Electroweak Restoration at the LHC and Beyond: The $Vh$ Channel

The LHC is exploring electroweak (EW) physics at the scale EW symmetry is broken. As the LHC and new high energy colliders push our understanding of the Standard Model to ever-higher energies, it will be possible to probe not only the breaking of but also the restoration of EW symmetry. We propose to observe EW restoration in double EW boson production via the convergence of the Goldstone boson equivalence theorem. This convergence is most easily measured in the vector boson plus Higgs production, $Vh$, which is dominated by the longitudinal polarizations. We define EW restoration by carefully taking the limit of zero Higgs vacuum expectation value (vev). EW restoration is then measured through the ratio of the $p_T^h$ distributions between $Vh$ production in the Standard Model and Goldstone boson plus Higgs production in the zero vev theory, where $p_T^h$ is the Higgs transverse momentum. As EW symmetry is restored, this ratio converges to one at high energy. We present a method to extract this ratio from collider data. With a full signal and background analysis, we demonstrate that the 14 TeV HL-LHC can confirm that this ratio converges to one to 40% precision while at the 27 TeV HE-LHC the precision will be 6%. We also investigate statistical tests to quantify the convergence at high energies. Our analysis provides a roadmap for how to stress test the Goldstone boson equivalence theorem and our understanding of spontaneously broken symmetries, in addition to confirming the restoration of EW symmetry.


I. INTRODUCTION
With the discovery of a Standard Model-like Higgs boson [1,2], we entered a new era of probing the nature of electroweak (EW) symmetry breaking. Through the measurement of many EW and Higgs boson processes, the LHC is exploring the nature of a spontaneously broken symmetry at and above its breaking scale. As the LHC continues to gather data, it pushes these precision measurements and our understanding of EW symmetry breaking (EWSB) to ever-higher energies. These higher energies are very interesting for precision EW measurements [3][4][5][6][7][8][9][10], in particular at future high energy colliders. As we get further above the EW scale, EW particles are essentially massless, and new interesting SM physics begins to appear. For example, the massive EW gauge bosons become partons and must be included in parton distribution functions as EW multiplets [11][12][13][14][15][16] and parton showers [6,[17][18][19]. While these effects are intrinsically interesting and necessary to our understanding of the SM, they have also been shown to significantly impact searches for beyond the SM physics [20].
In this paper, we propose a new study to test one of the central behaviors of the SM at high energy: restoration of EW symmetry. We will propose a systematic analysis to observe this restoration at high energy colliders. Our study will open new analysis methods to stress test our understanding of the SM and the spontaneous breaking of EW theory. The main ingredient of our analysis is that as massive bosons become massless, their longitudinal modes can be replaced by the associated Goldstone bosons via the Goldstone boson equivalence theorem (GBET). Indeed, the GBET is a central ingredient to our understanding of the quantum field theory of spontaneously broken symmetries. Hence, our analysis provides a roadmap for how to empirically test the GBET, deepen our knowledge of spontaneously broken symmetries, and confirm the restoration of EW symmetry at high energies.
In the SM, restoration of EW symmetry is equivalent to taking the limit where the Higgs vacuum expection value (vev), v, goes to zero 1 . In this limit the EW gauge bosons become massless. That is, only the transverse polarizations persist and the longitudinal polarizations are replaced by their associated Goldstone bosons, i.e. the GBET mentioned above. There is a long history [21][22][23][24][25][26][27][28][29][30][31][32][33][34] of trying to observe the GBET via longitudinal vector boson scattering. One of the interesting things about longitudinal vector boson scattering is that this process probes the quartic Goldstone boson coupling, which arises via the Higgs potential: is the Higgs doublet, G + , G 0 are the Goldstone bosons, h is the Higgs boson, and v = 246 GeV is the Higgs vev. Hence, longitudinal vector boson scattering probes the shape of the Higgs potential and the source of EWSB. Additionally, this process violated perturbative unitarity without a Higgs boson [22][23][24]. However, with the observation of a light Higgs boson with SM-like couplings to EW gauge bosons [35][36][37][38], longitudinal vector boson scattering is effectively unitarized with the violation of perturbative unitarity pushed to multi-TeV energies [23,34,[39][40][41][42][43], making it difficult to observe.
