Search for large missing transverse momentum in association with one top-quark in proton-proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector

TL;DR

This paper reports a search for events with a top-quark and large missing transverse momentum in proton-proton collisions at 13 TeV, using ATLAS data, to probe dark matter and vector-like quark models, setting upper limits on production cross-sections.

Contribution

It introduces a search strategy for top-quark plus missing energy events at 13 TeV and interprets results within dark matter and vector-like quark models, providing new constraints.

Findings

No significant deviation from Standard Model expectations.

Set 95% CL upper limits on production cross-sections.

Constrained parameter space of dark matter and vector-like quark models.

Abstract

This paper describes a search for events with one top-quark and large missing transverse momentum in the final state. Data collected during 2015 and 2016 by the ATLAS experiment from 13 TeV proton-proton collisions at the LHC corresponding to an integrated luminosity of 36.1 fb$^{-1}$ are used. Two channels are considered, depending on the leptonic or the hadronic decays of the $W$ boson from the top quark. The obtained results are interpreted in the context of simplified models for dark-matter production and for the single production of a vector-like $T$ quark. In the absence of significant deviations from the Standard Model background expectation, 95% confidence-level upper limits on the corresponding production cross-sections are obtained and these limits are translated into constraints on the parameter space of the models considered.

Tables

Table 1. Table 1: Overview of the event selections used to define the signal and control regions.

Selections (leptonic channel)	1L-DM-SR		1L-TCR	1L-WCR
Number of leptons	= 1		= 1	= 1
$p_{\text{T}}\left(\ell \right)$ [GeV]	> 30		> 30	> 30
Lepton charge	> 0		> 0	> 0
Number of jets	= 1		= 2	= 1
Number of $b\text{-tagged}$ jets	= 1		= 2	= 1
$p_{\text{T}}$($b\text{-tagged}$ jet) [GeV]	> 30		> 30	> 30
$E_{\text{T}}^{\text{miss}}$ [GeV]	> 50		> 50	> 50
$m_{\text{T}}^W+E_{\text{T}}^{\text{miss}}$ [GeV]	> 60		> 60	> 60
$m_{\text{T}}^W$ [GeV]	> 260
$\left|\Delta \varphi (\ell ,b)\right|$	< 1.2		-	-
Selections (hadronic channel)	0L-DM-SR	0L-VLT-SR	0L-TCR	0L-VCR
Number of forward jets	= 0	$\ge$ 1	-	-
Number of leptons	= 0		= 0	= 0
$E_{\text{T}}^{\text{miss}}$ [GeV]	$>$ 200		$>$ 200	$>$ 200
Number of large-$R$ jets	$\ge$ 1		$\ge$ 1	$\ge$ 1
Number of top-tagged jets	$\ge$ 1		$\ge$ 1	$\ge$ 1
$\Delta \Phi (E_{\text{T}}^{\text{miss}},J)$	$>\frac{\pi }{2}$		$>\frac{\pi }{2}$	$>\frac{\pi }{2}$
Number of track-jets	$\ge$ 1		$\ge$ 1	$\ge$ 1
Number of $b\text{-tagged}$ track-jets	= 1		$\ge$ 2	= 0
Veto jet (masked tile-calo)	-		applied	-
$\Omega =\frac{E_{\text{T}}^{\text{miss}}-p_{\text{T}}\left(J\right)}{E_{\text{T}}^{\text{miss}}+p_{\text{T}}\left(J\right)}$	$>$ –0.3		$>$ –0.3	$>$ –0.3
$\Delta {\Phi }_{\min }(E_{\text{T}}^{\text{miss}},\mathrm{calo}\mathrm{jets})$	$>$ 1.0		0.2 $\Delta {\Phi }_{\min }$ 1.0	$>$ 1.0

Table 2. Table 2: Relative effect (in %) of various sources of systematic uncertainty on the predicted background yields in the signal regions used for the dark-matter search, obtained after the fit to data. Individual sources of uncertainties are correlated, and their sum in quadrature is not necessarily equal to the total background uncertainty.

	1L-DM-SR		0L-DM-SR
	non-$t\overline{t}$	$t\overline{t}$	$t\overline{t}$	Single top	$W$+jets	$Z$+jets	Multijet	Other
$b$-tagging	$4.77$	$4.61$	$4.11$	$2.94$	$9.24$	$7.73$	-	$7.99$
$E_{\text{T}}^{\text{miss}}$	$11.7$.00	$1.14$	$2.2$	$2.12$	$2.2$	$2.16$	$2.99$	$1.99$
Large-$R$ jets	-	-	$9.04$	$9.5$	$13.2$.00	$12.5$.00	-	$11.8$.00
Small-$R$ jets	$9.9$	$6.96$	$1.27$	$2.93$	$1.01$	$0.511$	-	$0.968$
Lepton	$1.19$	$0.752$	< 0.1 <	< 0.1 <	< 0.1 <	< 0.1 <	-	< 0.1 <
Luminosity	$2.03$	$2.09$	$2.122$	$2.11$	$2.25$	$2.22$	-	$2.07$
Pile-up	$5.29$	$1.43$	$0.268$	$0.346$	$0.818$	$1.23$	-	$1.37$
Background modelling	$14.8$.00	$13.5$.00	$8.85$	$5.28$	$26.9$.00	$26.9$.00	$111$.000	$1.04$
Total systematic	$18.2$.00	$11.8$.00	$6.76$	$13.2$.00	$31.5$.00	$30.5$.00	$88.5$.00	$16.2$.00

Table 3. Table 3: Relative effect (in %) of various sources of systematic uncertainty on the predicted background yields in the signal region used for the vector-like T 𝑇 T -quark search, obtained after the fit to data. Individual sources of uncertainties are correlated, and their sum in quadrature is not necessarily equal to the total background uncertainty.

	0L-VLT-SR
	$t\overline{t}$	Single top	$W$+jets	$Z$+jets	Other
$b$-tagging	$3.93$	$3.16$	$11.0$.00	$7.69$	$8.4$
$E_{\text{T}}^{\text{miss}}$	$1.41$	$1.86$	$0.776$	$0.525$	$0.25$
Large-$R$ jets	$10.6$.00	$12.4$.00	$14.7$.00	$12.9$.00	$13.3$.00
Small-$R$ jets	$7.3$	$7.79$	$7.61$	$8.29$	$10.9$.00
Lepton	< 0.1 <	< 0.1 <	< 0.1 <	< 0.1 <	< 0.1 <
Luminosity	$2.18$	$2.29$	$2.40$	$2.12$	$2.24$
Pile-up	$1.27$	$2.91$	$4.65$	$4.09$	$5.12$
Background modelling	$14.2$.00	$6.06$	$28.4$.00	$26.4$.00	$2.68$
Total systematic	$7.89$	$15.5$0	$21.3$.00	$19.4$.00	$19.5$0

Table 4. Table 4: Numbers of events observed in the signal and control regions, together with the estimated SM backgrounds before the fit to data. The uncertainties include statistical and systematic uncertainties. The expected numbers of events for benchmark signals normalised to the theoretical prediction are also shown. The benchmark signals correspond to: the non-resonant (NR) DM model with m V = 1 subscript 𝑚 𝑉 1 m_{V}=1   TeV  and 2  TeV , m χ = 1 subscript 𝑚 𝜒 1 m_{\chi}=1   GeV , a = 0.5 𝑎 0.5 a=0.5 and g χ = 1 subscript 𝑔 𝜒 1 g_{\chi}=1 ; the resonant (R) DM model with m ϕ = 1 subscript 𝑚 italic-ϕ 1 m_{\phi}=1   TeV  and 2  TeV , m χ = 10 subscript 𝑚 𝜒 10 m_{\chi}=10   GeV , λ = 0.2 𝜆 0.2 \lambda=0.2 and y = 0.4 𝑦 0.4 y=0.4 ; and a VLT with a mass of 0.9  TeV .

