Search for W' signal in single top quark production at the LHC^*

Gauge sector extension is one of the promising new physics theories beyond the standard model (SM). Heavy charged gauge bosons (W'^±) are involved in a number of the new physics models, such as Extra Dimensions [1–7], Little Higgs [8–10], GUTs [11–13], etc. A simple but well-motivated scenario is the Left-Right symmetric model [14–18], which is based on the extended SU(2)_L × SU(2)_R × U(1) gauge group. Provided the current experimental constraints, a TeV-scaled charged gauge boson is allowed, which provides the opportunity for new physics searches at the LHC.

The leptonic decay W' → lν is the golden channel for searching for W' if the couplings to the SM leptons are not specifically suppressed. According to a M_T-distribution, determined by the transverse momentum of the charged leptons and missing transverse energy, lower mass limits of 5.1 (4.1) TeV for the sequential SM W' boson have been obtained by the ATLAS and CMS collaborations at $\sqrt{s}=13$ TeV LHC [19, 20]. Although the leptonic decay modes are experimentally clean and possibly may be the first observed, the other decay channels need to be studied in depth to understand the properties of the heavy bosons, especially in some leptonic branch ratio suppressed scenarios. Although the light quark decay modes of ${\rm{W}}^{\prime} \to {\rm{q}}\bar{{\rm{q}}}^{\prime}$ have a larger production rate than the ${\rm{W}}^{\prime} \to {\rm{t}}\bar{{\rm{b}}}$ channel, there is no advantage for searches for the W' boson due to the large QCD backgrounds at the LHC. Furthermore, the ${\rm{W}}^{\prime} \to {\rm{t}}\bar{{\rm{b}}}$ mode has a 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 its interactions with other particles. The chiral couplings to standard model fermions are crucial features which differ from the SM weak interactions in some specific models. It has been demonstrated 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 ${\rm{W}}^{\prime} \to {\rm{t}}\bar{{\rm{b}}}$ mode [21]. We have also found that the charged lepton angular distribution can be used to distinguish the chirality of W' in the decay mode of ${{\rm{W}}}^{{\prime} }\to {\rm{WH}}\to {\rm{b}}\bar{{\rm{b}}}{\rm{l}}\nu$ [22]. The investigation of the W' boson has also been extended to the associated production or exotic decay modes [21, 23–27].

Recently, the CMS collaboration has reported the latest results on the search for a resonance peak with ${\rm{W}}^{\prime} \to {\rm{t}}\bar{{\rm{b}}}$ [28]. The right-handed W' boson is excluded for mass less than 2.6 TeV with the top quark decaying hadronically and leptonically. Unfortunately, no evidence of the W' resonance peak can be observed directly up to now. Motivated by the reach of the W' investigation at the LHC, we provide various strategies to search for a significant excess from the standard model prediction in 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 (${p}_{{\rm{T}}}^{j}$), invariant mass of jets (M_jj), collision energy scale (H_T) and invariant mass of top and bottom quark (${M}_{{\rm{t}}\bar{{\rm{b}}}}$) are adopted to highlight the signal process. We find 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 Section 2 we briefly depict the theoretical framework and show the difference between the ${{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}$ and ${{{\rm{W}}}^{{\prime} }}_{{\rm{R}}}$ bosons. The detector simulation and numerical results with various schemes are presented in Section 3. Finally, a short summary is given in Section 4.

Heavy charged gauge bosons are predicted in many new physics theories. Provided that the SM is an approximation of the new physics in the low energy scale, the most direct detection for new physics should be via the decay of these heavy particles into the SM particles. The relevant gauge interactions between W' and fermions can be generalized in the formula

$$\begin{eqnarray}\begin{array}{ll} {\mathcal L} = & {g}_{{\rm{L}}}\frac{{g}_{2}}{\sqrt{2}}\bar{{\psi }_{u}^{i}}{\gamma }_{\mu }{V}_{{\rm{L}}}^{{\prime} ij}\frac{1}{2}(1-{\gamma }_{5}){\psi }_{j}^{d}{{\rm{W}}}_{{\rm{L}}}^{{\prime} }\\ & +{g}_{{\rm{R}}}\frac{{g}_{2}}{\sqrt{2}}\bar{{\psi }_{u}^{i}}{\gamma }_{\mu }{V}_{{\rm{R}}}^{{\prime} ij}\frac{1}{2}(1+{\gamma }_{5}){\psi }_{j}^{d}{{\rm{W}}}_{{\rm{R}}}^{{\prime} }+{\rm{H}}.{\rm{c}}.,\end{array}\end{eqnarray} \tag{ 1 }$$

where g₂ 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 ${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$ ) and g_L = 0, g_R = 1 the pure right-handed gauge interaction (labeled ${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$). V' is the flavor mixing matrix, the counterpart of the Cabibbo-Kobayashi-Maskawa matrix in the SM.

