throughout the parameter space <cit.>, whereas for the model considered here tt̅ + E_T,  miss,  mono-Z and mono-Higgs searches yield competitive bounds and often provide the leading constraints. See Figure <ref> for an illustration of the various E_T,  miss 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 Z+ E_T,  miss and h + E_T,  miss 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 300   fb^-1 of luminosity. The importance of di-top resonance searches <cit.> to probe  neutral spin-0 states with masses above the t t̅ 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 (b b̅) initiated production.
        
        This paper is structured as follows. In Section <ref> we describe the class of simplified DM models that we will study throughout our work, while Section <ref> contains a comprehensive review of the non-E_T,  miss 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 <ref>. The most important features of the resulting E_T,  miss  phenomenology are described in Section <ref>. In Section <ref> we finally present the numerical results of our analyses providing summary plots of the mono-X 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 <ref>. 
        
        
        
        § 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.
        
        
        
         §.§ Scalar potential
        
        
        The tree-level THDM scalar potential that we will consider throughout this paper is given by the following expression (see for example <cit.>)
        
        V_H    = μ_1 H_1^† H_1 + μ_2 H_2^† H_2 +  ( μ_3  H_1^† H_2 +  h.c. ) + λ_1   ( H_1^† H_1   )^2  + λ_2   ( H_2^† H_2   )^2 
           xx +  λ_3  ( H_1^† H_1   )  ( H_2^† H_2   ) + λ_4   ( H_1^† H_2   )  ( H_2^† H_1   ) +  [ λ_5    ( H_1^† H_2  )^2 +  h.c. ]   .
        
        Here we have imposed a Z_2 symmetry under which H_1 → H_1 and H_2 → -H_2 to suppress flavour-changing neutral currents (FCNCs), but allowed for this discrete symmetry to be softly broken by the term μ_3  H_1^† H_2 +  h.c. The vacuum expectation values (VEVs) of the Higgs doublets are given by ⟨ H_i ⟩ = (0,v_i/√(2))^T with v = √(v_1^2 + v_2^2)≃ 246   GeV and we  define tanβ = v_2/v_1. To avoid possible issues with electric dipole moments, we assume that the mass-squared terms μ_j, the quartic couplings λ_k and the VEVs are all real and as a result  the scalar potential as given in (<ref>) is CP conserving. The three physical neutral Higgses that emerge from V_H 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 P with the CP-odd Higgs that arises from (<ref>). This can be achieved by considering the following interaction terms 
        
        V_P  =  1/2 m_P^2  P^2 +  P  ( i  b_P    H_1^† H_2 +  h.c. ) + P^2  (  λ_P1  H_1^† H_1 +   λ_P2  H_2^† H_2  )   ,
        
        where m_P and b_P are parameters with dimensions of mass. We assume that V_P does not break CP and thus take b_P  to be real in the following. In this case P does not develop a VEV and remains a  pure CP eigenstate. Nevertheless, this term does lead to a soft breaking of the Z_2 symmetry. Notice that compared to <cit.> which include only the trilinear portal coupling b_P, we also allow for quartic portal interactions proportional to λ_P1 and λ_P2. A quartic self-coupling of the form P^4 has instead not been included in (<ref>), as it does not lead to any relevant effect in the observables studied in our paper.  
        
        The interactions in the scalar potential  (<ref>) mix the neutral CP-even weak eigenstates and we denote the corresponding mixing angle by α. The portal coupling b_P appearing in (<ref>) instead mixes the two neutral CP-odd weak eigenstates with θ representing the associated mixing angle. The resulting  CP-even mass eigenstates will be denoted by h and H, while in the CP-odd sector the states will be called A and a, where a  denotes the extra degree of freedom not present in THDMs. The scalar spectrum also contains two charged mass eigenstates H^± of identical mass. 
        
        Diagonalising the mass-squared matrices of the scalar states  leads to relations between the fundamental parameters entering V_H and V_P. These relations allow to trade the parameters m_P, μ_1, μ_2, μ_3, b_P, λ_1, λ_2, λ_4, λ_5 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 (<ref>) and (<ref>) is hence fully captured by the angles α, β, θ,  the EW VEV v, the quartic couplings λ_3, λ_P1, λ_P2 and the masses M_h, M_H, M_A, M_H^±, M_a. We will use these parameters as input in our analysis. 
        
        
        
         §.§ 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 <cit.>. This so-called natural flavour conservation hypothesis is automatically enforced by the aforementioned Z_2 symmetry acting on the doublets, if the right-handed fermion singlets transform accordingly. The Yukawa couplings are explicitly given by
         L_Y = - ∑_i=1,2 ( Q̅ Y_u^i H̃_i u_R  + Q̅ Y_d^i H_i d_R   + L̅ Y_ℓ^i H_i ℓ_R  +  h.c. )  .
        
        Here Y_f^i are Yukawa matrices acting on the three fermion generations and we have suppressed flavour indices, Q and L are left-handed quark and lepton doublets, while u_R, d_R and ℓ_R are right-handed up-type quark, down-type quark and charged lepton singlets, respectively. Finally, H̃_i = ϵ H_i^∗ with ϵ 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 H_2:
        
        Y_u^1     = Y_d^1 = Y_ℓ^1 =0  ,    (type I)  , 
        
        Y_u^1     = Y_d^2 = Y_ℓ^2 =0  ,    (type II)  , 
        
        Y_u^1     = Y_d^1 = Y_ℓ^2 =0 ,    (type III)  , 
        
        Y_u^1     = Y_d^2 = Y_ℓ^1 =0 ,    (type IV)  . 
        
        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  χ a separate Z_2 symmetry under which χ→ -χ can be used to forbid a coupling of the form L̅H̃_1 χ_R + h.c. At the level of renormalisable operators this leaves 
         L_χ = - i  y_χ P χ̅γ_5 χ ,
        
        as the only possibility to couple the pseudoscalar mediator P to DM. In order to not violate CP we require the dark sector Yukawa coupling y_χ to be real. The parameter y_χ and the DM mass m_χ are further input parameters in our analysis. 
        
        
        
        § 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. 
        
        
        
         §.§ Alignment/decoupling limit
        
        
        After EW symmetry breaking  the kinetic terms of the Higgs fields H_i  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  )    ( 2 M_W^2/v W_μ^+ W^- μ  + M_Z^2/v Z_μ Z^μ )  .
        
        In order to simplify the further analysis, we concentrate on the well-motivated alignment/decoupling limit of the THDM where α = β - π/2. In this case sin ( β - α ) = 1 meaning that  the field h has SM-like EW gauge boson couplings. It can therefore be identified with the boson of mass M_h ≃ 125   GeV 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 <cit.> are readily fulfilled. Notice that in the alignment/decoupling limit the scalar H does not interact with W-boson or Z-boson pairs at tree level because in this limit one has cos ( β - α ) = 0.
        
        
        
         §.§ Yukawa assignments
        
        
        Working in the alignment/decoupling limit the fermion-scalar interactions most relevant for the further discussion  are given by 
         L⊃     - y_t/√(2) t̅  [  h + ξ_f^ M H  - i   ξ_f^ M  ( cosθ  A - sinθ  a  )  γ_5  ]     t 
            -∑_f=b,τy_f/√(2) f̅  [  h + ξ_f^ M H  + i   ξ_f^ M  ( cosθ  A - sinθ  a  )  γ_5  ]    f   
            -  y_t/√(2) V_tb ξ_t^ M   H^+ t̅_R  b_L +  y_b/√(2)  V_tb ξ_b^ M   H^+ t̅_L  b_R +  h.c.
            - i   y_χ  ( sinθ  A + cosθ  a  )  χ̅γ_5 χ ,
        
        where y_f = √(2) m_f/v denote the SM Yukawa couplings and V_ij are the elements of the  Cabibbo-Kobayashi-Maskawa (CKM) matrix. The couplings ξ_f^ M encode the dependence on the choice of Yukawa sector (<ref>). In terms of tanβ one has 
           ξ_t^ I   =  ξ_b^ I = ξ_τ^ I = -β ,    (type I)  , 
           ξ_t^ II  = -β ,     ξ_b^ II  =  ξ_τ^ II  = tanβ ,    (type II)  , 
           ξ_t^ III  = ξ_b^ III  = -β ,     ξ_τ^ III  = tanβ ,    (type III)  , 
           ξ_t^ IV  =  ξ_τ^ IV  = -β ,     ξ_b^ IV  = tanβ ,    (type IV)  .
        
        
        Since the production of the pseudoscalar mediator a as well as pp → h, H, A is driven by top-quark loops that enter  the gluon-fusion (gg) channel at the LHC (see for instance <cit.> for a discussion in the context of E_T,  miss searches) we will  in the following focus on the region of small tanβ.  In this limit  the couplings of H,A,a  to down-type quarks and charged leptons in (<ref>)  are strongly Yukawa suppressed irrespectively of the chosen Yukawa assignment (<ref>). As a result existing bounds on the neutral scalar masses from flavour observables such as B_s →μ^+ μ^- that are known to receive tanβ enhanced corrections <cit.> are within experimental limits <cit.> even for a light scalar spectrum.
        
        
        
         §.§ Di-tau searches
        
        
        In order to understand whether the existing LHC searches for heavy neutral Higgses in fermionic final states such as f  f̅ = τ^+ τ^-, b b̅ pose constraints on the low tanβ region of our simplified model, it is important to  realise that while the pseudoscalars A and a couple both to DM, the heavy scalar H does not, as can be seen from (<ref>). If the channels A/a →χχ̅ are open, the discovery potential for H → f f̅ is therefore generically larger than that for the corresponding pseudoscalar modes. In fact, the constraints from pp → H → f f̅ are most stringent for model realisations with M_H < 2 m_t and M_a > max (M_H - M_Z, M_H/2  ), so that the decays H → tt̅, H → a a and H → a Z are kinematically forbidden and in consequence H is forced to decay to light SM fermions (see Section <ref>). 
        
        The typical restrictions that result from  LHC searches for heavy scalars can be illustrated by considering M_H = 300   GeV and  employing the 95% confidence level (CL) limit σ ( p p → H  )  BR ( H →τ^+ τ^-  ) < 0.4   pb <cit.> that is based on 13   fb^-1 of 13 TeV  data. Using the next-to-next-to-next-to-leading order results <cit.> for inclusive H production in gluon fusion, we then find that the current di-tau searches only exclude a narrow sliver of parameters in the M_a–tanβ plane with 0.55 ≲tanβ≲ 0.65 and M_a ≳ 210   GeV 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 0.2   pb would however improve the range of excluded tanβ values to 0.3 ≲tanβ≲ 1.2. As we will see in Section <ref>, such a constraint would be very valuable because probing models with tanβ =  O (1) and M_H ≃ M_a ≃ 300   GeV turns out to be   difficult  by other means. 
        
        
        
         §.§ Di-top searches
        
        
        Heavy scalar  and pseudoscalar bosons decaying dominantly into  top-quark pairs can be searched for by studying the resulting t t̅ invariant mass spectra m_t t̅. In contrast to di-top searches for spin-1 or spin-2 states, a peak in the m_t t̅ 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 gg → H/A signal will interfere with the QCD t t̅ background which at the LHC is mainly generated by the gluon-fusion channel gg → t t̅. 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 m_t t̅ spectrum <cit.>. The pp → H/A → t t̅ channel provides hence an interesting but challenging opportunity for hadron colliders to search for additional spin-0 bosons (see for instance <cit.> for recent phenomenological discussions). 
        
