Combination of the searches for pair-produced vector-like partners of the third-generation quarks at $\sqrt{s} =$ 13 TeV with the ATLAS detector

A combination of the searches for pair-produced vector-like partners of the top and bottom quarks in various decay channels ($T$$\rightarrow$$Zt/Wb/Ht$, $B$$\rightarrow$$Zb/Wt/Hb$) is performed using 36.1 fb$^{-1}$ of $pp$ collision data at $\sqrt{s}$ = 13 TeV with the ATLAS detector at the Large Hadron Collider. The observed data are found to be in good agreement with the Standard Model background prediction in all individual searches. Therefore, combined 95% confidence-level upper limits are set on the production cross-section for a range of vector-like quark scenarios, significantly improving upon the reach of the individual searches. Model-independent limits are set assuming the vector-like quarks decay to Standard Model particles. A singlet $T$ is excluded for masses below 1.31 TeV and a singlet $B$ is excluded for masses below 1.22 TeV. Assuming a weak isospin $(T,B)$ doublet and $|V_{Tb}| \ll |V_{tB}|$, $T$ and $B$ masses below 1.37 TeV are excluded.


Introduction
Naturalness arguments [1] suggest there should be a mechanism that cancels out the quadratically divergent contributions to the Higgs boson mass caused by radiative corrections from Standard Model (SM) particles.Several explanations are proposed in theories beyond the SM.Little Higgs [2,3] and Composite Higgs [4,5] models introduce a spontaneously broken global symmetry, with the Higgs boson emerging as a pseudo Nambu-Goldstone boson [6].Such models predict the existence of vector-like quarks (VLQs), color-triplet spin-1/2 fermions whose left-and right-handed chiralities transform in the same way under weak-isospin [7,8].In these models, VLQs are expected to couple preferentially to third-generation quarks [7,9] and can have flavor-changing neutral-current decays in addition to charged-current decays.An up-type VLQ T with charge +2/3 can decay into W b, Zt, or Ht.Similarly, a down-type quark B with charge −1/3 can decay into Wt, Z b, or Hb.In order to be consistent with results from precision electroweak measurements, the mass-splitting between VLQs belonging to the same SU (2) multiplet is required to be small [10], forbidding cascade decays such as T→W B. Couplings between the VLQs and the first-and second-generation quarks, although not favored, are not excluded [11,12].
At the Large Hadron Collider (LHC), VLQs with masses below approximately 1 TeV would mainly be pair-produced, a process dominated by the strong interaction.The corresponding predicted cross-section ranges from 195 fb to 2.0 fb for quark masses from 800 GeV to 1500 GeV [13] and depends only on the quark mass.Production of single VLQs via the electroweak interaction is also possible, but depends on the strength of the interaction between the new quarks and the weak gauge bosons.Representative Feynman diagrams for B B and T T production and decay are shown in Figure 1.The branching ratio (B) for each decay mode (T→W b, Zt, Ht and B→Wt, Z b, Hb) depends on the VLQ mass and weak-isospin quantum numbers, as calculated in Ref. [8].For a singlet T, all three decay modes have sizable branching ratios, while the charged-current decay mode T→W b is absent if T is either in a (X, T) doublet, where X is a VLQ with a charge of +5/3, or a (T, B) doublet with |V T b | |V t B |, where V i j are the elements of a generalized Cabibbo-Kobayashi-Maskawa matrix [8,14,15].Since the T quark branching ratios are identical in both doublets, no distinction is made between them when referring to the doublet T results.A singlet B will have a sizable branching ratio to all three decay channels, while the branching ratios in the doublet case depend on whether it is in a (T, B) doublet or (B, Y ) doublet, where Y is a VLQ with a charge of −4/3.For a (B, Y ) doublet, only neutral current couplings to SM quarks are allowed at leading order (LO), so the B→Wt decay is forbidden.Conversely, for a (T, B) doublet with B→Wt is the only allowed decay.Therefore, the specific B doublet scenario will be stated when interpreting the results.