	1L-DM-SR		1L-TCR		1L-WCR		0L-DM-SR		0L-VLT-SR		0L-TCR		0L-VCR
$t\overline{t}$	$391.078$ $\pm$	$138.203$	$\mathrm{12\hspace{0.17em}308.2}$ $\pm$	$3097.94$	$8432.74$ $\pm$	$1660.64$	$\mathrm{10\hspace{0.17em}187.5}$ $\pm$	$2851.74$	$3746.14$ $\pm$	$1231.66$	$6979.81$ $\pm$	$1721.48$	$6067.23$ $\pm$	$1834.51$
Single top	$65.6128$ $\pm$	$20.7592$	$2932.23$ $\pm$	$760.84$	$\mathrm{12\hspace{0.17em}174.8}$ $\pm$	$1725.53$	$1015.82$ $\pm$	$258.631$	$355.75$ $\pm$	$96.6554$	$273.875$ $\pm$	$70.6229$	$888.052$ $\pm$	$254.136$
$W$+jets	$34.2304$ $\pm$	$8.40418$	$1885.59$ $\pm$	$640.723$	$\mathrm{92\hspace{0.17em}109.8}$ $\pm$	$\mathrm{24\hspace{0.17em}086.8}$	$2239.95$ $\pm$	$897.389$	$769.55$ $\pm$	$306.128$	$147.147$ $\pm$	$87.4522$	$\mathrm{28\hspace{0.17em}470.4}$ $\pm$	$\mathrm{12\hspace{0.17em}134.7}$
$Z$+jets	$0.398342$ $\pm$	$0.858914$	$112.085$ $\pm$	$48.9708$	$3406.78$ $\pm$	$985.647$	$2655.56$ $\pm$	$1125.67$	$849.96$ $\pm$	$361.143$	$138.732$ $\pm$	$82.8244$	$\mathrm{27\hspace{0.17em}438}$ $\pm$	$\mathrm{11\hspace{0.17em}043.4}$
Other	$13.8398$ $\pm$	$14.9265$	$643.225$ $\pm$	$883.177$	$6672.99$ $\pm$	$\mathrm{10\hspace{0.17em}145.2}$	$531.821$ $\pm$	$194.494$	$88.81$ $\pm$	$28.03$	$1055.42$ $\pm$	$638.27$	$2731.51$ $\pm$	$763.502$
Total Background	$501.000$ $\pm$	$140.616$	$\mathrm{17\hspace{0.17em}882.4}$ $\pm$	$3371.7998$	$\mathrm{122\hspace{0.17em}818}$ $\pm$	$\mathrm{26\hspace{0.17em}262.108}$	$\mathrm{16\hspace{0.17em}560.6}$ $\pm$	$4507.61$	$5810.21$ $\pm$	$1733.06$	$8565.79$ $\pm$	$1960.04$	$\mathrm{65\hspace{0.17em}554.8}$ $\pm$	$\mathrm{22\hspace{0.17em}253}$
Data	511  $\pm$000		17 662  $\pm$00 000		127 286  $\pm$000 000		15 781  $\pm$00 000		5454  $\pm$0000		8493  $\pm$0000		62 304  $\pm$00 000
R DM $m_{\varphi }$ = 1 TeV	-		-		-		$\mathrm{11\hspace{0.17em}260.8}$ $\pm$	$1254.55$	-		$55.803$ $\pm$	$12.6643$	$8084.18$ $\pm$	$1579.65$
R DM $m_{\varphi }$ = 2 TeV	-		-		-		$468.977$ $\pm$	$83.2669$	-		$4.253$ $\pm$	$1.052$	$348.52$ $\pm$	$86.1307$
NR DM $m_{\varphi }$ = 1 TeV	$164.818$ $\pm$	$22.94$	$1.019$ $\pm$	$0.465$	$20.18$ $\pm$	$2.77$	$2090.29$ $\pm$	$283.251$	-		$28.9993$ $\pm$	$5.86888$	$1600.2$ $\pm$	$316.645$
NR DM $m_{\varphi }$ = 2 TeV	$6.49$ $\pm$	$2.66$	$0.0265$ $\pm$	$0.013$	$0.496$ $\pm$	$0.0965$	$94.6664$ $\pm$	$13.2137$	-		$1.07578$ $\pm$	$0.211743$	$75.178$ $\pm$	$15.2028$
VLT $m_{\mathrm{VLT}}$ = 0.9 TeV	-		-		-		-		$111.592$ $\pm$	$20.0503$	$21.011$ $\pm$	$5.331$	$76.1046$ $\pm$	$16.7274$

Table 5. Table 5: Numbers of events observed in the signal and control regions used for the non-resonant dark-matter search, together with the estimated SM backgrounds in the fit to data, under the background-only hypothesis. The uncertainties include statistical and systematic uncertainties. The uncertainties in the individual backgrounds are correlated, and do not necessarily add in quadrature to the total background uncertainty.

Leptonic channel	1L-DM-SR		1L-TCR		1L-WCR
$t\overline{t}$	$385.269$ $\pm$	$40.8536$	$\mathrm{12\hspace{0.17em}130.6}$ $\pm$	$1960.85$	$8467.17$ $\pm$	$798.86$
Non-$t\overline{t}$	$116.639$ $\pm$	$17.4398$	$5543.64$ $\pm$	$963.185$	$\mathrm{118\hspace{0.17em}664}$ $\pm$	$\mathrm{25\hspace{0.17em}865.4}$
Total	$501.908$ $\pm$	$62.3669$	$\mathrm{17\hspace{0.17em}674.2}$ $\pm$	$3055.84$	$\mathrm{127\hspace{0.17em}131}$ $\pm$	$\mathrm{25\hspace{0.17em}826.7}$
Data	511		17662		127286
Hadronic channel	0L-DM-SR		0L-TCR		0L-VCR
$t\overline{t}$	$9899.73$ $\pm$	$874.877$	$7155.79$ $\pm$	$618.559$	$5901.09$ $\pm$	$251.421$
Single top	$990.627$ $\pm$	$105.467$	$273.158$ $\pm$	$35.6454$	$879.088$ $\pm$	$98.3525$
$W$+jets	$2047.73$ $\pm$	$519.188$	$118.843$ $\pm$	$65.3012$	$\mathrm{23\hspace{0.17em}075.5}$ $\pm$	$4919.78$
$Z$+jets	$2455.82$ $\pm$	$461.385$	$135.397$ $\pm$	$60.6836$	$\mathrm{29\hspace{0.17em}915.4}$ $\pm$	$4607.27$
Multijet	$86.7035$ $\pm$	$90.1524$	$757.292$ $\pm$	$348.217$	$0$ $\pm$	$0$
Other	$328.204$ $\pm$	$40.8788$	$50.0684$ $\pm$	$5.55382$	$2667.09$ $\pm$	$305.944$
Total	$\mathrm{15\hspace{0.17em}808.8}$ $\pm$	$1170.09$	$8490.54$ $\pm$	$761.173$	$\mathrm{62\hspace{0.17em}438.1}$ $\pm$	$1485.9$
Data	15781		8493		62304

Table 6. Table 6: Numbers of events observed in the signal and control regions used for the resonant dark-matter search, together with the estimated SM backgrounds in the fit to data, under the background-only hypothesis. The uncertainties include statistical and systematic uncertainties. The uncertainties in the individual backgrounds are correlated, and do not necessarily add in quadrature to the total background uncertainty.

	0L-DM-SR		0L-TCR		0L-VCR
$t\overline{t}$	$9689.94$ $\pm$	$615.037$	$7109.08$ $\pm$	$460.894$	$5711.87$ $\pm$	$583.489$
Single top	$986.664$ $\pm$	$109.637$	$281.51$ $\pm$	$38.5556$	$869.132$ $\pm$	$107.291$
$W$+jets	$2066.4$ $\pm$	$539.021$	$120.77$ $\pm$	$67.1422$	$\mathrm{22\hspace{0.17em}993.4}$ $\pm$	$5015.16$
$Z$+jets	$2609.67$ $\pm$	$526.734$	$149.047$ $\pm$	$61.0072$	$\mathrm{30\hspace{0.17em}058.7}$ $\pm$	$4653.8$
Other	$330.293$ $\pm$	$44.1154$	$51.4163$ $\pm$	$6.11536$	$2674.89$ $\pm$	$314.522$
Multijet	$92.1532$ $\pm$	$88.2063$	$796.115$ $\pm$	$359.483$	$0$ $\pm$	$0$
Total	$\mathrm{15\hspace{0.17em}775.1}$ $\pm$	$365.001$	$8507.94$ $\pm$	$277.792$	$\mathrm{62\hspace{0.17em}307.9}$ $\pm$	$1416.3$
Data	15781		8493		62304

Table 7. Table 7: Numbers of events observed in the signal and control regions used for the vector-like T 𝑇 T -quark search, together with the estimated SM backgrounds in the fit to data, under the background-only hypothesis. The uncertainties include statistical and systematic uncertainties. The uncertainties in the individual backgrounds are correlated, and do not necessarily add in quadrature to the total background uncertainty.

