Impact of the 125 GeV Higgs boson on the singlet fermion dark matter searches at the LHC

The search for singlet fermion dark matter at high-energy colliders is commonly analyzed with a singlet scalar mediator, which however violates the standard model (SM) gauge invariance, renormalizability and unitarity. These problems can be cured by introducing a mixing between the singlet scalar $s$ and the SM Higgs boson $h$. Thus one has to consider two scalar mediators $h_1$ and $h_2$, where $h_1$ is identified as the discovered 125 GeV Higgs boson. As a specific example, we consider the dark matter (DM) search in the $t \bar{t} + E_{T}^{\text{miss}}$ channel. According to the masses of dark matter and two scalar mediators, we classify the process into four cases. By investigating the total cross sections and differential distributions, we find that the contribution of the 125~GeV Higgs boson $h_1$ cannot be neglected in all cases and can even dominate once $h_1$ is on-shell in dark matter production. Further, we study the impact of $h_1$ on the LHC bounds of dark matter searches in the hadronic and semileptonic channels with the integrated luminosity of $36~\text{fb}^{-1}$. Finally we make a brief comment that $h_1$ should be also considered in the vector DM search at high-energy colliders.


I. INTRODUCTION
The framework of the simplified model [1,2](see also recent reports [3,4]) has been widely used in studying the dark matter (DM) phenomenology at colliders, where the interaction energy scale can be much higher than the new physics scale so that the effective field theory approach is no longer valid [5][6][7][8][9]. In the simplified model for singlet fermion DM, a singlet scalar S is introduced, and its interaction with the standard model (SM) quarks and the fermionic DM χ are assumed to be described by the following Lagrangian [1,2]: where q(= u, d, s, c, b, t) are the SM quarks and v = 246 GeV. The couplings g q and g χ are usually chosen as g q = g χ = 1 − 5 [2,10] for simplicity and to guarantee a sufficient DM production rate at colliders. Since the coupling of the mediator to a heavier flavor is stronger, the DM production is dominated by the gluon fusion process with top quark involved at the Large Hadron Collider (LHC). Then, DM can be searched through the mono-jet signature [11,12] with the top quark in the loop or through the production in association with top quark(s). Although the process of DM in association with a top quark pair or single top quark has a small production cross section, it can provide cleaner signals and more information on the nature of DM and the interaction form and has been studied extensively in the literature [2,10,[13][14][15][16][17]. Especially, in Ref. [10] the tt + / E T signature is explored in the simplified model in a comprehensive way and is compared to the sensitivity with mono-jet signature as well as those without DM in the final state. In Refs. [13,14], it is found that the LHC search sensitivity can be significantly improved if one also includes DM production in association with a single top quark. In fact, recent experimental results at the LHC [18][19][20][21] already show that the tt + / E T channel has a comparable sensitivity with the mono-jet channel. Moreover, the shapes of angular separations between the two leptons from top quark decays are found to be not only useful in resolving the coupling among the mediator and SM quark [2,15,16] but also helpful in distinguishing DM spins [17].
However, one can easily find that the Lagrangian (1.1) is not gauge invariant under the SM gauge transformations, since the SM left-handed quarks are in SU (2) doublets while righthanded ones are singlets. In order that a singlet fermion DM can couple to SM particles in a renormalizable and gauge-invariant way, an economic way is to introduce a mixing between the singlet scalar s with the SM Higgs boson after the electroweak symmetry breaking. As a result, there will be two physical scalar states (h 1 and h 2 ) mediating the SM and DM interactions. These two scalars are identified as the 125 GeV Higgs boson and a second scalar boson that can be either lighter or heavier than 125 GeV [22][23][24], respectively. These two scalar bosons also appear in the Higgs portal vector dark matter (VDM) model [25].
The main phenomenological differences between the gauge-invariant Higgs portal DM models and the simplified model (1.1) originating from the existence of the 125 GeV Higgs boson has been discussed in Refs. [26][27][28], and references therein. Especially, in Ref. [27] we have found that the interference effect between two mediators can affect the LHC exclusion bounds considerably in some parameter space, which was already reflected in Ref. [26] to some extent.
