Detecting anomaly in vector boson scattering

Vector Boson Scattering (VBS) represents sensitive probe of both the Standard Model (SM) electroweak symmetry breaking (EWSB) and new physics Beyond-the-SM (BSM) Rauch:2016pai ; Green:2016trm . If the couplings of the Higgs boson to vector bosons deviate from the SM prediction, the cross sections of VBS processes will increase with center-of-mass energy up to the scale of new physics. In addition, many BSM models predict extended Higgs sector. The contribution from new resonances can also increase the VBS cross section in certain phase space.

Measuring the VBS processes at hadron collider is experimentally challenging due to their low signal yields and complex final states. The LHC experiments have built comprehensive searches for the VBS processes Alessandro:2018khj ; Baglio:2020bnc ; Gallinaro:2020cte . The same-sign $WW$ production with leptonic decay has the largest signal-to-background ratio among VBS processes. This channel was the first VBS process that has been observed during the run 1 of the LHC Aad:2014zda ; Khachatryan:2014sta and has been confirmed by the measurements at the LHC run II Aaboud:2019nmv ; Sirunyan:2017ret . The ATLAS and CMS Collaborations have also performed the measurements for other VBS channels, such as fully leptonic $ZZ$ Sirunyan:2017fvv ; Aad:2020zbq , fully leptonic $WZ$ Aaboud:2018ddq ; Sirunyan:2019ksz and semi-leptonic $WV$ or $ZV$ with the $V$ decaying hadronically Aad:2019xxo ; Sirunyan:2019der . New physics contributions to the VBS channels are usually parameterized by effective field theory (EFT) operators. Precision measurement of the VBS channels can be recast as constraints on the coefficient of the operators Fabbrichesi:2015hsa ; Liu:2018pkg ; Stolarski:2020qim .

Understanding the polarization of the gauge bosons is an important step after the measurements of the VBS processes. Vector bosons are unstable and can only be observed through their decay products. This lead to the interference among different polarizations, which cancels exactly only when the azimuthal angles of the decay products are integrated over. Even though selection cuts in analyses render the incompleteness of the cancellation, it is still possible to extract polarization fractions by fitting data with Monte Carlo simulated templates. There are studies aiming to determine the polarization of gauge bosons in the $W^{\pm}W^{\mp}$ channel Han:2009em ; Ballestrero:2017bxn , in fully leptonic $W^{\pm}W^{\pm}$ channel Ballestrero:2020qgv , in fully leptonic WZ/ZZ channels Ballestrero:2019qoy , in the SM Higgs decay Maina:2020rgd and in generic processes with boosted hadronically decaying $W$ boson De:2020iwq . Various kinematic observables have been proposed in these works to discriminate the longitudinal and transverse polarized gauge boson. Several recent studies have shown that deep neural network with input of final states momenta can be used for regression of the lepton angle in the gauge boson rest frame Searcy:2015apa ; Grossi:2020orx and classification of events from different polarizations Lee:2018xtt ; Lee:2019nhm .

Autoencoders have been widely used in model-agnostic searches at colliders, dubbed as anomaly detection or novelty detection. The main function of the autoencoder is that it learns to map an input to a latent compressed representation and then back to itself. The autoencoder which is trained on known SM processes could be able to identify the BSM events as anomalies Cerri:2018anq ; Collins:2018epr ; Collins:2019jip ; Blance:2019ibf ; Andreassen:2020nkr ; Nachman:2020lpy ; Collins:2019jip ; Farina:2018fyg ; Roy:2019jae . In other cases, when the anomaly can not be detected on a single event, density-based novelty evaluators DAgnolo:2018cun ; DeSimone:2018efk ; Hajer:2018kqm are proposed to detect discrepancies between two datasets in the latent space. Since the VBS processes are the perfect window to access any new physics related with EWSB, we can adopt autoencoders to detect possible new physics contributions to the process.