Both left- and right-handed W' bosons can exist in the left-right symmetric model, as well as the right-handed fermion doublets, which lead to a heavy neutrino (N). As discussed in Ref. [29], if the W' is heavier than N, the decay mode of W' → lN is open, which provides an interesting like-sign dilepton production process to learn the lepton number violation. Otherwise, we can only investigate the W' boson from its couplings to SM particles, with the W' → lN decay modes forbidden. Thus the three dominant decay modes are ${\rm{W}}^{\prime} \to {\rm{t}}\bar{{\rm{b}}}$, ${\rm{W}}^{\prime} \to {\rm{q}}\bar{{\rm{q}}}^{\prime}$, and W' → lν. The right-handed W' has the same decay modes as the left-handed one except for W' → lν, since the right-handed neutrino is absent in the SM. The ${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$ has a larger decay width than ${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$, which is expressed in the following formulae

$$\begin{eqnarray}&&{\varGamma }_{{{\rm{W}}}_{{\rm{R}}}^{{\prime} }}=\frac{{g}_{2}^{2}{g}_{{\rm{R}}}^{2}{m}_{{{\rm{W}}}^{{\prime} }}}{16{\rm{\pi }}}\left[2+\left(1-\frac{{m}_{{\rm{t}}}^{2}}{{m}_{{{\rm{W}}}^{{\prime} }}^{2}}\right)\left(1-\frac{{m}_{{\rm{t}}}^{2}}{2{m}_{{{\rm{W}}}^{{\prime} }}^{2}}-\frac{{m}_{{\rm{t}}}^{4}}{2{m}_{{{\rm{W}}}^{{\prime} }}^{4}}\right)\right],\\ &&{\varGamma }_{{{\rm{W}}}_{{\rm{L}}}^{{\prime} }}=\frac{{g}_{2}^{2}{g}_{{\rm{L}}}^{2}{m}_{{{\rm{W}}}^{{\prime} }}}{16{\rm{\pi }}}\left[3+\left(1-\frac{{m}_{{\rm{t}}}^{2}}{{m}_{{{\rm{W}}}^{{\prime} }}^{2}}\right)\left(1-\frac{{m}_{{\rm{t}}}^{2}}{2{m}_{{{\rm{W}}}^{{\prime} }}^{2}}-\frac{{m}_{{\rm{t}}}^{4}}{2{m}_{{{\rm{W}}}^{{\prime} }}^{4}}\right)\right],\end{eqnarray} \tag{ 2 }$$

where m_W' (m_t) is the mass of W' boson (top quark).

In this paper, we focus on the process

$$\begin{eqnarray}&&{\rm{pp}}\to {{\rm{W}}}^{{\prime} {\rm{+}}}{{\rm{/W}}}^{{\rm{+}}}\to \bar{{\rm{b}}}{\rm{t}}\to {\bar{{\rm{b}}}{\rm{bl}}}^{{\rm{+}}}\nu,\ \ \ \ {{\rm{l}}}^{{\rm{+}}}{{\rm{=e}}}^{{\rm{+}}},{{\rm{\ \mu }}}^{{\rm{+}}}.\end{eqnarray} \tag{ 3 }$$

The corresponding total cross section can be written as

$$\begin{eqnarray}&&\sigma =\displaystyle \int {f}_{q}({x}_{1}){f}_{\bar{{q}^{{\prime} }}}({x}_{2})\hat{\sigma }(\sqrt{{x}_{1}{x}_{2}S}\,){\rm{d}}{x}_{1}{\rm{d}}{x}_{2},\end{eqnarray} \tag{ 4 }$$

where ${f}_{q{\rm{\ /}}{\bar{q}}^{{\prime} }}({x}_{i})$ is the parton distribution function (PDF) with x_i the parton momentum fraction. $\sqrt{S}$ is the proton-proton collision center of mass energy. $\hat{\sigma }$ represents the partonic cross section of the process

$$\begin{eqnarray}\begin{array}{lll}q({p}_{1})+\bar{{q}^{{\prime} }}({p}_{2}) & \to & {{\rm{W}}}^{{\prime} +}{{\rm{/W}}}^{+}\to \bar{b}({p}_{3})+t({p}_{{\rm{t}}})\\ & \to & \bar{b}({p}_{3})+b({p}_{4})+{l}^{+}({p}_{5})+\nu ({p}_{6}),\end{array}\end{eqnarray} \tag{ 5 }$$

where p_i (i=1, 2, 3, 4, 5, 6) is the momentum of the corresponding particle, and p_t is the momentum of the top quark. The corresponding Feynman diagram is shown in Fig. 1 with the differential cross section

$$\begin{eqnarray}&&{\rm{d}}\hat{\sigma }=\frac{1}{2s}\overline{| {\mathcal M} {|}^{2}}{\rm{d}} {\mathcal L} ip{s}_{4},\end{eqnarray} \tag{ 6 }$$

where s = x₁x₂S, and ${\mathcal L} ip{s}_{4}$ denotes the Lorentz invariant phase space of the four final particles. $\bar{| {\mathcal M} {|}^{2}}$ represents the invariant amplitude of the partonic process (5) summed (averaged) over the final (initial) particle colors and spins, and can be written as,