Contributing analyses
Searches for pair-produced VLQ partners of the third-generation quarks have been performed by AT-LAS [16][17][18][19][20][21][22] and CMS [23][24][25] at the LHC at √ s = 13 TeV.This Letter presents the full combination of the ATLAS searches using 36.1 fb −1 of data collected in 2015 and 2016.The ATLAS detector is described in Ref. [26].Below is a brief description of each contributing analysis.
'H(bb)t + X' [16]: The primary targets of this analysis are T T events with at least one VLQ decaying into Ht, with H → b b.Events must have at least six jets [27] and either one lepton (electron [28] or muon [29]) or missing transverse momentum [30] E miss T > 200 GeV with zero leptons.The analysis uses b-tagging [31, 32] as well as dedicated top and Higgs jet tagging to classify the events into 22 and 12 search regions for the zero-lepton and one-lepton selections, respectively.The final discriminant is the scalar sum of the transverse momenta of the selected jets, lepton, and missing transverse momentum (S T ).The dominant background is the associated production of a t t pair with band c-quark jets, which is modeled via Monte Carlo (MC) simulation and assigned dedicated modeling uncertainties.
'W( ν)b + X' [17]: This analysis primarily targets T T→W bW b events with one W decaying leptonically and the other hadronically.Event selection requires one lepton, ≥3 jets, at least one of them being b-tagged, and a hadronically decaying W boson identified using jet substructure techniques [33].The final discriminant is the reconstructed mass of the T→W b→ νb candidate.The dominant background is from t t pair production, which is modeled using MC simulation with dedicated modeling uncertainties.
'W( ν)t + X' [18]: Very similar to the 'W( ν)b + X' analysis, this analysis is optimized to target B B signals, especially in the case where B→Wt.This analysis discriminates between the signal and the dominant t t background in the signal regions using either a boosted decision tree discriminant or the reconstructed mass of the B candidate.
'Z(νν)t + X' [19]: This analysis targets T T→Zt Zt events with an invisible Z decay.Events must have E miss T > 300 GeV, one charged lepton from the decay of a top quark, and ≥ 4 small-radius jets, which are reclustered [34] into large-radius jets.The analysis defines a single-bin signal region that capitalizes on various E miss T -based variables and requires at least two high-mass large-radius jets due to hadronically decaying top quarks and/or heavy bosons from the VLQ decays.The dominant backgrounds are t t+jets, W+jets and single-top events, which are estimated from MC simulation and normalized using dedicated control regions.'Z( )t/b + X' [20]: This analysis searches for T T and B B events containing a leptonically decaying Z boson (Z → + − ).The analysis requires at least two b-jets and contains opposite-sign dilepton and trilepton channels.The dominant backgrounds for the dilepton channels are t t and Z+jets, while the trilepton channels are dominated by diboson (W Z) and t t Z events, each modeled by MC simulation.
'Trilepton/same-sign dilepton' [21]: This analysis targets T T and B B decays with multilepton final states, with particular emphasis on events containing a pair of charged leptons with the same electric charge.Eight single-bin signal regions are defined in accord with the number of leptons and b-tagged jets.The background composition for this analysis varies between signal regions.Contributions from instrumental backgrounds (fake/non-prompt leptons and electrons with incorrectly measured charge) are estimated Table 1: The most sensitive decay channel for each analysis entering the combination.A '-' indicates that the analysis was not used for that signal process.

Analysis
T T decay B B decay HtH t WtW t Fully hadronic [22] HtH t HbH b using data-driven techniques, while background processes with prompt leptons, originating mostly from t t + W and diboson events, are modeled with MC simulations.
'Fully hadronic ' [22]: This analysis focuses on final states with zero leptons, low E miss T , and multiple jets and is the only analysis with significant sensitivity to B B → HbH b.Small-radius jets are reclustered into large-radius jets, which may be identified as top quarks, W/Z, or H bosons using a multi-class deep neural network [35].The final discriminant is the distribution of the signal likelihood calculated using the matrix-element method [36].The dominant background is from multijet production, which is estimated using a data-driven technique.
Most of the analyses were designed to be complementary.While each analysis provides sensitivity to various decay configurations, the most sensitive is shown in Table 1.All analyses use consistent definitions for the reconstructed physics objects, so only a few additional selection requirements were needed to suppress overlap.Compared to the standalone analyses, the W( ν)b + X and Z(νν)t + X analyses removed events with ≥6 jets and ≥3 b-jets to avoid overlap with the H(bb)t + X selection.The Z(νν)t + X analysis also requires S T < 1.8 TeV in a control region to mitigate the overlap with a signal region in the W( ν)b + X analysis.To reduce overlap with the Z( )t/b + X analysis, the trilepton/samesign dilepton analysis removed events with more than three leptons or events with a lepton pair having an invariant mass compatible with a Z boson (Z-veto).This Z-veto is the only added selection requirement with significant impact on the individual analysis sensitivity; however, that sensitivity is recovered by the Z( )t/b + X analysis.After applying these additional selection requirements, the remaining fraction of overlaping events between analysis regions was evaluated to be less than 1% between any two signal regions and less than 3% between any pair of signal or control regions and has negligible impact on the results.
The VLQ signal samples used by the analyses were generated with the LO generator P v2.2 [37] using the NNPDF2.3LO [38] set of parton distribution functions (PDF) and passed to P 8.186 [39] for parton showering and fragmentation.The samples are normalized using cross-sections computed with T ++ v2.0 [13] at next-to-next-to-leading order (NNLO) in QCD, including resummation of next-tonext-to-leading logarithmic soft gluon terms [40][41][42][43][44], and using the MSTW 2008 NNLO [45,46] PDF.Further information about simulated events and details of the background estimations for each analysis can be found in the respective publications.