As the above makes clear, the observation of EW symmetry restoration and the GBET is simplest in processes that are dominated by longitudinally polarized gauge bosons. Such a process is Higgs production in association with an EW gauge boson: qq → V h with V = W ± , Z (V h). In the GBET, the qq → V h production is equivalent to qq → G ±,0 h production (Gh) which arises from the Higgs kinetic term: The kinetic term contains the trilinear interactions (a) Z − G 0 − h, W ± − G ∓ − h and (b) where the subscript L indicates a longitudinally polarized vector boson. The interactions (b) contribute to pair production of longitudinally polarized gauge bosons qq where V = Z, W ± . However, the pair production of gauge bosons qq → V V is dominated by transverse polarizations to high energy [44][45][46]. For the V h channel, the contribution from transversely polarized vector bosons is suppressed since a portion of the Higgs doublet already exists in the final state.
From this discussion, Higgs production in association with W ± or Z is a prime candidate to observe EW restoration. In this paper we present an analysis strategy to do precisely this. While this may seem straightforward, complications immediately arise when trying to observe EW restoration at hadron colliders. Namely, the vector and Higgs bosons are intermediate states, and the collider observes their decay products. These decays occur at the EW scale and the GBET is not valid. This is clear by noting that while vector boson couplings to fermions are universal across generations, the Goldstone bosons couple like mass. Hence, their branching ratios are vastly different and it is necessary to unfold to the underlying two-to-two process.
As a proof of principle, we show that the convergence of the EW restoration can be observed in qq → V h in the Higgs transverse momentum distribution at the 14 TeV high luminosity LHC (HL-LHC) and the proposed 27 TeV high energy LHC (HE-LHC). We will define a signal strength that is a ratio of the vector boson V h and Goldstone boson Gh processes. In fact, numerically the signal strength is the same for both W ± h and Zh production. At high Higgs transverse momentum, we show that it is possible to observe the convergence of this signal strength to one, indicating that EW symmetry is restored. We will also explore various test statistics to determine how well the GBET converges. In particular, we propose a modified the Kullback-Leibler divergence to quantify the convergence.
In Sec. II we give the theoretical foundation for our work. Helicity amplitudes of diboson processes and polarized production rates are given in Sec. II(a), and in Sec.
we define what we mean by the v → 0 limit. In Sec. III we define a likelihood to perform the unfolding, and define the relevant signal strength. We present our collider analysis in Sec. IV, which is based on a deep neural network (DNN). In Sec. V, we present our results showing the convergence of our signal strength as well as the modified Kullback-Leibler divergence. Finally, in Sec. VI we conclude.

A. Amplitudes
To observe the convergence of the Goldstone boson equivalence theorem and restoration of EW symmetry, we need to look at EW gauge boson processes that are longitudinally dominated at high energy. To determine the channels to study, we first calculate di-boson helicity amplitudes in the high energy limit [45,[47][48][49]. The fully longitudinal double EW gauge boson production modes are where √ŝ is the partonic center of mass energy, the subscript L on EW gauge bosons indicates longitudinal polarization, and the subscripts on the quarks indicate quark helicity.
For W + W − production θ is the angle between W + and initial state quark, and for W Z production θ is the angle between the W and the initial state quark. Q q is the quark q's charge, T q 3 is the quark q's isospin, and c W = cos θ W , s W = sin θ W is the weak mixing angle. As expected, the fully longitudinal EW gauge boson pair production modes W + W − and W Z persist at high energy. However, so do transversely polarized gauge bosons with opposite helicities: where the subscript ± on EW gauge bosons indicate the transverse helicities, for ZZ final state θ is the angle between the initial state quark and Z + , and All other amplitudes are either zero or suppressed at high energies: From Eqs. (3) and (4) it is clear that double EW gauge boson production is not longitudinally dominated. Indeed, even though both fully longitudinal and transverse polarizations persist at high energy, as shown in Fig. 1(a,b) W W and W Z production are strongly dominated by the transverse polarizations. Here we use CTEQ6L1 parton distribution functions (pdfs) [50] implemented in LHAPDF [51] via ManeParse [52]. This is particularly striking in W W production where at high energies 90 − 95% of the W s are transversely polarized, while W Z production is 60 − 70% transversely polarized. Hence, to find the longitudinally polarized signal and observe EW restoration in qq → V V , either the differences in the angular distributions of the gauge bosons must be exploited or their polarizations must be tagged, which is very difficult [46,48,[53][54][55][56][57]. There is also an additional complication that the gauge bosons are not final state particles and different gauge boson polarizations interfere with each other [46,49,[53][54][55].