        The first LHC analysis that takes into account interference effects between the signal process gg → H/A → t t̅ and the SM background gg →  t t̅ is the ATLAS search <cit.>. It is based on 20.3   fb^-1 of 8 TeV LHC data and considers the m_t t̅ spectrum in final states with a single charged lepton (electron or muon), large E_T,  miss 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. sin ( β - α ) = 1. For a neutral scalar H (pseudoscalar A) with a mass of 500   GeV, the ATLAS analysis excludes the parameter values tanβ < 0.45 (tanβ < 0.85) at the 95% CL, while for the 750   GeV mass point  no meaningful constraint on tanβ can be set. Recasting these limits into bounds on the parameter space of spin-0 simplified DM models is straightforward <cit.> and we will analyse the resulting restrictions on our model in Section <ref>. 
        
        
        
         §.§ Flavour constraints
        
        
        Indirect constraints on the charged Higgs-boson mass M_H^± arise from Z → b b̅ <cit.>,  B → X_s γ <cit.> and B_q–B̅_q  mixing <cit.> since the latter processes receive corrections from the H^+ t̅_R  b_L +  h.c. and H^+ t̅_L  b_R +  h.c. terms in (<ref>). We find that B → X_s γ provides the strongest indirect constraint on M_H^± for small tanβ values in models of type I and III at present, while B_s–B̅_s oscillations represent the leading indirect constraint in the other two cases. For  M_H^± = 750   GeV we obtain the  bound tanβ≳ 0.8 from a combination of B-meson physics observables irrespective of the choice of the Yukawa sector. A model-independent lower limit of tanβ≳ 0.3 can also be obtained from the requirement that the top-quark Yukawa coupling remains perturbative <cit.>.  The latest LHC search limits on the charged Higgs mass in the pp → tbH^±  (H^±→ tb) channel <cit.> are satisfied for tanβ≳ 0.2 if M_H^± = 750   GeV is assumed, and therefore provide no relevant constraint. 
        
        
        
         §.§ EW precision constraints
        
         
        A scalar potential with two doublets such as the one introduced in (<ref>) 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 <cit.> that the tree-level potential V_H is custodially invariant for  M_A = M_H^± or M_H = M_H^±. Only in these two cases can H or A have a sizeable mass splitting from the rest of the Higgses without being in conflict with EW precision measurements, most importantly Δρ. Since the potential  (<ref>) mixes the  pseudoscalar degree of freedom in H_i with P, in the theory described by V_H + V_P there are however additional sources of custodial symmetry breaking compared to the case of the pure THDM. In the alignment/decoupling limit and taking  M_A = M_H^±, we find that the extended scalar sector gives rise to the following one-loop correction
        Δρ = 1/(4 π)^2M_H^±^2/v^2   [ 1 + f (M_H, M_a, M_H^±  ) + f  (M_a, M_H, M_H^±  )     ]  sin^2 θ ,
        
        with 
        
        f (m_1, m_2, m_3  ) = m_1^4  ( m_2^2 - m_3^2 )/m_3^2  (m_1^2 - m_2^2  )  ( m_1^2 - m_3^2  )ln ( m_1^2/m_3^2 )  .
        
        Notice that Δρ→ 0 in the limit sinθ→ 0 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 M_H = M_H^±,  the custodial symmetry is instead not broken by V_H + V_P  and as a result one has Δρ = 0  at the one-loop level. 
        
        From the above discussion it follows that only cases with M_A = M_H^± are subject to the constraints from the EW precision measurements, while scenarios with M_H = M_H^± 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 ,
        
        which corresponds to the value extracted in <cit.> from a simultaneous determination of the Peskin-Takeuchi parameters S, T and U. The fact that (<ref>) is proportional to the product of mass differences M_H^± - M_H and M_H^± - M_a as well as sin^2 θ implies that the existing EW precision data allow to set stringent bounds on sinθ if the relevant mass splittings in the scalar sector are sizeable. Taking for instance M_H^± = 750   GeV and M_a = 65   GeV, we find  that for M_H = 500   GeV (M_H = 300   GeV) the inequality sinθ < 0.35 (sinθ < 0.25) has to be satisfied in order to be compatible with (<ref>). We will see in Section <ref> that the  restrictions on sinθ can have a visible impact on the decay pattern of the scalar H, which in turn affects the mono-Z phenomenology discussed in Section <ref>. 
        
        
        
         §.§ Perturbativity and unitarity
        
         
        Perturbativity <cit.> and unitarity <cit.> 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 M_H, M_A, M_a ≤ M_H^± =  O ( 1   TeV) and always keep λ_3, λ_P1 and λ_P2 of O (1) or below.  For such input parameter choices all constraints discussed in this section are satisfied if tanβ is not too far below 1. We also only consider parameters for which the total decay widths of H and A are sufficiently small so that the NWA applies, i.e. Γ_i≲ M_i/3 for i=H,A. This requirement sets an upper limit on the mass of the charged Higgs boson that is often  stronger than bounds from perturbativity.
        
        
        
        § 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 <ref>. 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 tanβ and scalar spectra where the  new pseudoscalar a and the scalar h are the lightest degrees of freedom.
         
        
        
         §.§ Lighter pseudoscalar a
        
        
        As a result of CP conservation the field a has no couplings of the form aW^+W^-, aZZ and ahh.  In contrast the ahZ vertex is allowed by CP symmetry but vanishes  in the  alignment/decoupling  limit. At tree level the pseudoscalar a can thus only decay  into DM particles and SM fermions. The corresponding partial decay widths are given by
        Γ ( a →χχ̅ )     = y_χ^2/8π M_a  β_χ/acos^2 θ , 
        Γ ( a → f f̅ )     = N_c^f  ( ξ_f^ M)^2/8πm_f^2/v^2  M_a  β_f/asin^2 θ ,
        
        where  β_i/a = √(1 - τ_i/a) is the velocity of the particle i in the  rest frame of the final-state pair and we have defined  τ_i/a=4 m_i^2/M_a^2. Furthermore N_c^f = 3   (1) denotes the relevant colour factor for quarks (leptons) and the explicit expressions for the couplings ξ_f^ M  can be found in (<ref>). At the loop level the pseudoscalar a can also decay  to gauge bosons. The largest partial decay width is the one to  gluon pairs. It  takes the form 
        Γ (a → gg  ) = α_s^2/32 π^3 v^2  M_a^3   | ∑_q=t,b,cξ_q^ M f(τ_q/a)  |^2  sin^2 θ ,
        
        with 
        
        f (τ) = τarctan^2  ( 1/√(τ -1 ) )  .
        
        
        For small tanβ and non-zero values of sinθ the  couplings of a to DM and top quarks dominate over all other couplings. As a result, the decay pattern of a is in general very simple. This is illustrated in the panels of Figure <ref> for  two different  choices of parameter sets. The left panel shows the branching ratio of a for a very light DM particle with m_χ = 1   GeV.  One observes that below the t t̅ threshold one has BR (a →χχ̅) = 100 % while for M_a > 2 m_t both decays to DM and top-quarks pairs are relevant. In fact, sufficiently far above the t t̅ threshold one obtains   BR (a →χχ̅)/ BR (a → t t̅) ≃ 0.7  y_χ^2 tan^2 β/tan^2 θ independent of the specific realisation of the Yukawa sector. In the right panel we present our results for a DM state of m_χ = 100   GeV. In this case we see that below the χχ̅ threshold the pseudoscalar a 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 BR (a →χχ̅)/ BR (a → t t̅) is significantly larger for M_a > 2 m_t due to the different choice of sinθ. 
        
        
        
         §.§ Lighter scalar h
        
        
        For sufficiently heavy pseudoscalars a the decay pattern of h resembles  that of the  SM Higgs boson in the alignment/decoupling limit. For M_a < M_h/2 on the other hand decays to two on-shell a mediators are possible. The corresponding  partial decay width reads 
        Γ ( h → a a  ) = 1/32 π  g_haa^2   M_h  β_a/h ,
        
        with 
        
        g_haa    = 1/M_h v  [   ( M_h^2  - 2 M_H^2  + 4 M_H^±^2 - 2 M_a^2 -  2 λ_3  v^2   ) sin^2 θ
            - 2  ( λ_P1cos^2 β + λ_P2sin^2 β ) v^2  cos^2 θ ]  .
        
        Notice that the haa coupling contains terms proportional to both sin^2 θ and cos^2 θ. These contributions result from the trilinear and quartic couplings in the scalar potential (<ref>), respectively. In our THDM plus pseudoscalar extension, h → aa decays are even possible in the limit θ→ 0, which is not the case in the simplified model considered in <cit.>.
        
        
        
        Since the total decay width of the SM Higgs is only about 4   MeV, three-body decays of h into final states with a single a can also be relevant in the mass range M_h/2 < M_a ≲ M_h. Phenomenologically the most important three-body decay is the one where a is accompanied by a  pair  of DM particles but decays to an a and SM fermions are also possible. The corresponding partial decay widths are given by 
        Γ ( h → a χχ̅ )     = y_χ^2/32 π^3  g_haa^2   M_h  β_χ/a g(τ_a/h) cos^2 θ , 
        Γ ( h → a f f̅ )     = N_c^f  ( ξ_f^ M)^2/32 π^3m_f^2/v^2  g_haa^2   M_h  β_f/a g(τ_a/h) sin^2 θ , 
        
        with  <cit.>
        g ( τ  )     = 1/8 ( τ - 4   )  [ 4 -  ln ( τ/4 )   ] 
           xx - 5 τ - 4/4 √(τ-1) [ arctan ( τ-2/2 √(τ-1) ) - arctan ( 1/√(τ-1) ) ]   . 
        
        
        In Figure <ref> we show the branching ratios of the SM Higgs h for two different values of the DM mass. We observe that for a light pseudoscalar mediator a  one has in both cases BR ( h → a χχ̅ ) = 100 %. In fact, the total decay width of the lighter scalar h exceeds 3   GeV for masses M_a ≲ 70   GeV. Such large values of  Γ_h 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 <cit.>. Notice that since the pseudoscalar a decays with 100% to DM pairs for the considered values of m_χ one has BR ( h → a χχ̅ ) =  BR ( h → 2 χ 2 χ̅ ). This implies that for light DM the simplified model presented in Section <ref> is subject to the constraints arising from invisible decays of the Higgs boson <cit.>. We will analyse the resulting restrictions on the parameter space in Section <ref>. The right panel  finally illustrates that in cases where m_χ is close to a quarter of the SM Higgs mass also decays such as h → a b b̅ with a →χχ̅ can have branching ratios of a few percent (or more) for a narrow range of M_a values.  Notice that for the choice tanβ = 1 used in the figure the result for BR ( h → a b b̅ ) does not depend on the particular Yukawa assignment. 
        