$$\begin{eqnarray}\overline{| {\mathcal M} {|}^{2}}=\bigg\{\begin{array}{ll}\overline{|{ {\mathcal M} }_{{{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}}{|}^{2}}+\overline{|{ {\mathcal M} }_{{\rm{W}}}{|}^{2}}+2{\rm{Re}}({ {\mathcal M} }_{{{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}}^{* }{ {\mathcal M} }_{{\rm{W}}}), & \ {\rm{for}}\ {{{\rm{W}}}^{{\prime} }}_{{\rm{L}}};\\ \overline{|{ {\mathcal M} }_{{{{\rm{W}}}^{{\prime} }}_{{\rm{R}}}}{|}^{2}}+\overline{|{ {\mathcal M} }_{{\rm{W}}}{|}^{2}}, & \ {\rm{for}}\ {{{\rm{W}}}^{{\prime} }}_{{\rm{R}}},\end{array}\end{eqnarray} \tag{ 7 }$$

where $\overline{|{ {\mathcal M} }_{i}{|}^{2}}$ ($i={{{\rm{W}}}^{{\prime} }}_{{\rm{L}}},{{{\rm{W}}}^{{\prime} }}_{{\rm{R}}},{\rm{W}}$) is the corresponding invariant amplitude and $2{\rm{Re}}({ {\mathcal M} }_{{{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}}^{* }{ {\mathcal M} }_{{\rm{W}}})$ is the interference term between ${{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}$ and W,

$$\begin{eqnarray}&&\begin{array}{l}\overline{|{ {\mathcal M} }_{{{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}}{|}^{2}}=\frac{32{g}_{{\rm{L}}}^{4}{g}_{2}^{8}({p}_{1}\cdot {p}_{3})({p}_{4}\cdot {p}_{5})(2({p}_{2}\cdot {p}_{{\rm{t}}})({p}_{{\rm{t}}}\cdot {p}_{6})-{p}_{{\rm{t}}}^{2}({p}_{2}\cdot {p}_{6}))}{[{({p}_{{\rm{t}}}^{2}-{m}_{{\rm{t}}}^{2})}^{2}+{m}_{{\rm{t}}}^{2}{\varGamma }_{{\rm{t}}}^{2}][{({p}_{W}^{2}-{m}_{{\rm{W}}}^{2})}^{2}+{m}_{{\rm{W}}}^{2}{\varGamma }_{{\rm{W}}}^{2}][{(s-{m}_{{{\rm{W}}}^{{\prime} }}^{2})}^{2}+{m}_{{{\rm{W}}}^{{\prime} }}^{2}{\varGamma }_{{{\rm{W}}}^{{\prime} }}^{2}]},\\ \overline{|{ {\mathcal M} }_{{{{\rm{W}}}^{{\prime} }}_{{\rm{R}}}}{|}^{2}}=\frac{32{g}_{{\rm{R}}}^{4}{m}_{{\rm{t}}}^{2}{g}_{2}^{8}({p}_{1}\cdot {p}_{3})({p}_{2}\cdot {p}_{6})({p}_{4}\cdot {p}_{5})}{[{({p}_{{\rm{t}}}^{2}-{m}_{{\rm{t}}}^{2})}^{2}+{m}_{{\rm{t}}}^{2}{\varGamma }_{{\rm{t}}}^{2}][{({p}_{{\rm{W}}}^{2}-{m}_{{\rm{W}}}^{2})}^{2}+{m}_{{\rm{W}}}^{2}{\varGamma }_{{\rm{W}}}^{2}][{(s-{m}_{{{\rm{W}}}^{{\prime} }}^{2})}^{2}+{m}_{{{\rm{W}}}^{{\prime} }}^{2}{\varGamma }_{{{\rm{W}}}^{{\prime} }}^{2}]},\\ \overline{|{ {\mathcal M} }_{{\rm{W}}}{|}^{2}}=\frac{32{g}_{2}^{8}({p}_{1}\cdot {p}_{3})({p}_{4}\cdot {p}_{5})(2({p}_{2}\cdot {p}_{{\rm{t}}})({p}_{{\rm{t}}}\cdot {p}_{6})-{p}_{{\rm{t}}}^{2}({p}_{2}\cdot {p}_{6}))}{[{({p}_{{\rm{t}}}^{2}-{m}_{{\rm{t}}}^{2})}^{2}+{m}_{{\rm{t}}}^{2}{\varGamma }_{{\rm{t}}}^{2}][{({p}_{{\rm{W}}}^{2}{m}_{{\rm{W}}}^{2})}^{2}+{m}_{{\rm{W}}}^{2}{\varGamma }_{{\rm{W}}}^{2}][{(s-{m}_{{\rm{W}}}^{2})}^{2}+{m}_{{\rm{W}}}^{2}{\varGamma }_{{\rm{W}}}^{2}]},\\ 2{\rm{Re}}({ {\mathcal M} }_{{{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}}^{* }{ {\mathcal M} }_{{\rm{W}}})=\bigg\{\frac{2[(s-{m}_{{\rm{W}}}^{2})(s-{m}_{{{\rm{W}}}^{{\prime} }}^{2})+{m}_{{\rm{W}}}{\varGamma }_{{\rm{W}}}{m}_{{{\rm{W}}}^{{\prime} }}{\varGamma }_{{{\rm{W}}}^{{\prime} }}]}{[{(s-{m}_{{{\rm{W}}}^{{\prime} }}^{2})}^{2}+{M}_{{{\rm{W}}}^{{\prime} }}^{2}{\varGamma }_{{{\rm{W}}}^{{\prime} }}^{2}][{(s-{m}_{{\rm{W}}}^{2})}^{2}+{m}_{{\rm{W}}}^{2}{\varGamma }_{{\rm{W}}}^{2}]}\bigg\}\frac{32{g}_{{\rm{L}}}^{2}{g}_{2}^{8}({p}_{1}\cdot {p}_{3})({p}_{4}\cdot {p}_{5})(2({p}_{2}\cdot {p}_{{\rm{t}}})({p}_{{\rm{t}}}\cdot {p}_{6})-{p}_{{\rm{t}}}^{2}({p}_{2}\cdot {p}_{6}))}{[{({p}_{{\rm{t}}}^{2}-{m}_{{\rm{t}}}^{2})}^{2}+{m}_{{\rm{t}}}^{2}{\varGamma }_{{\rm{t}}}^{2}][{({p}_{{\rm{W}}}^{2}-{m}_{{\rm{W}}}^{2})}^{2}+{m}_{{\rm{W}}}^{2}{\varGamma }_{{\rm{W}}}^{2}]}.\end{array}\end{eqnarray} \tag{ 8 }$$

