Improving heavy Dirac neutrino prospects at future hadron colliders using machine learning

All the MC event samples are generated using MadGraph5_aMC@NLO v2.8.2 Alwall:2014hca , and then passed to PYTHIA8 Sjostrand:2014zea for parton showering, fragmentation, and hadronization. The detector response is evaluated by using DELPHES 3 deFavereau:2013fsa with an ATLAS detector card. In both background and signal processes the parton distribution function (PDF) data sets NN23LO1 PDF Ball:2012cx ; Ball:2013hta are chosen, and the MLM Mangano:2001xp ; Mangano:2006rw merging/matching procedure is adopted with xqcut set to be 40 GeV. FastJet Cacciari:2011ma is integrated by DELPHES for jet reconstruction, and the anti-$k_{T}$ algorithm Cacciari:2008gp is adopted with the parameter R set to be 0.6 and JetPTMin set to be 20 GeV. The decay tables of heavy neutrino $N$ and the $Z$ and $W$ bosons are computed by MadGraph5. The decay of $\tau$ lepton generally produces lots of jets in the detector, which makes the $\tau$ lepton signals suffer a severe contamination from the huge QCD backgrounds. Therefore, as mentioned above, in this paper we will consider only electrons and muons in the leptonic final states as our signal events.

At the detector level, the visible physical objects include jets, electrons and muons for our case, which can be reconstructed by their momenta $p_{T}$, pseudorapidities $\eta$ and azimuthal angles $\phi$. The MET $E_{T}^{\rm miss}$ can be reconstructed from all the visible objects via $E_{T}^{\rm miss}=|-\sum_{v_{i}}{\stackrel{{\scriptstyle\rightarrow}}{{p_{T}}}}(v_{i})|$, with $\stackrel{{\scriptstyle\rightarrow}}{{p_{T}}}(v_{i})$ the transverse momentum of the $i$th visible object. We will use the reconstructed jets, charged leptons, and MET to perform our signal and background analysis below. It has been examined that the total inclusive cross sections below are independent of the matching parameter. More explicitly, at the parton level we adopt the basic acceptance cuts for both signal and background events in Table 1. At $\sqrt{s}=$14 TeV and 27 TeV, the cuts are the same as those used in the ATLAS lepton analyses at $\sqrt{s}$ = 13 TeV in Ref. Aaboud:2018jiw . At $\sqrt{s}=$100 TeV, the $p_{T}$ cuts on charged leptons and jets are relatively larger than that at 14 TeV and 27 TeV to remove the soft processes. For concreteness, we set $p_{T}>$ 15 GeV for electrons and muons, and $p_{T}>$ 45 GeV for jets. In addition, we require at least one lepton has the transverse momentum larger than 20 GeV, i.e. $p^{\ell,\,{\rm leading}}_{T}>$ 20 GeV. The cuts on $|\eta|$ are the same, assuming that the detector geometry is identical to ATLAS.

The signal is generated based on the model file HeavyN for Dirac neutrinos Christensen:2008py ; Alloul:2013bka ; Ruiz:2020cjx ; Alva:2014gxa ; Degrande:2016aje ; Pascoli:2018heg . As stated in Section 2, for simplicity we have assumed that there is only one heavy neutrino $N$ with mass at the electroweak scale, while the other two heavy neutrinos are much heavier and decouple from our analysis at the high-energy colliders. The exclusive cross sections for the trilepton signal processes in Eq. (4) at $\sqrt{s}=$ 14 TeV, 27 TeV and 100 TeV are shown in the left, middle and right panels of Fig. 2, respectively, as function of the heavy neutrino mass $m_{N}$. For the sake of concreteness, we have set the lepton flavor $\ell=\mu$. For the exclusive processes, we have included also the $pp\to\ell N+j$ and $pp\to\ell N+2j$ processes. As shown in the middle and right panels of Fig. 1, there are not only $q\bar{q}^{\prime}$ processes, but also $gq$ (or $g\bar{q}$) and $gg$ processes. The exclusive cross sections for signals with 0, 1 and 2 jets are shown in Fig. 2, respectively, by the red, green and blue curves, and the acceptance cuts in Table 1 have been applied. It is noteworthy that for a fixed $m_{N}$ the cross sections for the processes with one jet are roughly three times smaller than those without any jet. A similar suppression factor can be observed when the cross sections with two jets are compared with those with only one jet. Given the exclusive cross sections in Fig. 2, the signal production cross sections can be modelled well by the first three exclusive data samples, and it is safe to neglect events with higher jet multiplicity $n_{j}\geq 3$ in the following study.