        
        
         §.§ Heavier scalar H
        
        
        In the alignment/decoupling limit of the pseudoscalar extensions of the THDM model the heavier scalar H does not couple to W^+ W^- and Z Z pairs. In addition the Hhh  vertex vanishes. Under the assumption that M_H > M_a and taking A to be sufficiently heavy, the scalar H can hence decay only to SM fermions or the aa and aZ final state at tree level. The corresponding partial decay widths are 
        Γ ( H → f f̅ )     =  N_c^f  ( ξ_f^ M)^2/8πm_f^2/v^2  M_H  β_f/H^3   , 
        Γ ( H → a a  )     =  1/32 π  g_Haa^2   M_H  β_a/H , 
        Γ ( H → a Z  )     =   1/16 πλ^3/2 (M_H, M_a, M_Z)/M_H^3 v^2 sin^2 θ ,
        
        with 
        
        g_Haa    = 1/M_H  v  [   ( 2 β)  (  2 M_h^2 - 4 M_H^2 + 4 M_H^±^2  - 2 λ_3 v^2  )  sin^2 θ
            + sin ( 2 β )  (λ_P1-λ_P2 ) v^2 cos^2 θ  ]  ,
        
        denoting the Haa coupling. We have furthermore introduced 
        λ (m_1, m_2, m_3) =  ( m_1^2 - m_2^2 - m_3^2  )^2 - 4   m_2^2  m_3^2  ,
        
        which characterises the two-body phase space for three massive particles. Notice that the appearance of λ_P1  and  λ_P2  in the partial decay width Γ ( H → a a  ) indicates again a qualitative difference between the scalar interactions considered  in <cit.> and the more general potential (<ref>). At the one-loop level the heavier scalar H 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 H as a function of M_a are displayed in Figure <ref> for two parameter sets. In the left panel the case of a scalar H with sinθ = 1/√(2) and M_H = 750   GeV is shown. One observes that for M_a ≲ 350   GeV the decay mode H → aZ has the largest branching ratio,  while  for heavier a the H → t t̅ channel represents the leading decay. Notice that for model realisations where the decay channel H → a Z dominates, interesting mono-Z signatures can be expected <cit.>. We will come back to this point in Section <ref>. The decay pattern of H is however strongly dependent on the mass of H since for M_H < M_H^± the mixing angle θ is constrained to be small by EW precision measurements (see Section <ref>). This behaviour is easy to understand from  (<ref>)  which in the  limit of small sinθ, tanβ =  O ( 1) and large M_H imply that Γ ( H → t t̅ ) ∝ m_t^2/(M_H tan^2 β), Γ ( H → a a  ) ∝ v^4/M_H^3  ( λ_P1 - λ_P2 )^2  and Γ ( H → a Z  ) ∝ M_H sin^2 θ. For M_H > 2 m_t  the decay mode H → t t̅ can hence  dominate over the whole M_a range of   interest. This feature is illustrated on the right-hand side of the figure for sinθ = 0.35 and M_H = 500   GeV. One also sees from this panel that    the  branching ratio of H → aa can be relevant as it does not tend to zero in the sinθ→ 0 limit if the combination λ_P1 - λ_P2 of quartic couplings is non-zero. For tanβ≳ 2 and λ_P1 - λ_P2≳ 1,  BR ( H → a a )   can even be the largest  branching ratio for M_a < M_H/2. This happens because the terms proportional to sin^2 θ and cos^2 θ in (<ref>) both give a sizeable contribution to the Haa coupling, while the H t t̅ coupling is suppressed by 1/tan^2 β. 
        
        
        
         §.§ Heavier pseudoscalar A
        
        
        For M_A > M_a  and assuming that decays to H are kinematically inaccessible, the pseudoscalar A can only decay to DM, SM fermions and the ah final state at tree level. In the alignment/decoupling limit the corresponding partial decay widths take the form 
        Γ ( A →χχ̅ )     = y_χ^2/8π M_A  β_χ/Asin^2 θ ,  
        Γ ( A → f f̅ )     =  N_c^f  ( ξ_f^ M)^2/8πm_f^2/v^2  M_A  β_f/Acos^2 θ , 
        Γ ( A → a h   )     =   1/16 πλ^1/2 (M_A, M_a, M_h)/M_A  g_Aah^2   ,
        
        with 
        
        g_Aah    = 1/M_A v  [  M_h^2  - 2 M_H^2  - M_A^2 + 4 M_H^±^2 -  M_a^2 - 2 λ_3 v^2  
            + 2  (  λ_P1cos^2 β + λ_P2sin^2 β ) v^2  ] sinθcosθ ,
        
        denoting the Aah coupling, and the analytic expression for the two-body phase-space  function λ (m_1, m_2, m_3) can be found in (<ref>). Like in the case of H, loop-induced decays of the heavier pseudoscalar A can be neglected for all practical purposes. 
        
        
        
        In Figure <ref> we present our results for the  branching ratios of the pseudoscalar A as a function of M_a for two different parameter choices. The left panel illustrates the case M_A = 750   GeV and one sees that for such an A the  branching ratios are all above 10% and  the hierarchy BR ( A → ah  ) >  BR ( A → t t̅ ) >  BR ( A →χχ̅ ) is observed for M_a ≲ 200   GeV. 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 A until the threshold M_a = M_A - M_h is reached. For larger M_a values only decays to χχ̅ and t t̅ final states matter and the ratio of their  branching ratios is approximately given by BR (A →χχ̅)/ BR (A → t t̅) ≃ 0.9  y_χ^2 tan^2 βtan^2 θ irrespective of the particular Yukawa assignment. Notice that a sizeable A → ah branching ratio is a generic prediction in the THDM plus pseudoscalar extensions with small tanβ, since the charged Higgs has to be quite heavy in this case in order to avoid the bounds from B → X_s γ and/or B_s-meson mixing.   Since a  →χχ̅ is typically the dominant decay mode of the lighter  pseudoscalar a, 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 <ref>. 
        
        
        
         §.§ Charged scalar H^±
        
        
        Since in the alignment/decoupling limit the H^+ h W^+ vertex vanishes, the  partial decay widths of the charged scalar H^+ that are relevant  in the small tanβ regime  read 
        Γ ( H^+ → t b̅ )     = N_c^t     |V_tb|^2  ( ξ_t^ M)^2/8πm_t^2/v^2 M_H^± ( 1 - m_t^2/ M_H^±^2 )^2  , 
        Γ ( H^+ → H W^+  )     = 1/16πλ^3/2 (M_H^±, M_H, M_W)/M_H^±^3 v^2 ,  
        Γ ( H^+ → A W^+  )     = 1/16πλ^3/2 (M_H^±, M_A, M_W)/M_H^±^3 v^2 cos^2 θ ,  
        Γ ( H^+ → a W^+  )     = 1/16πλ^3/2 (M_H^±, M_a, M_W)/M_H^±^3 v^2 sin^2 θ , 
        
        where in the case of H^+ → t b̅ we have  neglected terms  of O (m_b^2/M_H^±^2) in the expression for the partial decay width. Notice that in THDMs of type II and III also the decay H^+ →τ^+ ν_τ can be important if tanβ≫ 1. The result for Γ (H^+ →τ^+ ν_τ ) can be obtained from the expression given above for Γ ( H^+ → t b̅ ) by obvious replacements.
        
        
        
        The main branching ratios of the charged Higgs H^+ are displayed in Figure <ref>. On the left-hand side of the figure the case of sinθ = 0.35 and M_H = 500   GeV  is displayed and one observes that BR (H^+ → t b̅ ) >  BR (H^+ → HW^+  ) >  BR (H^+ → aW^+  ) for the shown values of M_a. Notice that for scenarios with M_H < M_H^± the hierarchy BR (H^+ → HW^+  ) >  BR (H^+ → aW^+  ) is a rather model-independent prediction since in such cases EW precision measurements require sinθ to be small and Γ ( H^+ → a W^+  )/ Γ ( H^+ → H W^+  ) ∝sin^2 θ. The same is not true for the hierarchy between BR (H^+ → t b̅ ) and BR (H^+ → HW^+  ) which depends sensitively on the choice of tanβ since Γ ( H^+ → t b̅ )/ Γ ( H^+ → H W^+  ) ∝ 1/tan^2 β. It follows that for values of tanβ > 1 the H^+ → H W^+ channel can also be the dominant decay mode. In model realisations with M_A < M_H^± there are no constraints from Δρ on sinθ and in turn the H^+ → aW^+ branching ratio can dominate for sufficiently large mixing in the pseudoscalar sector. This feature is illustrated by the right panel in the figure using sinθ = 1/√(2) and M_A= 500   GeV. For this choice of input parameters we find that BR (H^+ → aW^+  ) >  BR (H^+ → t b̅ ) for masses M_a ≲ 300   GeV. Since the  pseudoscalar a predominantly decays via a →χχ̅ it follows that THDM plus pseudoscalar extensions with M_A < M_H^± can lead to a resonant mono-W signal. We will discuss the LHC prospects for the detection of such a E_T, miss signature in Section <ref>. 
         
        
        
        § ANATOMY OF MONO-X SIGNATURES
        
        
        In this section we will discuss the most important features of the mono-X phenomenology of the pseudoscalar extensions of the THDM. We examine the mono-jet, the t t̅ + E_T,  miss, the mono-Z and the mono-Higgs signature. The b b̅ + E_T,  miss and mono-W  channel are also briefly considered. Our numerical analysis of the mono-X signals is postponed to Section <ref>.
        
        
        
         §.§ Mono-jet channel
        
        
        A first possibility to search for pseudoscalar interactions of the form (<ref>) consists in looking for a mono-jet signal, where the mediators that pair produce DM are radiated from heavy-quark loops <cit.>. Representative examples of the possible one-loop Feynman diagrams are shown in Figure <ref>. 
        
        For m_a > 2 m_χ and M_A≫ M_a only graphs involving the exchange of the light pseudoscalar a will contribute to the j + E_T,  miss 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 tanβ is small, the ratio of the fiducial cross sections in the two models is thus approximately given by the simple expression 
        σ (pp → j + E_T,  miss )/σ (pp → j + E_T,  miss )_ DMF≃ ( y_χsinθ/g_χ g_q tanβ )^2  .
        
        Here g_χ (g_q) 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 tanβ =  O (1) since bottom-quark loops have only an effect of a few percent on the j + E_T,  miss distributions  (see for instance <cit.> for a related discussion in the context of Higgs physics). Using the approximation (<ref>)  it is straightforward to recast existing mono-jet results on the DMF pseudoscalar model such as those given in <cit.> 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 j + E_T,  miss  cross sections including both top-quark and bottom-quark loops as well as the exchange of both a and A mediators. 
        
        
        
         §.§ t t̅/b b̅ + E_T,  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 t t̅ pairs <cit.>. Figure <ref> displays examples of tree-level diagrams that give rise to a t t̅ + E_T,  miss signature in the pseudoscalar extensions of the THDM model. 
        
        In the case that A is again much heavier than a, the signal strength for t t̅ + E_T,  miss 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 (<ref>). Using such a simple recasting procedure we find that the most recent  ATLAS <cit.> and CMS searches for t t̅+ E_T,  miss <cit.>  that are based on 13.2   fb^-1 and 2.2   fb^-1 of 13 TeV LHC data, respectively, only allow to set very weak bounds on tanβ.  For instance for M_a = 100   GeV, y_χ =1 and m_χ = 1   GeV  a lower limit of tanβ≳ 0.2 is obtained. The  t t̅+ E_T,  miss 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 <ref> are based on the search strategy developed recently in <cit.> which employs a shape fit to the difference in pseudorapidity of the two charged leptons in the di-leptonic channel of t t̅+ E_T,  miss.
        