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The couplings of g_L(R) are arbitrary in various models, while the Sequential W' model with the W' boson has the same couplings to quarks and leptons as the W boson. We have numerical results in the framework of the Sequential W' model. CTEQ6L1 [30] is set for PDF, with m_W = 80.4 GeV and m_t = 173.1 GeV [31]. The cross section of the process ${\rm{pp}}\to {{\rm{W}}}^{{\prime} {\rm{+}}}\to \bar{{\rm{b}}}{\rm{t}}\to {\bar{{\rm{b}}}{\rm{bl}}}^{{\rm{+}}}\nu \ ({{\rm{l}}}^{{\rm{+}}}{{\rm{=e}}}^{{\rm{+}}},{{\rm{\ \mu }}}^{{\rm{+}}})$ with respect to the W' mass at 13 and 14 TeV is shown in Fig. 2. There are more than ten events produced with a W' mass around 6 TeV with a luminosity of 300 fb⁻¹. The shaded region represents the uncertainty from the PDF with the energy scale varying from $\sqrt{S}/2$ to $2\sqrt{S}$. This uncertainty could affect the cross section by about 10%–20% with the tree level result. It will decrease with the higher order calculation, which is out of the scope of this work. So in the following work we focus on the investigation of W' at 14 TeV and assume a luminosity of 300 fb⁻¹ unless otherwise stated.

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Once the W' boson is produced at the LHC, the ${{\rm{W}}}^{{\prime} }\to {\rm{t}}\bar{{\rm{b}}}$ 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 the W' mass from ${\rm{t}}\bar{{\rm{b}}}$ production with the signal of $2\ {\rm{jets}}+1\ {\rm{lepton}}+{\rlap{/}{{{E}}}}_{{\rm{T}}}$.

The resonance peak through the invariant mass of ${M}_{{\rm{t}}\bar{{\rm{b}}}}$ can be reconstructed as shown in Fig. 3. The differential distributions with the invariant mass of ${M}_{{\rm{t}}\bar{{\rm{b}}}}$ between the ${{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}{\rm{+W}}$ and ${{{\rm{W}}}^{{\prime} }}_{{\rm{R}}}{\rm{+W}}$ differ from the interference term. The valley region is due to the negative contribution from the interference term in the mass region of ${m}_{{\rm{W}}}\lt {M}_{{\rm{t}}\bar{{\rm{b}}}}\lt {m}_{{{\rm{W}}}^{{\prime} }}$ for ${{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}$, whereas there is no interference term between ${{{\rm{W}}}^{{\prime} }}_{{\rm{R}}}$ and the W boson. This kind of phenomena can be used to distinguish ${{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}$ from ${{{\rm{W}}}^{{\prime} }}_{{\rm{R}}}$ if enough events are accumulated. Moreover, there are a large number of SM W bosons in the ${\rm{t}}\bar{{\rm{b}}}$ production compared with W', especially in the small ${M}_{{\rm{t}}\bar{{\rm{b}}}}$ region. It is therefore crucial to suppress the influence of W bosons in the search for the W' boson.

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Figure 4 shows the transverse momentum (p_T) distribution of the $\bar{{\rm{b}}}$ and b quarks related to the process ${\rm{pp}}\to {{\rm{W}}}^{{\prime} {\rm{+}}}\to \bar{{\rm{b}}}{\rm{t}}\to {\bar{{\rm{b}}}{\rm{bl}}}^{{\rm{+}}}\nu \ ({{\rm{l}}}^{{\rm{+}}}{{\rm{=e}}}^{{\rm{+}}},{{\rm{\ \mu }}}^{{\rm{+}}})$ with m_W' = 2 TeV. The $\bar{{\rm{b}}}$ quark distribution has a peak around 1 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 particles. Such distributions can be used to set cuts to suppress the backgrounds. In addition, the b quark distribution shows differences between ${{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}$ and ${{{\rm{W}}}^{{\prime} }}_{{\rm{R}}}$ because of the top quark spin correlation effects, which provides the opportunity to distinguish the chirality of the W' boson [21].

