Search for $W^{\prime}$ signal in single top quark production at LHC

The heavy charged gauge bosons were proposed in the theories beyond standard model. We explore the discovery potential for $W'\to t\bar{b}$ with top quark semi-leptonic decay at the LHC. We concentrate on the new physics signal search with the deviation from the standard model prediction if the resonance peak of $W'$ can not be observed directly. The events of signal with two jets plus one charged lepton and missing energy are simulated together with the dominant standard model backgrounds. In this paper, it is found that suitable cuts on the kinematic observables can effectively suppress the standard model backgrounds, so that it is possible to search for $W'$ signal at the LHC with its mass less than 6.6 TeV.

the properties of the heavy bosons, especially in some leptonic branch ratio suppressed scenarios.
Although the light quark decay modes of W ′ → qq ′ possess a larger production rate than W ′ → tb channel, there is no advantage for the searching of W ′ boson due to the large QCD backgrounds at the LHC. Furthermore, the W ′ → tb mode is featured by the characteristic jet-substructure with the top quark and a large number of events with single top quark production can be accumulated at the LHC.
If the W ′ is discovered at the LHC, it becomes imperative to investigate the details of its intrinsic properties and the interactions with other particles [20]. The chiral couplings to the standard model fermions are the crucial features which differ from the SM weak interactions in some specific models. It demonstrates that the angular distributions of the top quark and lepton resulting from top decay can be used to disentangle the chiral couplings of the W ′ to SM fermions with the W ′ → tb mode [20]. we also have found that the charged lepton angular distribution can be used to distinguish the chirality of W ′ in the decay mode of W ′ → WH → bblν [21]. Moreover, the investigating of W ′ boson is extended to the associated production or exotic decay modes [20,22].
Recently, The CMS collaboration has reported the latest results on the search of resonance peak with W ′ → tb [23]. The right-handed W ′ boson is excluded for the mass less than 2.6 TeV with the top quark hadronic and leptonic decaying. Unfortunately, no evidence on the W ′ resonance peak can be observed directly up to now. Motivated by the reach of W ′ investigation at the LHC, we provide various strategies to search for the significant excess from the standard model prediction in the kinematics distributions other than the new resonance peak. We propose four schemes based on different cuts to suppress the standard model backgrounds. Cuts on the transverse momentum of jets (p j T ), invariant mass of jets (M j j ), collision energy scale (H T ) and invariant mass of top and bottom quark (M tb ) are adopted to highlight the signal process respectively. One can obtain that the lower mass limit for the sequential W ′ boson is up to 3.7-6.6 TeV. This paper is arranged as follows. In Sec. II we briefly depict the theoretical framework and show the difference between W ′ L and W ′ R boson. The detector simulation and numerical results with various schemes are presented in III. Finally, a short summary is given in Sec. IV.

II. THEORETICAL FRAMEWORK
The heavy charged gauge bosons are predicted in many new physics theories. Provided that the theory of SM is the approximation of the new physics in the low energy scale, it will be the most directly detection for new physics via the decaying of the heavy particles into the SM particles.
The relevant gauge interactions between W ′ and fermions can be generalized in the formula of where g 2 is the SM electroweak coupling and g L (g R ) is the left-handed (right-handed) coupling constant, with g L = 1, g R = 0 the pure left-handed gauge interaction (labeled as W ′ L ) and g L = 0, g R = 1 the pure right-handed gauge interaction (labeled as W ′ R ). V ′ is the flavor mixing matrix as the Cabibbo-Kobayashi-Maskawa matrix in the SM.
Both handed W ′ bosons can be existed in the left-right symmetric model as well as the righthanded fermion doublets, which lead to a heavy neutrino(N). As discussed in [24], if the W ′ is heavier than N, the decay mode of W ′ → lN is open, which provides an interesting likesign dilepton production process to learn the lepton number violating. Otherwise, we can only investigate the W ′ boson from its couplings to SM particles with the W ′ → lN decay modes forbidden. Thus the dominant three decay modes are W ′ → tb, W ′ → qq ′ , and W ′ → ℓν. The right-handed W ′ has the same decay modes as the left-handed one except for the W ′ → ℓν since the right-handed neutrino is absent in SM. The W ′ L has a larger decay width than W ′ R which is expressed in the following formulae where m ′ W (m t ) is the mass of W ′ boson (top quark). In this paper, we focus on the the process of The corresponding total cross section can be written as where f q/q ′ (x i ) is the parton distribution function (PDF) with x i the parton momentum fraction. √ S is the proton-proton collision center of mass energy.σ represents the partonic cross section of   with p i (i=1,2,3,4,5,6) is the momentum of the corresponding particle, and p t is the momentum of top quark. The corresponding Feynman diagram is shown in Fig.1 with the differential cross where s = x 1 x 2 S , and Lips 4 denotes the Lorentz invariant phase space of four final particles. |M| 2 represents the invariant amplitude of the partonic process (5) summed (averaged) over the final (initial) particle colors and spins, and can be written as, where , The couplings of g L(R) are arbitrary in various models, while it is well known as the Sequential W ′ model with the W ′ boson has the same couplings to quarks and leptons as the W boson. We have the numerical results in the framework of Sequential W ′ model. CTEQ6L1 [25] is set for PDF, with m W = 80.4 GeV and m t = 173.1 GeV [26]. The cross section of process pp → W ′+ →bt → bbl + ν (l + = e + , µ + ) with respect to the W ′ mass at 13 and 14 TeV is shown in Fig. 2. There are even more than ten events produced with a W ′ mass around 6 TeV with a luminosity of 300 f b −1 .
So in the following work we focus on the investigation of W ′ at 14 TeV and suppose a luminosity