        Besides t t̅ + E_T,  miss also b b̅ + E_T,  miss production <cit.> has been advocated as a sensitive probe of spin-0 portal couplings to heavy quarks. Recasting the most recent 13 TeV LHC b b̅ + E_T,  miss searches <cit.> by means of a simple rescaling  similar to (<ref>) we find that no relevant bound on the parameter space of our simplified model can be derived unless the a b b̅ coupling is significantly enhanced. From (<ref>) 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 tanβ also direct searches for the light pseudoscalar a in final states containing bottom quarks or charged leptons are relevant (and naively even provide the leading constraints) we do not consider the b b̅ + E_T,  miss channel in what follows, restricting our numerical analysis to the parameter space with small tanβ. 
         
        
        
         §.§ Mono-Z channel
        
        
        A mono-X signal that is strongly suppressed in the case of the spin-0 DMF models <cit.> but will turn out to be relevant in our simplified DM scenario is the mono-Z channel <cit.>. A sample of one-loop diagrams that lead to such a signature are displayed in Figure <ref>. Notice that the left diagram in the figure allows for resonant Z + χχ̅ production through a HaZ vertex for a sufficiently heavy scalar H. Unlike the graph on the right-hand side it has  no counterpart in the spin-0 DMF simplified models. 
        
        
        
        As first emphasised in <cit.> the appearance of the contribution with virtual H and a exchange not only enhances the mono-Z cross section  compared to the spin-0 DMF models, but also leads to quite different kinematics in Z + χχ̅ production. In fact, for  masses M_H > M_a + M_Z the predicted E_T ,  miss spectrum turns out to be peaked at 
        
        E_T,  miss^ max≃λ^1/2 (M_H, M_a, M_Z)/2 M_H ,
        
        where the two-body phase-space  function λ(m_1, m_2, m_3) has been defined in (<ref>). Denoting the lower experimental requirement on E_T,  miss in a given mono-Z search by E_T,  miss^ cut the latter result can be used to derive a simple bound on M_H for which a significant fraction of the total cross section will pass the cut. We obtain the inequality 
        
        M_H ≳ M_a + √(M_Z^2+ ( E_T,  miss^ cut)^2) .
        
        Given that in the latest mono-Z analyses  <cit.>  selection cuts of E_T,  miss^ cut≃ 100   GeV are imposed it follows that the scalar H has to have a mass of M_H ≃ 500   GeV if one wants to be sensitive to pseudoscalars a with masses up to the t t̅ threshold M_a ≃ 350   GeV. 
        
        Our detailed Monte Carlo (MC) simulations of the Z + E_T,  miss signal in Section <ref> however reveals  that the above kinematical argument alone is insufficient to  understand the shape of the mono-Z exclusion in the M_a–tanβ plane in all instances. The reason for this is twofold. First, in cases where sinθ is small H → aZ is often not the dominant H decay mode and as a result the Z+ E_T,  miss measurements lose already sensitivity for masses M_a below the bound implied by the estimate (<ref>). Second, Z + χχ̅ production in gg → a Z  and  gg → A Z is also possible through box diagrams, and the interference between triangle and box graphs turns out to  be very relevant  in models that have a light scalar H or pseudoscalar A with a mass below the t t̅ threshold. We add that for tanβ >  O (10) also resonant mono-Z production via b b̅→ aZ and b b̅→ AZ can be relevant in models of type II and IV. In the context of the pure THDM such effects have been studied for instance in <cit.>.
         
        
        
         §.§ Mono-Higgs channel
        
        
        In certain regions of parameter space another possible smoking gun signature of the pseudoscalar extensions of the THDM turns out to be mono-Higgs production. As illustrated in Figure <ref> this signal can  arise from two different types of one-loop diagrams. For M_A > M_a + M_h the triangle graph with an Aah vertex depicted on the left-hand side allows for resonant mono-Higgs production and  thus dominates over the contribution of the box diagram displayed on the right. In consequence the mono-Higgs production cross sections in the THDM plus pseudoscalar extensions can exceed by far the small  spin-0 DMF model rates for the h + E_T,  miss signal <cit.>.
        
        
        
        Like in the case of the mono-Z signal the presence  of triangle diagrams with a trilinear scalar coupling also leads to a peak in the E_T,  miss distribution of h + χχ̅ production if the intermediate heavy pseudoscalar A can be resonantly produced. The peak position in the mono-Higgs case is obtained from  <cit.>
        E_T,  miss^ max≃λ^1/2 (M_A, M_a, M_h)/2 M_A .
        
        It follows that in order for events to pass the E_T,  miss cut necessary for a background suppression in mono-Higgs searches, the relation 
        
        M_A ≳ M_a + √(M_h^2 + ( E_T,  miss^ cut)^2) ,
        
        has to be fulfilled. A lesson to learn from (<ref>) is that mono-Higgs searches  in the h → b b̅ channel <cit.> are less suited to constrain the parameter space of our simplified model than those that focus on h →γγ<cit.>, because the minimal E_T,  miss requirements in the former analyses are always stricter than those in the latter. To give a relevant numerical example let us consider  E_T,  miss^ cut≃ 100   GeV, which represents a typical E_T,  miss cut imposed in the most recent h + χχ̅  (h →γγ) searches. From (<ref>) one sees that in such a case mono-Higgs analyses are very sensitive to masses up to M_a ≃ 330   GeV for M_A ≃ 500   GeV. 
        
        Like in the mono-Z case the above  kinematical argument however allows only for a qualitative understanding of the numerical results for the pp → h + χχ̅  (h →γγ) exclusions, since interference effects can be important in scenarios with a pseudoscalar A of mass M_A <2 m_t.   Notice that if M_a > M_A + M_h the role of A and a is interchanged and the h + E_T,  miss signal can receive large corrections from resonant  a exchanges,  as we will see explicitly  in Section <ref>.  Finally  in  type II and IV  models resonant mono-Higgs production from b b̅ initial states can also be important if tanβ is sufficiently large.
        
        
        
         §.§ Mono-W channel
        
        
        The last E_T,  miss signal that we consider is the mono-W channel <cit.>. Two representative Feynman graphs that lead to a resonant W + E_T,  miss signature in the pseudoscalar extension of the THDM are shown in Figure <ref>. These diagrams describe the single production of a charged Higgs H^± via the annihilation of light quarks followed by H^±→ a W^±  (a →χχ̅). One way to assess the prospects for detecting a mono-W signature consists in comparing the production cross sections of H^± to that of H and A. Using for instance tanβ = 1, we find σ ( pp → H^+  ) ≃ 1.0   fb for M_H^± = 500   GeV and σ ( pp → H^+  ) ≃ 0.2   fb for M_H^± = 750   GeV at the 13 TeV LHC. The corresponding cross sections in the case of the heavy  neutral spin-0 resonances read σ ( pp → H  ) ≃ 1.4   pb and σ ( pp → A  ) ≃ 3.1   pb and σ ( pp → H  ) ≃ 0.2   pb and σ ( pp → A  ) ≃ 0.3   pb, respectively. These numbers strongly suggest that an observation of a mono-W signal is compared to that of a mono-Z or mono-Higgs signature much less probable. We thus do not consider the W + E_T,  miss channel any further. 
        
        Let us finally add that besides a simple mono-W signature also Wt + E_T,  miss and Wtb+E_T,  miss signals can appear in the DM model introduced in Section <ref>.  For the relevant charged Higgs  production cross sections we find at 13 TeV the results σ ( g b̅→ H^+ t̅ ) ≃ 0.17   pb (σ ( g b̅→ H^+ t̅ ) ≃ 0.04   pb) and σ ( g g → H^+  t̅  b ) ≃ 0.10   pb (σ ( gg → H^+ t̅ b  ) ≃ 0.02   pb) using tanβ =1 and M_H^± = 500   GeV (M_H^± = 750   GeV). Given the small  H^±  production cross section in gb and gg fusion, we expect that searches for a Wt + E_T,  miss or a Wtb + E_T,  miss signal will  in practice provide no relevant constraint in the small tanβ regime. 
        
        
        
        § NUMERICAL RESULTS
        
        
        The numerical results of our mono-X analyses are presented in this section. After a brief description of the signal generation and the background estimates, we first study the impact of interference effects between the a and A contributions to  the j + χχ̅ and t t̅ + χχ̅ channels. Then the constraints on the parameter space of the THDM plus pseudoscalar extensions  are derived for several well-motivated benchmark scenarios. In the case of the mono-Z and mono-Higgs searches we also discuss the LHC Run II reach in some detail.  
        
        
        
         §.§ Signal generation
        
        
        The starting point of our MC simulations is a   UFO implementation <cit.> of the simplified model as described in Section <ref>. This implementation has been obtained by means of the  FeynRules 2 <cit.> and   NLOCT <cit.> packages. The generation of the j + E_T,  miss, Z + E_T,  miss  (Z →ℓ^+ ℓ^-) and h + E_T,  miss  (h →γγ) signal samples is performed at leading order (LO) with  MadGraph5_aMC@NLO <cit.> using  PYTHIA 8.2<cit.> for showering and  NNPDF2.3 <cit.> as parton distribution functions. The whole MC chain is steered with  CheckMATE 2 <cit.> which itself employs  FastJet <cit.> to reconstruct hadronic jets and  Delphes 3 <cit.> as a fast-detector simulation. The results of the   CheckMATE 2 analyses have been validated against  MadAnalysis 5 <cit.>.  The selection requirements imposed in our analyses resemble those used in the recent  LHC mono-jet <cit.>, mono-Z <cit.>  and mono-Higgs <cit.>  search, respectively. For what concerns our t t̅ + E_T,  miss  (t →ℓ b ν) recast we rely on the results of the sensitivity study <cit.>.  In this analysis the DM signal has been simulated at next-to-leading order (NLO) with  MadGraph5_aMC@NLO and  PYTHIA 8.2 using a  FxFx NLO jet matching prescription <cit.> and the final-state top quarks have been decayed with  MadSpin <cit.>.
        
        
        
         §.§ Background estimates
        
        
        For the j + E_T,  miss, Z + E_T,  miss  (Z →ℓ^+ ℓ^-) and h + E_T,  miss  (h →γγ)  recasts our background estimates rely on the background predictions obtained in the 13 TeV LHC analyses <cit.>, <cit.>  and <cit.>, respectively.  The given background numbers correspond to  3.2   fb^-1, 13.3   fb^-1, 2.3   fb^-1  and we extrapolate them to 40   fb^-1 of integrated luminosity to be able to assess the near-term reach of the different mono-X channels. Our extrapolations assume that while the relative systematic uncertainties remain the same, the relative statistical errors scale  as 1/√(L) with luminosity L. Depending on the signal region the relative systematic uncertainties amount to around 4% to 9% in the case of the mono-jet search, about 7% for the mono-Z analysis and approximately 20% for the mono-Higgs channel. 
        
        Since the j + E_T,  miss search is already systematics limited at 40   fb^-1 its constraining power  will depend sensitively on the assumption about the future systematic uncertainty on the associated SM background. This  should be kept  in mind when comparing the different exclusions presented below, because a better understanding of the backgrounds can have a visible impact on the obtained results. Since the t t̅ + E_T,  miss  (t →ℓ b ν) search will still be statistically limited for  40   fb^-1, we  base our forecast in this case  on a data set of  300   fb^-1  assuming that the relevant SM background is known to 20%. In the mono-Z and mono-Higgs cases we will  present below, besides 40   fb^-1 projections, results for 100   fb^-1 and 300   fb^-1 of data. From these results one can assess if the existing Z + E_T,  miss and h + E_T,  miss search strategies will at some point become systematics limited  in LHC Run II.
        
