Constraints on spin-0 dark matter mediators and invisible Higgs decays using ATLAS 13 TeV pp collision data with two top quarks and missing transverse momentum in the final state

Abstract

This paper presents a statistical combination of searches targeting final states with two top quarks and invisible particles, characterised by the presence of zero, one or two leptons, at least one jet originating from a b-quark and missing transverse momentum. The analyses are searches for phenomena beyond the Standard Model consistent with the direct production of dark matter in pp collisions at the LHC, using 139 fb$^{-\text {1}}$ of data collected with the ATLAS detector at a centre-of-mass energy of 13 TeV. The results are interpreted in terms of simplified dark matter models with a spin-0 scalar or pseudoscalar mediator particle. In addition, the results are interpreted in terms of upper limits on the Higgs boson invisible branching ratio, where the Higgs boson is produced according to the Standard Model in association with a pair of top quarks. For scalar (pseudoscalar) dark matter models, with all couplings set to unity, the statistical combination extends the mass range excluded by the best of the individual channels by 50 (25) GeV, excluding mediator masses up to 370 GeV. In addition, the statistical combination improves the expected coupling exclusion reach by 14% (24%), assuming a scalar (pseudoscalar) mediator mass of 10 GeV. An upper limit on the Higgs boson invisible branching ratio of 0.38 ($\text {0.30}^{+\text {0.13}}_{-\text {0.09}}$) is observed (expected) at 95% confidence level.

1 Introduction

The existence of a non-luminous component of matter in the universe, dark matter (DM), is supported by compelling astrophysical evidence [1, \(1\sigma \)). More details are given in Appendix A.2.

The expected numbers of events estimated in a simultaneous profile likelihood fit to all tt0L-low CRs are shown in Table 2. The observed data are compatible with the prediction, agreeing to within $2\sigma$ in each signal region.

The results presented in this paper show the final combination of the tt0L-low and tt0L-high analyses, estimated in a simultaneous fit of all CRs and SRs. The details of this combination and the single-channel individual limits are discussed in Appendix A.3.

The tt1L analysis This analysis requires exactly one lepton (e or $\mu$), at least four jets, two of which must be $b\text {-tagged}$ , and $E_{\text {T}}^{\text {miss}}\,> 230\,\text {GeV}\,$, and was designed to target spin-0 DM models. The $E_{\text {T}}^{\text {miss}}$ significance $\mathcal {S}$ must be above 15 and, only for this analysis, it considers only jets and leptons in the events and their resolution, as described in Ref. [109]. A recursive variable-radius reclustering algorithm [110] is applied to the jets to identify at least one large-variable-radius jet loosely consistent with a top quark ($m_{\textrm{top}^{\textrm{reclustered}}} > 150\,\text {GeV}\,$). The use of a variable-radius algorithm, instead of a fixed-radius one, increases the acceptance of both highly boosted events and less boosted events when no explicit categorisation is performed. In addition, a requirement on the ‘topness’ likelihood variable [111] is used to distinguish between the signal and dileptonic decays in SM $t\bar{t}$ events where one of the leptons is misidentified or outside the acceptance. This variable quantifies how well each event satisfies the dileptonic $t\bar{t}$ hypothesis, using the top quark and W boson mass constraints and a requirement that the centre-of-mass energy of the event is minimised. The $E_{\text {T}}^{\text {miss}}$  triggers were used to select data that then populate this SR. This region is divided into four disjoint regions according to the azimuthal distance between the $E_{\text {T}}^{\text {miss}}$ and the lepton momentum, $\Delta \phi ({\varvec{p}}_{\text {T}}^{\text {miss}},\ell )$, which is presented in Fig. 2b and is found to be larger for pseudoscalar mediator models. The binning also maximises the sensitivity for scalar mediator models, which are more similar to the background but are characterised by a larger production cross section at low masses. An additional requirement of $\Delta \phi ({\varvec{p}}_{\text {T}}^{\text {miss}},\ell ) > 1.1$ is applied to suppress the SM background. The dominant backgrounds, $t\bar{t}$ and $t\bar{t}Z$, are estimated by means of dedicated CRs.

The tt2L analysis The last analysis considers events with two opposite-charge leptons (e or $\mu$), at least one $b\text {-tagged}$ jet and large values of $E_{\text {T}}^{\text {miss}}$ significance ($\mathcal {S} > 12$), exploiting events collected with dilepton triggers. Events are then separated into two categories depending on whether the two leptons have the same or different flavour, and in the same-flavour selection an additional requirement of $|m_{\ell \ell }-m_Z|>20$ $\text {GeV}$  is added to suppress the $Z$+jets background. In this selection, the main discriminating variable is the leptonic stransverse mass $m_{\text {T2}}$ [105, 106], which is used to bound the individual masses of a pair of identical particles that are each presumed to have decayed into one visible and one invisible particle. This quantitiy is used to bound dileptonic top pair decays. To maximise the search sensitivity, the $m_{\text {T2}}$ spectrum is divided into six bins, starting from 110 $\text {GeV}$ . The $m_{\text {T2}}$ distribution for selected events with two leptons with the same flavour is presented in Fig. 2c. In this search, the main backgrounds are from $t\bar{t}$, $t\bar{t}Z$, single-top-quark tW-channel, Z+jets, and diboson processes. These backgrounds are estimated with MC simulations and normalised with data in orthogonal CRs for the dominant contributions ($t\bar{t}$ and $t\bar{t}Z$), while the background arising from fake/non-prompt leptons is estimated directly from the data.