	0L-VLT-SR		0L-TCR		0L-VCR
$t\overline{t}$	$3556.56$ $\pm$	$277.848$	$7160.43$ $\pm$	$374.516$	$5312.44$ $\pm$	$739.719$
Single top	$322.749$ $\pm$	$44.9592$	$278.286$ $\pm$	$36.603$	$818.348$ $\pm$	$123.198$
$W$+jets	$664.53$ $\pm$	$198.605$	$125.799$ $\pm$	$71.8749$	$\mathrm{21\hspace{0.17em}856.4}$ $\pm$	$5871.73$
$Z$+jets	$832.087$ $\pm$	$182.492$	$159.518$ $\pm$	$63.8904$	$\mathrm{31\hspace{0.17em}770}$ $\pm$	$4769.39$
Other	$82.0945$ $\pm$	$13.8548$	$801.2797$ $\pm$	$316.9771$	$2590.03$ $\pm$	$341.838$
Total	$5458.02$ $\pm$	$161.483$	$8525.31$ $\pm$	$273.325$	$\mathrm{62\hspace{0.17em}347.2}$ $\pm$	$1440.4$
Data	5454		8493		62304

Equations

$${\mathcal{L}}_{\mathrm{int}}=a{V}_{\mu }\bar{u}{\gamma }^{\mu }{P}_{\mathrm{R}}t+g_{\chi }{V}_{\mu }\bar{\chi }{\gamma }^{\mu }\chi +h.c.,$$

$${\mathcal{L}}_{\mathrm{int}}=\lambda \varphi {\bar{d}}^c{P}_{\mathrm{R}}s+y\varphi \bar{\chi }{P}_{\mathrm{R}}t+h.c.,$$

Full text

\AtlasTitle

Search for large missing transverse momentum in association with one top-quark in proton–proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector \AtlasAbstract This paper describes a search for events with one top-quark and large missing transverse momentum in the final state. Data collected during 2015 and 2016 by the ATLAS experiment from 13 TeV proton–proton collisions at the LHC corresponding to an integrated luminosity of 36.1 fb*-1* are used. Two channels are considered, depending on the leptonic or the hadronic decays of the $W$ boson from the top quark. The obtained results are interpreted in the context of simplified models for dark-matter production and for the single production of a vector-like $T$ quark. In the absence of significant deviations from the Standard Model background expectation, 95% confidence-level upper limits on the corresponding production cross-sections are obtained and these limits are translated into constraints on the parameter space of the models considered.

\AtlasRefCodeEXOT-2017-16 \PreprintIdNumberCERN-EP-2018-301 \AtlasJournalJHEP \AtlasJournalRefJHEP 05 (2019) 41 \AtlasDOI10.1007/JHEP05(2019)041

\size@chapter\sectfont

Contents

@afterheading@starttoc

toc

1 Introduction

In spite of its successes in describing the phenomenology of the fundamental particles and the corresponding interactions, the Standard Model (SM) can be considered as a low-energy approximation of a more fundamental theory with new degrees of freedom and symmetries that would become manifest at a higher energy.

One argument supporting the idea that new particles beyond the SM might exist arises from astrophysical measurements, such as the rotational speed of stars in galaxies and gravitational lensing [1, 2, 3]. These observations point to the existence of non-light-emitting matter, a dominant fraction of which is of non-baryonic form, usually referred to as dark matter (DM). Even if there are no viable candidates in the SM for particles which could explain DM, proton–proton collisions at the Large Hadron Collider (LHC) may possibly produce new particles that couple both to SM particles and to these DM candidates. While such candidates are not expected to interact significantly with detectors, the SM particles produced in association with the unobserved DM particles could allow these processes to be detected. Search strategies depend on the type of particle or system that is recoiling against the unseen particle. Both ATLAS and CMS have carried out searches for invisible particles produced in association with jets [4, 5, 6, 7], photons [8, 9], $W$ or $Z$ bosons [10, 5, 11] and Higgs bosons [12, 13, 14, 15], significantly constraining the allowed parameter space for different classes of models predicting DM candidates.

This paper describes a search for the production of invisible particles in association with a single top-quark in proton–proton collisions produced at the LHC with a centre-of-mass energy of $\sqrt{s}=13$ TeV and detected using the ATLAS detector. Such a final state, commonly referred to as “mono-top”, is characterised by a top-quark and significant missing transverse momentum, which is due to the undetected particles. Background contributions from SM processes [16] are expected to be small. In addition, this search is sensitive to specific DM models, since the presence of top-quarks in the final state constrains the flavour structure of the considered couplings [17, 18]. Similar searches were previously conducted by the CDF Collaboration using 7.7 fb*-1* of Tevatron $p\bar{p}$ collisions at $\sqrt{s}=1.96$ TeV [19] and by the ATLAS and CMS collaborations using $\sqrt{s}=8$ TeV [20, 21] and $\sqrt{s}=13$ TeV [22] LHC data. Searches for new phenomena in events with same-charge leptons and $b$-tagged jets [23] provide information complementary to the results from mono-top searches and exclude new vector resonances with masses up to 3 TeV, assuming a dark-sector coupling of 1.0 and a coupling to SM particles above 0.3.

A final state with a top-quark and missing transverse momentum can also originate from the single production of new vector-like quarks if these decay into a top-quark and a $Z$ boson that decays invisibly into two neutrinos. Vector-like quarks are colour-triplet spin-$1/2$ fermions in which, in contrast to the SM quarks, the left- and right-handed components have the same properties under transformations of the electroweak symmetry group $\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{U}(1)_{\mathrm{Y}}$. Such new particles are predicted in SM extensions, such as Little Higgs [24, 25] and Composite Higgs [26, 27] models, and are expected to mix with SM quarks [28]. In order to preserve gauge invariance, only a limited set of possible representations exist [29, 30] and their electric charge can be $+2/3e$ ($T$ quark), $-1/3e$ ($B$ quark), $+5/3e$ ($X$ quark) or $-4/3e$ ($Y$ quark), with $e$ being the elementary charge. In this paper, only the single production of vector-like $T$ quarks (VLT) via an electroweak interaction is considered. Although couplings of $T$ quarks to first- and second-generation SM quarks are not excluded [31, 32], it is common to assume that they couple exclusively to third-generation SM quarks [33]. Such couplings can be described in terms of $\sin\theta_{\mathrm{L}}$ [34], where $\theta_{\mathrm{L}}$ is the mixing angle of the $T$ quark with the top-quark, or in terms of a generalised coupling $\kappa_{T}$ [35, 36]. The $T$ quarks can decay either via the charged current, i.e. $T\rightarrow Wb$, or via flavour-changing neutral currents [37], i.e. $T\rightarrow Zt$ and $T\rightarrow Ht$. The $T\rightarrow Zt\to\nu\bar{\nu}Wb$ decay is considered in the present search.

The ATLAS and CMS collaborations have sought pair production of $T$ quarks decaying into third-generation quarks in $pp$ collisions at a centre-of-mass energy of 8 TeV [38, 39, 40, 41], targeting all three possible decay modes. Searches at 13 TeV have aimed at final states with leptons, targeting the $T\rightarrow Zt$ decay [42, 43], the $T\rightarrow Wb$ decay [44, 45], as well as general single-lepton and fully hadronic final states with boosted bosons [46, 47] and multiple $b$-tagged jets [46, 48, 49]. The most stringent mass limit for an isospin singlet $T$ is 1.3 TeV [50]. For such large $T$ masses, the cross-section for single $T$ production may be larger than the pair-production cross-section because of the larger available phase space. Nonetheless, the comparison of single- and pair-production cross-sections depends on the assumed coupling to the SM quarks. Single production of $T$ quarks was sought at 8 TeV [51, 52, 40] by the ATLAS Collaboration. At 13 TeV, the ATLAS and CMS collaborations have sought the decays $T\rightarrow Wb$ [53, 54], $T\rightarrow Ht$ [55, 56] and $T\rightarrow Zt$ [57, 58, 43].

In this paper, two channels for the mono-top signature are considered, targeting the case in which the $W$ boson originating from the top-quark decays into an electron or muon and a neutrino (leptonic channel) and the case in which it decays into a pair of quarks (hadronic channel). These analyses define different signal regions, maximising the signal discovery sensitivity, and control regions, enriched with the dominant background processes. The statistical interpretation of the results is based on a simultaneous fit to the signal and control regions to determine a possible signal contribution and constrain the main backgrounds with data, taking into account experimental and theoretical systematic uncertainties.

The paper is organised as follows. The signal models are introduced in Section 2. After a brief introduction to the ATLAS detector, given in Section 3, the data samples and samples of simulated signal and background events are described in Section 4. The algorithms for the reconstruction and identification of final-state particles are summarised in Section 5. Section 6 describes the criteria for the selection of candidate signal events. This section also describes the estimation of the background contribution with the help of dedicated control regions in data. The experimental and theoretical systematic uncertainties (Section 7) are taken into account in the statistical interpretation of data, with the results presented in Section 8. Concluding remarks are given in Section 9.

2 Signal phenomenology

This paper presents a search for two different signals: DM candidates produced in association with top-quarks and single production of vector-like $T$ quarks decaying into a top-quark and a $Z$ boson.

2.1 DM candidates associated with top-quarks

In this search the resonant and non-resonant production of DM particles associated with a top-quark are considered. The non-resonant case, represented in Figure 1(a) and Figure 1(b), corresponds to a flavour-changing neutral-current interaction, producing a top-quark and a new vector particle $V$, which in turn decays invisibly into a pair of DM particles. Such a process can be parameterised through a general Lagrangian [59, 16]:

[TABLE]

where a massive vector boson $V$ is coupled to a DM particle (represented by a Dirac fermion $\chi$) with a strength controlled by the parameter $g_{\chi}$. The term $P_{\mathrm{R}}$ is the right-handed chirality projector. The parameter $a$ stands for the coupling constant between the massive vector boson $V$ and the $t$- and $u$-quarks, and $\gamma^{\mu}$ are the Dirac matrices. Another possibility is the resonant case, corresponding to the production of a coloured charge-2/3 scalar ($\phi$) decaying into a top-quark and a spin-1/2 DM particle ($\chi$) [60]. This process, represented in Figure 1(c), is described by the following Lagrangian [59, 16]:

[TABLE]

where the parameters $\lambda$ and $y$ represent the couplings of the charged scalar to the $d$- and $s$-quarks and to the top-quark and the DM particle $\chi$, respectively.