In this work, we will investigate the impact of h 1 (with the mass of 125 GeV) in the singlet fermionic DM (SFDM) model on the sensitivity of the LHC search for tt + / E T .
Depending on the masses of DM and mediators, four different cases (named Case A,B,C and D) shall be classified and discussed. The collider bounds obtained from the simplified model framework are applicable only in certain parameter space of Case C as we will show below. In general, we will find that the simplified model cannot reproduce the results derived

II. SFDM MODEL
The renormalizable and gauge-invariant Lagrangian that describes the SM extended with a gauge singlet fermion χ ∼ (1, 1, 0) 1 and a real singlet scalar S, i.e., the SFDM model, is [22,23,26,27,31,32] where y u and y d are the SM Yukawa couplings to the up-and down-type quarks, respectively, with suppressed generation indices. The interactionQ LH χ R is forbidden by the U (1) Y symmetry while the interactionL LH χ R can be discarded by a Z 2 symmetry under which only χ is odd, i.e., χ → −χ [27] or by global U (1) symmetry [33].
The scalar potential of the SFDM is given by [27,34] V The fields h and s are introduced after electroweak symmetric breaking as where v H and v S are the vacuum expectation values of H and S, respectively. The mass matrix of the scalar fields is so that the mass eigenstates h 1 and h 2 can be defined as 1 For χ in a nontrival representation of the SM gauge groups, see Refs. [29,30] It is well known [30,[35][36][37][38] that, for the most general Lagrangian that describes a real singlet scalar extension of the SM, there is a shift symmetry S → S + ∆ S , which holds in the SFDM model even though χ is introduced [32]. Thus we can freely choose S = 0 without loss of generality in this paper.
The minimal conditions In the basis of S = 0, the mixing between S and H comes solely from the term µ 1 SH † H in Eq. (2.2), and the mass matrix is simplified to Introducing the variable [35] y ≡ −2µ 2 the eigenvalues of the mass matrix can be expressed as where the sign +(−) corresponds to m h 1 (m h 2 ), and the mixing angle tan θ = y 1 + 1 + y 2 , tan(2θ) = y.
In terms of mass eigenstates, the interaction Lagrangian of interest can be written as The couplings of h 1 and h 2 to the SM fermion pair (ff ) or weak gauge boson pair (V V ) with V = W or Z are given by where xx = ff , V V , g SM hxx is the corresponding SM coupling, c θ ≡ cos θ and s θ ≡ sin θ. The couplings of h 1 and h 2 to the DM pair χχ are respectively.
In the SFDM model with v S = 0, the free parameters were chosen to be m h 1 , m h 2 , θ, λ HS , µ 2 , λ S , v H , m χ and g χ [32]. Identifying m h 1 and v H as m h 1 = 125 GeV, v H = 246 GeV, we however choose the following parameters for the convenience of collider phenomenology where λ 1 and λ 2 are normalized triple scalar couplings defined as The production cross sections of h 1 and h 2 at the LHC can be expressed as The total widths of h 1 and h 2 are 2 where Γ SM h (m h 1 ) and Γ SM h (m h 1 ) correspond to the total decay width of the SM Higgs boson [44] with the mass being m h 1 and m h 2 , respectively. The decay widths of h 1 and h 2 into where the Heaviside step function θ(x) = 1 for x > 0 and θ(x) = 0 for x ≤ 0 and m χ is the DM mass.
respectively. It is clear that the above widths are not sensitive to the signs of λ 1 and λ 2 .
Therefore we shall concentrate on the magnitudes of λ 1 and λ 2 in the collider study for the SFDM model.
Since h 2 may also decay into an extended dark sector other than χχ, similar to the simplified models [1][2][3][4] we can introduce the minimal total width of h 2 in the SFDM model [26,27]: It should be emphasized that the minimal total width of h 2 in the SFDM model also includes the partial decay width into W W * and ZZ * , without which as in the simplified model the cancellation [45] between diagrams with h 2 W W and h 2 tt interactions in DM production in association with single top quark pp → tjχχ does not occur and the sensitivity in pp → tjχχ can be even comparable to that in pp → ttχχ [13,14].