3.2 Orthogonalisation

In order to combine the results of the different searches, the searches are required to be statistically independent and any possible overlaps of kinematic regions were investigated and removed as described in the following. The three analysis channels are disjoint because of their requirements on lepton multiplicity. The tt0L-high and tt0L-low channels are kept orthogonal by the requirements on the large-radius jet as well as on the $E_{\text {T}}^{\text {miss}}$ and its significance, $\mathcal {S}$. In addition, one of the Z+jets CRs in the tt0L-high analysis, denoted by CRZAB-T0 in Ref. [28], is not considered and a single control region, CRZAB-TTTW, is used to normalise the Z+jets process in all SRs of the tt0L-high analysis. This has negligible impact on the tt0L-high analysis results and it is done to ensure orthogonality between the Z+jets CRs in the tt0L-high and tt0L-low analyses, as those events are used to normalise the Z+jets background in the tt0L-low analysis. To the same end, the Z+jets CR in the tt0L-low analysis only selects events with either $N_{\text {large-radius jet}} < 2$ or subleading large-radius jet mass $< 60$ $\text {GeV}$ .

The CRs used to normalise the $t\bar{t}Z$ background overlap. The three analysis channels share a common strategy to determine the amount of $t\bar{t}Z$ (with $Z\,\rightarrow \nu \nu$) background in their SRs. The strategy is to construct CRs requiring three charged leptons in order to maximise their $t\bar{t}Z$ (with $Z\,\rightarrow \ell \ell$) event content, which once determined can be scaled by the ratio of $Z\,\rightarrow \nu \nu$ to $Z\,\rightarrow \ell \ell$ branching fractions. These control regions differ only in minor selections adapted to the SR of each specific channel. In the combination, the $t\bar{t}Z$ estimation is harmonised by using the most inclusive CR$_{t\bar{t}Z}$, from the tt2L analysis [30], as a common CR across all channels. The fitted normalisation parameter value obtained in the combination is consistent within 1% with the one published in Ref. [29].

4 Statistical combination and uncertainties

The statistical combination of the analyses considered in this paper consists of maximising a profile likelihood ratio [102] constructed from the product of the individual analysis likelihoods:

$$\begin{aligned} \Lambda (\alpha ;\theta ) = \frac{L\left( \alpha ,\hat{\hat{\theta }}(\alpha )\right) }{L\left( \hat{\alpha },\hat{\theta }\right) }. \end{aligned}$$

The $\alpha$ and $\theta$ parameters represent, respectively, the parameter of interest and the nuisance parameters. In the numerator, the nuisance parameters are set to their profiled values $\hat{\hat{\theta }}(\alpha )$, which maximise the likelihood function for fixed values of the parameter of interest $\alpha$. In the denominator, both the parameter of interest and the nuisance parameters are set to the values that jointly maximise the likelihood: $\hat{\alpha }$ and $\hat{\theta }$, respectively.

For the DM signal model interpretations, upper limits on the signal cross section are calculated following the $\mathrm {CL_s}$ formalism, using the profile likelihood ratio as a test statistic. The parameter of interest is the overall signal strength, defined as a scale factor multiplying the cross section predicted by the signal hypothesis, and it is bounded from below by zero. The final result is provided as a ratio of the lowest excluded signal cross section to the predicted cross section with all couplings set to unity. For the $H\rightarrow \text {inv}$  signal model interpretation, the branching fraction $\mathcal {B}_{H\rightarrow \text {inv}}$ is considered as the parameter of interest $\alpha$, following the implementation described in Refs. [112, 113].

As described in Sect. 3, for each channel the estimation of the dominant SM backgrounds is aided by means of dedicated control regions that constrain free-floating normalisation factors for each of these backgrounds.

Systematic uncertainties are modelled in the likelihood function as nuisance parameters $\theta$ constrained by Gaussian or log-normal probability density functions [114].