        
        
         §.§ Interference effects
        
        
        Our simplified model contains two pseudoscalar mediators a and A that are admixtures of the  neutral CP-odd weak eigenstates entering  (<ref>) and (<ref>). In mono-jet production the two contributions interfere and the resulting  LO matrix element  takes the following schematic form
         M ( p p → j + χχ̅ ) ∝1/m_χχ̅^2 -M_a^2 - i M_a Γ_a  - 1/m_χχ̅^2 -M_A^2 - i M_A Γ_A ,
        
        where m_χχ̅ denotes the invariant mass of the DM pair and Γ_a and Γ_A are  the total decay widths of the two pseudoscalar mass eigenstates.
        The same results hold for instance also in the case of the pp → t t̅ +  χχ̅ amplitude. Notice that the contributions from virtual a and A exchange have opposite signs in (<ref>) resulting from the transformation from the weak to the mass eigenstate basis. Such a destructive interference of  two contributions also appears in fermion scalar singlet models with Higgs mixing and has there  shown to be phenomenologically relevant <cit.>. 
        
        The impact of interference effects on the predictions of the mono-jet and t t̅ + E_T,  miss cross sections is illustrated in Figure <ref> for three different values of the mass of the pseudoscalar A.  Both plots display partonic LO results at 13 TeV LHC energies.  In the left  panel the basic selection requirements E_T,  miss > 250   GeV and |η_j| < 2.4  are imposed with η_j denoting the pseudorapidity of the jet, while in the right figure only the cut E_T,  miss > 150   GeV is applied. Focusing first on  the cross sections for M_A = 750   GeV (red curves), one observes that in this case interference effects do not play any role since the pseudoscalar A is too heavy and effectively decouples. One also sees  that at M_a ≃ 350   GeV the cross sections of both mono-jet and t t̅ + E_T,  miss production are enhanced due to t t̅ threshold effects. Notice furthermore that the enhancement is more pronounced for the j + E_T,  miss signal because the top-quark loops develop an imaginary piece once the internal tops can go on-shell. 
        
        
        
        The results for M_A = 500   GeV (green curves) resemble closely those for M_A = 750   GeV until M_a ≃ M_A - M_h ≃ 375   GeV at which point one observes an enhancement of the rates compared to the case of very heavy A. This feature is a consequence of  the fact that for M_a < M_A - M_h the A → ah channel is the dominant decay mode of A, as can be seen from the right plot in Figure <ref>. For larger masses of a the phase space of A → ah closes  and in turn BR (A →χχ̅ ) increases. This leads to  constructive interference between the two terms in (<ref>) until M_a ≃ M_A = 500   GeV where the interference becomes destructive. Notice furthermore that the same qualitative explanations apply to the case of M_A = 300   GeV (blue curves) where the constructive and destructive interference takes place at M_a ≃ M_A - M_h ≃ 175   GeV and M_a ≃ M_A = 300   GeV, respectively. Comparing the left and right panel of Figure <ref>, one finally sees that the observed interference pattern is at the qualitative level independent of the choice of sinθ. 
        
        
        
         §.§ Summary plots
        
        
        Below we study four different benchmark scenarios that exemplify the rich E_T, miss phenomenology of the simplified DM model introduced in Section <ref>. Throughout our analysis we work in the alignment/decoupling limit, adopting the parameters M_H^± = 750   GeV, λ_3 = λ_P1 = λ_P2 = 0, y_χ = 1 and m_χ= 1   GeV and consider a Yukawa sector of type II.  The shown results however also hold in the case of the other Yukawa sectors (<ref>) since for tanβ =  O (1) effects of bottom-quark loops in mono-jet, mono-Z and mono-Higgs production amount to corrections  of a few percent only. The model-dependent contributions from b b̅-initiated production also turn  out to be small for such values of tanβ. The constraints on all benchmark scenarios will be presented in the M_a–tanβ plane, in which  the  parameter regions that are excluded at 95% CL  by the various searches will be indicated.
        
        
        
          §.§.§ Benchmark scenario 1: sinθ = 0.35, M_H = 500   GeV
        
        
        In the first benchmark scenario we choose sinθ = 0.35, M_H = 500   GeV and  M_A = 750   GeV, where the choice of sinθ guarantees that EW precision measurements are satisfied for  all  values of M_a that we consider (see Section <ref>). The upper left panel in Figure <ref> summarises the various 95% CL exclusions. One first observes that the constraint from invisible decays of the Higgs (pink region) excludes all shown values of tanβ for mediator masses of M_a ≲ 100   GeV. This constraint has been obtained by imposing the 95% CL limit BR ( h → invisible ) < 25% set  by ATLAS <cit.>. Notice that in the THDM plus pseudoscalar extensions one has BR ( h → invisible ) ≃ BR ( h → 2 χ 2 χ̅ ) ≃ 100% for a DM mass of m_χ = 1   GeV largely independent of sinθ, M_H and M_A, and as a result the h → invisible constraint is roughly the same in all of our benchmark scenarios. One furthermore sees that taken together the existing limits  from flavour physics (dotted black line) and di-top searches (dashed black curve)  exclude the parameter region with tanβ≲ 0.8. Here the di-top constraint is obtained from the results <cit.> by rescaling the limit quoted by ATLAS using the t t̅ branching ratio of the heavy scalar mediator H (see Section <ref>). 
        
        Turning ones attention to the constraints that arise from DM searches, one observes that even with an integrated luminosity of 300   fb^-1, t t̅ + E_T,  miss measurements (green region) should be able to exclude only a small part of the M_a–tanβ plane. For pseudoscalar masses M_a around the EW scale values of tanβ≲ 0.6  can be tested, while t t̅ + E_T,  miss searches have essentially no sensitivity to the parameter region with M_a ≳ 2 m_t  since the decay channel a → t t̅ opens up. The weakness of the t t̅ + E_T,  miss constraint  is expected (see (<ref>)) since  the t t̅ + a production cross section is suppressed by sin^2 θ≃ 0.1 in our first benchmark. This suppression is also the reason for our finding that with 40   fb^-1 of 13 TeV data, mono-jet searches will not lead to any relevant restriction on tanβ, if one assumes that these near-future measurements are plagued by systematic uncertainties at the 5% level in the low-E_T,  miss signal regions.  
        
        The hypothetical mono-Z search (blue region) based on 40   fb^-1 of data provides the strongest  constraint for M_a ≲ 250   GeV, excluding tanβ values slightly above 2 for light mediators a. This strong bound is a result of the resonant enhancement of Z + χχ̅ production in our first benchmark scenario. Notice furthermore the sharp cut-off  of the Z + E_T,  miss exclusion at M_a ≃ 260   GeV. For larger pseudoscalar masses M_a one finds that BR (H → aZ  ) ≲ 10%  (see the right panel in Figure <ref>) and as a result  mono-Z production through triangle graphs is strongly reduced. This explains why the Z + E_T,  miss search looses sensitivity already before M_a ≃ 350   GeV as one would naively expect from (<ref>) for the E_T,  miss^ cut = 120   GeV  high-mass signal region  requirement imposed in <cit.>. We finally see that with 40   fb^-1 of integrated luminosity mono-Higgs searches (orange region) can  cover only a small part of the parameter space compared to mono-Z measurements. 
        
        
        
          §.§.§ Benchmark scenario 2: sinθ = 0.25, M_H = 300   GeV
        
        
        In our second benchmark scenario, the sine of the mixing angle is sinθ =0.25 and the masses of H and A are taken to be M_H = 300   GeV and  M_A = 750   GeV. The corresponding exclusion contours are depicted in the upper right panel of  Figure <ref>. The constraints from h → invisible (pink region) and flavour physics (dotted black line) resemble the exclusions that apply in the first benchmark case. The recent ATLAS di-top search does instead not lead to a constraint since, on the one hand, t t̅ decays of the scalar H are kinematically forbidden, and on the other hand, the  ATLAS sensitivity to very heavy pseudoscalars A is not sufficient to set a bound on tanβ.  
        
        Given the smallness of sinθ, we find that our hypothetical   t t̅ + E_T,  miss search  only probes the parameter region with M_a ≲ 2m_t and tanβ≲ 0.4. Mono-jet measurements are expected to provide even weaker restrictions and in consequence  we do not show  the corresponding bounds. As in the case of the first benchmark scenario, the mono-Z exclusion (blue region) is the most stringent constraint for a large range of M_a values, excluding values of tanβ≲ 1.5 for  M_a ≃ M_h. The dip of the exclusion limit at M_a ≃ 170   GeV coincides with the bound derived in (<ref>) if the low-mass signal region requirement E_T,  miss^ cut = 90   GeV <cit.> is imposed.  One also observes that for larger mediator masses the mono-Z exclusion strengthens until the point where M_a ≃ 220   GeV. This is a result of the constructive interference  between triangle and box graphs (see Figure <ref>). The bound that follows from our  40   fb^-1 mono-Higgs projection (orange region) is compared to the mono-Z exclusion again rather weak. 
        
        
        
          §.§.§ Benchmark scenario 3: sinθ = 1/√(2), M_A = 500   GeV
        
        
        Our third benchmark scenario employs sinθ = 1/√(2), M_A = 500   GeV and  M_H  = 750   GeV. Notice that for M_H = M_H^±  the mixing in the pseudoscalar sector can be  large since there are no constraints on sinθ and M_a from Δρ. The  constraints on the M_a–tanβ plane corresponding to these parameter choices  are presented in the lower left panel of Figure <ref>. The bounds from h → invisible decays (pink region) and flavour physics (dotted black line) are essentially unchanged with respect to the previous benchmarks. The shown di-top constraint (dashed black curve) differs from the one displayed in the upper left panel since it follows from the bound  provided in <cit.> for a pseudoscalar with a mass of 500   GeV. 
        
        In the case of the mono-jet constraint (red region) one sees that it should now be possible to exclude tanβ≲ 0.4 values for M_a ≲ 350   GeV. One furthermore observes that future t t̅ + E_T,  miss searches (green region) are expected to extend the parameter space excluded by the non-E_T,  miss constraints to tanβ values above 1  for  M_a ≲ 200   GeV. Although the scalar H is very heavy, we find that the  mono-Z projection (blue region) still provides relevant constraints in the  M_a–tanβ for masses below the a → t t̅ threshold, because the mixing angle θ is maximal in our third benchmark. The strongest E_T,  miss constraint  is however provided by the mono-Higgs search (orange region),  which should be able to exclude values of tanβ≲ 2 for pseudoscalars a with masses at the EW scale. Notice that the mono-Higgs exclusion has a sharp cut-off at M_a ≃ 350   GeV, as expect from the inequality (<ref>) for E_T,  miss^ cut = 105   GeV <cit.>. 
        
        
        
          §.§.§ Benchmark scenario 4: sinθ = 1/√(2), M_A = 300   GeV
        
        
        In the fourth benchmark we consider the parameters sinθ = 1/√(2), M_A = 300   GeV and  M_H =  750   GeV. As can be seen from the lower right panel of Figure <ref>, the regions excluded by  Higgs to invisible decays (pink region) and flavour physics (dotted black line) are close to identical to those arising in all the other  scenarios. In contrast, di-top searches do not lead to a restriction because the pseudoscalar A is too light to decay to two on-shell top quarks, while the ATLAS search <cit.> is not yet sensitive to very heavy scalars H. 
        