2.2 Single production of vector-like $T$ quarks

The single production of $T$ quarks can occur via a charged $WTb$ or a neutral $ZTt$ vertex. However, $ZTt$ production is suppressed because of the required top quark in the initial state. For this reason, $ZTt$ production is not considered in this analysis and single VLT production refers to $T$ production via the $WTb$ vertex throughout this paper. The $T$ quarks can decay into $bW$, $tH$ and $tZ$, with the corresponding branching ratios ($\mathcal{B}$) depending on the specific model considered [33, 36].

The specific case of single production of vector-like $T$ quarks decaying into $tZ$, followed by the $Z$ boson decaying into neutrinos, results in a mono-top signature. As can be seen in Figure 1(d), one important difference between DM production and vector-like $T$-quark production is the presence of additional quarks in the single production of $T$ quarks, which will lead to at least one jet being detected at a small angle relative to the beam line. Similarly to the DM case, the topology of the VLT signal has a distinctive signature, characterised by the presence of a top-quark and missing transverse momentum, arising from the $Z\to\nu\bar{\nu}$ decay (and from the $t\to bW\to\ell\nu$ decay in the single-lepton channel case).

3 ATLAS detector

The ATLAS experiment [61] at the LHC is a multipurpose particle detector with nearly $4\pi$ coverage around the collision point.111 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the $z$-axis along the beam pipe. The $x$-axis points from the IP to the centre of the LHC ring, and the $y$-axis points upwards. Cylindrical coordinates $(r,\phi)$ are used in the transverse plane, $\phi$ being the azimuthal angle around the $z$-axis. The pseudorapidity is defined in terms of the polar angle $\theta$ as $\eta=-\ln\tan(\theta/2)$. Angular distance is measured in units of $\Delta R\equiv\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$. It consists of an inner tracking detector surrounded by a thin superconducting solenoid providing a $2\text{\,}\mathrm{T}$ axial magnetic field, electromagnetic and hadron calorimeters, and a muon spectrometer. The inner tracking detector covers the pseudorapidity range $|\eta|<2.5$. It consists of a silicon pixel detector, including the insertable B-layer [62, 63] installed after Run 1 of the LHC, a silicon microstrip detector, and a transition-radiation tracking detector. Lead/liquid-argon (LAr) sampling calorimeters provide electromagnetic (EM) energy measurements with high granularity for $|\eta|<3.2$. A steel/scintillator-tile hadron calorimeter covers the central pseudorapidity range ($|\eta|<1.7$). The endcap and forward regions are instrumented with LAr calorimeters for both the EM and hadronic energy measurements up to $|\eta|=4.9$. The outer part of the detector includes a muon spectrometer with high-precision tracking chambers providing coverage up to $|\eta|=2.7$, fast detectors for triggering over $|\eta|<2.4$, and three large air-core toroidal superconducting magnets with eight coils each. A two-level trigger system [64], using custom hardware followed by a software-based trigger level, is used to select events of interest at an average rate of $1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$.

4 Data and simulation samples

This analysis is performed using $pp$ collision data recorded at a centre-of-mass energy of $\sqrt{s}=$13\text{,}\mathrm{TeV}$$ with the ATLAS detector during 2015 and 2016 in the periods when the LHC was operating with $25\text{\,}\mathrm{n}\mathrm{s}$ bunch spacing and with an average number of collisions per bunch crossing $\langle\mu\rangle$ of around 23. Only periods in which all detector components necessary for this analysis were functional are considered, resulting in a data sample with a total integrated luminosity of $36.1\text{\,}{\mathrm{fb}}^{-1}$.

In the single-lepton channel, events are required to pass at least one of the single-muon or single-electron triggers [64]. The triggers require a $p_{\text{T}}$ of at least $20\text{\,}\mathrm{GeV}$ ($26\text{\,}\mathrm{GeV}$) for muons and $24\text{\,}\mathrm{GeV}$ ($26\text{\,}\mathrm{GeV}$) for electrons for the 2015 (2016) data sets, and also have requirements on lepton reconstruction and isolation. These are complemented by triggers with higher $p_{\text{T}}$ thresholds and relaxed isolation and identification requirements to ensure maximum efficiency at higher lepton $p_{\text{T}}$. In the hadronic channel, events are considered if they are accepted by triggers that select events with high missing transverse momentum, with online thresholds of $70\text{\,}\mathrm{GeV}$ in 2015 and $90\text{\,}\mathrm{GeV}$ to $110\text{\,}\mathrm{GeV}$ in 2016.

For all signal and background processes of interest, Monte Carlo (MC) events were simulated.

Signal events for both the resonant and non-resonant DM scenarios were generated according to a simplified model [65] described in Section 1, implemented in MadGraph5_aMC@NLOv2.3.2 [66] through FeynRules 2.0 [67, 68]. Such generation was done at leading order (LO) using the NNPDF3.0LO [69] parton distribution function (PDF) set. Parton showering, hadronisation and underlying-event modelling were handled using the Pythia 8.212 [70] event generator with the A14 [71] set of tuned parameters, using the NNPDF2.3LO PDF set [72]. Signal samples for the resonant model were generated assuming a DM mass of $m_{\chi}=10$ GeV and a range of the new scalar masses, $m_{\phi}$, between $1$ TeV and $5$ TeV, representing two different kinematic regimes. The kinematic distributions predicted by the model have only a small dependence on the coupling parameters and therefore all samples were generated using a coupling constant of $\lambda=0.2$ and a mixing parameter of $y=0.4$. The remaining kinematic dependence on the different couplings and masses was accounted for by means of a reweighting procedure (see Section 8 for details). Similarly, the signal samples for the non-resonant model were generated for values of $m_{V}$ between $500$ GeV and $3$ TeV, corresponding to the expected sensitivity of the analysis, and a benchmark DM mass $m_{\chi}=1$ GeV. The values of the couplings were set to $a=0.5$ and $g_{\chi}=1.0$. The kinematic effect of changing the coupling values was taken into account by using the previously mentioned reweighting procedure. The samples were normalised to the theoretical LO cross-sections, computed with MadGraph5_aMC@NLO.

The single production of $T$ quarks was generated using the Feynrules 2.0 implementation of a general model [35] interfaced to MadGraph5_aMC@NLOv2.3.2. The NNPDF2.3 LO PDF set and Pythia 8.212 with the A14 set of tuned parameters were used. Since the current analysis targets a final state with large missing transverse momentum, only the $T\rightarrow Zt$ decay, with $Z$ decaying invisibly, was considered, as represented in Figure 1(d). Samples were generated for $T$ masses in the range from 700 to 2000 GeV with a benchmark coupling of $\kappa_{T}=0.5$ in the $WTb$ production vertex. Additional samples were generated with alternative values of $\kappa_{T}=0.1$ and $1.0$ in order to study the effect of a varying $T$-quark width on kinematic distributions. The samples were normalised to the next-to-leading-order (NLO) cross-section by correcting the LO cross-sections calculated with MadGraph5_aMC@NLO for the difference between the NLO and LO cross-sections reported for the neutral single-$T$ production process via a $ZTt$ coupling [36]. For large values of the coupling the narrow-width approximation used in the cross-section calculation does not apply, so the cross-sections were corrected to include width effects, using a reweighting procedure similar to that previously mentioned, in order to account for the corresponding kinematic effects.

For the background samples, several matrix element (ME) event generators were combined with parton shower and hadronisation programs. Powheg-Box v2 [73, 74, 75, 76, 77, 78, 79] interfaced to Pythia 8.210 using the A14 set of tuned parameters was used to simulate $t\bar{t}$ production at NLO. Single top production was generated at NLO with Powheg-Box v1 for the $t$-, $Wt$- and $s$-channels and at LO with MadGraph5_aMC@NLO for the $tZq$ process, interfaced to Pythia 6.428 [80]. The CTEQ6L1 PDF set [81] and the Perugia 2012 set of tuned parameters [82] were used in the parton shower, hadronisation, and underlying-event simulation. The CT10f4 (CT10) PDF set [83] was used in the matrix element calculations for the $t$-channel ($Wt$- and $s$- channels). To model the $W\text{+\,jets}$ and $Z\text{\,+\,jets}$ background the Sherpa v2.2.1 [84] generator was used. Matrix elements were calculated for up to two partons at NLO and up to four partons at LO using the Comix [85] and OpenLoops [86] ME generators, and merged with the Sherpa parton shower [87] according to the ME+PS@NLO prescription [88]. The NNPDF3.0 next-to-NLO (NNLO) PDF set [89] was used in conjunction with a Sherpa parton shower tuning from the authors. Diboson processes were simulated with Powheg-Box v2 interfaced to Pythia 8.186. The CT10nlo PDF set was used for the hard process while the CTEQ6L1 PDF set was used for the parton shower. For the simulation of $t\bar{t}$ events with additional bosons $t\bar{t}$ + $X$ ($X=W$, $Z$, Higgs), MadGraph5_aMC@NLO v2.3.2 interfaced to Pythia 8.186 was used at NLO in QCD. Non-perturbative effects were modelled with the AZNLO set of tuned parameters [90].