These complications do not arise in EW gauge boson production in association with a Higgs: The longitudinal polarizations persist at high energy while transverse polarizations decrease with energy. This is even more clear in Figs     v = 0, the µ 2 parameter in Eq. (1) must be zero or negative. Hence, in principle the Higgs field could have a non-zero mass. We will enforce the tree level relationships between the Higgs mass m h , the µ 2 parameter, and the vev then take the limit v → 0: That is, we consider a massless Higgs doublet field consistent with the parameter relationships in the SM.
Once the SU ( As expected from the Goldstone boson equivalence theorem, the Goldstone boson production amplitudes agree with high energy longitudinal gauge boson amplitudes in Eqs. (3) and (7).
To observe how quickly the Goldstone boson equivalence theorem converges in V h production, we define signal strengths as ratios of Higgs transverse momentum, p h T , distributions: While √ŝ is the relevant quantity for the convergence of the GBET, we use p h T since it is more easily reconstructable when there is missing energy from gauge boson decays. The signal strengths are shown in Fig. 2 Then both W ± h and Zh distributions can be fit to the same parameter, making the combination of these measurements straightforward.

III. SIGNAL STRENGTH AND LIKELIHOOD
We now turn to how to observe EW restoration in the EW gauge boson plus Higgs boson production via the signal strength in Eq. (11). As discussed before, one immediate issue is that the vector and scalar bosons are not final state particles, and their decay products are detected. The complication is that although the production of V h occurs at high energies, the decays of the vector boson and Higgs occur at the EW scale ∼ 100 GeV where the Goldstone boson equivalence theorem is not a good approximation.
To extract the signal strength [Eq. (11)], the detector level events need to be unfolded [58][59][60] to the partonic qq → V h level. There are many modern machine learning [61][62][63][64] methods to unfold events. However, we are primarily interested in only the Higgs transverse momentum distribution. Hence, we adapt the unfolding method in Ref. [65] and use a simple one-dimensional likelihood function method. For each bin of p h T we define a likelihood function: where i labels each p h T bin; B i is the expected number of background events and n obs,i the total number of observed events in the ith bin; L is the integrated luminosity; ∆σ V h i is the partonic cross section in each bin; and ij is an efficiency matrix. The efficiency matrix takes into account detector effects, branching ratios, parton showering, and hadronization. Using the uniform signal strength in Eq. (11), the binned likelihood function is where µ j V h is the signal strength and ∆σ Gh j the Goldstone boson plus Higgs production rate in the jth Higgs transverse momentum bin. For Zh production the relevant Goldstone The efficiency matrix takes care of the probability that a parton level event in the ith bin is in the jth bin at the detector level. To calculate ij we generate V h events in MadGraph5 aMC@NLO [66] with parton showers and hadronization via PYTHIA8 [67] and detector effects via DELPHES3 [68]. By comparing detector level reconstructed p h T to the parton level information from MadGraph5 aMC@NLO, the efficiency matrix ij can be determined.
Finally, we use the global likelihood across all bins where Pois(x|y) is a conditional Poisson distribution and S i is the expected number of signal events in the ith bin. Now, given a number of observed events n obs,i , Eq. (14) is maximized to determine the binned signal strengths µ i V h .

IV. COLLIDER ANALYSIS
We now turn to extracting our signal from background. Note, for each signal with different multiplicities of jets, the efficiency matrix ij in Eq. (13) must be recalculated to map onto the partonic qq → V h event.
The major backgrounds are: QCD production of V + ll, V +HF, V + cl as well as top pair, single top and vector boson pair. Here l = u, d, s, g, and HF indicates "heavy flavor": bb, bc, cc, bl. For the zero and one-lepton signals, we include backgrounds from missing leptons. The missing lepton rate is estimated by using the default setting of DELPHES3.

A. Simulation
We use MadGraph5 aMC@NLO [66] for parton level generation and heavy particle decay, PYTHIA8 [67] for parton showers and hadronization, DELPHES3 [68] for fast detector simulation. Finally, MLM jet matching [69] up to one additional jet is used for background simulation.
In the detector simulation, we use the default CMS card with some modifications. The basic acceptance level cuts at both 14 TeV HL-LHC and 27 TeV HE-LHC are: • Lepton transverse momentum, p T : • Lepton, η , and jet, η j , rapidity: • Minimum separation between jets, j, and leptons, : • Electron isolation: PTRatioMax= 0.43 2 considering particles with p T > 0.5 GeV and within a cone of radius ∆R < 0.3.
• Muon isolation: PTRatioMax= 0.25 considering particles with p T > 0.5 GeV and within a cone of radius ∆R < 0.4.