        The shapes of the exclusions from the j + E_T,  miss (red region) and t t̅ + E_T,  miss (green region) measurements display an interference pattern that is very similar to the one seen in Figure <ref>. In turn  future mono-jet (t t̅ + E_T,  miss) searches are expected to be able to exclude tanβ≲ 0.4  (tanβ≲ 1) values for mediator masses M_a above the t t̅ threshold. Focusing  our attention on the mono-Z projection (blue region) we observe that the corresponding exclusion curve has a pronounced dip at M_a ≃ 180   GeV. It originates from the interference of triangle diagrams with box graphs  that correspond to gg → AZ → Z+χχ̅ (see Figure <ref>). This interference is destructive and maximal when the decay channel A → aZ starts to close,  leading to Br (A →χχ̅ ) ≃ 100% for the considered value of M_A. 
        
         Like in the third benchmark  the mono-Higgs search (orange region) is again the most powerful E_T,  miss constraint as it allows to exclude tanβ≲ 3.7 values for M_a ≃ 100   GeV. We also note that the mono-Higgs search maintains sensitivity for M_a values well above the estimate presented in (<ref>). The reason is that for sufficiently light pseudoscalars A,  triangle diagrams with resonant a exchange (see Figure <ref>) can provide a sizeable contribution to mono-Higgs production. This resonant enhancement allows one to probe values of tanβ above 1 for M_a ≳ 300   GeV. Notice finally that at M_a ≃ M_A  = 300   GeV the a and A  contributions  interfere destructively leading to a visible dip in the h + E_T,  miss exclusion.  
        
        
        
         §.§ LHC Run II reach
        
        
        The future prospects of the mono-Z (blue regions) and mono-Higgs (orange regions) constraints are illustrated in Figure <ref> for  our four benchmark scenarios. We find that by collecting more data the reach of the Z + E_T,  miss measurements are expected to strengthen, but that the actual improvement depends sensitively on the assumption about the systematic uncertainty on the irreducible SM backgrounds. Assuming a systematic error of 7%, we observe that the limits on tanβ will improve by a mere 10% when going from 40   fb^-1 to 300   fb^-1 of data. In order to further exploit the potential of mono-Z searches, advances in the modelling of ZZ production within the SM would hence be very welcome. 
        
        
        
        In contrast to mono-Z searches it turns out that in the case of the h + E_T,  miss measurements systematic uncertainties will not be a limiting factor even at the end of LHC Run II. By increasing the amount of data to 100   fb^-1 and  300   fb^-1, we anticipate that it should be possible to improve the 40   fb^-1 mono-Higgs limits on tanβ by typically 25% and 50%, respectively. Notice that larger data sets will be most beneficial in our first and second benchmark scenario in which sinθ is small.  In these cases  the resulting h + E_T,  miss  (h →γγ)  event rates are so low that the sensitivity in the mono-Higgs channel is limited largely by statistics for 40   fb^-1 of luminosity. 
        
        As explained earlier in Section <ref>, we expect that forthcoming searches for  spin-0 resonances in the τ^+ τ^- final state should allow to set relevant constraints on tanβ in model realisations with a light scalar H of mass M_H< 2 m_t. In  the case of our second benchmark scenario this means that it should be possible to test and to exclude the parameter space with tanβ≲ O (1)  and M_a ≳ 210    GeV at LHC Run II. Such an exclusion would indeed be precious,  because as illustrated by the upper right panel of  Figure <ref>, this part of the  M_a–tanβ plane is notoriously difficult to  constrain through E_T,  miss searches. 
        
        
        
        Finally, let us comment on an effect already mentioned  briefly in Sections <ref> and <ref>. In pseudoscalar extensions of the THDM that feature a  tanβ enhancement of the bottom-quark Yukawa coupling it is possible in principle to obtain relevant contributions to mono-X signals not only from the gg → Z/h + E_T, miss transitions but also from the b b̅→ Z/h + E_T, miss channels. In Figure <ref> only the model-independent contribution from gg production was taken into account, because the exclusion bounds remain essentially unchanged if also the b b̅-initiated channels are included.
        
        With 300   fb^-1 of integrated luminosity this situation is however expected to change. Searches for mono-Z signals, for example, should be able to exclude values of tanβ≳ O (8) for certain ranges of M_a  in all four benchmarks. In the third and fourth benchmark scenario there are particularly relevant changes to the projected sensitivity of mono-Higgs searches, as illustrated in Figure <ref>. For M_A = 500   GeV (left panel) we observe that, after including both gg and b b̅ initiated production, model realisations with tanβ≳ 10 for M_a ≲ 220   GeV are excluded. The impact of b b̅→ h + E_T,  miss is even more pronounced for a light A with M_A  = 300   GeV (right panel).  In this case we see that it should be possible to exclude masses M_a ≲ 170   GeV for any value of tanβ. The results displayed in Figure <ref> have been obtained in the context of a Yukawa sector of type II. Almost identical sensitivities are found in models of type IV, while in pseudoscalar THDM extensions of type I and III bottom-quark initiated contributions are irrelevant, since they are tanβ suppressed (see (<ref>)).
        
        
        
        § CONCLUSIONS
        
        
        We have proposed a new framework of renormalisable simplified models for dark matter searches at the LHC, namely single-mediator extensions of two Higgs doublet models containing a fermionic dark matter candidate. The mediator can have both scalar or pseudoscalar quantum numbers and all amplitudes are unitary as long as the mediator couplings are perturbative. Constraints from Higgs coupling measurements are averted by mixing the mediator with the heavy scalar or pseudoscalar partners of the Standard Model Higgs. This framework unifies previously established simplified  spin-0 models, while avoiding their shortcomings, and can reproduce several of their features in the appropriate limit. 
        
        In this work we have focused on the case of a pseudoscalar mediator a. We have considered the alignment/decoupling limit, in which some of the Higgs partners have masses close to the TeV scale, while either the neutral scalar H or pseudoscalar A is lighter with a mass as low as 300   GeV. For the mass of the new pseudoscalar mediator we have considered the range of half the Higgs-boson mass to 500   GeV.  These parameter choices are well motivated by Higgs physics, LHC searches for additional spin-0 states, electroweak precision measurements and quark-flavour bounds such as those arising from B → X_s γ and B-meson  mixing. Limits on the quartic couplings that arise from perturbativity, unitarity and the requirement that the  total decay widths of H and A are sufficiently small for the  narrow-width approximation to be valid have also been taken into account in our analysis.
        
        By studying the partial decay widths and branching ratios of the spin-0 particles, we have found that the total decay width of the heavier scalar H can be dominated by the H → a Z channel, while the heavier pseudoscalar A generically decays with large probability through A → a h.  In consequence, the production cross sections for mono-Z and mono-Higgs final states are resonantly enhanced and the obtained limits are competitive with mono-jet searches and even impose the dominant constraints for most of the parameter space at 40   fb^-1 of 13 TeV LHC data. This surprising result is a consequence of a consistent implementation of the scalar sector and is therefore not predicted by previously considered simplified models (such as the ATLAS/CMS Dark Matter Forum pseudoscalar model). Our findings underline the importance of a complementary approach to searches for dark matter at the LHC and are in qualitative agreement with the conclusions drawn in <cit.>. 
        
        We have furthermore  emphasised, that searches for associated production of dark matter with a tt̅ pair will profit from improved statistics unlike the mono-jet search, for which the reach seems systematics limited. We have therefore extrapolated the corresponding constraints to a dataset of 300    fb^-1, where  t t̅ + E_T,  miss searches are  expected  to be more powerful than j +  E_T,  miss measurements for large parts of the parameter space.
        
        The rich structure of the  two Higgs doublet plus pseudoscalar models has been exemplified by an analysis of four different parameter scenarios. The specific benchmarks have been chosen  to capture different aspects of the mono-X phenomenology that are of interest for future LHC searches.  The results for all scenarios are presented in the form of  M_a–tanβ planes, in which  the  parameter regions that are excluded at 95% confidence level  by the various E_T,  miss and non-E_T,  miss searches have been indicated  (see Figure <ref>). We found that the constraining power of mono-Z and mono-Higgs searches depends sensitively on the mass hierarchies between M_a and M_A or M_H, while the sensitivity to the other model parameters such as the amount of mixing in the CP-odd sector is less pronounced. It has also been shown that as a result of the interference of a and A contributions the bounds in the  M_a–tanβ plane that result from  the j + E_T,  miss and t t̅ + E_T,  miss channels strengthen above the threshold M_a ≃ M_A in model realisations with a light pseudoscalar A. In addition the reach of the 13 TeV LHC in the mono-Z and mono-Higgs channel has been explored (see Figure <ref>). While mono-Higgs searches are  found  not to be limited by systematic uncertainties even at the end of LHC Run II, in the case of the mono-Z measurements the systematic error can become a limiting factor.  This feature makes the h + E_T,  miss signal particularly interesting in the context of two Higgs doublet  plus pseudoscalar extensions. 
        
        It has moreover been pointed out that  constraints from di-top resonance searches and flavour observables provide further important handles to test the considered simplified dark matter models. Because the former signature allows to look for neutral spin-0 states with masses above the t t̅ threshold, to which E_T,  miss searches have only limited access if the dark matter mediators are top-philic, the development of more sophisticated  strategies to search for heavy neutral Higgses  in t t̅ events seems particularly timely. We have also highlighted the possibility to constrain benchmark scenarios featuring  a  light scalar H by forthcoming searches for heavy spin-0 states  in the τ^+ τ^- final state, and finally illustrated the impact of bottom-quark initiated production in the case of h + E_T,  miss (see Figure <ref>).
        
        To conclude, we stress that meaningful bounds from LHC searches for dark matter can only be extracted if the underlying models are free from theoretical inconsistencies, such as  non-unitary scattering amplitudes or couplings that implicitly violate gauge symmetries. Future ATLAS and CMS analyses of spin-0 mediator scenarios should therefore be based on consistent embeddings of the established ATLAS/CMS Dark Matter Forum  simplified models. For any effort in this direction,  standalone  UFO implementation of the dark matter models discussed in this article can be obtained from the authors on request. 
        
         
        We thank all participants of the fourth LHC Dark Matter Working Group public meeting, in particular Nicole Bell, Giorgio Busoni and Jose Miguel No, for interesting discussions. We are grateful to Stefan Liebler for pointing out the potential relevance of bottom-quark initiated production processes. UH acknowledges partial support by the ERC Consolidator Grant HICCUP (No. 614577) and thanks the CERN Theoretical Physics Department for hospitality. 
        
        10Abdallah:2014hon 
          J. Abdallah  et al.,
          
          arXiv:1409.2893 [hep-ph].
          
          Abdallah:2015ter 
          J. Abdallah  et al.,
          
          Phys. Dark Univ.  9-10, 8 (2015)
          
          [arXiv:1506.03116 [hep-ph]].
          
          
          
        Abercrombie:2015wmb 
          D. Abercrombie  et al.,
          
          arXiv:1507.00966 [hep-ex].
          
          
          
        Chala:2015ama 
          M. Chala, F. Kahlhoefer, M. McCullough, G. Nardini and K. Schmidt-Hoberg,
          
          JHEP  1507, 089 (2015)
          
          [arXiv:1503.05916 [hep-ph]].
          
          
          
        Bell:2015sza 
          N. F. Bell, Y. Cai, J. B. Dent, R. K. Leane and T. J. Weiler,
          
          Phys. Rev. D  92, no. 5, 053008 (2015)
          
          [arXiv:1503.07874 [hep-ph]].
          