For simplicity, in the following simulations, we assume there is only either $V_{eN}\neq 0$ or $V_{\mu N}\neq 0$ in the neutrino sector, which induces the combinations of charged lepton flavors $eee$ and $ee\mu$ or $\mu\mu e$ and $\mu\mu\mu$, with the first, second and third one corresponding, respectively, to the flavor indices $\alpha$, $\beta$ and $\gamma$ in Eq. (4). In the charge space, the three leptons of an event can be either $+-+$ or $-+-$. All the resultant trilepton states are tabulated in Table 2. Our trilepton observables below will be constructed in both the charge and flavor spaces. Note that the flavor combinations $\ell_{\alpha}^{\pm}\ell_{\beta}^{\mp}\ell_{\alpha}^{\pm}$ ($\alpha\neq\beta$) imply that we have both nonzero $V_{eN}$ and $V_{\mu N}$, which is a clear signature of lepton flavor violation (LFV). For instance, the states $e^{\pm}\mu^{\mp}e^{\pm}$ can originate from the production and decay chain:

where the process $pp\to eN$ is induced by the neutrino mixing $V_{eN}$ while the decay $N\to\mu W$ is from the mixing $V_{\mu N}$. Such a LFV process is clearly absent in the SM.

The main background events for our signal are from $pp\to ZW^{\pm}$ process in the SM, where both $Z$ and $W^{\pm}$ bosons decay into leptonic final states, i.e.

with potentially one or two additional jets. The SM backgrounds have the same particles in the final state as the signal events. To select the signal events, we demand the following pre-selection conditions for every event:

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  The total number of energetic charged leptons is exactly 3, i.e. $n_{e}+n_{\mu}=3$.

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  In order to suppress the background events from the low mass DY process and hadronic decays, the invariant mass of the two leading leptons should be larger than 12 GeV, i.e. $m_{\ell_{1}\ell_{2}}$ $>$ 12 GeV.

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  The number of jets is not larger than 2, i.e. $n_{j}\leq 2$.

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  The missing transverse momentum $E_{T}^{\rm miss}=|-\sum_{v_{i}}\stackrel{{\scriptstyle\rightarrow}}{{p_{T}}}(v_{i})|$ is larger than 20 GeV, i.e. $E^{\rm miss}_{T}>20$ GeV.

The SM backgrounds receive also contributions from the off-shell processes $pp\to Z^{\ast}W,\,\gamma^{\ast}W\to\ell\ell\ell\nu$ Pascoli:2018heg , with the photon process interfering with the $Z$-boson mediated diagrams. To estimate the cross section for the off-shell processes and interference terms, we first obtain the cross section for the purely on-shell $ZW\to\ell\ell\ell\nu$ process, and then the total cross section for the same final state, combining all the contributions from the on-shell $ZW$, off-shell $Z^{\ast}W$, virtual photon $\gamma^{\ast}W$ and the interference terms. Subtracting the on-shell $ZW$ contribution from the total cross section, we obtain the off-shell and interference contributions to the backgrounds. As a demonstration, the cross sections for the on-shell and off-shell plus interference contributions with $\ell=\mu$ are shown in the second and third rows of Table 3, where we have applied the geometrical acceptance cuts in Table 1 and the pre-selection cuts above. As seen in this table, for the 0-jet and 1-jet processes, the off-shell plus interference contributions are only roughly 6% and 4% of the cross sections for the corresponding on-shell processes.