Statistical analysis
The statistical analysis is the same as in the individual analyses and is based on a binned likelihood function constructed as the product of the Poisson probabilities of all bins entering the combination.This function depends on the signal-strength parameter µ, a factor multiplying the theoretical signal cross-section (µ ≡ σ/σ theory ), and a set of nuisance parameters that encode the effect of the systematic uncertainties on the signal and background expectations.These parameters are included with Gaussian or log-normal constraints.Additional unconstrained nuisance parameters are included to control the normalization of the main backgrounds, following the settings used in the standalone searches.The combination is achieved by performing a fit with all bins from all the regions considered from each analysis.
The analysis is limited by statistical uncertainties, and the precise correlation model for the systematic uncertainties was found to not significantly affect the results.The detector-related uncertainties are treated as fully correlated across analyses, with the following exceptions.The central values and uncertainties of the b-tagging and the luminosity measurement were updated after the publication of the Z(νν)t + X and W( ν)b + X analyses.Therefore, to avoid propagating constraints caused by the change in the method, these uncertainties are correlated between the Z(νν)t + X and W( ν)b + X analyses, but uncorrelated with the other searches, which are correlated among themselves.The modeling uncertainties and background normalization parameters are treated as uncorrelated between analyses.Although some background processes are common to multiple analyses, the phase space and the techniques used to estimate those backgrounds can be quite different.Residual correlations are therefore expected to be negligible.

Results
The behavior of the combination is consistent with the fits from the individual analyses.The post-fit values of all nuisance parameters are compatible with the standalone analyses, with the constraints generally determined by the analysis most sensitive to the given nuisance parameter.Similarly, the background predictions in each analysis after the combined fit are very close to the results from the standalone analyses.After the combination, no significant excess is observed in the data, so 95% confidence level (CL) limits are set on the cross-section of a VLQ signal.To increase the applicability and usefulness of this combination, limits are evaluated both for benchmark scenarios with specific branching ratios and for general combinations of branching ratios.
For an assumed set of branching ratios, upper limits are set on the production cross-sections for T T and B B as a function of the VLQ mass using the CL s method [47,48] with the asymptotic approximation [49].Observed and expected upper limits on the T T cross-sections as a function of mass are shown in Figure 2 for the benchmark scenarios of an isospin singlet or doublet T. Analogous limits on the B B cross-section are shown in Figure 3 for an isospin singlet B or B in a (T, B) doublet.The observed limits from the individual analyses, after the additional selections defined in this letter, are also shown.For a singlet T, masses below 1.31 TeV are excluded, while a T in an isospin doublet is excluded for masses below 1.37 TeV.A singlet B is excluded for masses below 1.22 TeV, B in a (T, B) doublet is excluded for masses below 1.37 TeV, and B in a (B, Y ) doublet is excluded for masses below 1.14 TeV.
The combination is significantly more sensitive than any one analysis.For example, in the case of the SU(2) singlet, the limit on the T T cross-section is improved by up to a factor of ∼ 1.7, which translates to an increase of 110 GeV in the mass limit.Figure 2: Observed (solid line) and expected (dashed line) 95% CL upper limits on the T T cross-section versus mass for the combination and the standalone analyses in black and colored lines, respectively.The (a) singlet and (b) doublet scenarios [8] are displayed.The shaded bands correspond to ±1 and ±2 standard deviations around the combined expected limit.The rapidly falling thin red line and band show the theory prediction and corresponding uncertainty [13], respectively.
In addition, model-independent lower limits are set on the VLQ mass for all combinations of branching ratios, assuming B(T→Ht) + B(T→Zt) + B(T→W b) = 1 and B(B→Hb The resulting lower limits on the VLQ mass as a function of branching ratio are presented in Figure 4. Limits corresponding to B(T→W b) = 1 and B(B→Wt) = 1 are found to also be applicable to Y Ȳ →W bW b and X X→WtWt, respectively.The high degree of complementarity between the analyses is clearly demonstrated in Figure 4.For any combination of branching ratios, the combination leads to observed (expected) lower mass limits of 1.31 (1.22) TeV for T and 1.03 (0.98) TeV for B. Limits on the signal strength, which can be used to interpret the results in scenarios with additional VLQ decays that escape detection [50], are available in the HEPData repository [51,52].
[GeV] B m 700 800 900 1000 1100 1200 1300 1400 Figure 3: Observed (solid line) and expected (dashed line) 95% CL upper limits on the B B cross-section versus mass for the combination and the standalone analyses in black and colored lines, respectively.The (a) singlet and (b) (T, B) doublet scenarios [8] are displayed.The shaded bands correspond to ±1 and ±2 standard deviations around the combined expected limit.The rapidly falling thin red line and band show the theory prediction and corresponding uncertainty [13], respectively.