In this work, focusing on the fully leptonic and semi-leptonic channels of the $W^{\pm}W^{\mp}$+jets process, we propose a neural network based on the Transformer architecture vaswani2017attention to learn the features of the VBS process. Those features are not only useful in separating the VBS process from the SM backgrounds but also capable of discriminating different polarizations of the $W$ bosons in the VBS process. An autoencoder is trained on the features to reduce the dimensionality so that only the most relevant features are kept. Eventually, we perform binned log-likelihood test in the latent space to find out whether the distributions of the feature is coincide with the SM prediction. The EFT and Two Higgs Doublet Model (2HDM) are considered as examples to demonstrate that this method is able to test a wide class of BSM physics.

The paper is organized as follows. The analysis framework is introduced in Sec. II, including the event generation, architecture of neural network and binned log-likelihood analysis. Discrimination of different polarization modes of the $WWjj$ production is discussed in Sec. III. In Sec. IV and Sec. V, we consider the applications of our method to effective field theory and two Higgs Doublet Model, respectively. Our conclusions are presented in Sec. VI.

Among polarization modes of the VBS processes, the longitudinally polarized component is most closely related to the unitarity issue, i.e. the property of the Higgs boson and possible new physics. There have been extensive studies on separating the polarization of the gauge boson in the VBS process, exploiting various kinematic variables. The lepton angular distribution in the gauge boson rest frame is known to be sensitive to the vector boson polarization,

where the $f_{L,R,0}$ is the fraction of the corresponding helicity and the $\theta$ is the angle between the vector boson flight direction in a certain frame and the lepton flight direction in the vector boson rest frame. Even though the shape of the angular distribution is a good discriminating variable, it can not be reconstructed precisely for the most of the time. In the dileptonic channel of $W^{\pm}W^{\mp}jj$, there are two missing neutrinos in the final state. One can not reconstruct the rest frame for individual $W$ boson. As for the semi-leptonic channel, even though the neutrino momentum can be solved up to a twofold ambiguity (thus the full momenta of all particles can be calculated), there are usually large uncertainties in measuring the jets momenta and identifying the forward-backward jets and jets from $W$ boson decay. Moreover, the shape of the $\theta$ distribution can be distorted by kinematic cuts that need to be used to separate VBS from its backgrounds Stirling:2012zt .

In this section, we demonstrate that our network is capable of discriminating different polarization modes of the electroweak $W^{\pm}W^{\mp}jj$ production with the low-level inputs.

In absence of direct observations of new states, a practical way for investigating the new physics lies in a description based on the EFT, which is valid up to the scale of new physics. The EFT contains a complete set of independent gauge-invariant operators made up by the SM fields. There have been numerous studies on constraining the coefficients of these operators with precision measurements at experiments DiVita:2017eyz ; Ellis:2018gqa ; Grojean:2018dqj ; Biekotter:2018rhp ; Almeida:2018cld . Most of the operators are tightly constrained by the elctroweak precision tests (EWPT) of the SM. We will consider the following operator Giudice:2007fh ; Contino:2013kra

since it is less constrained by the EWPT. The $\Phi$ field is Higgs doublet and $h$ denotes the Higgs boson field with the vacuum expectation value $v=246.2$ GeV. The $\mathcal{O}_{H}$ operator contributes to the Higgs boson kinetic term, and an appropriate field redefinition is required to bring back the kinetic term to its canonical form

It leads to the following changes to the Higgs couplings

The up-dated global fit to the EFT coefficients constrains $\bar{c}_{H}\lesssim 0.4$ (marginalizing over all other operators) Dawson:2020oco . Future lepton colliders, such as the ILC, will constrain the $\bar{c}_{H}$ to the 1% level Jung:2020uzh .