In addition, the four-lepton production $4\ell$ contributes to the backgrounds, if one of the charged leptons is missed by the detectors Pascoli:2018heg . However, after the geometrical acceptance cuts and pre-selection cuts in particular the missing energy cut, such a process can be effectively removed. As demonstrated in the last row of Table 3, for the 0-jet and 1-jet cases, the four-lepton process contributes only roughly 5% and 6% to the backgrounds, respectively. Taking into account the contributions of the sub-dominant off-shell and four-lepton processes to the SM backgrounds, we scale the cross sections of the $ZW^{\pm}$ background, e.g. by, respectively, a factor of 1.11 and 1.10 for the cases of 0-jet and 1-jet with $\ell=\mu$ in Table 3.

We use MadGraph5 to generate the matrix elements of the signal processes $pp\to\ell N$, $pp\to\ell N+j$, and $pp\to\ell N+2j$ and the background processes $pp\to WZ$, $pp\to WZ+j$, and $pp\to WZ+2j$. The MLM merging/matching procedure is adopted to produce $10^{4}$ events as an inclusive data sample, and three values are taken for the jet parameter $xqcut$, i.e. $xqcut=40$ GeV, 80 GeV, and 100 GeV. As an explicit example, the cross sections $\sigma/|V_{\ell N}|^{2}$ for the signal of $m_{N}=500$ GeV with 0, 1 and 2 jets at $\sqrt{s}=14$ TeV for the three $xqcut$ values are shown in Fig. 3. As clearly seen in this figure, the signal cross sections are almost the same, differing by $\sim 1\%$ for the $xqcut$ values above. For the matched cross sections for the signal processes with 0, 1 and 2 jets at $\sqrt{s}=14$ TeV, 27 TeV and 100 TeV in Fig. 2, we have taken $xqcut=40$ GeV.

To demonstrate the task of background and signal event discrimination, we present in Table 4 the cross sections of signal events with $m_{N}=150$ GeV and $m_{N}=500$ GeV, and that of the SM background process $pp\to ZW^{\pm}+\textrm{jets}$ at $\sqrt{s}=$ 14 TeV, 27 TeV and 100 TeV. All the signal cross sections are normalized to $|V_{\ell N}|^{2}=1$. The exclusive cross sections for $0$, $1$, and $2$ jets are all presented in Table 4, and we have applied the acceptance cuts in Table 1. It can be observed in Table 4 that when $m_{N}$ is small, say 150 GeV, the cross sections of signal can be comparable with that of the SM background processes at $\sqrt{s}=14$ TeV, 27 TeV and 100 TeV. When $m_{N}$ is large, say 500 GeV, the signal cross sections can be roughly 50 times smaller than that for the SM backgrounds at 14 TeV, and down to roughly 30 times smaller at the 100 TeV collider.

In order to enhance the efficiency of discriminating the background and signal events, in this subsection we define some feature observables. To reconstruct these observables from charged leptons in the final state, it is crucial to identify their origins in terms of the background or signal hypothesis. To this end, we use the background $W^{-}Z$ and the signal $e^{-}N$ as explicit examples to describe how to identify the origins of charged leptons in our reconstruction algorithm.