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To be as realistic as possible, we simulate the detector performance by smearing the lepton and jet energies based on the assumption of Gaussian resolution parametrization

$$\begin{eqnarray}&&\frac{\delta (E)}{E}=\frac{a}{\sqrt{E}}\oplus b\end{eqnarray} \tag{ 9 }$$

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 [32]. In order to identify an isolated jet or lepton, we define the angular separation between particle i and particle j as

$$\begin{eqnarray}&&\bigtriangleup {R}_{ij}=\sqrt{\bigtriangleup {\phi }_{ij}^{2}+\bigtriangleup {\eta }_{ij}^{2}},\end{eqnarray} \tag{ 10 }$$

where △ϕ_ij and η_ij are the difference in azimuthal angle and rapidity between the related particles.

For the process in Eq. (5), W' decays to two particles which are back to back in the transverse plane. The W boson and bottom quark are collimated, because the top quark is highly boosted, so 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

$$\begin{eqnarray}&&\begin{array}{l}\quad\bigtriangleup {R}_{{\rm{lb}}}\gt 0.3,\ \bigtriangleup {R}_{{\rm{bb}}}\gt 0.4,\ {P}_{{\rm{T}}}^{l}\gt 20\ {\rm{GeV}},\\ \ {P}_{{\rm{T}}}^{j}\gt 50\ {\rm{GeV}},\ \eta (j)\lt 3.0,\ {\rm{\ }}\rlap{/}{{\rm{E}}}\gt 25\ {\rm{GeV}}.\end{array}\end{eqnarray} \tag{ 11 }$$

Figure 5 shows the invariant mass (${M}_{{\rm{t}}\bar{{\rm{b}}}}$) distribution for m_W' = 2, 3, 4, 5 TeV at 14 TeV with the basic cuts. Compared with Fig. 3, the discrepancy of the peak between W' and W boson is weakened after the basic cuts due to more events with small transverse momentum being generated in the W boson process.

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Besides the W boson intermediate process, the dominant backgrounds include the ${{\rm{W}}}^{{\rm{+}}}{\rm{jj}}$, ${{\rm{W}}}^{{\rm{+}}}{\rm{b}}\bar{{\rm{b}}}$, ${{\rm{W}}}^{{\rm{+}}}{\rm{g}}\to {\rm{t}}\bar{{\rm{b}}}$, bq → tj and ${\rm{t}}\bar{{\rm{t}}}$ processes. To suppress these backgrounds, we first require a bottom quark (b-tagging) in the final jets, with tagging efficiency 0.6 and the mis-tagging efficiency neglected. Then we attempt to use various kinematics variables to highlight the excess over the standard model prediction in the observation of final states with $2\ {\rm{jets}}+1\ {\rm{lepton}}+{\rlap{/}{{{E}}}}_{{\rm{T}}}$. In the following we investigate the excluded W' mass region from four strategies, i.e., the ${P}_{{\rm{T}}}^{j}$-Scheme, M_jj-Scheme, H_T-Scheme, and ${M}_{{\rm{t}}\bar{{\rm{b}}}}$-Scheme.

3.1.  ${P}_{{\rm{T}}}^{j}$-scheme

We set cuts on the jet transverse momenta ${P}_{{\rm{T}}}^{ji}$ (i = 1,2) with ${P}_{{\rm{T}}}^{j1}\gt {P}_{{\rm{T}}}^{j2}$, provided the signal process has a larger number of high P_T events than the W boson process. Figure 6 illustrates the invariant mass ${M}_{{\rm{t}}\bar{{\rm{b}}}}$-distribution for various W' masses with the basic cuts and ${P}_{{\rm{T}}}^{j1}\gt \frac{1}{5}{m}_{{{\rm{W}}}^{{\prime} }}$, ${P}_{{\rm{T}}}^{j2}\gt 100$ GeV. The lower peak in each curve is the remnant contribution from the 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 ${P}_{{\rm{T}}}^{j}$ cut, the SM W boson effects will be suppressed so that it would be possible to observe the excess in the ${M}_{{\rm{t}}\bar{{\rm{b}}}}$-distribution plots. The other backgrounds are investigated as well. In Table 1 we list the remaining cross sections after the ${P}_{{\rm{T}}}^{j}$ cuts. The cross section of W⁺jj is the largest of the backgrounds, and decreases sharply with the increasing of the ${P}_{{\rm{T}}}^{j1}$ cuts since most of the jets are soft. The background cross sections decrease with the increasing of the ${P}_{{\rm{T}}}^{j1}$ cuts as well as the signal process, thus we adopt varying cuts of ${P}_{{\rm{T}}}^{j1}\gt \frac{1}{5}{m}_{{{\rm{W}}}^{{\prime} }}$ and ${P}_{{\rm{T}}}^{j2}\gt 100$ GeV. The cross sections of the total background cross section and signal are listed in Table 2 as well as the significance $S/\sqrt{B}$. We display the significance with respect to the W' mass in Fig. 8 to illustrate the detectable mass region at the LHC with $\sqrt{S}=14$ TeV. It shows that the upper limit can reach 3.8 (4) TeV with a 3σ significance for left-handed (right-handed) W'. Furthermore, the significance for ${{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}$ is slightly lower than ${{{\rm{W}}}^{{\prime} }}_{{\rm{R}}}$ because of the negative effects on the cross section from the interference with the W boson.