Conclusion
The ATLAS Collaboration has performed a combination of seven analyses searching for pair-produced VLQs.Upper limits on the cross-section are calculated and used to set lower limits on the VLQ mass for various benchmark scenarios and for general combinations of branching ratios.This combination results in the most stringent limits to date on VLQ pair production.Due to the high degree of complementarity between the analyses, the combination has significantly better sensitivity than the standalone analyses, for the first time excluding T (B) masses below 1.31 (1.03) TeV for any combination of decays into SM particles.
[19] ATLAS Collaboration, Search for pair production of vector-like top quarks in events with one lepton, jets, and missing transverse momentum in √ s = 13 TeV pp collisions with the ATLAS detector, JHEP 08 (2017) 052, arXiv: 1705.10751[hep-ex].
[20] ATLAS Collaboration, Search for pair-and single-production of vector-like quarks in

Figure 1 :
Figure 1: Representative leading-order Feynman diagrams for (a) T T and (b) B B pair production.The studied VLQ decays are also displayed.

Figure 4 :
Figure 4: Observed lower limits at 95% CL on the mass of the (a) T and (b) B as a function of branching ratio assuming B(T→Ht) + B(T→Zt) + B(T → W b) = 1 and B(B→Hb) + B(B→Z b) + B(B→Wt) = 1.The yellow markers indicate the branching ratios for the SU(2) singlet and doublet scenarios for VLQ masses above 800 GeV where the branching ratios become approximately independent of the VLQ mass[8].
Also at Department of Physics, University of Michigan, Ann Arbor MI; United States of America.r Also at Giresun University, Faculty of Engineering, Giresun; Turkey.s Also at Graduate School of Science, Osaka University, Osaka; Japan.t Also at Hellenic Open University, Patras; Greece.u Also at Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest; Romania.v Also at II.Physikalisches Institut, Georg-August-Universität Göttingen, Göttingen; Germany.w Also at Institucio Catalana de Recerca i Estudis Avancats, ICREA, Barcelona; Spain.Also at Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Budapest; Hungary.aa Also at Institute of Particle Physics (IPP); Canada.ab Also at Institute of Physics, Academia Sinica, Taipei; Taiwan.ac Also at Institute of Physics, Azerbaijan Academy of Sciences, Baku; Azerbaijan.ad Also at Institute of Theoretical Physics, Ilia State University, Tbilisi; Georgia.ae Also at Istanbul University, Dept. of Physics, Istanbul; Turkey.a f Also at LAL, Université Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay; France.ag Also at Louisiana Tech University, Ruston LA; United States of America.ah Also at Manhattan College, New York NY; United States of America.ai Also at Moscow Institute of Physics and Technology State University, Dolgoprudny; Russia.a j Also at National Research Nuclear University MEPhI, Moscow; Russia.ak Also at Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Freiburg; Germany.al Also at School of Physics, Sun Yat-sen University, Guangzhou; China.am Also at The City College of New York, New York NY; United States of America.an Also at The Collaborative Innovation Center of Quantum Matter (CICQM), Beijing; China.ao Also at Tomsk State University, Tomsk, and Moscow Institute of Physics and Technology State University, Dolgoprudny; Russia.
b Also at Centre for High Performance Computing, CSIR Campus, Rosebank, Cape Town; South Africa.c Also at CERN, Geneva; Switzerland.d Also at CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille; France.e Also at Département de Physique Nucléaire et Corpusculaire, Université de Genève, Genève; Switzerland.f Also at Departament de Fisica de la Universitat Autonoma de Barcelona, Barcelona; Spain.g Also at Departamento de Física Teorica y del Cosmos, Universidad de Granada, Granada (Spain); Spain.h Also at Department of Applied Physics and Astronomy, University of Sharjah, Sharjah; United Arab Emirates.i Also at Department of Financial and Management Engineering, University of the Aegean, Chios; Greece.j Also at Department of Physics and Astronomy, University of Louisville, Louisville, KY; United States of America.k Also at Department of Physics and Astronomy, University of Sheffield, Sheffield; United Kingdom.l Also at Department of Physics, California State University, Fresno CA; United States of America.m Also at Department of Physics, California State University, Sacramento CA; United States of America.n Also at Department of Physics, King's College London, London; United Kingdom.o Also at Department of Physics, St. Petersburg State Polytechnical University, St. Petersburg; Russia.p Also at Department of Physics, University of Fribourg, Fribourg; Switzerland.q x Also at Institut für Experimentalphysik, Universität Hamburg, Hamburg; Germany.y Also at Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen/Nikhef, Nijmegen; Netherlands.z ap Also at TRIUMF, Vancouver BC; Canada.aq Also at Universita di Napoli Parthenope, Napoli; Italy.* Deceased