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  For the $W^{-}Z\to e^{-}e^{+}e^{-}+E_{T}^{\rm miss}$ background events, it is known that the charged lepton with positive charge, i.e. $e^{+}$, must come from the $Z$ boson, labeled as $\ell_{2}^{+}$. To determine the second lepton $e^{-}$ from $Z$ decay, there is a two-fold ambiguity. In order to choose it out, we demand that the lepton $\ell_{1}^{-}$ has the minimal $|m_{\ell_{2}^{+}\ell_{1}^{-}}-M_{Z}|$. Then the last lepton $e^{-}$ is from $W$ boson decay and can be labelled as $\ell_{3}^{-}$. This would fix the background kinematics $W^{-}Z\to\ell_{1}^{-}\ell_{2}^{+}\ell_{3}^{-}\bar{\nu}=e^{-}e^{+}e^{-}\bar{\nu}$. This can be easily generalized to the cases of $\mu^{-}\mu^{+}\mu^{-}+E_{T}^{\rm miss}$ and the charge conjugate $W^{+}Z$ background.

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  For the background mode $e^{-}e^{+}\mu^{-}+E_{T}^{\rm miss}$, the determination of kinematics is rather trivial: the opposite-sign same-flavor lepton pair is from the $Z$ boson decay, while the third one is from the $W$ boson, i.e. $W^{-}Z\to\mu^{-}\bar{\nu}e^{+}e^{-}$. It is equally trivial for the background $\mu^{-}\mu^{+}e^{-}\bar{\nu}$.

Generally speaking, in the background hypothesis, we need to first reconstruct the $Z$ boson from a pair of charged leptons $\ell_{1}^{\pm}\ell_{2}^{\mp}$, and then the kinematic properties of $W$ boson can be reconstructed from the third charged lepton $\ell_{3}^{\pm}$ and the missing energy. On the other hand, for the signal hypothesis, the kinematic variables of the heavy neutrino are related to $\ell_{2}^{\mp}$, $\ell_{3}^{\pm}$ and the missing energy. Note that DELPHES is unable to handle fakes for charge identification. In the detector, bremsstrahlung and straight-like tracks produced at high $p_{T}$ are the main sources of lepton charge mis-identification. The mis-identification portion is only about 0.3%, which will not significantly affect our results FernandezPretel:2791516 .

After identifying the origins of these leptons in terms of background and signal hypotheses, we can reconstruct the feature observables, which are divided into three categories, i.e. global kinematic observables, observables for background event vetoing, and observables favoring signal events. Since there are at least 4 particles in the phase space of our signal and background events, we need at least 12-dimension phase space, or equally at least 12 independent observables, to describe the collider events. Since the longitudinal component of the momentum of neutrino in the final state is missing, we can use the hypothesis that missing energy is from $W$ boson decay and derive its expected value.

Let us first consider the events without any jet, and list the definitions of the global kinematic observables for background and signal discrimination. For illustration purpose, the distributions of SM backgrounds and signals with $m_{N}=500$ GeV and 1000 GeV at $\sqrt{s}=14$ TeV are presented in Fig. 4 as the red, blue and green lines, respectively.

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  The scalar sum of transverse momenta $H_{T}=\sum_{v_{i}}|\stackrel{{\scriptstyle\rightarrow}}{{p_{T}}}(v_{i})|$, where $v_{i}$ runs over all the reconstructed visible particles. This quantity is related to the mass parameter $m_{N}$ for signal events; for background events, it related to the transverse momenta of the $W$ and $Z$ bosons. The distributions for the SM backgrounds and signals are shown in the left panel of Fig. 4.

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  The missing transverse momentum $E_{T}^{\rm miss}=|-\sum_{v_{i}}\stackrel{{\scriptstyle\rightarrow}}{{p_{T}}}(v_{i})|$. The corresponding distributions for backgrounds and signals are presented in the middle panel of Fig. 4.

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  The invariant mass $m_{3\ell}$ of trilepton final state. This quantity is related to the thresholds of the signal and background processes. As seen in the right panel of Fig. 4, the distributions of $m_{3\ell}$ are significantly different for the backgrounds and signals.

Six feature observables are introduced to describe the phase space in favor of the signal events, with the distributions for backgrounds and signals shown in Fig. 5.