Three types of sources of systematic uncertainty are considered: detector-related (experimental) uncertainties, uncertainties related to the modelling of SM background processes, and uncertainties related to the modelling of the signal processes. Regarding the experimental and SM modelling uncertainties, all details are given in Refs. [28,29,30] respectively for the zero-, one- and two-lepton channels. The tt0L-low channel includes the same uncertainties as the tt0L-high channel and, in addition, uncertainties associated with the b-jet trigger efficiencies. The typical size of these uncertainties is a few percent. All analyses use common event-quality criteria and object reconstruction and identification definitions. For this reason, all experimental systematic uncertainties are treated as correlated across channels in the statistical combination. The dominant sources of experimental systematic uncertainty in the combination are the uncertainties related to the jet energy scale and resolution, followed by either flavour-tagging uncertainties or uncertainties related to the missing transverse momentum, depending on the analysis channel.

Fig. 4

Summary of the total uncertainty in the background prediction for each SR of the tt0L-low, tt0L-high, tt1L, and tt2L analysis channels in the statistical combination after the profiled likelihood fit. Their dominant contributions are indicated by individual lines. Individual uncertainties can be correlated, and do not necessarily add up in quadrature to the total background uncertainty

Uncertainties in the modelling of the SM background processes in MC simulation and their theoretical cross-section uncertainties are also taken into account. All modelling uncertainties are treated as uncorrelated across different channels as they probe different regions of the available phase space.

Uncertainties related to the MC modelling of the DM signals include fragmentation and renormalisation scale uncertainties, and the uncertainties related to the modelling of the parton shower. The impact of these uncertainties varies from 10 to 25%. Uncertainties related to the $t\bar{t}H$ with $H\rightarrow \text {inv}$  signal modelling also include fragmentation and renormalisation scale uncertainties, parton shower uncertainties and PDF uncertainties. Among these, scale uncertainty effects, which are evaluated in the simplified template cross-section formalism [36, 115], are the dominant contribution and range between 7 and 17%. Signal modelling uncertainties are treated as fully correlated across analysis channels.

All sources of uncertainty in the SM backgrounds are summarised in Fig. 4. In most of the SRs, the dominant systematic uncertainties are the ones related to theory predictions and MC modelling, while jet uncertainties are the dominant experimental ones. No significant difference from either the composition or the value of the total uncertainty presented in the published individual analyses is observed.

5 Exclusion limits

Exclusion limits at 95% CL are presented in Fig. 5a and b for DM models with a spin-0 scalar or pseudoscalar mediator particle, respectively. The three individual channels are also presented for comparison. The tt0L limits are the result of the statistical combination of the tt0L-low and tt0L-high SRs. The tt0L-low selection improves the expected scalar (pseudoscalar) mediator stand-alone cross-section limit of the tt0L-high by up to 15% (5%) and it is strongest for mediator masses values around $10\,\text {GeV}\,$. Details of the comparison can be found in Appendix A.3.

The signal generation considered in these results includes both the top-quark-pair final states (DM+$t\bar{t}$) and single-top-quark final states (DM+tW and DM+tj). The limits are expressed in terms of the ratio of the excluded cross section to the nominal cross section for a coupling of $g = g_q = g_{\chi } = 1$. With these assumptions, scalar DM models are characterised by a higher cross section than for pseudoscalar DM models with low mediator masses [20], while the two models have very similar cross sections beyond the top quark decay threshold ($m(\phi )$ or $m(a) \sim 2\cdot m_t$). A DM particle mass of $1\,\text {GeV}\,$ is considered, although the results are valid as long as the mass of the mediator is larger than twice the mass of the DM particle. The solid (dashed) lines show the observed (expected) exclusion limits for each individual analysis and their statistical combination. For scalar (pseudoscalar) DM models, the combination extends the excluded mass range by $50~(25)\,\text {GeV}\,$ beyond that of the best of the individual analyses, excluding mediator masses up to $370\,\text {GeV}\,$. In addition, the combination improves the expected cross-section limits by 14% and 24%, for low-mass scalar and pseudoscalar DM mediators, respectively. This directly translates into more stringent exclusion limits on the couplings. When only the associated production of DM and two top quarks is considered in the interpretation of the results, the excluded scalar (pseudoscalar) mediator mass range obtained from the combination is reduced by $70\,(20)\,\text {GeV}\,$ relative to the sensitivity of the combination as reported in Fig. 5a and b. As the production of DM in association with a single top quark is most relevant for higher masses in the scalar mediator models [26], the impact of this process for masses below $50\,\text {GeV}\,$ is negligible. In contrast, for the pseudoscalar mediator models, the ratio of single-top-quark channel to $t\bar{t}$ channel cross sections is relatively constant [26]. When considering only DM+$t\bar{t}$ associated production, the cross-section upper limit weakens by about 18% over the whole mass range.

