Hadronic Mono-W ′ Probes of Dark Matter at Colliders

Particle collisions at the energy frontier can probe the nature of invisible dark matter via production in association with recoiling visible objects. We propose a new potential production mode, in which dark matter is produced by the decay of a heavy dark Higgs boson radiated from a heavy W ′ boson. In such a model, motivated by left-right symmetric theories, dark matter would not be pair produced in association with other recoiling objects due to its lack of direct coupling to quarks or gluons. We study the hadronic decay mode via W ′ → tb and estimate the LHC exclusion sensitivity at 95% confidence level to be 10 2 − 10 5 fb for W ′ boson masses between 250 and 1750 GeV.


INTRODUCTION
Though dark matter (DM) represents the majority of the matter density of the universe, its particle nature remains a mystery.A variety of DM candidates have been considered, many with connections to the electroweak symmetry-breaking scale [1,2], which would also explain the observed relic density [3].A robust program of dedicated experiments search for DM interactions [4][5][6][7], which have not yet detected a signal.
In this paper we describe a new search mode, in which DM recoils against a heavy W ′ boson that decays to a hadronically decaying top quark and a b quark (referred to as the tb final state).This mode provides a statistically independent and theoretically distinct probe of DM production from other p miss T +X searches.In addition to probing DM, this channel also probes the W ′ boson itself.While the LHC already sets very stringent limits on high-mass W ′ bosons [32,33], searches at lower masses (m W ′ ⪅ 1 TeV) are more challenging due to the stringent trigger requirements on the decay products of the W ′ boson.The recoiling DM allows p miss Tbased triggers to be used, which opens up the possibility to push W ′ boson searches to lower masses.
The paper is organized as follows.A model of DM production in association with a W ′ boson is presented.Selection and reconstruction strategies are proposed and the expected sensitivity of the LHC dataset is described.The final section puts the expected sensitivity in experi-

MODEL
We present a model of a heavy W ′ boson that can be produced in association with invisible DM particles via the radiation of a dark Higgs boson, which decays to DM particles.Such a model would not produce a signature in other p miss T +X search modes. 1he W ′ boson is a new gauge boson that commonly arises in models of new physics, such as extended gauge theories [34][35][36][37][38] or composite Higgs [39].In this work, we consider extending the electroweak gauge group known as the left-right symmetric model [34,35].
In this model, the SU (2) R ×U (1) B−L symmetry is broken to U (1) Y which results in one new massive charged gauge boson, the W ′ boson, and one new massive neutral gauge boson, the Z ′ boson.This symmetry breaking is accomplished through the vacuum expectation value of an additional scalar multiplet.
The fermion content of the theory is the same as the SM, with the addition of a right-handed neutrino, N R .The gauge representations of the fermions under .
(1) The scalar content consists of a bi-doublet ϕ, which contains the SM Higgs doublet, and an SU (2) R triplet ∆ R : We assume the potentials are engineered such that these scalars have the vacuum expectation values: where v 2 = κ 2 1 + κ 2 2 = (246 GeV) 2 and v R is a free parameter.
After accounting for the states that give mass to the W ′ and Z ′ bosons, the triplet ∆ R contains one neutral state, one charged state, and one doubly-charged state.We call the neutral state a dark Higgs boson (h D ) due to its lack of direct interactions with the SM quarks, preventing it from being produced in an s-channel process at the LHC.It is given by δ For simplicity we assume no mixing between scalars.In principle the SM Higgs boson and the h D boson can mix, however, experimentally a non-zero mixing would still need to be small, ≲ O(10%) [40,41].
In this model the mass of the W ′ is where g R is the gauge coupling of SU (2) R .Using Eq. ( 5), the mass of the W ′ boson (m W ′ ) can be specified, rather than the value of v R , such that the relevant parameter space of this model is m W ′ and g R .The gauge coupling of U (1) B−L is determined by the choice of g R since The hadronic decay of W ′ → tb is mediated by the interaction while the production cross section of pp When m W ′ is fixed and v R ≫ v, the cross-section scaling becomes proportional to g 4 R .The mass of the Z ′ boson in this model is where g BL is the gauge coupling of U (1) B−L .The Z ′ boson couples to leptons, which would be visible unless the Z ′ boson is heavy enough to avoid experimental bounds.When m W ′ ≈ 800 GeV and g BL ≳ 2.5, the mass of the In this minimal version of a left-right model, the h D boson can dominantly decay to right-handed neutrinos N R .If these are sufficiently light, less than of order keV, then they could comprise the majority of the DM in the universe [42].More generally, the DM could be any new stable particle.Our search, like other collider searches, is agnostic to the identity of the DM.