The current 95% C.L. upper limits on the invisible decay branching ratio and the total width of the 125 GeV Higgs boson are 0.24 [46,51,52] and 0.13 GeV [53], respectively. In the left panel of Fig. 1, we show the constraints in the g χ − m χ plane with sin θ = 0.2 and Γ(h 1 → h 2 h 2 ) = 0, which indicates that the constraint from the invisible decay branching ratio of the 125 GeV Higgs boson is stronger than that from its total width. If m h 1 > 2m h 2 , the decay channel h 1 → h 2 h 2 is kinematically allowed. In the right panel of Fig On the other hand, searches for di-Higgs production play a key role in the determination of the triple scalar coupling λ 2 . The cross section of pp → h 1 h 1 can be parameterized as 5 (2.34) Figure 2 shows the di-Higgs production cross section in the SFDM model (assuming Γ(h 2 → χχ) = 0 and sin θ = 0.2) alongside the combined upper limit at the 13 TeV 4 We find that light boson direct searches [54] can also directly constrain the triple scalar coupling λ 1 . But as found in Ref [55], the constraint from the light boson direct searches is much weaker than that from the BSM decay branching ratio. 5 In reality, the coupling of h 1 − h 1 − h 1 can contribute to the non-resonant production of h 1 h 1 , which is neglected here. LHC [56], which implies that λ 2 < 7. On the other hand, the theoretical constraint on λ 2 can be found in Ref. [38], which is λ 2 10. For Γ(h 2 → χχ) = 0, larger λ 2 could be allowed depending on the mass m χ and also the coupling g χ , which will not be studied in details in this paper.
In Fig. 3, the total width of h 2 for sin θ = 0.2, m χ = 65 GeV (for a larger m χ , the total width is smaller) is displayed. We can find that for g χ 1, the ratio Γ h 2 /m h 2 is below 10% and Γ h 2 /m h 2 < 1 can still be satisfied even with g χ = 5. To evaluate the impact of the decay h 2 → h 1 h 1 , we further show the branching ratio of h 2 → χχ for various parameter choices in Fig. 4. One can find that including h 2 → h 1 h 1 can decrease Br(h 2 → χχ) especially for smaller g χ . However it does not affect much the behavior of interplay between h 1 and h 2 in the DM search in Section III, so we will keep λ 2 = 0 for simplicity hereafter.
In the SFDM model, the DM pair can annihilate into either SM gauge bosons/fermions through h 1 /h 2 mediation or scalar bosons through t-channel/s-channel process if it is kinematically allowed. The annihilation cross section of the former is proportional to g 2 χ while that of the latter is proportional to g 4 χ (g 2 χ ) for t-channel (s-channel) annihilation. So a large g χ is required to annihilate the DM effectively, which renders the DM direct detections quite stringent (except in the resonant region of the s-channel process m χ ∼ m h 1,2 /2). The details of the DM thermal relic density and the spin-independent DM-nucleon scattering cross section for benchmark points in each case are provided in the Appendix. In order to guarantee that the DM has a relic density below the observation and keep consistent with the DM direct detections, the SFDM model should be generalized beyond the minimal setup of the 400 600 800 1000   [61]. Then the dark matter direct detection constraints can be weakened or even completely evaded.
Moreover, the particle χ discussed in the current paper may correspond to a heavier dark state in the dark sector that can decay into the genuine DM candidate. Then, as long as the heavier dark state(s) does not leave any signal at the detector (due to a long lifetime or invisible decay), it will produce the same collider phenomenology as the SFDM model 6 .
Now we are ready to study the impact of the 125 GeV Higgs boson h 1 in the DM search with tt + / E T signature. Previous studies in the mono-jet, VBF and mono-V signatures can be found in Refs. [26,27]. The tt+ / E T channel at the 8 TeV LHC was preformed in Ref. [26], and compared with the CMS results.
There are two mediators in pp → ttχχ in the SFDM model, which are shown in Fig. 5.
The total amplitude is then proportional to  Depending on the relations of m h 1 , m h 2 and m χ , the process pp → ttχχ in the SFDM model can be categorized into four cases, namely, • Case A: m h 1 , m h 2 > 2m χ , 6 Work in progress.