Fig. 5

Exclusion limits for colour-neutral a scalar or b pseudoscalar mediator dark matter models as a function of the mediator mass $m(\phi )$ or m(a) for a DM mass $m_{\chi } = 1~\text {GeV}\,$. Associated production of DM with both single top quarks (tW and tj channels) and top quark pairs is considered. The limits are calculated at 95% CL and are expressed in terms of the ratio of the excluded cross section to the cross section for a coupling assumption of $g = g_q = g_{\chi } = 1$. The solid (dashed) lines show the observed (expected) exclusion limits for each individual channel and their statistical combination

Fig. 6

a The expected negative logarithmic profile likelihood ratios $-2\,\Delta \!\ln (\Lambda )$ as a function of $\mathcal {B}_{H\rightarrow \text {inv}}$ for each of the three channels and their statistical combination and b these likelihood ratios for the observed data

Table 3 Summary of results from direct searches for invisible decays of the 125 $\text {GeV}$  Higgs boson in the $t\bar{t}H$ topology using 139 fb$^{-1}$  of Run 2 data, and their statistical combination. Shown are the best-fit values of $\mathcal {B}_{H\rightarrow \text {inv}}$ , computed as not being bounded below by zero, for consistency with previous results [114]. Observed and expected upper limits on $\mathcal {B}_{H\rightarrow \text {inv}}$ at the 95% CL are computed with the $\mathrm {CL_s}$ method and are new results with respect to the individual analysis papers quoted in the last table column. The corresponding Asimov datasets for the expected results are constructed using nuisance parameter values from a fit to data with $\mathcal {B}_{H\rightarrow \text {inv}}\,= 0$, and the quoted uncertainty corresponds to the 68% confidence interval

The negative logarithmic profile likelihood ratios $-2\,\Delta \!\ln (\Lambda )(\mathcal {B}_{H\rightarrow \text {inv}}\,;\theta )$ as a function of $\mathcal {B}_{H\rightarrow \text {inv}}$ from the individual analyses and their combination are shown in Fig. 6.

Expected results are obtained using the Asimov dataset technique and calculated from asymptotic formulae [102]. The best-fit values of $\mathcal {B}_{H\rightarrow \text {inv}}$ for the individual analyses are compatible, agreeing to within one standard deviation. Their statistical combination yields a best-fit value of $0.08^{+0.15}_{-0.15}$, consistent with the SM prediction of 0.12%. The combined observed 95% CL upper limit on $\mathcal {B}_{H\rightarrow \text {inv}}$ is 0.38 while the expected value is $0.30^{+0.13}_{-0.09}$. The individual analysis results are presented in Table 3, while the details of the tt0L combination are reported in Appendix A.3. The overall uncertainty is dominated by the statistical uncertainty of the data and, to a lesser extent, by systematic uncertainties associated with the modelling of the SM processes and jet-related uncertainties. Higgs boson invisible decays represent a specific case of the DM simplified models considered in the previous section, where the mass of the scalar mediator is assumed to be $125\,\text {GeV}\,$. The two results are compatible with each other, when taking into account the different order of accuracy used in event generation for the $H\rightarrow \text {inv}$  model.

6 Conclusion

In summary, a statistical combination of three analyses using 139 fb$^{-1}$ of pp collisions delivered by the LHC at a centre-of-mass energy of 13 $\text {TeV}$  and collected by the ATLAS detector is presented. The three analyses are all designed to select events with two top quarks and invisible particles, and consider all possible light lepton multiplicities arising from the decays of the two top quarks.

The statistical combination is used to set 95% confidence-level constraints on spin-0 simplified dark matter models. All production modes with top quarks in the final state (DM+$t\bar{t}$, DM+t) are considered. For scalar (pseudoscalar) dark matter models, the combination extends the excluded mass range by $50~(25)\,\text {GeV}\,$ beyond that of the best of the individual channels, excluding mediator masses up to $370\,\text {GeV}\,$ with all couplings set to unity. In addition, the combination improves the observed coupling exclusion limit by 24%, assuming a pseudoscalar mediator mass of $10\,\text {GeV}\,$.

The specific case where the mediator corresponds to the SM $125~\text {GeV}\,$ Higgs boson is also considered when interpreting the results presented in this paper. An upper limit on the Higgs boson invisible branching ratio of 0.38 ($0.30^{+0.13}_{-0.09}$) is observed (expected) at 95% confidence level.

Appendices

Appendix

A The tt0L-low analysis

The tt0L-low analysis aims to enhance the sensitivity to DM+$t\bar{t}$signals with low mediator masses ($m(\phi ),m(a)<100~\text {GeV}\,$). Two main discriminating variables, $\cosh _{\text {max}}$  and $\chi _{t\bar{t}\text {, had}}^2$ , are defined in order to reduce the most dominant top quark backgrounds. Angular separations between b-tagged jets, $E_{\text {T}}^{\text {miss}}$ or large-radius jets are used to further reduce the contamination from Standard Model processes. To ensure orthogonality with the tt0L-high selections, additional orthogonalisation requirements are also applied, as detailed in Sect. 3.1.

1.1 A.1 Discriminating variables

The full event selections performed in the signal regions can be found in Table 1. The discriminating variables are described in more detail below.