The minimum jet transverse momentum, p j T , requirement is different between 14 and 27 TeV: • At 14 TeV: • At 27 TeV [70]: Finally, since our signal is rich in b-quarks, we also use a b tagging rate of 0.70 with mis-tag rates of 0.125 for charm jets and 0.003 for light jets [71].

B. Classification
To classify signal from background, we use "pre-cuts" followed by a DNN. The pre-cuts are basic multiplicity and invariant mass cuts to help separate signal and background: • For the two lepton signals (n = 2) we require exactly two same flavor, opposite sign leptons that reconstruct the Z mass |m − m Z | ≤ 10 GeV, where m is the di-lepton invariant mass. In addition, we require at least two jets (n j ≥ 2) passing the cuts in Eqs. (16,18,19) • For both the zero (n = 0) and one lepton (n = 1) signal we require either two or three jets (n j = 2, 3) to pass the cuts in Eqs. (16,18,19).
For all signals we require exactly two b-tagged jets (n b = 2).
After events pass the pre-cuts, a DNN is used to further classify signal and background.
The inputs of the DNN are high-level reconstructed variables and are detailed in Appendix A.
The DNN is a binary classifier consisting of three hidden layers with 2 10 , 2 12 , and 2 10 nodes.
We adopt LeakyReLU [72] for non-linearity, use batch normalization between layers, and the output layer uses softmax to create a probability. We use cross entropy as the loss function with an L2 penalty: where y s is the signal indicator with y s = 1 for signal and y s = 0 for background, p is the predicted signal probability, and W 2 is the matrix norm of the weight matrices. While the same DNN structure is used for all six categories, the L2 penalty value λ changes.
Cut flow tables and signal significances are given in Tab  HL-LHC and 15 ab −1 for the HE-LHC. We use the the asymptotic formula for a discovery significance with Poisson statistics where N s , N b are the number of signal and background events, respectively. It is clear that background and signal are well separated.
In Fig. 3 we show the reconstructed vector boson p T distributions after the DNN selection for all six categories. The background is cumulative, and the signal is overlaid. At high energies the signal and background separation is better. This is precisely where we expect to see EW restoration.

V. RESULTS
To fit the signal strengths in Eq. (11) we perform pseudo-experiments to sample the binned p h T distribution. After the collider analysis of the previous section, we have a sample of signal and background events. That sample is used to create a probability density function   (PDF) for the signal and background p h T distribution. The total number of events is sampled according to a Gaussian distribution with the mean ν = S tot + B tot and standard deviation σ = √ ν, where the total number of expected signal and background events are respectively, and S i , B i are the expected number of signal and background events in the ith bin after the DNN, respectively. The total number of events is then distributed according to the p h T PDF. In practice, instead of the Higgs transverse momentum, we use the di-lepton p T for two lepton categories. At tree level, this is equivalent to p h T for the V h signal. For the zero and one lepton categories, we do use the reconstructed Higgs p T .
For each of the six categories, we perform these pseudo-experiments at 14 TeV and 27 TeV. Then the p h T distribution is repeatedly sampled for each pseudo-experiment. These samples determine the number of observed events n obs,i for each bin in Eqs. (12,13,14). In   Eqs. (12,13,14), S i and B i are the same as used to set the mean and standard deviation for n obs,i sampling. For each pseudo-experiment, we maximize the likelihood function Eq. (14) to find the best fit value for the signal strength and then determine the 68% CL on µ V h . For each category, we average the the best fit values and error bars over all pseudo-experiments.
In Fig. 4   In an optimistic scenario, the systematic uncertainty on V h production is expected to be 5% [70]. The red uncertainty bands show the statistical uncertainty, and the green bands show statistical and a 5% systematic uncertainty added in quadrature.
At low p h T , the signal strength is significantly far from one and then converges to one at higher energies, as expected. Indeed, in the last overflow bin, we find the central value of the signal strength and 68% CL to be: That is, the signal agrees with the EW restoration prediction at 40% at the HL-LHC and 6% at the HE-LHC. Hence, the V h rate converges to the expected rate with EW symmetry restored. This measurable convergence indicates empirically that the longitudinal modes can be replaced with the Goldstone bosons, and EW restoration can be observed at high energies.