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  The two-fold longitudinal momentum solutions $p^{\nu,i}_{z}$ (with $i=0,1$) of the light neutrino in the final state. In our analysis, the transverse momentum of light neutrino is defined by $p_{T}(\nu)=E^{\rm miss}_{T}=|-\sum_{v_{i}}\stackrel{{\scriptstyle\rightarrow}}{{p_{T}}}(v_{i})|$, where $v_{i}$ denotes charged leptons and jets. Since the light neutrino masses $m_{\nu}\simeq 0$, we have $E^{2}_{\nu}=p^{2}_{T}(\nu)+p^{2}_{z}(\nu)$, where $E_{\nu}$ and $p_{z}(\nu)$ are the energy and longitudinal momentum of light neutrino, respectively ATLAS:2012byx . For the hypothesis that light neutrino is from the on-shell $W$ boson decay, we can have the relation $(p_{\ell}+p_{\nu})^{2}=m^{2}_{W}$, where $p_{\ell}$ is the four-momentum of the charged lepton from $W$ decay. Then we can have an quadratic equation for $p_{z}(\nu)$, which yields two solutions $p_{z}^{\nu,0}$ and $p_{z}^{\nu,1}$, with $p_{z}^{\nu,0}$ labeling the one with the larger absolute value. More details can be found in the Appendix. The distributions of $p_{z}^{\nu,0}$ and $p_{z}^{\nu,1}$ are shown, respectively, as the solid and dashed lines in the upper left panel of Fig. 5.

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  The angular separation $\Delta\Theta(\ell_{3},\nu)$ between the charged lepton $\ell_{3}$ and the neutrino from $W$ boson decay. As the longitudinal momentum of neutrino has a two-fold ambiguity, the separation $\Delta\Theta(\ell_{3},\nu)$ is also two-folded, as seen in the upper middle panel of Fig. 5. When the heavy neutrino mass $m_{N}$ is large, the daughter $W$ boson will be highly boosted, and the charged lepton and neutrino from its decay will be highly collimated. Therefore the separation $\Delta\Theta(\ell_{3},\nu)$ can be viewed as a measurement of the Lorentz boost factor of the $W$ boson from $N$ decay.

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  The transverse momentum $p_{T}^{W}$ of the reconstructed $W$ boson. The corresponding distributions are presented in the upper right panel of Fig. 5. Obviously, when $m_{N}$ is large, $p_{T}^{W}$ tends also to be large.

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  The transverse momentum $p_{T}^{\ell_{3}}$ of the lepton $\ell_{3}$ originating from $W$ boson decay. As seen in the lower left panel of Fig. 5, $p_{T}^{\ell_{3}}$ tends to be large for the signal.

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  The transverse mass $m_{T}^{N}$ of the heavy neutrino. The proper mass $m_{N}$ of $N$ can not be fully reconstructed due to the missing information of longitudinal momentum of the neutrino in the final state. For each event, once the lepton $\ell_{1}$ is identified, the transverse mass can be defined via

  $\displaystyle m_{T}^{N}=\sqrt{2p_{T}^{\ell_{2}\ell_{3}}E_{T}^{\rm miss}(1-\cos{\Delta\phi({\ell_{2}\ell_{3},E_{T}^{\rm miss}}}))}\,,$

  (12)

  where $p_{T}^{\ell_{2}\ell_{3}}$ is the transverse momentum of the $\ell_{2}\ell_{3}$ subsystem, and $\Delta\phi(\ell_{2}\ell_{3},E_{T}^{\rm miss})$ denotes the azimuthal angle between the $\ell_{2}\ell_{3}$ subsystem and the MET $E_{T}^{\rm miss}$. The distributions of $m_{T}^{N}$ are shown in the lower middle panel of Fig. 5.