The considered cross-sections for the dominant backgrounds, $t\bar{t}$ and $W$/$Z$ + jets, were evaluated at NNLO in quantum chromodynamics (QCD) [91, 92]. The calculation for $t\bar{t}$ also includes next-to- next-to-leading logarithmic soft gluon terms.

The EvtGen v1.2.0 program [93] was used to simulate properties of the bottom and charmed hadron decays except for samples generated with Sherpa. All simulated samples except the DM non-resonant signal in the leptonic channel and $t\bar{t}$ + $X$ processes were processed with the full simulation of the ATLAS detector [94] using Geant4 [95]. Additional samples used in the estimation of systematic uncertainties were instead produced using Atlfast2 [96], in which a parameterised detector simulation was used for the calorimeter responses. This simulation was also used for the generation of the DM non-resonant signal in the leptonic channel and $t\bar{t}$ + $X$ processes. All samples were simulated with a varying number of minimum-bias interactions generated with Pythia 8.186 using the A2 set of tuned parameters [97], overlaid on the hard-scattering event to account for the multiple $pp$ interactions in the same or nearby bunch crossings (pile-up). Simulated events were corrected using per-event weights to describe the distribution of the average number of interactions per proton bunch-crossing as observed in data

5 Event reconstruction and object selection

Events are required to have at least one vertex candidate with at least two tracks with $p_{\text{T}}>$400\text{,}\mathrm{MeV}$$. The primary vertex is taken to be the vertex candidate with the largest sum of squared transverse momenta of all associated tracks.

Electron candidates are reconstructed from an isolated electromagnetic calorimeter energy deposit matched to a track in the inner detector passing tight likelihood-based requirements [98]. They are required to have a transverse energy $E_{\text{T}}>$30\text{,}\mathrm{GeV} and pseudorapidity $|\eta|<2.47$, with the transition region between the barrel and endcap electromagnetic calorimeters, $1.37<|\eta|<1.52$, excluded. Electron candidates must have a track satisfying requirements of $|d_{0}|/\sigma_{d_{0}}<5$ for the transverse impact parameter significance relative to the beamline and $|\Delta z_{0}\sin\theta|<$0.5\text{\,}\mathrm{m}\mathrm{m} for the longitudinal impact parameter calculated relative to the primary vertex. Furthermore, electrons must satisfy isolation requirements based on inner detector tracks and topological clusters in the calorimeter [99], with an isolation efficiency of $90\%$ ($99\%$) for electrons from $Z\rightarrow ee$ decays with $p_{\text{T}}=$25(60)\text{,}\mathrm{GeV}$$. Correction factors are applied to simulated electrons to take into account the small differences in reconstruction, identification, and isolation efficiencies between data and MC simulation.

Muon candidates are reconstructed by combining tracks reconstructed in the inner detector with matching tracks reconstructed in the muon spectrometer, and are required to satisfy $p_{\text{T}}>$30\text{,}\mathrm{GeV} and $|\eta|<2.5$  [[100](#bib.bibx100)]. Muon candidates must satisfy requirements of $|d_{0}|/\sigma_{d_{0}}<3$ and $|\Delta z_{0}\sin\theta|<$0.5\text{\,}\mathrm{m}\mathrm{m} for the transverse impact parameter significance and the longitudinal impact parameter, respectively. An isolation requirement based on inner detector tracks and topological clusters in the calorimeters is imposed, which achieves an isolation efficiency of $90\%$ ($99\%$) for muons from $Z\rightarrow\mu\mu$ decays with $p_{\text{T}}=$25(60)\text{,}\mathrm{GeV}$$. Similarly to electrons, correction factors are applied to muons to account for the small differences between data and simulation [100].

Jets are reconstructed from topological clusters of energy deposited in the calorimeter [99] using the anti-$k_{t}$ algorithm [101] with a radius parameter of 0.4 (1.0) for small-$R$ (large-$R$) jets, as implemented in the FastJet package [102].

Small-$R$ jets are calibrated using an energy- and $\eta$-dependent simulation-based calibration scheme with corrections derived from data [103]. Jets are accepted within the fiducial region $|\eta|<2.5$ and $p_{\text{T}}>$30\text{,}\mathrm{GeV} ($p_{\text{T}}>$25\text{\,}\mathrm{GeV}) for the leptonic (hadronic) analysis. In the hadronic channel this threshold has been relaxed to increase forward-jet acceptance. Forward jets in the region $2.5<|\eta|<4.5$ are also considered in the vector-like $T$-quark search analysis. Quality criteria are imposed to reject events that contain any jets arising from non-collision sources or detector noise [104]. To reduce the contribution from jets associated with pile-up, jets with $p_{\text{T}}<$60\text{,}\mathrm{GeV}$$ and $|\eta|<2.4$ must satisfy a criterion that matches them to the hard-scatter vertex using information from tracks reconstructed in the inner tracking detector [105].

To prevent double counting of electron energy deposits as small-$R$ jets, the closest jet with distance $\Delta R_{y,\phi}\equiv\sqrt{{(\Delta y)}^{2}+{(\Delta\phi)}^{2}}<0.2$ from a reconstructed electron is removed. If the nearest surviving jet is within $\Delta R_{y,\phi}=0.4$ of the electron, the electron is discarded to ensure it is sufficiently separated from nearby jet activity. Jets with fewer than three tracks and distance $\Delta R_{y,\phi}<0.2$ from a muon are removed to reduce the number of jet fakes from muons depositing energy in the calorimeters. Muons with a distance $\Delta R_{y,\phi}<0.4$ from any of the surviving jets are removed to avoid contamination due to non-prompt muons from heavy-flavour hadron decays.

Large-$R$ jets are trimmed [106] to mitigate the impact of initial-state radiation, underlying-event activity and pile-up. The jet energy and pseudorapidity are further calibrated to account for residual detector effects using energy- and $\eta$-dependent calibration factors derived from simulation, with uncertainties derived from data [107]. Trimmed large-$R$ jets are considered if they fulfil $p_{\text{T}}>$250\text{,}\mathrm{GeV}$$ and $|\eta|<2.0$. To identify large-$R$ jets that are more likely to have originated from hadronically decaying top-quarks than from the fragmentation of other quarks and gluons, jet substructure information is exploited.

In the trimming procedure, sub-jets, with radius $R_{\text{sub}}$ $=0.2$, are clustered starting from the large-$R$ jet constituents using a $k_{t}$ algorithm. A sub-jet is retained only if it contains at least $5\%$ of the total large-$R$ jet transverse momentum, thereby removing the soft constituents from the large-$R$ jet. A top-tagging algorithm [108] is applied, corresponding to a loose working point with an approximately constant top-tagging efficiency of $80\%$ above $p_{\text{T}}$ of $400\text{\,}\mathrm{GeV}$. The algorithm depends on the calibrated jet mass, measured from clusters in the calorimeter, and the $N$-subjetiness ratio $\tau_{32}$ [109]. The $N$-subjetiness $\tau_{N}$ [109] expresses how well a jet can be described as containing $N$ or fewer sub-jets. The ratio $\tau_{32}$ = $\tau_{3}$/$\tau_{2}$ allows discrimination between jets containing a three-prong structure and jets containing a two-prong structure.

In addition to calorimeter-based jets, jets reconstructed from inner detector tracks using the anti-$k_{t}$ algorithm with a radius parameter of $0.2\text{\,}\mathrm{a}$re also used in the hadronic channel, following a similar strategy as in [110]. They are referred to as track-based jets and are required to satisfy $p_{\text{T}}>$10\text{,}\mathrm{GeV}$$ and $|\eta|<2.5$.

Small-$R$ calorimeter-based and track-based jets with $|\eta|<2.5$ are $b$-tagged as likely to contain $b$-hadrons using multivariate techniques which exploit the long lifetime of $b$-hadrons and large invariant mass of their decay products relative to $c$- and light hadrons [111, 112]. The working point used provides an average tagging efficiency of $70\%$ for $b$-jets and a rejection factor of 12.2 (7.1) against calorimeter-based (track-based) jets initiated by $c$-quarks and 381 (120) against calorimeter-based (track-based) jets initiated by light-flavour quarks, in simulated $t\bar{t}$ events. Correction factors are derived and applied to correct for the small differences in $b$-quark selection efficiency between data and MC simulation [111, 113, 114].