1.1.1 $\cosh _{\text {max}}$ 

The $\cosh _{\text {max}}$  variable is designed to distinguish signal events from single-top events in the tW channel and $t\bar{t}$ events with a lepton missed by the reconstruction algorithms (top-with-lost-lepton), which are among the main backgrounds in the analysis. Such events may enter the signal regions because of high $E_{\text {T}}^{\text {miss}}$ originating from the $t \rightarrow bW \rightarrow b\ell \nu$ decay, and the lost lepton.

The reconstruction of events containing a top quark with a lost lepton is attempted by assuming that the $E_{\text {T}}^{\text {miss}}$ is equal to the $p_{\text {T}}$ of the leptonically decaying W boson with a lost lepton, $E_{\text {T}}^{\text {miss}} \sim p_{\text {T}}^{W}$.

The top-with-lost-lepton background can then be reconstructed by combining the missing transverse momentum with the correct b-tagged jet ($t\rightarrow bW$). In practice, a four-vector with $p_{\text {T}}$ and $\phi$ corresponding to the ${\varvec{p}}_{\text {T}}^{\text {miss}}$ vector and its mass equal to the W boson mass is built, while its pseudorapidity $\eta _W$ (or equivalently $p_{z}^{W}$) remains unknown. Choosing the x-axis to be in the direction of $p_{\text {T}}\,^{W}$ and adopting $(E, p_x, p_y, p_z)$ coordinates:

$$\begin{aligned} {\varvec{p}}_{W}&= \left( \sqrt{(p_{\text {T}}\,^{W})^2+(p_{z}^{W})^2+m_{W}^2},p_{\text {T}}\,^{W}, 0, p_{z}^{W}\right) , \end{aligned}$$

(1)

$$\begin{aligned} {\varvec{p}}_{b}&= \left( \sqrt{(p_{\text {T}}\,^{b})^2+(p_{z}^{b})^2+m_{b}^2},\right. \nonumber \\&\left. p_{\text {T}}\,^{b}\cdot \cos (\phi _{W}-\phi _{b}), p_{\text {T}}\,^{b}\cdot \sin (\phi _{W}-\phi _{b}), p_{z}^{b}\right) , \end{aligned}$$

(2)

$$\begin{aligned} m_{t}^2&= ({\varvec{p}}_{W} + {\varvec{p}}_{b})^2, \end{aligned}$$

(3)

where the b superscript and subscript refer to one of the selected b-tagged jets. Substituting Eqs. (1) and (2) in Eq. (3), and assuming the massless limit for the b-tagged jet, the equivalence below is formed:

$$\begin{aligned}&\sqrt{1+\left( \frac{m_{W}}{p_{\text {T}}\,^{W}\cdot \cosh \eta _{W}}\right) ^2}\cdot \cosh \eta _{W}\cdot \cosh \eta _{b}\nonumber \\&\quad -\sinh \eta _{W}\cdot \sinh \eta _{b} = \frac{m_{t}^2-m_{W}^2}{2{p_{\text {T}}\,^{W}}{p_{\text {T}}\,^{b}}}+\cos (\phi _{W}-\phi _{b}), \end{aligned}$$

(4)

where $\eta _{W}$ is unknown. Given that $E_{\text {T}}^{\text {miss}}\,\sim p_{\text {T}}\,^{W} > 160~\text {GeV}\,$ in the signal regions and $\cosh \eta _{W}\ge 1$, it may be assumed that $m_{W} \sim 80~\text {GeV}\,\ll p_{\text {T}}\,^{W}\cdot \cosh \eta _{W}$, such that:

$$\begin{aligned} \sqrt{1+\left( \frac{m_{W}}{p_{\text {T}}\,^{W}\cdot \cosh \eta _{W}}\right) ^2} \sim 1. \end{aligned}$$

Equation (4) can thus be simplified:

$$\begin{aligned} \cosh (\eta _{W}-\eta _{b})&\sim \frac{m_{t}^2-m_{W}^2}{2{p_{\text {T}}\,^{W}}{p_{\text {T}}\,^{b}}}+\cos (\phi _{W}-\phi _{b}) \nonumber \\&\sim \frac{m_{t}^2-m_{W}^2}{2E_{\text {T}}^{\text {miss}}\,p_{\text {T}}\,^{b}}+\cos (\phi _{E_{\text {T}}^{\text {miss}}\,}-\phi _{b}). \end{aligned}$$

(5)

By definition, $\cosh (x) \ge 1$ so that the right-hand side of Eq. (5) is expected to be larger than 1 in the case of successful leptonic top reconstruction. The discriminating observable $\cosh _{\text {max}}$ is therefore defined as:

$$\begin{aligned} \cosh _{\text {max}}\,= \max \{\cosh \Delta \eta ^{1}_{W,b}, \cosh \Delta \eta ^{2}_{W,b}\}, \end{aligned}$$

where $\Delta \eta ^{1}_{W,b}$ and $\Delta \eta ^{2}_{W,b}$ represent the pseudorapidity difference between the W boson candidate and either of the two leading b-tagged jets selected in the event. Events with high $\cosh _{\text {max}}$  values are likely to contain a top quark with a lost lepton and are excluded from the signal regions.