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  Two reconstructed heavy neutrino masses $m_{N}^{{\rm rc},i}$ (with $i=0,\,1$), which has a two-fold ambiguity as a result of the two solutions $p_{z}^{\nu,0}$ and $p_{z}^{\nu,1}$. For signal events, one of them must be close to the real $m_{N}$. While for background events, neither of them is expected to be condensed in the mass window of $m_{N}$. As seen in the lower right panel of Fig. 5, $m_{N}^{\rm rc}$ can well distinguish the signal from backgrounds.

In order to further suppress the SM background events, six more feature observables are introduced, and the corresponding distributions are shown in Fig. 6.

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  The invariant mass $m_{\ell_{1}\ell_{2}}$ of the two leptons $\ell_{1}$ and $\ell_{2}$. For background events, $\ell_{1}$ and $\ell_{2}$ are from $Z$ boson decay, and $m_{\ell_{1}\ell_{2}}$ forms a peak at the $Z$ mass $m_{Z}$; for signal events such an observable is rather broad, as seen in the upper left panel of Fig. 6. Therefore, this observable should be the main killer for the background events.

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  The transverse momentum $p_{T}^{\ell_{1}\ell_{2}}$ of the $\ell_{1}\ell_{2}$ subsystem. For the background events, it is equivalently the transverse momentum $p_{T}^{Z}$ of the $Z$ boson. As the $W$ and $Z$ boson masses are very close, they fly back-to-back in the background events. In other words, $p_{T}^{W}$ and $p_{T}^{Z}$ are correlated for background events. In contrast, for the faked $Z$ boson in the signal events, there should be no such a correlation, and the distribution $p_{T}^{\ell_{1}\ell_{2}}$ is significantly broader, as shown in the upper middle panel of Fig. 6.

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  The angular distance $\Delta R(\ell_{1}\ell_{2},\ell_{3})$ between the $\ell_{1}\ell_{2}$ subsystem and the charged lepton $\ell_{3}$ from $W$ boson decay. For the SM backgrounds, when the energies of $W$ and $Z$ bosons are large, they are Lorentz boosted, which can be measured to some extent by the separation $\Delta R(\ell_{1}\ell_{2},\ell_{3})$. As seen in the upper right panel of Fig. 6, this separation has a relatively sharper peak for the signal events.

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  The angular distance $\Delta R(\ell_{1},\ell_{2})$ between the two leptons $\ell_{1}$ and $\ell_{2}$ reconstructing the $Z$ boson. For background events, these two leptons are from the $Z$ boson decay. When $Z$ boson is produced near its peak, most of $Z$ bosons have small boost factors. Then the pair of leptons from $Z$ decay are expected to fly back-to-back. The signal events have a relatively broader distribution of $\Delta R(\ell_{1},\ell_{2})$, as shown in the lower left panel of Fig. 6.

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  The rapidity difference $\Delta y(\ell_{1}\ell_{2},\ell_{3})$ between the $\ell_{1}\ell_{2}$ subsystem and the charged lepton $\ell_{3}$ originating from $W$ boson. As presented in the lower middle panel of Fig. 6, the signal events have a relatively smaller $|\Delta y(\ell_{1}\ell_{2},\ell_{3})|$ than the background events.

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  The transverse momentum $p_{T}^{\ell_{2}\ell_{3}}$ of the $\ell_{2}\ell_{3}$ subsystem. As in the signal events these two leptons are from $N$ decay, $p_{T}^{\ell_{2}\ell_{3}}$ tends to be larger for the signals than for backgrounds, as seen in the lower right panel of Fig. 6.

For the pre-selected events with one jet, i.e. the 1-jet channel, three more feature observables are introduced, and their corresponding distributions are shown in Fig. 7.

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  The transverse momentum $p_{T}^{j}$ of jet. The corresponding distributions for the SM backgrounds and signals are shown in the left panel of Fig. 7.

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  The angular distance $\Delta R(j,A)$ between the jet and the rest of the products, i.e. the three charged leptons and neutrino. As the longitudinal momentum of neutrino has a two-fold ambiguity, $\Delta R(j,A)$ is also two-folded, as seen in the right panel of Fig. 7.