• Case B: m h 1 > 2m χ and m h 2 < 2m χ , • Case C: m h 1 < 2m χ and m h 2 > 2m χ , and In the following, we will denote the cross section of diagrams with each scalar mediator (h 1 /h 2 ) as σ h 1 and σ h 2 , while the total cross section that includes the interference effect between diagrams with different mediators is denoted as σ h 1 +h 2 .
For Case A, both h 1 and h 2 can be on-shell in DM production, so that the cross sections can be described as with the narrow width approximation (NWA), where σ prod (m h i ) denotes the cross section of pp → tth i for on-shell h i , i = 1, 2 with SM couplings. In this case, the interference effect is small unless m h 2 m h 1 ; thus, the cross section with two mediators is approximately equal to the sum of the cross sections with one mediator: In the left panel of Fig. 6, we show the leading-order (LO) cross section of pp → ttχχ at the 13 TeV LHC for Case A with g χ = 0.08, m χ = 1 GeV, Γ h 2 = Γ min h 2 and m h 2 65 GeV satisfying the constraint from the invisible decay branching ratio of h 1 . We find that the h 2 provides a larger cross section than the h 1 when m h 2 70 GeV. With increasing m h 2 , the contribution from h 2 decreases dramatically and becomes negligible for m h 2 300 GeV, in which scenario the process pp → ttχχ is effectively described by a single mediator h 1 . Note that here we assume Γ h 2 = Γ min h 2 . If we consider the decay of h 2 into h 1 h 1 , which is possible if m h 2 > 2m h 1 , the branching ratio of h 2 → χχ will be suppressed. On the other hand, h 1 can also decay into h 2 h 2 if m h 2 < m h 1 /2 so that the branching ratio of h 1 → χχ is suppressed.
This fact will bring the m h 2 dependence to the cross section of σ h 1 , which is not shown in the Fig. 6.
For Case B, only h 1 can be on-shell so that the contribution of h 1 is dominant and (3.5) For Case C, only h 2 can be on-shell. If Γ h 2 /m h 2 1, which implies that the NWA can be applied, one obtains If Γ h 2 = Γ min h 2 , as shown in Fig. 4 [60], the total width of h 2 can be significantly enhanced as compared to Γ min h 2 . Consequently, the branching ratio of h 2 into χχ is possibly small. Thus although h 1 is off-shell in DM production for Case C, its contribution should be taken into account, which however has not been paid much attention to. It should be noted that the wide width of h 2 may make the NWA in Eq. (3.7) invalid.
For Case D, both h 1 and h 2 are off-shell, so that the cross sections are small but the interference effect between diagrams with different mediators in Fig. 5 is significant and always destructive, as can be seen from Eq. (3.1).
To consider the wide width effects, we assume that the total width of h 2 is rescaled by a factor of 15 irrespective of its mass, i.e., Γ h 2 = 15 × Γ min h 2 , which satisfies Γ h 2 /m h 2 < 1 as shown in Fig. 3. Then, the cross section of pp → ttχχ with h 2 mediation is reduced by a factor of about 15. On the other hand, a larger coupling g χ can also increase the h 2 decay width as well as enhance the σ h 1 . To compare with the simplified model study in Refs. [2,10], we choose g χ = 4. In Fig. 7 the left panel and g χ = 4, m χ = 80 GeV, and Γ h 2 = Γ min h 2 in the right panel. We find that σ h 1 +h 2 < σ h 1 + σ h 2 in the region of m h 2 2m χ (Case D) irrespective of g χ and Γ h 2 . This is due to the destructive interference between diagrams with h 1 and h 2 in case D. The destructive interference effect is most significant at m h 2 m h 1 . Experimentally, the invariant mass of χχ cannot be reconstructed directly. However, its feature can be reflected in the distribution of the transverse momentum of χχ (missing transverse momentum), i.e., p χχ T . In Fig. 10 we show the parton-level distributions of p χχ T for Case C with m χ = 80 GeV, m h 2 = 200, 300, 500 GeV, g χ = 1 and Γ h 2 = 15 × Γ min h 2 (similar for g χ = 4 and Γ h 2 = Γ min h 2 ). One can obtain that p χχ T distribution in the SFDM model with two mediators is always softer than that in the simplified model with h 2 , the effect of which on the cut efficiency of a concrete experimental search will be discussed in Section IV.