Figure 7 illustrates the modelling of the shape of $\cosh _{\text {max}}$  in SRWX and SRTX. The $\cosh _{\text {max}}$  distribution in SR0X is shown in Fig. 3.

Fig. 7

Distributions of $\cosh _{\text {max}}$ in a SRWX and b SRTX events passing all the SR requirements except those on $\cosh _{\text {max}}$ itself (which are indicated by the arrows). The contributions from all SM backgrounds are shown after the profile likelihood simultaneous fit to all tt0L-low CRs, with the hatched bands representing the total uncertainty. The category ‘$t\bar{t}$ (other)’ represents $t\bar{t}$ events without extra jets or events with extra light-flavour jets. ‘Other’ includes contributions from $t\bar{t}{+}W$, tZ and tWZ processes. The expected distributions for selected signal models are shown as dashed lines. The underflow (overflow) events are included in the first (last) bin. The bottom panels show the ratio of the observed data to the total SM background prediction, with the hatched area representing the total uncertainty in the background prediction and the red arrows marking data outside the vertical-axis range

Fig. 8

Distributions of $\chi _{t\bar{t}\text {, had}}^2$ in a SRWX and b SRTX events passing all the SR requirements except those on $\chi _{t\bar{t}\text {, had}}^2$ itself (which are indicated by the arrows). The contributions from all SM backgrounds are shown after the profile likelihood simultaneous fit to all tt0L-low CRs, with the hatched bands representing the total uncertainty. The category ‘$t\bar{t}$ (other)’ represents $t\bar{t}$ events without extra jets or events with extra light-flavour jets. ‘Other’ includes contributions from $t\bar{t}{+}W$, tZ and tWZ processes. The expected distributions for selected signal models are shown as dashed lines. The underflow (overflow) events are included in the first (last) bin. The bottom panels show the ratio of the observed data to the total SM background prediction, with the hatched area representing the total uncertainty in the background prediction and the red arrows marking data outside the vertical-axis range

1.1.2 $\chi _{t\bar{t}\text {, had}}^2$ 

The $\chi _{t\bar{t}\text {, had}}^2$ observable approximately quantifies how likely an event is to include two hadronically decaying top quarks. It is used primarily to reject backgrounds containing no hadronic top quark decays, such as Z+jets events. It is defined as follows:

$$\begin{aligned} \chi _{t\bar{t}\text {, had}}^2\,&= \left( \frac{m_{W_1}-m_{W_{\text {ref}}}}{\sigma _{m_{W}}}\right) ^2 \nonumber \\&\quad + \left( \frac{(m_{t_1}-m_{W_1})-(m_{t_{\text {ref}}}-m_{W_{\text {ref}}})}{\sigma _{m_{t}-m_{W}}}\right) ^2 \nonumber \\&\quad + \left( \frac{(m_{t_2}-m_{W_2})-(m_{t_{\text {ref}}}-m_{W_{\text {ref}}})}{\sigma _{m_{t}-m_{W}}}\right) ^2. \end{aligned}$$

(6)

Up to seven jets, including the two selected b-tagged jets, are considered in the calculation. The first W boson candidate, $W_1$, is built from two non-b-tagged jets, while the first top quark candidate, $t_1$, combines $W_1$ and one of the b-tagged jets, $b_1$, such that $t_1\rightarrow W_1 b_1$. According to Monte Carlo simulations, the second W boson candidate, $W_2$, is in more than 50% of the cases too soft to lead to two individual jets satisfying the reconstruction criteria. Hence, it is built from a single non-b-tagged jet to which the mass of the W boson is attributed. As a result, the second top quark candidate, $t_2 \rightarrow W_2 b_2$, contains only one non-b-tagged jet and the remaining b-tagged jet, $b_2$.

The first term in Eq. (6) corresponds to the invariant mass constraint from $W_1$. The values $m_{W_{\text {ref}}}$ and $\sigma _{m_{W}}$ are respectively the mean and the standard deviation of the experimental invariant mass distribution expected for hadronically decaying W bosons. The second and third terms correspond to the invariant mass constraints from $t_1$ and $t_2$, respectively. Since $m_{W_1}$ and $m_{t_1}$ ($m_{W_2}$ and $m_{t_2}$) are strongly correlated, the W boson mass is subtracted from the top quark mass to decouple these two terms from the first one. The values of $m_{t_{\text {ref}}}$ and $\sigma _{m_{t}-m_{W}}$ are respectively the mean of the experimental top quark mass distribution and the standard deviation of the $m_{t}-m_{W}$ distribution expected for reconstructed hadronic top quark decays. The values of $m_{W_{\text {ref}}}$, $\sigma _{m_{W}}$, $m_{t_{\text {ref}}}$ and $\sigma _{m_{t}-m_{W}}$ are taken from Ref. [117]:

• $m_{W_{\text {ref}}} = 80.51~\text {GeV}\,$, $\sigma _{m_{W}} = 12.07~\text {GeV}\,$,

• $m_{t_{\text {ref}}}-m_{W_{\text {ref}}} = 85.17~\text {GeV}\,$, $\sigma _{m_{t}-m_{W}} = 16.05~\text {GeV}\,$.