The missing transverse momentum is calculated as the negative vector sum of the transverse momenta of particles in the event, and its magnitude is denoted $E_{\text{T}}^{\text{miss}}$. In addition to the identified jets, electrons, muons, hadronically decaying $\tau$-leptons and photons, a track-based soft term is included in the $E_{\text{T}}^{\text{miss}}$ calculation by considering tracks associated with the hard-scattering vertex in the event which are not also associated with an identified jet, electron, muon, hadronically decaying $\tau$-lepton, or photon [115, 116].

6 Event selection and background estimation

The experimental signature of mono-top events expected in the DM (resonant and non-resonant) and vector-like $T$-quark models considered is the presence of a top-quark and significant missing transverse momentum, as seen in Section 2. For the case of single VLT production, at least one additional forward jet is also expected.

The leptonic channel is only considered in order to target the non-resonant DM model. In this model, the $u$-quark-initiated production of top-quarks is favoured over anti-top-quark production, due to the PDF structure of the proton. Therefore, positively charged leptons are favoured in the final state. Events that pass preselection are required to contain exactly one positively charged lepton and one $b\text{-tagged}$ jet with $p_{\text{T}}>$30\text{,}\mathrm{GeV}. In order to reduce the number of multijet background events, which are characterised by low $E_{\text{T}}^{\text{miss}}$ and low $W$ boson transverse mass222The transverse mass of the lepton and $E_{\text{T}}^{\text{miss}}$ system is defined as $m_{\text{T}}^{W}=\sqrt{2p_{\text{T}}(\ell)E_{\text{T}}^{\text{miss}}(1-\cos\Delta\phi(p_{\text{T}}(\ell),E_{\text{T}}^{\text{miss}}))}$, where $p_{\text{T}}(\ell)$ denotes the modulus of the lepton transverse momentum, and $\Delta\phi(p_{\text{T}}(\ell),E_{\text{T}}^{\text{miss}})$ the azimuthal angle between the missing transverse momentum and the lepton directions. $m_{\text{T}}^{W}$, it is also required that $E_{\text{T}}^{\text{miss}}>$50\text{\,}\mathrm{GeV} and $m_{\text{T}}^{W}+E_{\text{T}}^{\text{miss}}>$60\text{,}\mathrm{GeV}$$.

In the hadronic channel, because of the large expected Lorentz boost of the top-quarks produced in the signal events, the top-quark decay products can be collimated into a large-$R$ jet. This signature is used in both the non-resonant and resonant DM models and the VLT models. Preselected events are then required to contain zero leptons, one large-$R$ jet with $p_{\text{T}}>$250\text{,}\mathrm{GeV} and $|\eta|<2.0$. In order to suppress the multijet background contribution, $E_{\text{T}}^{\text{miss}}>$200\text{\,}\mathrm{GeV} is also required.

6.1 Signal region definition

The signal region selection is optimised for the different considered benchmarks with simulated data, using variables tested and found to be well-modelled. In the optimisation the sensitivity is estimated by performing a fit to the shape of the most discriminating observable including systematic uncertainties (see Section 8 for details). These observables are $E_{\text{T}}^{\text{miss}}$ in the leptonic channel and the transverse mass of the top-tagged large-$R$ jet ($J$) and the $E_{\text{T}}^{\text{miss}}$ system, $m_{\text{T}}(E_{\text{T}}^{\text{miss}},J)$ 333The transverse mass of the large-$R$ jet and $E_{\text{T}}^{\text{miss}}$ is defined as $m_{\mathrm{T}}(E_{\text{T}}^{\text{miss}},\mathrm{J})=\sqrt{m(\mathrm{J})^{2}+2E_{\text{T}}^{\text{miss}}\cdot\left(E_{\mathrm{T}(\mathrm{J})}-p_{\text{T}}(\mathrm{J})\cos(\Phi(\mathrm{J})-\Phi(E_{\text{T}}^{\text{miss}}))\right)}$, where $m_{\mathrm{T}}(\mathrm{J})$ is the reconstructed invariant mass of the calibrated calorimeter-cluster constituents of a large-$R$ jet and $E_{\mathrm{T}}(\mathrm{J})$ is the projection of its energy in the transverse plane., in the hadronic channel. For the tested mass hypothesis, the resulting best-performing selections lead to three signal regions: 1L-DM-SR for the non-resonant DM search in the leptonic channel and 0L-DM-SR and 0L-VLT-SR targeting the search in the hadronic channel for DM and VLT quarks, respectively.

In the leptonic channel, the mono-top signal is enhanced in regions of phase space characterised by high $m_{\text{T}}^{W}$ values. In addition, the lepton and $b\text{-tagged}$ jet are closer to each other when originating from the decay of a top-quark than in the case of $W\text{+\,jets}$ and multijet background events. Hence, in addition to the preselection described previously, the region 1L-DM-SR is defined by requiring $m_{\text{T}}^{W}>$260\text{,}\mathrm{GeV}$$ and $|\Delta\phi(\ell,b)|<1.2$.

In the hadronic channel, events in 0L-DM-SR and 0L-VLT-SR are required to contain exactly one top-tagged large-$R$ jet with $p_{\text{T}}>$250\text{,}\mathrm{GeV} and one $b\text{-tagged}$ track-based jet, in addition to the preselection criteria. The distance between the top-tagged large-$R$ jet and the $E_{\text{T}}^{\text{miss}}$ in the transverse plane, $\Delta\Phi(E_{\text{T}}^{\text{miss}},J)$, is required to fulfil $\Delta\Phi(E_{\text{T}}^{\text{miss}},J)\geq\pi/2$ since for signal events they are more likely to be produced back-to-back. In order to suppress background events due to fake $E_{\text{T}}^{\text{miss}}$ mostly coming from jet mis-reconstruction in multijet production, the asymmetry between $E_{\text{T}}^{\text{miss}}$ and the $p_{\text{T}}$ of the top-tagged large-$R$ jet defined as $\Omega=(E_{\text{T}}^{\text{miss}}-p_{\text{T}}(J))/(E_{\text{T}}^{\text{miss}}+p_{\text{T}}(J))$ is required to be $\Omega>-0.3$. The multijet background is additionally suppressed by requiring the minimum distance between the $E_{\text{T}}^{\text{miss}}$ and any small-$R$ jet in the transverse plane to be $\Delta\Phi_{\mathrm{min}}>1.0$. The signal region 0L-VLT-SR is defined by requiring in addition at least one forward jet with $p_{\text{T}}>$25\text{\,}\mathrm{GeV}. The signal region requirements are summarised in Table 1.

6.2 Background estimation

Dedicated control regions enriched in the dominant backgrounds are included in the fit to constrain these backgrounds with data. Multijet production background is estimated from data, while the rest of background processes are taken from simulation.

The dominant background in the signal regions is due to $t\bar{t}$ production in both channels, representing 78% of the total background in the leptonic and 55% (64%) in the DM (VLT) hadronic channels. This is followed by contributions from $W\text{+\,jets}$ (13%) and single top production (6.8%) in the leptonic channel and from $W\text{+\,jets}$ and $Z\text{\,+\,jets}$ production, at the level of 12% (13%) for $W\text{+\,jets}$ and 14% (15%) for $Z\text{\,+\,jets}$ in the DM (VLT) signal regions, in the hadronic channel. A minor background in the signal region with a non-negligible contribution in the control regions is multijet production. The rest of the backgrounds considered in the analysis are diboson production as well as $t\bar{t}$ production in association with a $Z$, $W$ or Higgs boson.

The estimation of the multijet background is in particularly important in the control regions used to estimate the main backgrounds. In the leptonic channel the multijet background originates from either misidentification of a jet as a lepton candidate (fake lepton) or from the presence of a non-prompt lepton (e.g., from a semileptonic $b$- or $c$-hadron decay) that passes the isolation requirement. The shape and the normalisation of the relevant distributions in multijet events and related systematic uncertainties are estimated using a matrix method in the electron channel and the anti-muon method in the muon channel [117]. The matrix method exploits differences in efficiencies to pass loose or tight quality requirements [98] between prompt leptons, obtained from $W$ and $Z$ decays, and non-prompt or fake lepton candidates, from the misidentification of photons or jets. These efficiencies are measured in dedicated control regions. The prompt lepton efficiencies are measured as a function of the $p_{\text{T}}$ of the leading jet and the angular distance between the lepton and its nearest jet, while the non-prompt or fake efficiencies are parameterised in terms of the $p_{\text{T}}$ of the leading jet, the angle in the transverse plane between the lepton and the $E_{\text{T}}^{\text{miss}}$ and the $b\text{-tagged}$ jet multiplicity. Multijet background events containing non-prompt muons are modelled with the anti-muon method using a sample of events enriched in non-isolated muons [117]. Most of these events originate from $b$- or $c$-hadron decays in jets. These events pass the kinematic requirements of the selections described in Section 5. Only some of the muon identification criteria are modified, ensuring there is no overlap with the signal selection. The normalisation is determined using a binned maximum-likelihood fit to the number of events observed in data in a control region dominated by multijet events. This region is defined with the preselection criteria, but removing the requirement on $E_{\text{T}}^{\text{miss}}$ and requiring $m_{\text{T}}^{W}<$60\text{,}\mathrm{GeV}$$.