III. NUMERICIAL RESULTS AND DISCUSSION
Once the W ′ boson is produced at the LHC, the W ′ → tb channel will play an important role in the search for W ′ signal in the large W ′ mass region. In this work, we provide various strategies to investigate the lower limit on W ′ mass from the tb production with the signal of The resonance peak through the invariant mass of M tb can be reconstructed as shown in Fig. 3.
The differential distributions with the invariant mass of M tb between the W ′ L + W and W ′ R + W differ from the interference term. The valley region dues to the negative contribution from the interference term in the mass region of m W < M tb < m W ′ for W ′ L , whereas no interference term between W ′ R and W boson. This kind of phenomena can be used to distinguish W ′ L from W ′ R if enough events are accumulated. Moreover, the SM W boson takes over a large number in the tb production comparing with W ′ especially in the small M tb region. Thus it is crucial to suppress the influence from W boson in the search of W ′ boson.  anti-quark has a peak around one TeV since for a parent particle of mass M decaying to two light particles, there is a Jacobian peak near M/2 in the transverse momentum distribution of final state particle. Such distributions can be used to set cuts to suppress the backgrounds. In addition, the distribution of bottom quark shows difference between W ′ L and W ′ R because of the top quark spin correlation effects, which provides the opportunity to distinguish the chirality of W ′ boson [20].
To be as realistic as possible, we simulate the detector performance by smearing the leptons and jets energies according to the assumption the Guassian resolution parametrization where δ(E)/E is the energy resolution, a is a sample term, b is a constant term, and ⊕ denotes a sum in quadrature. We always use a = 5%, b = 0.55% for leptons and a = 100%, b = 5% for jets respectively [27]. In order to identify the isolate jet (lepton), we define the angular separation between particle i and particle j as where △φ i j (η i j ) is the difference of azimuthal angles (rapidity) between the related particles.
For the process of (5), W ′ decays to two particles which are back to back in the transverse plane. The W boson and bottom quark are collimated due to the top quark is highly boosted, so that the angular separation △R ℓb between the charged lepton and bottom quark is peaked at a low value and the angular separation △R bb between the bottom quark and bottom anti-quark is peaked near π. Therefore, we impose the basic cuts as We set cuts on the jet transverse momenta P ji T (i= 1,2) with P j1 T > P j2 T , provided the signal process with a larger number of high P T than the W boson process. Fig. 6 illustrates the invariant mass M tb -distribution for various W ′ masses with the basic cuts and P j1 T > 1 5 m W ′ , P j2 T > 100 GeV.
The lower peak in each curve is the remnant contribution from SM W boson which is about one order of magnitude less than the signal peak. The number of events in each bin is displayed in Fig.7. Taking m W ′ = 3 TeV as an example, there remain hundreds of events after the cuts. If we set the proper P j T cut, the SM W boson effects will be suppressed so that it would be possible to observe the excess in the M tb -distribution plots. The other backgrounds are investigated as well.
In Table I we list the remaining cross section after the P j T cuts. The cross section of W + j j is the largest in the backgrounds while it decreases sharply with the increasing of the P j1 T cuts since most of the jets are soft. The background cross sections decrease with the increasing of the P j1 T cuts as well as the signal process, thus we adopt a varying cuts as P j1 T > 1 5 m W ′ and P j2 T > 100 GeV. The cross sections of the total background cross section and signal are listed in Table II as well as the significance S / √ B. We display the significance with respect to the W ′ mass in Fig.8 The distribution of the invariant mass of the two jets M j j for the signal is different from the backgrounds. We show the number of events in each bin with respect to the invariant mass M j j in Fig.9, where the basic cuts are required as well as M j j > 1 2 m W ′ . The W boson influence can be neglected in the M j j distribution after cuts. Comparing with the M tb distribution in Fig.7, there is no clear peak in the curves, while the excess is obvious. Moreover, the upland is broaden but lower with the W ′ mass increasing. The cross section of backgrounds with the basic cuts and varying M j j cuts is listed in Table III . After we set M j j > 3000 GeV, the main background is W + j j with the cross section of 0.26 f b. As well as the cross section of signal and backgrounds is listed in Table IV with different W ′ masses. Suppose the W ′ mass is 4 TeV, after we set a cut of M j j > 2000 GeV, there remain 170 (120) events for W ′ L (W ′ R ) at 14 TeV LHC with the luminosity of 300 f b −1 . Fig. 10 illustrates the detectable W ′ mass region at 14 TeV with the basic cuts and The W ′ mass should be larger than 3.7 (4) TeV with a 3σ significance