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Table 1.  The cross sections of SM backgrounds at 14 TeV with the basic cuts, ${P}_{{\rm{T}}}^{j2}\gt 100$ GeV and various ${P}_{{\rm{T}}}^{j1}$ cuts.

σ/fb	W+jj/fb	${{\rm{W}}}^{{\rm{+}}}{\rm{b}}\bar{{\rm{b}}}$/fb	${{\rm{W}}}^{{\rm{+}}}{\rm{g}}\to {\rm{t}}\bar{{\rm{b}}}$/fb	bq → tj/fb	${\rm{t}}\bar{{\rm{t}}}$	W/fb
${P}_{{\rm{T}}}^{j1}\gt 400$ GeV	114.0	1.159	7.726	10.85	86.9	2.006
${P}_{{\rm{T}}}^{j1}\gt 600$ GeV	26.44	0.2313	1.189	1.505	14.23	0.4858
${P}_{{\rm{T}}}^{j1}\gt 800$ GeV	9.254	0.0608	0.1971	0.1967	1.84	0.1433
${P}_{{\rm{T}}}^{j1}\gt 1000$ GeV	3.173	0.0152	0.0435	0.0393	0.25	0.0479
${P}_{{\rm{T}}}^{j1}\gt 1200$ GeV	0	0.0005	0.0108	0.0098	0	0.0172

Table 2.  The cross sections of signal (σ_S) and SM backgrounds (σ_B) at 14 TeV with the basic cut, ${P}_{{\rm{T}}}^{j1}\gt \frac{1}{5}{m}_{{{\rm{W}}}^{{\prime} }}$ and ${P}_{{\rm{T}}}^{j2}\gt 100$ GeV.

	mW' = 2 TeV			mW' = 3 TeV			mW' = 4 TeV			mW' = 5 TeV
	${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$	${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$	${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$	${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$
σS/fb	23.800		31.180	3.220		4.060	0.480		0.580	0.060		0.085
σB/fb	222.64			44.08			11.7			3.29
$S/\sqrt{B}$	27.6		36.2	8.4		10.6	2.4		2.9	0.6		0.8

3.2. M_jj-scheme

The distribution of the invariant mass of the two jets M_jj for the signal is different from the backgrounds. We show the number of events in each bin with respect to the invariant mass M_jj in Fig. 9, where the basic cuts are required as well as ${M}_{{\rm{jj}}}\gt \frac{1}{2}{m}_{{{\rm{W}}}^{{\prime} }}$. The influence of the W boson can be neglected in the M_jj distribution after cuts. Compared with the ${M}_{{\rm{t}}\bar{{\rm{b}}}}$ distribution in Fig. 7, there is no clear peak in the curves, while the excess is obvious. Moreover, the plateau is broader but lower with increasing W' mass.

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The cross sections of backgrounds with the basic cuts and varying M_jj cuts are listed in Table 3. After we set M_jj > 3000 GeV, the main background is W⁺jj, with a cross section of 0.26 fb. The cross sections of signal and backgrounds are listed in Table 4 with different W' masses. Supposing the W' mass is 4 TeV, after we set a cut of M_jj > 2000 GeV, there remain 132 (93) events for ${{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}$ (${{{\rm{W}}}^{{\prime} }}_{{\rm{R}}}$) at 14 TeV LHC with luminosity of 300 fb⁻¹.

Table 3.  The cross sections of SM backgrounds at 14 TeV with basic cuts and M_jj cut.

σ/fb	W+jj	${{\rm{W}}}^{{\rm{+}}}{\rm{b}}\bar{{\rm{b}}}$	${{\rm{W}}}^{{\rm{+}}}{\rm{g}}\to {\rm{t}}\bar{{\rm{b}}}$	${\rm{bq}}\to {\rm{tj}}$	${\rm{t}}\bar{{\rm{t}}}$	W
Mjj > 1000 GeV	104.7	0.4459	13.19	1.495	8.710	0.506
Mjj > 1500 GeV	28.56	0.0725	2.952	0.2065	1.090	0.084
Mjj > 2000 GeV	7.932	0.0018	0.7667	0.0197	0.080	0.0184
Mjj > 2500 GeV	3.173	0.0039	0.2281	0.0098	0	0.0468
Mjj > 3000 GeV	0.2644	0.0010	0.0590	0	0	0.0013

Table 4.  The cross sections of signal (σ_S) and SM backgrounds (σ_B) at 14 TeV with the basic cuts and ${M}_{{\rm{jj}}}\gt \frac{1}{2}{m}_{{{\rm{W}}}^{{\prime} }}$.

	mW' = 2 TeV			mW' = 3 TeV			mW' = 4 TeV			mW' = 5 TeV
	${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$	${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$	${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$	${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$
σS/fb	21.420		21.100	2.831		2.400	0.440		0.310	0.050		0.042
σB/fb	129.04			32.97			8.81			3.4
$S/\sqrt{B}$	32.7		33.2	8.5		7.2	2.6		1.8	0.5		0.4

Figure 10 illustrates the detectable W' mass region at 14 TeV with the basic cuts and ${M}_{{\rm{jj}}}\gt \frac{1}{2}{m}_{{{\rm{W}}}^{{\prime} }}$ for $S/\sqrt{B}\gt 3$. The W' mass should be larger than 3.6 (3.9) TeV with a 3σ significance for ${{{\rm{W}}}^{{\prime} }}_{{\rm{R}}}$ (${{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}$) if there is no excess in the M_jj distribution.