In the hadronic channel the estimation of the multijet background is performed using a set of control regions (B,C and D) dominated by multijet background and defined to be orthogonal to the considered signal region (0L-DM-SR or 0L-VLT-SR). The shape of the multijet background is estimated from the control region B, which differs from the signal region by requiring zero top-tagged large-$R$ jets. This shape is normalised by a factor which is calculated as the ratio of the numbers of multijet events in regions C and D. Region C (region D) differs from the signal region (B region) by requiring $\Delta\Phi(E_{\text{T}}^{\text{miss}},J)<\pi/2$.

In regions B and D, with zero top-tagged large-$R$ jets, $J$ is a large-$R$ jet chosen randomly from the selected large-$R$ jets. The multijet contribution in these control regions is determined from the difference between data and the residual contribution of other background processes evaluated from simulation assuming the theoretical predictions for the corresponding cross-sections.

The control regions are defined to be orthogonal to each other and to the signal region. They are required to fulfil the preselection criteria. In the leptonic channel, control regions enriched in $t\bar{t}$ and $W\text{+\,jets}$ processes are used (referred to as 1L-TCR and 1L-WCR, respectively). In the hadronic channel, a control region enriched in $t\bar{t}$ production (referred to as 0L-TCR) and a region enriched in both the $W\text{+\,jets}$ and $Z\text{\,+\,jets}$ processes (referred to as 0L-VCR) are defined.

The control regions in the leptonic channel, 1L-TCR and 1L-WCR, are defined by modifying the requirement on $m_{\text{T}}^{W}$ to a window around the $W$ mass, $60$ GeV $<m_{\text{T}}^{W}<100$ GeV, and removing the requirement on $|\Delta\phi(\ell,b)|$. For the 1L-TCR, events are also required to contain a second $b\text{-tagged}$ jet. The $t\bar{t}$ control region in the hadronic channel, 0L-TCR, is defined by requiring two $b\text{-tagged}$ track-based jets and the minimum distance between the $E_{\text{T}}^{\text{miss}}$ and any small-$R$ jet in the transverse plane to satisfy $\Delta\Phi_{\mathrm{min}}<1$ (in order to reduce the signal contribution) and $\Delta\Phi_{\mathrm{min}}>0.2$ in order to suppress the multijet background). Events with calorimeter-based jets located close to disabled modules of the hadronic calorimeter are vetoed. In the 0L-VCR control region a veto on $b\text{-tagged}$ track-based jets is applied.

Table 1 details the control region selection in comparison with the signal region requirements. A comparison of the observed and expected distributions for $E_{\text{T}}^{\text{miss}}$ and $m_{\text{T}}(E_{\text{T}}^{\text{miss}},J)$ in the control regions is shown in Figure 2 for the leptonic and hadronic channel, respectively. The expectations in the leptonic (hadronic) channel are obtained from a fit of the background-only hypothesis to data in the 1L (0L) control regions, where the normalisations of the $t\bar{t}$ and $W\text{+\,jets}$ ($t\bar{t}$ and $W$/$Z$ + jets) processes are treated as nuisance parameters in the fit (see Section 8 for details of the fit).

7 Systematic uncertainties

The normalisation and shapes of the signal and background estimates are affected by systematic uncertainties from experimental sources and theoretical predictions. Each source of uncertainty is included as a nuisance parameter in the likelihood fit that determines the possible signal contribution. The analysis is limited by statistical uncertainties, thus the inclusion of systematic uncertainties leads to only a small degradation of the expected sensitivity.

The sources of experimental uncertainty include the uncertainty in the lepton trigger, identification and isolation efficiencies, the lepton energy and momentum scale and resolution [98, 100, 99], the $E_{\text{T}}^{\text{miss}}$ trigger and track-based soft-term scale and resolution [115, 116], the jet pile-up rejection requirement, energy scale and resolution [118], resolutions for relevant large-$R$ jet properties (mass, transverse momentum and the $N$-subjetiness ratio $\tau_{32}$), the $b$-tagging efficiency [111, 112], the pile-up reweighting, and the luminosity.

The uncertainty in the combined 2015+2016 integrated luminosity is 2.1%, derived following a methodology similar to that detailed in Ref. [119], using a calibration of the luminosity scale through $x$–$y$ beam-separation scans and using the LUCID-2 detector for the baseline luminosity measurements [120]. This systematic uncertainty is applied to all backgrounds and signals that are estimated using MC events, which are normalised to the measured integrated luminosity.

Theoretical cross-section uncertainties are applied to the normalisation of the simulated processes. Additional shape uncertainties stemming from theoretical estimations are calculated by comparing samples simulated with different assumptions and are estimated for the dominant backgrounds.

Uncertainties in the modelling of the $t\bar{t}$ and $t$-channel single top background come from the choice of NLO-matching method, the parton shower and hadronisation modelling, and the amount of additional gluon radiation. The NLO-matching uncertainty is estimated by comparing events produced with Powheg-Box and MadGraph5_aMC@NLO [66], both interfaced with Herwig++ [121]. The parton shower, hadronisation, and underlying-event model uncertainty is estimated by comparing two parton shower models, Pythia and Herwig++, while keeping the same hard-scatter matrix element generator. Variations of the amount of additional gluon radiation are estimated by comparing simulated samples with enhanced or reduced radiation and different values of tunable parameters related to additional radiation [122]. The choice of scheme to account for the interference between the $Wt$ and $t\bar{t}$ processes constitutes another source of systematic uncertainty that is estimated by comparing samples using either the diagram removal scheme or the diagram subtraction scheme [123].

Modelling uncertainties affecting the shape of the $W$/$Z$ + jets background are estimated in the hadronic channel, where these processes constitute an important background. An uncertainty in the modelling of $W$/$Z$ + jets is estimated by comparing the nominal simulation with a MadGraph5_aMC@NLO simulation in which matrix elements were calculated at LO for up to four partons. In addition, the effects of independently varying the scales for the renormalisation, factorisation, and resummation by factors of 0.5 and 2 are used. Since the $W$/$Z$ + jets background is constrained by a control region with a veto on $b\text{-tagged}$ jets, an additional uncertainty related to the $b$-flavour content in the $W$/$Z$ + jets background is taken into account by varying the number of events containing $b$-hadrons by 50% [124, 125]. Uncertainties in the modelling of the signal samples have been evaluated for signal points close to the expected exclusion mass limits and found to be negligible.

The effects of parton distribution function (PDF) uncertainties on the acceptance of the $t\bar{t}$ and $W$/$Z$ + jets backgrounds are estimated following the PDF4LHC prescription [126].

The systematic uncertainty of 50% associated with the data-driven modelling of the multijet events is estimated in the leptonic channel, based on comparisons of the rates obtained using alternative methods, as described in previous analyses [117]. In the case of the hadronic channel, this systematic is derived from a closure test of the data-driven method in a multijet-dominated validation region using simulated dijet samples.

A breakdown of the effects of the various sources of systematic uncertainty on the background prediction is presented in Table 2 and Table 3 for the two searches. The relative effects on the background yields in the signal region after the simultaneous fit to data in the signal and control regions are shown. The dominant background modelling uncertainties are due to the modelling of single top $Wt$ production for the leptonic channel and the modelling of the $b$-flavour content in the $W$/$Z$ + jets backgrounds.

8 Results

In order to test for the presence of a signal, a simultaneous fit to data in the signal and control regions is performed. The fit is based on a profile-likelihood technique, where systematic uncertainties are allowed to vary as Gaussian-distributed nuisance parameters (NP) and subsequently acquire their best-fit values. Additionally, the dominant backgrounds are constrained by treating their normalisation as NP in the fit. The calculation of confidence intervals and hypothesis testing is performed using a frequentist method as implemented in RooStats [127] using the asymptotic approximation [128].

The $E_{\text{T}}^{\text{miss}}$ distribution is used in the 1L signal region and the number of events is used instead in the control regions, while for the case of the 0L regions the distribution of the transverse mass of the top-tagged large-$R$ jet and $E_{\text{T}}^{\text{miss}}$ system, $m_{\text{T}}(E_{\text{T}}^{\text{miss}},J)$, is used in signal and control regions. For each of the three fits the binning of the distributions is optimised separately to obtain the highest expected sensitivity. For the testing of the non-resonant DM signal, both the 1L and 0L regions are used simultaneously in the fit (two signal regions and four control regions). For the resonant DM and VLT tests the fits are performed in the corresponding 0L regions, one signal region and two control regions for each fit. Uncertainties due to the limited size of the simulated samples are taken into account in each bin of the fitted distributions. Nuisance parameters accounting for systematic uncertainties are not considered in the fit if they have an impact on either normalisation or shape which is below 1%. The systematic uncertainties are symmetrised and also smoothed if the bin-to-bin statistical variation is significant. Most uncertainties are found to be neither significantly constrained nor pulled from their initial values. Small variations are observed in the $t\bar{t}$ modelling and multijet background uncertainties due to the mis-modelling observed in the shape of the transverse momentum distribution of top-quarks [129, 130]. Small variations are also observed in the large-$R$ jet and $Z$/$W$ + jets modelling uncertainties. The results of the fit show that the data are compatible with the background-only hypothesis.