EXPERIMENTAL SENSITIVITY
The model described above includes interactions which can generate a final state with a top quark, a bottom quark, and missing transverse momentum, see Fig. 1.We estimate the sensitivity of the LHC dataset to these hypothetical signals using samples of simulated pp collisions at √ s = 13 TeV with an integrated luminosity of 300 fb −1 .
Simulated signal and background samples are used to model the reconstruction of the W ′ boson candidates, estimate selection efficiencies, and expected signal and background yields.Collisions and decays are simulated with Madgraph5 v3.4.1 [43], and Pythia v8.306 [44] is used for fragmentation and hadronization.The model for the W ′ boson was adapted in Feyn-Rules [45] from Ref. [46].The detector response is simulated with Delphes v3.5.0 [47] using the standard CMS card, extended to include an additional reconstruction of wide-cone jets, and root version 6.2606 [48].
Selected narrow-cone (wide-cone) jets are clustered using the anti-k T algorithm [49] with radius parameter R = 0.4 (R = 1.2) using FastJet 3.1.2[50] and are required to have p T ≥ 20 GeV and 0 ≤ |η| ≤ 2.5.Widecone jets with mass within [50,110] ([125, 225]) GeV are tagged as W -boson (top-quark) jets.Events are required to have no reconstructed isolated photons, muons, or electrons with p T ≥ 10 GeV and |η| ≤ 2.5; isolation requires that less than 12% (25%) of the p T of the electron or photon (muon) be deposited in a cone with ∆R < 0.5 centered on the particle.To satisfy a trigger requirement and suppress backgrounds, events must have at least 200 GeV of p miss T .Candidate W ′ bosons are reconstructed in one of three approaches: The three approaches are illustrated in Fig. 2. If several reconstruction approaches are available for a single event, preference is given to t + b and then W + b + b.If several jets are available within one approach, preference is given to the jets that minimize the difference between the reconstructed and known top-quark and Wboson masses.Distributions of reconstructed W ′ boson candidate masses are shown in Fig. 3 for two choices of W ′ boson mass, and the selection efficiency is shown in TABLE I: Expected yields in 300 fb −1 of LHC data for background and signal (W ′ → tb) processes.Cross sections for backgrounds are at NLO in QCD [53]; cross sections for signal are set to the expected 95% CL upper limit.The calculations are described in the text.Shown are the cross section (σ), the trigger and selection efficiency (ε), and the expected yield (N ).Collider searches for signals with this event topology typically focus on the hadronic decay of the W boson [51,52] as it has a larger branching fraction and its decay can be efficiently reconstructed using large-radius jets and state-of-the-art tagging algorithms.The background from multijet events can be accurately modelled using data driven methods.The dominant backgrounds are the production of topquark pairs (t t) or the production of a single top quark in association with a b quark (t b or tb).Additional backgrounds are due to production of a heavy vector boson (W or Z bosons), which decays invisibly or whose decay products are not reconstructed, in association with two b quarks and two additional hard quarks or gluons.Radiation of additional gluons is modeled by Pythia.Contributions from QCD multi-jet production is suppressed by the p miss T requirement.Distributions of the expected reconstructed W ′ boson masses for the background and signal processes are shown in Fig 5 and the expected yields in 300 fb −1 are shown in Table I.
We also consider the 3-body decay of the W ′ boson, in which the h D boson is a W ′ boson decay product rather than radiation from an on-shell W ′ boson.Reconstruction of the W ′ boson, in principle, requires knowledge of the invisible h D boson's four-momentum.We reconstruct the 3-body decay using the same techniques as for the 2-body decay, but with p miss T added to the W ′ boson candidate as an estimate of the h D boson transverse momentum.No estimate is made of the longitudinal momentum of the h D boson.Distributions of expected background and signals are shown in Fig. 6.
Expected limits are calculated at 95% CL using a profile likelihood ratio [54] with the CLs technique [55,56] with pyhf [57,58] for a binned distribution in the reconstructed mass of the hypothetical W ′ boson, with 20 bins, where bins without simulated background events have been merged into adjacent bins.The background is Expected limits as functions of the W ′ boson mass are shown in Fig. 7 and translated into limits on the coupling (g R ) in Fig. 8.