A. Statistical Test of EW Restoration
To test how well EW restoration is being observed, one needs to measure how the convergence is improving by using higher and higher energy bins. At low p h T bins, although the statistical error is small, the Goldstone and gauge boson distributions do not agree. As one moves towards higher p h T bins, while the two distributions converge, the statistical errors also increase, as shown in Figs. 4 and 5. In this section we explore statistical measures of the restoration and discuss their implications, taking into account both the theory convergence as well as the experimental uncertainties. The goal is, assuming that the SM is a good description of the data, we want to test the agreement between the qq → V h and qq → Gh (µ j V h = 1) production as a function of p h T . As a first choice, using the language that the high energy physics community is more familiar with, we consider using "χ 2 per degree of freedom" as a function of p T bins. One generically anticipates this quantity to decrease as an indicator of better convergence. After using the method in the previous sections in separating signal and background, we now have six-category samples, post-selection cuts, that have the significance of our analysis as a function of p h T . One can define "χ 2 per degree of freedom" 3 : log Pois(n obs,l | j ∆σ Gh j lj L + B l ) Pois(n obs,l |S l + B l ) , where we sum over the m ranked p h T bins (from low to high). Using the methods of the previous section, we perform 10,000 pseudo-experiments. The results are shown in the left panel of Fig. 6. We show the median over all pseudo-experiments as well as the band where 68% and 95% of the pseudo-experiments lie. From the figure we can see, as anticipated, the ∆χ 2 m decreases as one includes more high p h T bins. However, we note here that ∆χ 2 m has some disadvantages in measuring restoration. First, for the low p h T bins, each bin contributes to a sizable ∆χ 2 since the Gh and V h hypothesis are in poor agreement and statistical uncertainty is small. At high p h T , the statistical uncertainties increase. Hence, even if the Gh and V h distributions do not converge, as more bins are averaged over ∆χ 2 m will decrease. In other words, even if the higher bins contain no separation power, e.g. the background uncertainty being infinitely larger than the signal strength, the ∆χ 2 m decreases. This reflects that ∆χ 2 m measures the agreement between two hypotheses: as the uncertainties increase, the error bars overlap, and the hypotheses are in "good agreement." However, to measure EW restoration, the convergence of V h and Gh must be measured and ∆χ 2 m is not a good measure of convergence. As can be seen, the measurement of the EW restoration is not a typical particle physics test. The issue is that we want to measure the convergence of two hypotheses with energy, not just determine how well they agree globally. Ideally, the measure should contrast different hypotheses for a given experimental data set with proper weight for each bin according to the "information" contained there. We turn to Shannon's information theory and find that generically −p log p measures the information of a distribution p. While there might be an equivalent or better definition outside of our scope, we use a modified Kullback-Leibler (KL) divergence. KL divergence is a commonly used quantity contrasting the information between two different distributions, and often plays the role of loss function for machine learning. The KL divergence tests the information difference between two hypotheses. To do this, for each pseudo-experiment we first define properly normalized probability for each bin for the V h hypothesis where the i ≤ m are bin numbers with increasing p h T . We have assumed independent event samples, and so have taken a product of probabilities across all signal categories. Restricting where the sum over j is over all bins and not restricted to bins less than p h T,m . The KL divergence for the first m bins is then: Now the interpretation of the KL-divergence is clear. If the two hypotheses describe the data equally well, the log goes to zero and the KL divergence is zero. The KL-divergence has a similar property as the Gibbs free energy, being positive definite. Hence, when the agreement of the two hypotheses is worse, KL m is larger. As more bins are included, we expect the EW restoration to describe data better and the KL divergence should approach zero.
When the two hypotheses do not agree, the weighted sum in Eq. (27) guarantees that the largest contributions come from bins for the conditional probabilities p ≤m i are largest.
Hence, the KL divergence contains more information from ∆χ 2 m and is expected to be a better measure of convergence. In Fig. 6 we show the differential KL divergence, KL m . We show the median over all pseudo-experiments as well as the band where 68% and 95% of the pseudo-experiments lie. As can be clearly seen, whereas the χ 2 per degree of freedom test began to plateau at high energies, the KL-divergence decrease more steadily. This more readily shows that the agreement of the V h and Gh hypotheses continues to get better at high p h T and we observe EW restoration. We want to emphasize here that the convergence between V h and Gh distributions is directly represented by the fact that ∆χ 2 m and KL m decrease as higher and higher p h T bins are included. We would like to note that somewhat counter-intuitively the 14 TeV statistical tests seem to be "better" than the 27 TeV results. That is, the 14 TeV values are lower. This is due to larger statistical uncertainties at 14 TeV, which suppresses both the χ 2 per degree of freedom and KL m relative to 27 TeV. Hence, although the uncertainties between different machines can be compared to determine which machine is more sensitive, the convergence as given by these statistical tests should be considered on a machine-by-machine basis.