The correlations among the variables above can be quantified by their linear correlation coefficients, which are shown in Figs. 8 and 9, respectively, for backgrounds and the signal with $m_{N}=500$ GeV at $\sqrt{s}=14$ TeV. It turns out that most of the coefficients are close to zero. Variable decorrelation via the square-root of the covariance matrix is the first step of the TMVA training processes, and ML methods take the correlation matrices into account during the training processes.

It is very clear in Fig. 7 that the distributions of $p_{T}^{j}$ and $\Delta R(j,A)$ are rather similar for the SM backgrounds and signal. In other words, the ML analysis in Section 3.3 will be dominated by the events without any jet. Since events with jets contribute little to the discrimination of signal from backgrounds in our analysis and the simulations with two (or more) jets cost a lot of computing resources, we will not consider the channels with two (or more) jets from now on. Here we describe the procedure how we perform our analysis.

1. 1.

   In the MC generation, we produce 0-jet and 1-jet events together. In the multi-variate analyses, we divide our data sample into two separate subgroups, i.e. 0-jet and 1-jet data samples.

2. 2.

   For each heavy neutino mass, we perform ML training for the 0-jet and 1-jet groups, respectively, and obtain the best performance to separate signal and background events.

3. 3.

   Finally, we combine these two subgroups of data samples to get the prospects of $|V_{\ell N}|^{2}$ at the high-energy colliders.

It is noteworthy that the separation power of the variables above varies for different beam energies and heavy neutrino masses. For example, for most center-of-mass energies and heavy neutrino mass $m_{N}$, it is found that the most powerful observable is $m_{\ell_{1}\ell_{2}}$. As shown in the upper left panel of Fig. 6, the SM background events form a sharp peak near the narrow window at the $Z$ mass of 91 GeV, while for the signal events, the peak value of $m_{\ell_{1}\ell_{2}}$ moves in term of the heavy neutrino mass. It is remarkable in the left panel of Fig. 10 that when $m_{N}$ is around 150 GeV, the observable $m_{\ell_{1}\ell_{2}}$ for the signal peaks at around 90 GeV, which makes the separation of signal and background events rather challenging. It is found that other observables have also less separation power for the case of $m_{N}=150$ GeV. As an explicit example, the distributions of $m_{N}^{\rm rc,0}$ for backgrounds and the signal cases $m_{N}=150$ GeV and 500 GeV are shown in the right panel of Fig 10, which are denoted, respectively, by the red, dashed blue and solid blue lines. It is clear that for the case of $m_{N}=150$ GeV, the distribution of $m_{N}^{\rm rc,0}$ for signal almost overlaps with that for the SM backgrounds.

As mentioned in the introduction, we will use ML to achieve the optimal separation power between the background and signal events. In particular, we adopt two multi-variate analysis methods, i.e. the MLP Fukushima:2013b and BDTG 10.2307/2699986 techniques, which have been encoded in the TMVA package 2007physics…3039H . A portion of MC samples are used for ML training. Input variables are prepossessed for decorrelation via the square-root of the covariance matrix (cf. Figs. 8 and 9). Nodes are constructed in the ML algorithms. The number of nodes is related to the number of trees in BDTG, while that is related to the number of hidden layers and the number of nodes on each layer in MLP. There are weights connecting those nodes. In ML methods, more useful variables have larger weights such as $m_{\ell_{1}\ell_{2}}$ and $m_{N}^{\rm rc}$, while less useful variables have small weights, for instance $p_{T}^{j}$, $\Delta R(j,A)$, $\Delta R(\ell_{1},\ell_{2})$ and $\Delta y(\ell_{1}\ell_{2},\ell_{3})$. A loss function is defined to describe the sum of difference between the machine prediction (the probability to be a signal event) and data (1 for signal and 0 for background). The goal of training process is to adjust the weights to minimize the loss function. Since the weights are free parameters in the ML, the number of weights can not be too large compared to the sizes of the samples, otherwise the ML will be over-fitted. In our case, the number of trees is set to be 1000 in BDTG. In MLP there is one layer, where there are 25 nodes.