The $\chi ^2$ is recomputed for each possible jet combination and the final $\chi _{t\bar{t}\text {, had}}^2$  corresponds to the minimum value obtained. Events with high $\chi _{t\bar{t}\text {, had}}^2$  values are less likely to contain two hadronic top quark decays and are therefore excluded from the signal regions.

Table 4 Selection criteria for the top-with-lost-lepton control regions used in the tt0L-low analysis

Table 5 Selection criteria for the Z+jets control regions used in the tt0L-low analysis

Figure 8 illustrates the modelling of the shape of $\chi _{t\bar{t}\text {, had}}^2$  in SRWX and SRTX. The $\chi _{t\bar{t}\text {, had}}^2$  distribution in SR0X is shown in Fig. 3.

1.2 A.2 Background estimation

The event topologies in the signal regions and control regions are kept as similar as possible to reduce any bias originating from differences between their kinematic phase spaces. For this purpose, control regions with one or more leptons in the final state are split according to the mass of the heaviest large-radius jet, as is done for the signal regions, while all $E_{\text {T}}^{\text {miss}}$ -related variables are recalculated by treating the selected leptons as invisible, denoted by the subscript ‘no lepton’ in the variable names.

One of the most prominent sources of background in the signal regions is semileptonic $t\bar{t}$ decay where the lepton is misreconstructed or outside the detector acceptance, while the contribution from the dileptonic $t\bar{t}$ decay is negligible. Control regions selecting events with exactly one lepton (e or $\mu$) are defined in order to estimate the background originating from a top quark decay with a lost lepton, which includes single-top events in the tW channel, and $t\bar{t}{+}b$ and $t\bar{t}$ (other) events.

Fig. 9

$E_{\text {T}}^{\text {miss}}$ distributions in a SR0X, b SRWX and c SRTX events passing all the SR requirements. The contributions from all SM backgrounds are shown after the profile likelihood simultaneous fit to all tt0L-low CRs, with the hatched bands representing the total uncertainty. The category ‘$t\bar{t}$ (other)’ represents $t\bar{t}$ events without extra jets or events with extra light-flavour jets. ‘Other’ includes contributions from $t\bar{t}{+}W$, tZ and tWZ processes. The expected distributions for selected signal models are shown as dashed lines. The overflow events are included in the last bin. The bottom panels show the ratio of the observed data to the total SM background prediction, with the hatched area representing the total uncertainty in the background prediction and the red arrows marking data outside the vertical-axis range

A $\chi ^2$-based observable [118], $\chi _{t\bar{t}\text {, lep}}^2$ , taking into account the kinematic properties of $E_{\text {T}}^{\text {miss}}$ , lepton, jets and the b-tagging information, is used to reconstruct semileptonic $t\bar{t}$events and separate them from tW and $t\bar{t}{+}b$ events. It follows an approach similar to that for the $\chi _{t\bar{t}\text {, had}}^2$  variables by placing constraints on the masses of the hadronically decaying W boson, the hadronically decaying top quark and the leptonically decaying top quark. The presence of extra b-tagged jets is used to select $t\bar{t}{+}b$ over single-top processes. Tighter $\cosh _{\text {max, no lepton}}$ selections are required in the single-top control regions to reduce the contamination from semileptonic $t\bar{t}$events failing the $\chi _{t\bar{t}\text {, lep}}^2$ reconstruction and attain high purity in tW events. Table 4 presents the full event selections applied to define the top-with-lost-lepton control regions.