          
          
        Kahlhoefer:2015bea 
          F. Kahlhoefer, K. Schmidt-Hoberg, T. Schwetz and S. Vogl,
          
          JHEP  1602, 016 (2016)
          
          [arXiv:1510.02110 [hep-ph]].
          
          
          
        Bell:2015rdw 
          N. F. Bell, Y. Cai and R. K. Leane,
          
          JCAP  1601, no. 01, 051 (2016)
          
          [arXiv:1512.00476 [hep-ph]].
          
          
          
        Haisch:2016usn 
          U. Haisch, F. Kahlhoefer and T. M. P. Tait,
          
          Phys. Lett. B  760, 207 (2016)
          
          [arXiv:1603.01267 [hep-ph]].
          
          
          
        Englert:2016joy 
          C. Englert, M. McCullough and M. Spannowsky,
          
          Phys. Dark Univ.  14, 48 (2016)
          
          [arXiv:1604.07975 [hep-ph]].
          
          
          
        Duerr:2016tmh 
          M. Duerr, F. Kahlhoefer, K. Schmidt-Hoberg, T. Schwetz and S. Vogl,
          
          JHEP  1609, 042 (2016)
          
          [arXiv:1606.07609 [hep-ph]].
          
          
          
        Bauer:2016gys 
          A. Albert  et al.,
          
          Phys. Dark Univ.  16, 49 (2017)
          
          [arXiv:1607.06680 [hep-ex]].
          
          
          
        Deshpande:1977rw 
          N. G. Deshpande and E. Ma,
          
          Phys. Rev. D  18, 2574 (1978).
          
          
          
          
        Ma:2006km 
          E. Ma,
          
          Phys. Rev. D  73, 077301 (2006)
          
          [hep-ph/0601225].
          
          
          
        Barbieri:2006dq 
          R. Barbieri, L. J. Hall and V. S. Rychkov,
          
          Phys. Rev. D  74, 015007 (2006)
          
          [hep-ph/0603188].
          
          
          
         LopezHonorez:2006gr
          L. Lopez Honorez, E. Nezri, J. F. Oliver and M. H. G. Tytgat,
          
          JCAP  0702 (2007) 028
          
          [hep-ph/0612275].
          
          
          
        ATLAS-CONF-2015-044 
          The ATLAS and CMS Collaborations,
          
          https://cds.cern.ch/record/2052552/files/ATLAS-CONF-2015-044.pdfATLAS-CONF-2015-044.
          
          
         
        partII
          M. Bauer, U. Haisch and F. Kahlhoefer, in preparation. 
          
        Bell:2016ekl 
          N. F. Bell, G. Busoni and I. W. Sanderson,
          
          JCAP  1703, no. 03, 015 (2017)
          
          [arXiv:1612.03475 [hep-ph]].
          
          
          
        Ipek:2014gua 
          S. Ipek, D. McKeen and A. E. Nelson,
          
          Phys. Rev. D  90, no. 5, 055021 (2014)
          
          [arXiv:1404.3716 [hep-ph]].
          
          
          
        No:2015xqa 
          J. M. No,
          
          Phys. Rev. D  93, no. 3, 031701 (2016)
          
          [arXiv:1509.01110 [hep-ph]].
          
          
          
        Goncalves:2016iyg 
          D. Goncalves, P. A. N. Machado and J. M. No,
          
          Phys. Rev. D  95, no. 5, 055027 (2017)
          
          [arXiv:1611.04593 [hep-ph]].
          
          
          
        Aaboud:2016tnv 
          M. Aaboud  et al. [ATLAS Collaboration],
          
          Phys. Rev. D  94, no. 3, 032005 (2016)
          
          [arXiv:1604.07773 [hep-ex]].
          
          
          
        CMS:2016pod 
          The CMS Collaboration,
          
          https://cds.cern.ch/record/2205746/files/EXO-16-037-pas.pdfCMS-PAS-EXO-16-037.
          
          
          
        ATLAS:2016ljb 
          The ATLAS Collaboration,
          
          https://cds.cern.ch/record/2206132/files/ATLAS-CONF-2016-050.pdfATLAS-CONF-2016-050.
          
          
         
         CMS:2016mxc 
          The CMS Collaboration,
          
          https://cds.cern.ch/record/2204933/files/EXO-16-005-pas.pdfCMS-PAS-EXO-16-005.
          
          
          
        ATLAS-CONF-2016-086  
        The ATLAS Collaboration,
          
           http://cds.cern.ch/record/2206279/files/ATLAS-CONF-2016-086.pdfATLAS-CONF-2016-086.  
           
        CMS:2016uxr 
          The CMS Collaboration,
          
          http://cds.cern.ch/record/2138506/files/B2G-15-007-pas.pdfCMS-PAS-B2G-15-007.
          
          
          
        ATLAS:2016bza 
          The ATLAS Collaboration,
          
          https://cds.cern.ch/record/2206138/files/ATLAS-CONF-2016-056.pdfATLAS-CONF-2016-056.
          
          
          
        CMS:2016hmx 
          The CMS Collaboration,
          
          https://inspirehep.net/record/1479665/files/EXO-16-038-pas.pdfCMS-PAS-EXO-16-038.
          
          
          
        Sirunyan:2017onm 
          A. M. Sirunyan  et al. [CMS Collaboration],
          
          JHEP  1703, 061 (2017)
          
          [arXiv:1701.02042 [hep-ex]].
          
          
          
        Aaboud:2016obm 
          M. Aaboud  et al. [ATLAS Collaboration],
          
          Phys. Lett. B  765, 11 (2017)
          
          [arXiv:1609.04572 [hep-ex]].
          
          
            
        CMS:2016mjh 
          The CMS Collaboration,
          
          https://cds.cern.ch/record/2202804/files/EXO-16-012-pas.pdf  CMS-PAS-EXO-16-012.
          
          
          
        ATLAS-CONF-2016-011 
          The ATLAS Collaboration,
          
          http://cds.cern.ch/record/2139812/files/ATLAS-CONF-2016-011.pdfATLAS-CONF-2016-011.
          
          
          
        CMS:2016xok 
          The CMS Collaboration,
          
          https://cds.cern.ch/record/2204916/files/EXO-16-011-pas.pdfCMS-PAS-EXO-16-011.
          
          
          
        Aaboud:2016zkn 
          M. Aaboud  et al. [ATLAS Collaboration],
          
          Phys. Lett. B  762, 334 (2016)
          
          [arXiv:1606.03977 [hep-ex]].
          
          
          
        Khachatryan:2016jww 
          V. Khachatryan  et al. [CMS Collaboration],
          
          arXiv:1612.09274 [hep-ex].
          
          
          
        Aad:2015pla 
          G. Aad  et al. [ATLAS Collaboration],
          
          JHEP  1511, 206 (2015)
          
          [arXiv:1509.00672 [hep-ex]].
          
          
          
        Khachatryan:2016whc 
          V. Khachatryan  et al. [CMS Collaboration],
          
          arXiv:1610.09218 [hep-ex].
          
          
          
        Haisch:2015ioa 
          U. Haisch and E. Re,
          
          JHEP  1506, 078 (2015)
          
          [arXiv:1503.00691 [hep-ph]].
          
          
          
        ATLAS:2016pyq 
          The ATLAS Collaboration,
          
          http://cds.cern.ch/record/2206229/files/ATLAS-CONF-2016-073.pdfATLAS-CONF-2016-073.
          
          
          
        HaischTeVPA16
        U. Haisch,
          https://indico.cern.ch/event/469963/contributions/2277614/attachments/1334210/2008314/TeVPA16_Uli.pdftalk “Dark matter at the LHC: Effective field theories, simplified models
          & beyond” at TeVPA,  2016.      
        
        Gunion:1989we 
          J. F. Gunion, H. E. Haber, G. L. Kane and S. Dawson,
          
          Front. Phys.  80, 1 (2000).
          
          Branco:2011iw 
          G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and J. P. Silva,
          
          Phys. Rept.  516, 1 (2012)
          
          [arXiv:1106.0034 [hep-ph]].
          
          
          
        Glashow:1976nt 
          S. L. Glashow and S. Weinberg,
          
          Phys. Rev. D  15, 1958 (1977).
          
          
          
          
        Paschos:1976ay 
          E. A. Paschos,
          
          Phys. Rev. D  15, 1966 (1977).
          
          
          
          
        Haisch:2012kf 
          U. Haisch, F. Kahlhoefer and J. Unwin,
          
          JHEP  1307, 125 (2013)
          
          [arXiv:1208.4605 [hep-ph]].
          
          
          
        ATLAS-CONF-2016-085
          The ATLAS Collaboration,
          
           http://cds.cern.ch/record/2206278/files/ATLAS-CONF-2016-085.pdfATLAS-CONF-2016-085.
          
          
          
        CMS:2016rjp 
          The CMS Collaboration,
          
          https://cds.cern.ch/record/2231507/files/HIG-16-037-pas.pdfCMS-PAS-HIG-16-037.
          
          
        Bobeth:2001sq 
          C. Bobeth, T. Ewerth, F. Krüger and J. Urban,
          
          Phys. Rev. D  64, 074014 (2001)
          
          [hep-ph/0104284].
          
          
          
        CMS:2014xfa 
          V. Khachatryan  et al. [CMS and LHCb Collaborations],
          
          Nature  522, 68 (2015)
          
          [arXiv:1411.4413 [hep-ex]].
          
          
          
        Anastasiou:2016hlm 
          C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog, A. Lazopoulos and B. Mistlberger,
          
          JHEP  1609, 037 (2016)
          
          [arXiv:1605.05761 [hep-ph]].
          
          
          
        Dicus:1994bm 
          D. Dicus, A. Stange and S. Willenbrock,
          
          Phys. Lett. B  333, 126 (1994)
          
          [hep-ph/9404359].
          
          
          
        Frederix:2007gi 
          R. Frederix and F. Maltoni,
          
          JHEP  0901, 047 (2009)
          
          [arXiv:0712.2355 [hep-ph]].
          
          
          
        Djouadi:2015jea 
          A. Djouadi, L. Maiani, A. Polosa, J. Quevillon and V. Riquer,
          
          JHEP  1506, 168 (2015)
          
          [arXiv:1502.05653 [hep-ph]].
          
          
          
        Craig:2015jba 
          N. Craig, F. D'Eramo, P. Draper, S. Thomas and H. Zhang,
          
          JHEP  1506, 137 (2015)
          
          [arXiv:1504.04630 [hep-ph]].
          
          
          
        Denner:1991ie 
          A. Denner, R. J. Guth, W. Hollik and J. H. Kühn,
          
          Z. Phys. C  51, 695 (1991).
          
          
          
          
        Haisch:2007ia 
          U. Haisch and A. Weiler,
          
          Phys. Rev. D  76, 074027 (2007)
          
          [arXiv:0706.2054 [hep-ph]].
          
          
          
        Freitas:2012sy 
          A. Freitas and Y. C. Huang,
          
          JHEP  1208, 050 (2012)
          Erratum: [JHEP  1305, 074 (2013)]
          Erratum: [JHEP  1310, 044 (2013)]
          
          [arXiv:1205.0299 [hep-ph]].
          
          
          
        Hermann:2012fc 
          T. Hermann, M. Misiak and M. Steinhauser,
          
          JHEP  1211, 036 (2012)
          
          [arXiv:1208.2788 [hep-ph]].
          
          
          
        Misiak:2015xwa 
          M. Misiak  et al.,
          
          Phys. Rev. Lett.  114, no. 22, 221801 (2015)
          
          [arXiv:1503.01789 [hep-ph]].
          