These two methods combine the multi-dimensional feature observables in Figs. 4 to 7 into a single likelihood variable. The 0-jet and 1-jet events above are combined based on their production cross sections (cf. Fig. 2). However, the jet relevant distributions in Fig. 7 do not contribute much to the separation power in the ML analysis, as compared to the distributions of charged leptons and MET in Figs. 4 to 6. The distributions of one-dimensional likelihood variables in MLP and BDTG for the backgrounds and signal with $m_{N}=500$ GeV at $\sqrt{s}=14$ TeV are shown in the left and right panels of Fig. 11, respectively. The background and signal distributions are denoted, respectively, by the red and blue lines. It is clear that both the two methods can well distinguish the signal from backgrounds.

After optimizing the ML estimator cuts, we can achieve good background rejections (i.e. the portion of rejected background w.r.t. the total) and high efficiencies to pick out signal events (i.e. the portion of signal left w.r.t. the total). As an explicit example, the rejection curves for the two methods BDTG and MLP with $m_{N}=500$ GeV are demonstrated in the right panel of Fig. 12, as functions of the signal efficiency. As a comparison, the rejection curves for the case $m_{N}=150$ GeV are shown in the left panel of Fig. 12. In both the two panels, the BDTG and MLP methods are depicted in red and black, respectively. Comparing the two panels of Fig. 12, the separation performance of the ML estimators is significantly better at $m_{N}=500$ GeV than that at 150 GeV (cf. Fig. 10). For both the two heavy neutrino scenarios in Fig. 12, the BDTG analysis gives slightly better discrimination performance than MLP, especially at high signal efficiency. In general, MLP and BDTG methods yield compatible results. When the number of input observables is the same, the BDTG method can be about 7.5 times faster than the MLP method in training processes. Therefore, in this work we will use the results of the BDTG method and take the MLP method as a cross check.

It is remarkable in Figs. 4 to 6 that the most distinguishing feature observables are $m_{\ell_{1}\ell_{2}}$ and $m_{N}^{\rm rc}$. To show the separation power of these observables, the rejection curves for the observables $m_{\ell_{1}\ell_{2}}$ and the $m_{N}^{\rm rc}$, the other 15 observables in Figs. 4 to 6 and all the 18 observables are shown in Fig. 13, respectively, as the red, blue and magenta lines. The left and right panels are for the cases of $m_{N}=150$ GeV and 500 GeV, respectively. As clearly seen in Fig. 13, the discrimination power is dominated by $m_{\ell_{1}\ell_{2}}$ and $m_{N}^{\rm rc}$ for both the two scenarios. For the challenging case $m_{N}=150$ GeV, including the other 15 observables can undoubtedly improve the yield, where the distributions of $m_{\ell_{1}\ell_{2}}$ and $m_{N}^{\rm rc}$ overlap for the signal and backgrounds (cf. Fig. 10).

Compared to the cut-based analysis, the ML methods can not only use the full information of the distributions in Figs. 4 to 7, but also take into account the correlations among the variables (cf. Figs. 8 and 9). Optimizing the discrimination power of the signal and background data sets, the ML analysis can significantly improve the prospects of $|V_{\ell N}|^{2}$ at the future high-energy hadron colliders. As seen in Fig. 14 below, the mixing angles $|V_{eN}|^{2}$ and $|V_{\mu N}|^{2}$ can be probed down to ${\cal O}(10^{-6})$ to ${\cal O}(10^{-5})$ for $m_{N}=100$ GeV at the future hadron colliders, which is roughly one order of magnitude better than the current sensitivities in the literature.