VI. CONCLUSIONS
In this paper, we have studied the potential of the HL-LHC and HE-LHC to observe EW restoration in pp → V h production. Discussions of EW symmetry restoration have traditionally been limited to the longitudinal vector boson scattering. Using the Goldstone boson equivalence theorem, it can be seen that this scattering occurs via the quartic term in the Higgs potential. However, Goldstone bosons also have interactions via the Higgs kinetic terms. These terms contribute to the production of longitudinal gauge bosons in qq → V V and qq → V h channels.
As we showed, the qq → V V production is dominated by transverse polarizations to very high energies. Hence, it is difficult to observe Goldstone boson production in this channel.
Since qq → V h is a purely s-channel process with a component of the Higgs doublet, it is longitudinally dominated starting at relatively low energies. In Sec. II, we defined EW symmetry restoration by taking the limit of the Higgs vev going to zero and enforcing the SM tree-level relations for the Higgs potential parameters. This results in a massless Higgs doublet in the EW restored theory. From this, we defined a differential signal strength µ V h as the ratio of the p h T distributions of V h and Gh production. As shown, this signal strength is the same for Zh and W ± h, allowing for easy extraction of global signal strength in all V h channels. This convergence can be seen in Fig. 2.
As EW symmetry is restored, the longitudinal gauge bosons are replaced with their Goldstone boson counterparts. Hence, the signal strength µ V h is expected to converge to the Goldstone calculation at high energies. Using a sophisticated collider analysis, we showed that by performing a fit to these signal strengths, it can be observed that the V h channel converges to Gh at high energies. Indeed, for p h T ∼ 400 GeV, the V h and Gh distributions agree at around 80%. Additionally, as can be seen in Figs. 4 and 5, the extracted signal strength in all V h signal categories agrees with the partonic level prediction. Finally, to quantify the agreement between the V h and Gh hypothesis, we defined a differential Kullback-Leibler divergence. If two hypotheses agree, the KL divergence is small. As shown in Fig. 6, the V h and Gh hypothesis agree well at high energy, and EW restoration can be well observed.
To summarize, we demonstrated the EW restoration could be observed in the V h channel.
Indeed, EW restoration can be confirmed to 40% precision at the HL-LHC and 6% precision at the HE-LHC. Our study can be further extended to other future colliders, as well as the other di-boson production modes highlighted in the theory discussion. Our study clearly highlights the possibility of studying the physics phenomena of electroweak restoration at high energy colliders as well as electroweak breaking. First, note the Higgs is always reconstructed from the two leading bottom tagged jets. In the 2-lepton categories the Z is reconstructed from the two leptons, and in the zero lepton category the Z is reconstructed from the missing transverse energy. Note for the zero lepton category, we only reconstruct the p T of the Z. In the 1-lepton case, there is a missing neutrino. Its transverse momentum is assumed to be the missing transverse energy of the event. The neutrino's longitudinal momentum is found by requiring that the neutrino plus lepton system reconstruct the W -mass. This leads to a two-fold ambiguity, and we choose the neutrino momentum that is closer to the lepton. Hence, in the 1-lepton case the W is reconstructed. Additionally, we label the leading bottom jet at b 0 , the next to leading bottom jet as b 1 , and the leptons as 0 , 1 similarly. For 3-jet categories, we label the leading non-b jet as j.
The following definitions are used: • The invariant mass of two objects i, j is M ij .
• The reconstructed mass of an object i is M recon i .
• For a final state system i, the transverse mass is defined at M T,i = E 2 i − p 2 Z,i , where E i is the total energy of the objects i and p Z,i is the z-component of their momentum.
• The transverse momentum of an object i is p i T .
• The difference between the rapidities of two objects i, j is ∆η ij = |η i − η j |.
• The scalar sum of all transverse momentum is H T .
• The number of non-b jets is n j .
The input variables for the DNN for each event category are: -MET, p h T , p Z T , p b 0 T , p b 1 T , p 0 T , p 1 T , H T .
-Transverse mass of reconstructed Higgs and Z, M T,Zh .
n j .
-Transverse mass of reconstructed Higgs and W, M T,W h .
-Transverse mass of W : M T,W .
-Transverse mass of W + b 0 : M T,W b 0 .
-Transverse mass of W + b 1 : M T,W b 1 .
-Transverse momentum of reconstructed Higgs and Z, M T,Zh .