DISCUSSION
Studying the 2-body case alone, we find a cross section limit that ranges from ≈ 3 pb, when both the W ′ boson and the h D boson are light, down to ≈ 20 fb, when both W' mass [GeV] 200 400 600 800 1000 1200 1400 1600 1800 the W ′ boson and the h D boson are relatively heavy.
As either the W ′ boson or the h D boson becomes more massive, the √ ŝ of the system is pushed to larger values and leads to better sensitivity.
In the 3-body case, the √ ŝ of the system only depends on m W ′ .The sensitivity, however, still depends on m h D through the efficiency, as seen in Fig. 4. The limits here range from ≈ 30 pb at low mass to ≈ 20 fb at high mass.Generally, the 2-body and 3-body cases are produced simultaneously and can be considered in a combined limit, however, we make conservative bounds and separate them for the purpose of clarity.
In terms of the left-right model used, Fig. 8 shows the expected limits on the SU (2) R gauge coupling g R .At low masses, coupling values are probed down to ≈ 0.6, while at higher masses, the limits are expected to be marginally weaker.Even though the limits are calculated using a particular left-right model, they will roughly correspond to the limits found in other models with a W ′ boson.This is because, generically, such models are parameterized by a mass that scales roughly as ∼ g R v R and a cross section that scales roughly as ∼ g 4 R .If DM particles are exclusively produced in the decay of a h D boson, this model provides a unique opportunity to detect them.In the context of this model, the h D boson does not couple with any SM particles, making it impossible to produce directly at the LHC, as opposed to other scalar mediators predicted by traditional DM simplified models [30,31].While the model does include a Z ′ boson, which could provide an additional signature via p miss T +Z ′ [24], here we assume the Z ′ is too heavy to  When analyzing the discovery potential for a W ′ boson in the final state predicted by this model, the presence of significant p miss T provides a boost to the W ′ boson, allowing for an increased sensitivity to lower masses when compared to searches for W ′ bosons produced at rest [33,59,60].A similar argument can be made for the case when the W ′ boson is boosted by initial state +W ′ channel to the jet+W ′ channel, considerations on the effect of the trigger have to be made.At the LHC, searches that look at final states with hadronic jets, and no other objects, have to rely on datasets that are collected online by triggers, which require high thresholds on the jet transverse momenta.To select a jet+W ′ final state, a requirement of ≈ 500 GeV is placed on the recoiling jet p T to be in trigger efficiency plateau.The p miss T +W ′ channel can rely on events selected online by triggers that require large p miss T .Thresholds are typically lower for those triggers.For example, to select a p miss T +W ′ final state, a requirement of ≈ 200 GeV is place on the p miss T to be in the trigger efficiency plateau.This provides higher signal efficiency at low mass, and therefore, a stronger limit on the couplings.Previous dedicated searches for W ′ → tb in the low-mass region were performed at the Tevatron, with the results obtained by the CDF experiment in 2015 [61] still yielding the strongest limits in the 300-900 GeV range , at the level of g W ′ /g W < 0.1(0.2) for m W ′ = 300(500) GeV.

Missing transverse momentum [GeV]
For the simplest case of a right-handed neutrino as dark matter, the decays of W ′ → ℓN R where the N R is invisible is another relevant search channel [62,63].