          
          
        Czakon:2015exa 
          M. Czakon, P. Fiedler, T. Huber, M. Misiak, T. Schutzmeier and M. Steinhauser,
          
          JHEP  1504, 168 (2015)
          
          [arXiv:1503.01791 [hep-ph]].
          
          
          
        Abbott:1979dt 
          L. F. Abbott, P. Sikivie and M. B. Wise,
          
          Phys. Rev. D  21, 1393 (1980).
          
          
          
          
        Geng:1988bq 
          C. Q. Geng and J. N. Ng,
          
          Phys. Rev. D  38, 2857 (1988)
          Erratum: [Phys. Rev. D  41, 1715 (1990)].
          
          
          
          
        Buras:1989ui 
          A. J. Buras, P. Krawczyk, M. E. Lautenbacher and C. Salazar,
          
          Nucl. Phys. B  337, 284 (1990).
          
          
          
          
        Eberhardt:2013uba 
          O. Eberhardt, U. Nierste and M. Wiebusch,
          
          JHEP  1307, 118 (2013)
          
          [arXiv:1305.1649 [hep-ph]].
          
          
          
        Khachatryan:2015qxa 
          V. Khachatryan  et al. [CMS Collaboration],
          
          JHEP  1511, 018 (2015)
          
          [arXiv:1508.07774 [hep-ex]].
          
          
          
        ATLAS:2016qiq 
          The ATLAS Collaboration,
          
          https://inspirehep.net/record/1480448/files/ATLAS-CONF-2016-089.pdfATLAS-CONF-2016-089.
          
          
          
        Haber:1992py 
          H. E. Haber and A. Pomarol,
          
          Phys. Lett. B  302, 435 (1993)
          
          [hep-ph/9207267].
          
          
          
        Pomarol:1993mu 
          A. Pomarol and R. Vega,
          
          Nucl. Phys. B  413, 3 (1994)
          
          [hep-ph/9305272].
          
          
          
        Gerard:2007kn 
          J.-M. Gerard and M. Herquet,
          
          Phys. Rev. Lett.  98, 251802 (2007)
          
          [hep-ph/0703051].
          
          
          
         Grzadkowski:2010dj 
          B. Grzadkowski, M. Maniatis and J. Wudka,
          
          JHEP  1111, 030 (2011)
          
          [arXiv:1011.5228 [hep-ph]].
          
          
          
        Haber:2010bw 
          H. E. Haber and D. O'Neil,
          
          Phys. Rev. D  83, 055017 (2011)
          
          [arXiv:1011.6188 [hep-ph]].
          
          
          
        Olive:2016xmw 
          C. Patrignani  et al. [Particle Data Group Collaboration],
          
          Chin. Phys. C  40, no. 10, 100001 (2016).
          
          
          
          
        Gunion:2002zf 
          J. F. Gunion and H. E. Haber,
          
          Phys. Rev. D  67, 075019 (2003)
          
          [hep-ph/0207010].
          
          
          
        Barroso:2013awa 
          A. Barroso, P. M. Ferreira, I. P. Ivanov and R. Santos,
          
          JHEP  1306, 045 (2013)
          
          [arXiv:1303.5098 [hep-ph]].
          
          
          
        Kanemura:1993hm 
          S. Kanemura, T. Kubota and E. Takasugi,
          
          Phys. Lett. B  313, 155 (1993)
          
          [hep-ph/9303263].
          
          
          
        Akeroyd:2000wc 
          A. G. Akeroyd, A. Arhrib and E. M. Naimi,
          
          Phys. Lett. B  490, 119 (2000)
          
          [hep-ph/0006035].
          
          
          
        Ginzburg:2005dt 
          I. F. Ginzburg and I. P. Ivanov,
          
          Phys. Rev. D  72, 115010 (2005)
          
          [hep-ph/0508020].
          
          
          
        Grinstein:2015rtl 
          B. Grinstein, C. W. Murphy and P. Uttayarat,
          
          JHEP  1606, 070 (2016)
          
          [arXiv:1512.04567 [hep-ph]].
          
          
          
        Djouadi:1995gv 
          A. Djouadi, J. Kalinowski and P. M. Zerwas,
          
          Z. Phys. C  70, 435 (1996)
          
          [hep-ph/9511342].
          
          
          
        Aad:2014aba 
          G. Aad  et al. [ATLAS Collaboration],
          
          Phys. Rev. D  90, no. 5, 052004 (2014)
          
          [arXiv:1406.3827 [hep-ex]].
          
          
          
        Khachatryan:2014jba 
          V. Khachatryan  et al. [CMS Collaboration],
          
          Eur. Phys. J. C  75, no. 5, 212 (2015)
          
          [arXiv:1412.8662 [hep-ex]].
          
          
          
        Fox:2012ru 
          P. J. Fox and C. Williams,
          
          Phys. Rev. D  87, no. 5, 054030 (2013)
          
          [arXiv:1211.6390 [hep-ph]].
          
          
          
        Haisch:2013ata 
          U. Haisch, F. Kahlhoefer and E. Re,
          
          JHEP  1312, 007 (2013)
          
          [arXiv:1310.4491 [hep-ph]].
          
          
          
        Haisch:2013fla 
          U. Haisch, A. Hibbs and E. Re,
          
          Phys. Rev. D  89, 034009 (2014)
          
          [arXiv:1311.7131 [hep-ph]].
          
          
          
        Buckley:2014fba 
          M. R. Buckley, D. Feld and D. Goncalves,
          
          Phys. Rev. D  91, 015017 (2015)
          
          [arXiv:1410.6497 [hep-ph]].
          
          
          
        Harris:2014hga 
          P. Harris, V. V. Khoze, M. Spannowsky and C. Williams,
          
          Phys. Rev. D  91, 055009 (2015)
          
          [arXiv:1411.0535 [hep-ph]].
          
          
          
        Mattelaer:2015haa 
          O. Mattelaer and E. Vryonidou,
          
          Eur. Phys. J. C  75, no. 9, 436 (2015)
          
          [arXiv:1508.00564 [hep-ph]].
          
          Arina:2016cqj 
          C. Arina  et al.,
          
          JHEP  1611, 111 (2016)
          
          [arXiv:1605.09242 [hep-ph]].
          
          
          
        Bishara:2016jga 
          F. Bishara, U. Haisch, P. F. Monni and E. Re,
          
          Phys. Rev. Lett.  118, no. 12, 121801 (2017)
          
          [arXiv:1606.09253 [hep-ph]].
          
          
          
        Lin:2013sca 
          T. Lin, E. W. Kolb and L. T. Wang,
          
          Phys. Rev. D  88, no. 6, 063510 (2013)
          
          [arXiv:1303.6638 [hep-ph]].
          
          
          
        Artoni:2013zba 
          G. Artoni, T. Lin, B. Penning, G. Sciolla and A. Venturini,
          
          arXiv:1307.7834 [hep-ex].
          
          
          
        Backovic:2015soa 
          M. Backovic, M. Krämer, F. Maltoni, A. Martini, K. Mawatari and M. Pellen,
          
          Eur. Phys. J. C  75, no. 10, 482 (2015)
          
          [arXiv:1508.05327 [hep-ph]].
          
          
          
        Haisch:2016gry 
          U. Haisch, P. Pani and G. Polesello,
          
          JHEP  1702, 131 (2017)
          
          [arXiv:1611.09841 [hep-ph]].
          
          
              
        Harlander:2013mla 
          R. V. Harlander, S. Liebler and T. Zirke,
          
          JHEP  1402, 023 (2014)
          
          [arXiv:1307.8122 [hep-ph]].
          
          
            
        Degrande:2011ua 
          C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer and T. Reiter,
          
          Comput. Phys. Commun.  183, 1201 (2012)
          
          [arXiv:1108.2040 [hep-ph]].
          
          
          
        Alloul:2013bka 
          A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks,
          
          Comput. Phys. Commun.  185, 2250 (2014)
          
          [arXiv:1310.1921 [hep-ph]].
          
          
          
        Degrande:2014vpa 
          C. Degrande,
          
          Comput. Phys. Commun.  197, 239 (2015)
          
          [arXiv:1406.3030 [hep-ph]].
          
          
          
        Alwall:2014hca 
          J. Alwall  et al.,
          
          JHEP  1407, 079 (2014)
          
          [arXiv:1405.0301 [hep-ph]].
          
          
          
        Sjostrand:2014zea 
          T. Sjöstrand  et al.,
          
          Comput. Phys. Commun.  191, 159 (2015)
          
          [arXiv:1410.3012 [hep-ph]].
          
          
          
        Ball:2012cx 
          R. D. Ball  et al.,
          
          Nucl. Phys. B  867, 244 (2013)
          
          [arXiv:1207.1303 [hep-ph]].
          
          
          
        Dercks:2016npn 
          D. Dercks, N. Desai, J. S. Kim, K. Rolbiecki, J. Tattersall and T. Weber,
          
          arXiv:1611.09856 [hep-ph].
          
          
        Cacciari:2011ma 
          M. Cacciari, G. P. Salam and G. Soyez,
          
          Eur. Phys. J. C  72, 1896 (2012)
          
          [arXiv:1111.6097 [hep-ph]].
          
          
          
        deFavereau:2013fsa 
          J. de Favereau  et al. [DELPHES 3 Collaboration],
          
          JHEP  1402, 057 (2014)
          
          [arXiv:1307.6346 [hep-ex]].
          
          
            
        Conte:2012fm 
          E. Conte, B. Fuks and G. Serret,
          
          Comput. Phys. Commun.  184, 222 (2013)
          
          [arXiv:1206.1599 [hep-ph]].
          
          
          
        Conte:2014zja 
          E. Conte, B. Dumont, B. Fuks and C. Wymant,
          
          Eur. Phys. J. C  74, no. 10, 3103 (2014)
          
          [arXiv:1405.3982 [hep-ph]].
          
          
          
        Frederix:2012ps 
          R. Frederix and S. Frixione,
          
          JHEP  1212, 061 (2012)
          
          [arXiv:1209.6215 [hep-ph]].
          
          
          
        Artoisenet:2012st 
          P. Artoisenet, R. Frederix, O. Mattelaer and R. Rietkerk,
          
          JHEP  1303, 015 (2013)
          
          [arXiv:1212.3460 [hep-ph]].
          
          
          
         Kim:2008pp 
          Y. G. Kim, K. Y. Lee and S. Shin,
          
          JHEP  0805, 100 (2008)
          
          [arXiv:0803.2932 [hep-ph]].
          
          
          
        Baek:2011aa 
          S. Baek, P. Ko and W. I. Park,
          
          JHEP  1202, 047 (2012)
          
          [arXiv:1112.1847 [hep-ph]].
          
          
          
        LopezHonorez:2012kv 
          L. Lopez-Honorez, T. Schwetz and J. Zupan,
          
          Phys. Lett. B  716, 179 (2012)
          
          [arXiv:1203.2064 [hep-ph]].
          
          
          
        Ko:2016ybp 
          P. Ko and J. Li,
          
          Phys. Lett. B  765, 53 (2017)
          
          [arXiv:1610.03997 [hep-ph]].
          
          
          
        Baek:2017vzd 
          S. Baek, P. Ko and J. Li,
          
          Phys. Rev. D  95, no. 7, 075011 (2017)
          
          [arXiv:1701.04131 [hep-ph]].