C. H T -Scheme
Due to the large mass of W ′ boson, the signal process can happen only if a great energy transferring in the collision. Thus we can use the great energy scale H T to discrete the signal and backgrounds. H T is the scalar sum of the transverse momentum for the final state, which is defined as Fig. 11 shows the number of events per bin with respect to H T with the basic cuts and H t > 1 2 m W ′ . It has a broaden upland in each curves like in the M j j distribution, while the upper mass limit for W ′ is up to 5 TeV for two events remaining. The cross section of backgrounds is listed in where P jT is the transverse momentum of j particle. While its longitudinal momentum can not be detected, we can obtain it by solving the equation which implies the neutrino and charged lepton are generated by the on-shell W boson. Solving this quadratic equation for the neutrino longitudinal momentum leads to a twofold ambiguity.
Furthermore, we can use the solution to reconstruct the top quark invariant mass through We adopt the cuts on the top reconstruction, Provided that all the final state momentum is confirmed, then we can reconstruct the whole process.
The invariant mass of M tb could obtain from Fig. 13 displays the number of events per bin with basic cuts and M tb > 3 4 M W ′ for W ′ mass varying from 2 to 6 TeV. It is easy to find that the mass peak is clear in the M tb distribution due to the whole process reconstruction. The cross section of backgrounds is listed in Table VII with the basic cuts and varying M tb cuts. One can find that if a strict M tb cut is adopted, all the backgrounds effects can be neglected except for the W boson process. Table VIII shows the total cross section for signal and backgrounds as well as the significance. The number of the signal events is more than one with m W ′ = 6 TeV at 14 TeV with a luminosity of 300 f b −1 . The corresponding significance     distribution with respect to the W ′ mass is displayed in Fig.14a. The upper mass limit can be up to 6.2 (6.6) TeV with a 3σ significance after we require M tb > 3 4 m W ′ for W ′ L (W ′ R ) if there is no excess to be observed.
Currently, the integrated luminosity is 36.1 f b −1 reported by the ATLAS collaboration and 35.9 f b −1 [28] by the CMS collaboration with the collision energy of 13 TeV [29], so we investigate the process of (3) at 13 TeV as well. Fig.14b displays the significance distribution with respect to the W ′ mass with the basic cuts and a loose cut of M tb > 2 3 m W ′ . If there is no excess to be observed, the W ′ can be excluded with the mass less than 4.9 (5.5) TeV for W ′ L (W ′ R ) with 3σ significance.

IV. SUMMARY
We investigate the process of pp → W ′ /W → tb → bblν for the W ′ signal via the kinematic distributions. As the signal events are characterized by 2 jets + 1 lepton + / E T , the dominant standard model backgrounds are W + j j, W + bb, W + g → tb and bq → t j. To reduce the backgrounds and improve the significance, we adopt four schemes, i.e., the transverse momentum of jets, the invariant mass of jets, the scalar sum of the transverse momentum as well as the missing transverse energy and the invariant mass of tb with the top quark reconstruction. By applying suitable cuts, it is possible to search for W ′ signal at the LHC. For example, at 14 TeV with a luminosity of