Fig. 10

Exclusion limits for colour-neutral a scalar or b pseudoscalar mediator dark matter models as a function of the mediator mass $m(\phi )$ or m(a) for a DM mass $m_{\chi } = 1~\text {GeV}\,$. Associated production of DM with both single top quarks (tW and tj channels) and top quark pairs is considered. The limits are calculated at 95% CL and are expressed in terms of the ratio of the excluded cross section to the nominal cross section for a coupling assumption of $g = g_q = g_{\chi } = 1$. The solid (dashed) lines show the observed (expected) exclusion limits for the tt0L-high and tt0L-low analyses and their statistical combination

Table 6 Results from the tt0L-low and tt0L-high searches for invisible decays of the 125 $\text {GeV}$  Higgs boson in the $t\bar{t}H$ topology using 139 fb$^{-1}$  of Run 2 data, and their statistical combination. Shown are the best-fit values of $\mathcal {B}_{H\rightarrow \text {inv}}$ , as well as observed and expected upper limits on $\mathcal {B}_{H\rightarrow \text {inv}}$ at the 95% CL. The corresponding Asimov datasets for the expected results are constructed using nuisance parameter values from a fit to data with $\mathcal {B}_{H\rightarrow \text {inv}}\,= 0$, and the quoted uncertainty corresponds to the 68% confidence interval

Fig. 11

a The expected negative logarithmic profile likelihood ratios $-2\,\Delta \!\ln (\Lambda )$ as a function of $\mathcal {B}_{H\rightarrow \text {inv}}$ for each of the two tt0L analyses and their statistical combination and b these likelihood ratios for the observed data

Another major background component in the signal regions contains $Z\rightarrow \nu \nu$ produced in association with jets. Control regions selecting events with two leptons with opposite charge and the same flavour (ee or $\mu \mu$) are defined in order to estimate the $Z(\nu \nu )$+jets background. The invariant mass and transverse momentum of the dilepton system, $m_{\ell \ell }$ and $p_{\text {T}}\,^{\ell \ell }$ respectively, and the missing transverse momentum significance $\mathcal {S}$ serve as the major discriminants to suppress the contamination from dileptonic $t\bar{t}$ events. To obtain enough events, several selections applied in the signal regions are omitted in the corresponding CRs. Table 5 presents the full event selections applied to define the Z+jets control regions.

Validation regions are not included in the statistical model and serve only to validate the extrapolation over lepton multiplicity when going from the control regions to the signal regions. The event selections for the validation regions therefore require zero leptons, while being orthogonal to the signal region selections.

In the $t\bar{t}$-enriched validation regions, $t\bar{t}$events are selected by inverting the tight $\cosh _{\text {max}}$ requirement applied in the signal regions and adding a looser upper bound. The validation regions for $t\bar{t}{+}b$, single-top and Z+jets are merged into a single $t\bar{t}$-suppressed validation region because of the limited number of events in the 0-lepton phase space. In these regions the $\chi _{t\bar{t}\text {, had}}^2$ selection applied in the signal regions is inverted. The $p_{\text {T}}\,^{t\bar{t}}/E_{\text {T}}^{\text {miss}}\,$ requirements are discarded because they become irrelevant when the value of $\chi _{t\bar{t}\text {, had}}^2$ is too large. Tight $\Delta R\left( b_1,b_2\right)$ selections are imposed to minimise the contamination from $W\text {+jets}$ events, with their thresholds optimised in each region to provide a number of events similar to that in the $t\bar{t}$-enriched VRs. All the background predictions in the VRs agree with the data to within $1\sigma$.

1.3 A.3 Results

All tt0L-low signal and control regions are included in a statistical model based on the combined likelihood fit. The normalisations of the $t\bar{t}{+}b$, $t\bar{t}$ (other), single-top and Z+jets background processes are free-floating. For the $t\bar{t}$ background, the normalisation factors are decorrelated in the three kinematic regimes (CR0X, CRWX and CRTX) to account for a possible top quark $p_{\text {T}}$  dependence of the normalisation factor. The yield results are presented in Table 2.

Figure 9 shows the $E_{\text {T}}^{\text {miss}}$ distributions in the three tt0L-low signal regions. The background contributions are obtained from the profile likelihood simultaneous fit to all tt0L-low CRs with a background-only hypothesis.

Exclusion limits at 95% CL are presented in Fig. 10a and b for DM models with a spin-0 scalar or pseudoscalar mediator particle, respectively. The tt0L-low analysis, the tt0L-high analysis and the full tt0L combination are presented separately in order to quantify the improvement gained by adding the tt0L-low channel to the tt0L search. As they were designed to do, the tt0L-low signal regions extend the sensitivity to low-mass mediator models, with an improvement of up to about $15\%$ in the cross-section limit for scalar mediator particles.

In addition, the negative logarithmic profile likelihood ratios $-2\,\Delta \!\ln (\Lambda )(\mathcal {B}_{H\rightarrow \text {inv}}\,;\theta )$ as a function of $\mathcal {B}_{H\rightarrow \text {inv}}$ for the tt0L-low and tt0L-high analyses, and their combination, are illustrated in Fig. 11.

Table 6 presents the best-fit value, and the observed and expected upper limits on $\mathcal {B}_{H\rightarrow \text {inv}}$ at the 95% CL for the tt0L-low analysis, the tt0L-high analysis and their statistical combination. Since the tt0L-low selection was designed to target mediator masses below 100 $\text {GeV}$ , the improvement in the expected upper limit at the Higgs boson mass is found to be relatively small.