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3.3. H_T-scheme

Due to the large mass of the W' boson, the signal process can happen only if a lot of energy is transferred in the collision. Thus we can use a high energy scale H_T to distinguish the signal and backgrounds. H_T is the scalar sum of the transverse momentum for the final state, which is defined as

$$\begin{eqnarray}&&{H}_{{\rm{T}}}={P}_{{\rm{T}}}^{j1}+{P}_{{\rm{T}}}^{j2}+{P}_{{\rm{T}}}^{l}+{\rlap{/}{{{E}}}}_{{\rm{T}}},\end{eqnarray} \tag{ 12 }$$

Figure 11 shows the number of events per bin with respect to H_T with the basic cuts and ${H}_{{\rm{t}}}\gt \frac{1}{2}{m}_{{{\rm{W}}}^{{\prime} }}$. It has a broad plateau in each curve, like in the M_jj distribution, while the upper mass limit for W' is up to 5 TeV for twenty events remaining.

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The cross sections of backgrounds are listed in Table 5 with the basic cuts and varying M_jj cut. As shown in the table, the W⁺jj and bq → tj processes are cut down to zero after the H_T > 3000 GeV cut, while other backgrounds have a tiny cross section left. A suitable H_T cut is therefore an effective way to suppress the SM backgrounds.

Table 5.  The cross sections of SM backgrounds at 14 TeV with the basic cuts and H_T cut.

σ/fb	W+jj	${{\rm{W}}}^{{\rm{+}}}{\rm{b}}\bar{{\rm{b}}}$	${{\rm{W}}}^{{\rm{+}}}{\rm{g}}\to {\rm{t}}\bar{{\rm{b}}}$	bq → tj	${\rm{t}}\bar{{\rm{t}}}$	W
HT > 1000 GeV	85.4	0.9222	3.734	5.035	54	1.276
HT > 1500 GeV	14.81	0.1490	0.3508	0.354	4.355	0.240
HT > 2000 GeV	3.437	0.0294	0.0357	0.0393	0.17	0.060
HT > 2500 GeV	0.2644	0.0074	0.0109	0.0197	0	0.017
HT > 3000 GeV	0	0.0020	0.0031	0	0	0.005
HT > 3500 GeV	0	0.0010	0.0031	0	0	0.0012

The total cross sections for signal and backgrounds are summarized in Table 6. For m_W' = 4 TeV, there are about 156 (177) events for the${{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}$ (${{{\rm{W}}}^{{\prime} }}_{{\rm{R}}}$) process and 1080 events for backgrounds with a cut of${H}_{{\rm{T}}}\gt \frac{1}{2}{m}_{{{\rm{W}}}^{{\prime} }}$. As shown in Fig. 12, the W' can be detectable with mass below 4.5 TeV in the H_T distribution for 3σ significance at 14 TeV.

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Table 6.  The cross sections of signal (σ_S) and SM backgrounds (σ_B) and the significance at 14 TeV with the basic cuts and ${H}_{T}\gt \frac{1}{2}{M}_{{{\rm{W}}}^{{\prime} }}$

	mW' = 2 TeV			mW' = 3 TeV			mW' = 4 TeV			mW' = 5 TeV
	${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$	${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$	${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$	${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$
σS/fb	25.490		36.550	3.380		4.360	0.520		0.590	0.076		0.083
σB/fb	150.36			20.26			3.77			0.31
$S/\sqrt{B}$	36.0		51.6	13.0		16.8	4.6		5.3	2.4		2.6

3.4.  ${M}_{{\rm{t}}\bar{{\rm{b}}}}$-scheme

In the${{\rm{W}}}^{{\prime} }\to {\rm{t}}\bar{{\rm{b}}}$ channel, the most effective way to reconstruct the W' mass peak is by the momentum of the top and bottom quarks. Provided the top quarks decay semi-leptonically, all the momenta of the final state can be detected in the detector, except the neutrino. However, we can obtain the transverse momentum of neutrino from the conservation of transverse momentum, using the formula

$$\begin{eqnarray}&&{P}_{\nu {\rm{T}}}=-({P}_{{\rm{lT}}}+{P}_{{\rm{j1}}}+{P}_{{\rm{j2}}}),\end{eqnarray} \tag{ 13 }$$

where P_jT is the transverse momentum of particle j. While its longitudinal momentum cannot be detected, we can obtain it by solving the equation

$$\begin{eqnarray}&&{m}_{{\rm{W}}}^{2}={({P}_{\nu }+{P}_{{\rm{l}}})}^{2},\end{eqnarray} \tag{ 14 }$$

which implies the neutrino and charged lepton are generated by an 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

$$\begin{eqnarray}\begin{array}{ll}{M}_{{\rm{rt}}} & =\sqrt{{({P}_{\nu }+{P}_{{\rm{l}}}+{P}_{{\rm{j}}})}^{2}}.\end{array}\end{eqnarray} \tag{ 15 }$$