IV. IMPACT ON THE UPPER LIMITS
Having observed the distinct difference between the total cross sections and differential distributions of pp → ttχχ with two mediators and with one mediator, we will investigate the impact of the Higgs boson h 1 on the 95% C.L. upper limits of searches for DM produced with top quark pair at the 13 TeV LHC, which have been performed by the CMS and ATLAS Collaborations in the hadronic, semileptonic and dileptonic channels [19,21,62,63]. Similar to Ref. [13], we closely follow the CMS analyses [21,62,63] and concentrate on the hadronic and semileptonic channels since the dileptonic are typically less sensitive in these analyses.
On the other hand, although recent search using events with 35.9 fb −1 [21] has improved the upper limit significantly as compared to that with 2.2 fb −1 [62, 63], the background event numbers in signal regions are not provided [21]. Therefore we take the strategy that we first recast the results with 2.2 fb −1 and then project them to the integrated luminosity of 36 fb −1 with the assumption that the signal and background uncertainties scale as the integrated luminosity and the square root of the integrated luminosity, respectively. Another reason for projection is that a multivariate discriminant "resolved top tagger" was used in the analysis [21] without giving any details for recasting. 7 In our analysis, we only consider the inclusive hadronic channel without using "resolved top tagger" similar to Ref. [13].
Having imposed the selection cuts [62,63], the number of signal events with integrated luminosity L is then where σ before cut denotes the cross section of pp → ttχχ without any cut and denotes the cut efficiency in the hadronic or semileptonic channel.
The LHC search sensitivity to our models can be measured by the signal strength µ = σ/σ th , where σ denotes the observed production cross section of pp → ttχχ at the 13 TeV LHC and σ th is the theoretical signal cross section. The 95% C.L. upper limit on µ is investigated using the CLs method [68][69][70], with inputs of production cross section and cut efficiencies of a concrete model from our simulation as well as the number of background events and their uncertainties provided in the experimental papers [62, 63].
Then we will investigate the impact of the mediator h 1 on the upper limits µ in the SFDM model. From Eq. (4.1), we know that the number of signal events after selections depend on the total cross section of pp → ttχχ as well as the cut efficiency. As having been discussed in Section III, for Case A and Case B, the signal production is dominated by the process with mediator h 1 . So adding h 1 will change both the production rate and the final state kinematics (i.e., cut efficiency) significantly. On the other hand, for Case C and Case D, both h 1 and h 2 have an impact on the signal production cross section and cut efficiency.
In Fig. 11, we show the 95% C.L. upper limits on µ in the inclusive hadronic (jets) and semileptonic ( + jets) channels [62,63]   Parameter space of m h 2 denoted as thick green curves is excluded by the measurements of Higgs BSM decay branching ratio.
For Case C and Case D, both h 1 and h 2 play important roles in pp → ttχχ in the SFDM model. Figure 12 shows the cut efficiencies for Case C and Case D in the inclusive hadronic and semileptonic channels in the SFDM model with two mediators as well as one mediator h 2 . Since the p χχ T distribution in the SFDM model is softer than that in the simplified model (see Fig. 10), a lower cut efficiency is achieved in the former scenario. On the other hand, in both scenarios the cut efficiencies are lowest when m h 2 is around 180 GeV. This is because the DM pairs χχ are mostly produced through the on-shell h 2 mediation while events with m χχ > m h 2 are suppressed by the destructive interference between h 1 and h 2 in the SFDM model and the h 2 propagator in the simplified model. As a result, the p χχ T distribution is softer for smaller m h 2 for m h 2 > 2m χ . On the other hand, when m h 2 < 2m χ , the DM pair can only be produced through off-shell h 2 mediation. Then the relative suppression on event rates with higher m χχ is weaker for lighter h 2 , leading to harder p χχ T spectra for lighter h 2 . This can be seen from Fig. 13: For m h 2 < 180 GeV, the p χχ T spectrum decreases with m h 2 . On the other hand, for m h 2 > 180 GeV, p χχ T gets increased for larger m h 2 . Finally, we show the upper limits on µ for Case C and Case D in the inclusive hadronic and semileptonic channels with the integrated luminosity of 36 fb −1 in Fig. 14 the destructive interference between diagrams with h 1 and h 2 mediation, the LHC search sensitivities on the SFDM