The numbers of events observed in the signal and control regions are presented in Table 4, together with the backgrounds estimated prior to the simultaneous fit. The distribution of the observable used in the fit ($E_{\text{T}}^{\text{miss}}$ or $m_{\text{T}}(E_{\text{T}}^{\text{miss}},J)$) in the signal regions for data and the fitted SM expectation under the background-only hypothesis are shown in Figure 3. In these plots, the expected contribution from a benchmark signal is also shown for comparison. No significant excess above the SM expectation is found in any of the signal regions.

Since there is no evidence of a signal, expected and observed upper limits on the signal cross-section as a function of the $V$ mass for the non-resonant model, the mass of the scalar particle $\phi$ for the resonant model and the VLT mass are derived at 95% confidence level (CL) and are shown in Figure 4. Comparing the cross-section limits with the theoretical expectation, lower limits on the invisible particle and VLT masses can be derived. The LO values of the cross-section for non-resonant (resonant) DM production are evaluated using MadGraph5_aMC@NLO, as detailed in Section 4, assuming $m_{\chi}=1$ GeV, $a=0.5$ and $g_{\chi}=1$ ($m_{\chi}=10$ GeV, $\lambda=0.2$ and $y=0.4$). For the VLT interpretation, the single-$T$ production cross-section is taken from the NLO calculations for $c_{W}=1$, with the coupling $c_{W}$ defined in Ref. [36]. The narrow-width approximation is used [36]. For the current analysis, it was checked using dedicated Monte Carlo samples that the chirality of the coupling has negligible impact on the considered observables. The cross-section is also corrected for width effects calculated with MadGraph5_aMC@NLO, assuming that the ratio of NLO to LO cross-sections remains approximately the same for a non-vanishing $T$-quark width. The computed cross-section is then multiplied by the value of ${\mathcal{B}}(T\to Zt)$ in the singlet model, which is $\approx 25$% in the range of VLT masses investigated in this analysis. The considered benchmark coupling of $\kappa_{T}=0.5$ corresponds to $c_{W}=0.45$. The observed (expected) mass limits at 95% CL are 2.0 (1.9) TeV and 3.4 (3.3) TeV for the non-resonant and resonant dark-matter models, respectively. For the VLT case, there is no observed or expected mass exclusion for the considered reference benchmark coupling.

Two-dimensional exclusion regions in the planes formed by the mediator masses, the DM particle mass, and couplings between the DM, the new heavy particle and the SM fermions are obtained by reweighting the events using the transverse momentum from the vector sum of the momenta of the DM candidates. This procedure is validated with dedicated samples and allows reproduction of the correct event kinematics for the masses and couplings required for the multidimensional scans. The observed (expected) 95% CL upper limit contours for the signal strength $\sigma/\sigma_{\text{theory}}$ are shown in Figures 5(a)–5(c) for the non-resonant model, in which $\sigma$ is the observed (expected) limit on the model cross-section at a given point of the parameter space and $\sigma_{\text{theory}}$ is the predicted cross-section in the model at the same point. The corresponding results for the resonant model are shown in Figures 5(d) and 5(e). Since a reweighting procedure was used to obtain the required signal points, the results shown in Figure 5 include a systematic uncertainty in the signal normalisation associated with this procedure. This uncertainty was estimated from dedicated MC samples to be 10% and 25% for the non-resonant and resonant case, respectively, by comparing reweighted samples with those generated with the corresponding signal masses and couplings.

The limited sensitivity of the current analysis to single VLT production for low $T$ masses (cf. Figure 4(c)) implies that there is also less sensitivity to the corresponding coupling. This can be seen in Figure 6(a), which shows the expected and observed 95% CL upper limits on $c_{W}$, taken as the sum in quadrature of the left- and right-handed couplings $c_{{\mathrm{L}},W}$ and $c_{{\mathrm{R}},W}$, as a function of the VLT mass. Nonetheless, the sensitivity remains approximately constant for masses up to 1.4 TeV. A singlet $T$, which corresponds to ${\mathcal{B}}(T\to Zt)\approx 25$% over the mass range studied in this analysis, was assumed. The obtained limits on $c_{W}$ can also be translated into expected and observed 95% CL upper limits for the mixing angle of a singlet $T$ with the SM top-quark, as shown in Figure 6(b). For these results, a signal reweighting procedure was adopted in order to take into account the width effects induced by the variation of the $c_{W}$ coupling. The systematic uncertainty in the signal normalisation was estimated to be 3% from dedicated MC samples and was considered when deriving the limits shown in Figure 6. In the range $m(T)>1.1$ TeV, the obtained exclusion limit on the $c_{W}$ coupling improves on the previous results [43].

9 Conclusions

This analysis seeks anomalous production of events with large $E_{\text{T}}^{\text{miss}}$ and a single top-quark in LHC $pp$ data at $\sqrt{s}=13$ TeV collected by the ATLAS detector in 2015 and 2016, corresponding to an integrated luminosity of 36.1 fb*-1*. No deviations with respect to SM predictions are observed and 95% CL upper limits on the production cross-section of three BSM processes are obtained: resonant and non-resonant production of dark matter (DM) in association with single top-quarks, and single production of vector-like $T$ quarks decaying into $tZ(\to\nu\bar{\nu})$.

These limits are also interpreted in terms of the excluded regions in the parameter space of the considered BSM scenarios. For DM production in the non-resonant scenario, masses of a new vector particle coupling to the DM candidate up to 2 TeV are excluded at 95% CL for $m_{\chi}=1$ GeV, $g_{\chi}=1.0$ and $a=0.5$, while in the resonant case, masses of a new scalar particle coupling to DM up to 3.4 TeV are excluded at 95% CL for $m_{\chi}=10$ GeV, $y=0.4$ and $\lambda=0.2$. For the production of $T$ singlets, couplings of these new quarks to top-quarks and $W$ bosons, $c_{W}$, above 0.7 are excluded for $m_{T}=1.4$ TeV and below.

Appendix A Event yields in the signal and control regions after the fit to data

The numbers of events observed in the signal and control regions are presented in Tables 5, 6 and 7, together with the backgrounds estimated in the simultaneous fit to data in the corresponding regions under the background-only hypothesis. In Table 5, 1L and 0L DM signal regions as well as the 1L and 0L control regions are included in the fit. Table 6 includes 0L DM signal and control regions, while VLT signal and control regions are considered for Table 7.

Acknowledgements

We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently.

We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC, Australia; BMWFW and FWF, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP, Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT, Chile; CAS, MOST and NSFC, China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic; DNRF and DNSRC, Denmark; IN2P3-CNRS, CEA-DRF/IRFU, France; SRNSFG, Georgia; BMBF, HGF, and MPG, Germany; GSRT, Greece; RGC, Hong Kong SAR, China; ISF and Benoziyo Center, Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; NWO, Netherlands; RCN, Norway; MNiSW and NCN, Poland; FCT, Portugal; MNE/IFA, Romania; MES of Russia and NRC KI, Russian Federation; JINR; MESTD, Serbia; MSSR, Slovakia; ARRS and MIZŠ, Slovenia; DST/NRF, South Africa; MINECO, Spain; SRC and Wallenberg Foundation, Sweden; SERI, SNSF and Cantons of Bern and Geneva, Switzerland; MOST, Taiwan; TAEK, Turkey; STFC, United Kingdom; DOE and NSF, United States of America. In addition, individual groups and members have received support from BCKDF, CANARIE, CRC and Compute Canada, Canada; COST, ERC, ERDF, Horizon 2020, and Marie Skłodowska-Curie Actions, European Union; Investissements d’ Avenir Labex and Idex, ANR, France; DFG and AvH Foundation, Germany; Herakleitos, Thales and Aristeia programmes co-financed by EU-ESF and the Greek NSRF, Greece; BSF-NSF and GIF, Israel; CERCA Programme Generalitat de Catalunya, Spain; The Royal Society and Leverhulme Trust, United Kingdom.

The crucial computing support from all WLCG partners is acknowledged gratefully, in particular from CERN, the ATLAS Tier-1 facilities at TRIUMF (Canada), NDGF (Denmark, Norway, Sweden), CC-IN2P3 (France), KIT/GridKA (Germany), INFN-CNAF (Italy), NL-T1 (Netherlands), PIC (Spain), ASGC (Taiwan), RAL (UK) and BNL (USA), the Tier-2 facilities worldwide and large non-WLCG resource providers. Major contributors of computing resources are listed in Ref. [131].