We adopt cuts on the top reconstruction of

$$\begin{eqnarray}&&|{M}_{{\rm{rt}}}-{m}_{{\rm{t}}}|\leqslant 20\ {\rm{GeV}}.\end{eqnarray} \tag{ 16 }$$

Provided that all the final state momenta are confirmed, then we can reconstruct the whole process. The invariant mass of M_tb can be obtained from

$$\begin{eqnarray}\begin{array}{ll}{M}_{{\rm{t}}\bar{{\rm{b}}}} & =\sqrt{{({P}_{\nu }+{P}_{{\rm{l}}}+{P}_{{\rm{j1}}}+{P}_{{\rm{j2}}})}^{2}}.\end{array}\end{eqnarray} \tag{ 17 }$$

Figure 13 displays the number of events per bin with basic cuts and${M}_{{\rm{t}}\bar{{\rm{b}}}}\gt \frac{3}{4}{M}_{{{\rm{W}}}^{{\prime} }}$ for W' mass varying from 2 to 5 TeV. It is easy to find that the mass peak is clear in the${M}_{{\rm{t}}\bar{{\rm{b}}}}$ distribution due to the whole process reconstruction.

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The cross sections of backgrounds are listed in Table 7 with the basic cuts and varying${M}_{{\rm{t}}\bar{{\rm{b}}}}$ cuts. One can find that if a strict${M}_{{\rm{t}}\bar{{\rm{b}}}}$ cut is adopted, all the background effects can be neglected except for the W boson process. Table 8 shows the total cross sections for signal and backgrounds as well as the significance. The number of signal events is more than 1 for m_W' = 6 TeV at 14 TeV with a luminosity of 300 fb⁻¹. The corresponding significance distribution with respect to the W' mass is displayed in Fig. 14(left). The upper mass limit can be up to 6.2 (6.6) TeV with a 3σ significance after we require${M}_{{\rm{t}}\bar{{\rm{b}}}}\gt \frac{3}{4}{m}_{{{\rm{W}}}^{{\prime} }}$ for${{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}$ (${{{\rm{W}}}^{{\prime} }}_{{\rm{R}}}$) if there is no excess observed.

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Table 7.  The cross sections of SM backgrounds at 14 TeV with reconstruction and${M}_{{\rm{t}}\bar{{\rm{b}}}}\gt \frac{3}{4}{m}_{{{\rm{W}}}^{{\prime} }}$.

σ/fb	W+jj	${{\rm{W}}}^{{\rm{+}}}{\rm{b}}\bar{{\rm{b}}}$	${{\rm{W}}}^{{\rm{+}}}{\rm{g}}\to {\rm{t}}\bar{{\rm{b}}}$	bq → tj	${\rm{t}}\bar{{\rm{t}}}$	W
mW' = 2 TeV	4.495	0.0034	0.4454	0.9146	0.3498	1.675
mW' = 3 TeV	0.2644	0	0.0372	0.0197	0.0486	0.083
mW' = 4 TeV	0	0	0.0016	0.0098	0.0085	0
mW' = 5 TeV	0	0	0	0	0.0016	0
mW' = 6 TeV	0	0	0	0	0.0003	0

Table 8.  The cross sections of signal (σ_S) and SM backgrounds (σ_B) at 14 TeV with basic cut, reconstruction and${M}_{{\rm{t}}\bar{{\rm{b}}}}\gt \frac{3}{4}{m}_{{{\rm{W}}}^{{\prime} }}$.

	mW' = 2 TeV			mW' = 3 TeV			mW' = 4 TeV			mW' = 5 TeV			mW' = 6 TeV
	${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{L}}}^{{\prime} }$		${{\rm{W}}}_{{\rm{R}}}^{{\prime} }$
σS/fb	17.002		22.910	2.000		3.127	0.268		0.452	0.034		0.063	0.004		0.0086
σB/fb	7.87			0.453			0.02			0.0016			0.0003
$S/\sqrt{B}$	105.0		141.4	51.5		80.5	33.0		56.8	14.7		27.3	4.0		8.6

Currently, the integrated luminosity is 36.1 fb⁻¹ reported by the ATLAS collaboration and 35.9 fb⁻¹ [19] by the CMS collaboration, with a collision energy of 13 TeV [33], so we investigate the process of Eq. (3) at 13 TeV as well. Figure 14(right) displays the significance distribution with respect to the W' mass with the basic cuts and a loose cut of${M}_{{\rm{t}}\bar{{\rm{b}}}}\gt \frac{2}{3}{m}_{{{\rm{W}}}^{{\prime} }}$. If there is no excess observed, the W' can be excluded for a mass less than 4.9 (5.6) TeV for${{{\rm{W}}}^{{\prime} }}_{{\rm{L}}}$ (${{{\rm{W}}}^{{\prime} }}_{{\rm{R}}}$) with 3σ significance.

4. Summary

We would like to thank the members of the particle groups at the University of Jinan and Shandong University for their helpful discussions and comments. HL and WS thank the University of Arizona for their hospitality during the writing of this paper.