model are extremely weak in the region of m h 2 < 2m χ . Without the destructive interference effects as in the simplified model, the sensitivities in the same region can be more than order of magnitude better, but still way below the LHC probe at the current stage. For m h 2 2m χ , the interference effects on the total cross section can be destructive or constructive in the SFDM model depending on m h 2 as shown in the right panel of Fig. 8  Specifically, we find that when both h 1 and h 2 are off-shell (Case D), the destructive interference makes the total cross section much smaller than that in the simplified model without h 2 . If only h 2 is on-shell (Case C), the effect of h 1 on the total cross section becomes more important for larger m h 2 . Besides, with a larger total width of h 2 , which may come from a large coupling g χ or dominant decay of h 2 into the extra dark sector particles, the relative contribution of h 1 (h 2 ) to the total cross section for Case C is further increased (decreased). It is found that irrespective of g χ and Γ h 2 the interference effect for Case C is destructive in the region of 2m χ m h 2 380 GeV and constructive for m h 2 380 GeV with sin θ = 0.2 and m χ = 80 GeV. In addition to the total cross section, h 1 can also affect the differential distribution of the DM+tt process. Especially, the p χχ T in the SFDM model is always soften as compared to that in the simplified model for Case C and Case D.
Finally, we study the impact of h 1 on the LHC bounds of the DM+tt search in the inclusive hadronic and semileptonic channels with the integrated luminosity of 36 fb −1 . We find that the upper limit on the signal strength µ for Case A in the SFDM model is smaller than 10, which is almost independent of m h 2 . For Case B, the sensitivity also depends on the triple scalar coupling λ 1 of h 1 − h 2 − h 2 . Roughly, the upper limit is below 50 for the benchmark values discussed. For Case C, the sensitivity in the SFDM model is extremely weak as compared to that in the simplified model due to the destructive interference between the SM Higgs boson and the singlet scalar, which were largely ignored in theoretical and experimental papers except in Refs. [17,26,27,31,39,40,55]. For Case D, the upper limit in the SFDM model is better than that in the simplified model in the region of m h 2 300 GeV and becomes opposite for m h 2 300 GeV.
Before closing, we would like to point out that the 125 GeV Higgs boson is also important for the VDM search at high-energy colliders. If one generates the vector DM mass by a dark Higgs mechanism, then there will be a mixing between the dark Higgs boson and the SM Higgs boson [25], resulting in two scalar propagators that can produce interesting interference [26,27]. 8 Then the amplitude for the VDM pair production at high-energy colliders will take a form similar to Eq. (3.1). Effects of these two scalar propagators have been studied in the context of characterizing the mass and the spin of the Higgs portal scalar, fermion and vector DM at the ILC [39,40] and at the LHC and 100 TeV pp collider [17].
In conclusion, we would like to emphasize that the contribution of the 125 GeV Higgs boson should be properly included to interpret correctly the LHC dark matter searches in case of the s-channel scalar mediators: It is important not only for the gauge invariance and renormalizability at the high-energy scale, but also for the quantitative difference of the upper limits and kinematic distributions.
Then, the DM can only annihilate through the s-channel h 1,2 mediation as in Case C. For m h 2 = 110 and 130 GeV, the relic densities becomes much larger because of the cancellation between the contributions from the mediators h 1 and h 2 .
For Case A and Case B, since m h 1 > 2m χ the coupling g χ is severely constrained by the measurements of Higgs invisible decay branching ratio. In Tab. III, we show the relic densities for benchmark points in Case B with g χ = 0.15 and m χ = 50 GeV (the relic densities for Case A, which are larger, are not shown here). The relic densities of all benchmark points are larger than the measured DM relic density. As we explained in Section II, this can be weakened with the opening of new DM annihilation channels such as χχ → Z Z or coannihilation within a richer dark sector.
We can find that the DM-nucleon scattering cross section is well described by Benchmark points in all case are challenged by current DM direct detections [73][74][75][76] (for comparison, the σ SI p of points with Ωh 2 < 0.120 should be rescaled by a factor Ωh 2 /0.120). This indicates that there will be other DM annihilation mechanisms if our DM indeed comprises a component of a full DM sector.