CONCLUSIONS
In this work we study the search channel of pp → W ′ (tb)h D where the h D boson decays invisibly.This channel plays a dual role of extending the mono-X pro- gram of looking for DM at the LHC through the recoil of the visible object X and extending the searchable range of W ′ bosons to lower masses.Expanding the mass range for W ′ boson searches is especially novel and is accomplished through use of p miss T triggers, which have a lower threshold than comparable hadronic jet triggers.
We estimate that the current LHC dataset could be sensitive to W ′ boson production in the range from 20 FIG.8: Expected limits on the coupling (g R ) for an integrated luminosity of 300 fb −1 as functions of the W ′ boson mass for three choices of the h D boson mass.Results are calculated from the expected limits on the cross section in Fig. 7. fb to 30 pb, depending on the W ′ boson mass.These translate to limits on the coupling (g R ) as low as 0.6, which can be interpreted across a fairly generic set of W ′ boson models.
Future directions include improved reconstruction algorithms for the W ′ boson, perhaps using machine learning [64,65] to improve the accuracy of the jet-parton assignment and reconstruction of the missing z component of the invisible decay of the h D boson.

FIG. 1 :
FIG. 1: Feynman diagram describing the production of a heavy W ′ boson recoiling against a dark Higgs boson (h D ) and decaying to a top and a bottom quark.If the s-channel W ′ boson is virtual (real), the decay is 2-body (3-body).

FIG. 2 :
FIG. 2:Three possible W ′ boson reconstruction strategies, using wide-cone and narrow-cone jets, which can be b-tagged, W -tagged or top-tagged.See text for details.
Fig 4.  Collider searches for signals with this event topology typically focus on the hadronic decay of the W boson[51,52] as it has a larger branching fraction and its decay can be efficiently reconstructed using large-radius jets and state-of-the-art tagging algorithms.The background from multijet events can be accurately modelled using data driven methods.The dominant backgrounds are the production of topquark pairs (t t) or the production of a single top quark in association with a b quark (t b or tb).Additional backgrounds are due to production of a heavy vector boson (W or Z bosons), which decays invisibly or whose decay products are not reconstructed, in association with two b quarks and two additional hard quarks or gluons.Radiation of additional gluons is modeled by Pythia.Contributions from QCD multi-jet production is suppressed by the p miss

FIG. 3 :
FIG.3: Top (bottom): Distribution of the reconstructed W ′ boson candidate mass in simulated events with m ′ W = 1000 (1500) GeV, for each of the three reconstruction strategies (seeFig 2), where the selected objects are angled-matched (∆R < 0.4) and -unmatched (∆R > 0.4) to the correct parton-level objects

FIG. 4 :
FIG. 4: Efficiency of the selection in each approach (t + b, W + b + b, jj + b + b, see Fig 2) as a function of the W ′ boson mass, for two choices of the dark Higgs boson mass (m h D ).

. 5 :
FIG. 5: Top: Distribution of the missing transverse momentum (p miss T ) for the expected background and selected signals normalized to an integrated luminosity of 300 fb −1 after all requirements other than p miss T > 200 GeV are met.Bottom: Distribution of the reconstructed W ′ boson 2-body candidate mass for the expected background and selected signals normalized to an integrated luminosity of 300 fb −1 after the full selection.

FIG. 6 :
FIG.6: Distribution of the reconstructed W ′ boson 3body candidate mass for the expected background and selected signals normalized to an integrated luminosity of 300 fb −1 after the full selection.

FIG. 7 :
FIG.7: Top: Summary of expected upper limits at 95% CL on the h D W ′ production cross section and the 2body decay branching fraction of W ′ as a function of the W ′ boson mass normalized to an integrated luminosity of 300 fb −1 for three choices of the h D boson mass.Also shown are expected theoretical cross sections and branching fractions at leading order for a coupling value of g R = 0.36.Bottom: The same distributions as above except for the 3-body decay of the W ′ boson.