We study its effects on the EW $W^{+}W^{-}jj$ production at the LHC. As the polarization vector $\epsilon_{L}^{\mu}\sim\frac{p^{\mu}}{m_{V}}+\mathcal{O}(\frac{m_{V}}{E})$ grows with momentum $p$, the longitudinally polarized gauge boson scattering ($W_{L}W_{L}\to W_{L}W_{L}$) is dominant at high energy. In the high energy limit, the amplitude for the longitudinal $W$ boson scattering without Higgs contribution is

which cancels with the amplitude from Higgs exchange

leaving terms not rising with energy. Here, $s,t,u$ are Mandelstam variables. However, the cancellation only holds if the Higgs boson couplings to gauge bosons are exactly SM-like. The $\mathcal{O}_{H}$ operator modifies the Higgs boson couplings as shown in Eq. IV.3, leading to an incomplete cancelation up to the scale where new physics states come in. As a result, the fraction of the $W^{+}_{L}W^{-}_{L}jj$ is increased and the kinematic properties of final states are changed.

We adopt the UFO model as implemented in Ref. Alloul:2013naa to generate the EW $W^{+}W^{-}jj$ events in the EFT. All of the coefficients except the $\bar{c}_{H}$ are set to zero. Both the dileptonic channel and the semi-leptonic channel are considered. Only those events that pass through the preselection cuts as listed in Sec. II.1 will be fed into the network for further analyses. The production cross section of the EW $W^{+}W^{-}jj$ process (with different choices of $\bar{c}_{H}$) before and after the preselections are given in Tab. 2. The $\bar{c}_{H}=0$ case corresponds to the SM. We can find the fraction of the longitudinal $W$ production increases with $|\bar{c}_{H}|$ as the cancellation become less exact. And our preselection cuts can raise the fraction of the longitudinal $W^{+}_{L}W^{-}_{L}jj$, especially for the dileptonic channel. After the preselections, the production rate of the semi-leptonic channel is an order of magnitude large than that of the dileptonic channel.

In this and the next section, the same network that is trained on the labeled SM background processes as well as the SM $W^{\pm}W^{\mp}jj$ with different polarizations is used for testing. Events of the new physics are not used for training the network, in order to show that our method is model agnostic. Analyzing the preselected events of both SM background processes and the EFT processes with the pre-trained network, we can obtain the distributions of those processes in the 3-dimensional latent space. The normalized distributions are presented in Fig. 7, where the background corresponds to the weighted sum of all SM processes (including $tt_{\ell}$, $tW_{\ell}$/$t_{\ell}W$, $W_{\ell}Wjj^{\rm QCD}$, $W_{\ell}Zjj^{\rm QCD}$ and $W_{\ell}Zjj^{\rm EW}$) as discussed in Sec. II.1. Since the network is trained to classify the SM background processes with the SM $WWjj^{\rm EW}$, it is not surprised to find that the background events are well separated from the signal events (EW $WWjj$ production in the EFT). Moreover, there are visible differences among the distributions of EW $WWjj$ production with different $\bar{c}_{H}$. This feature can be used to constrain the value of $\bar{c}_{H}$.

To measure the consistency of the SM and EFT with non-zero $\bar{c}_{H}$, we perform the binned log-likelihood test in the latent space. As have been discussed in Sec. II.3, only ten bins with highest signal to background ratios are used. According to our simulation, this will select $\sim$ 30% signal events and $\sim$ 0.5% background events after the preselection. The null hypothesis is the SM backgrounds plus SM EW $W^{+}W^{-}jj$ and the test hypothesis is the SM backgrounds plus EFT EW $W^{+}W^{-}jj$ with a non-zero $\bar{c}_{H}$. The required integrated luminosity to achieve 95% Confidence Level (C.L.) probing for different $\bar{c}_{H}$ are presented in Fig. 8. It can be seen that the semi-leptonic channel outperforms the dileptonic channel if the systematic uncertainty can be controlled below $\sim$ 5%. Due to higher backgrounds in the semi-leptonic channel, the sensitivity drop quickly when the systematic uncertainty is larger than 5%. With systematic uncertainty around 5%, our method will be able to constrain the $\bar{c}_{H}$ to [-0.2,0.1] at high luminosity LHC.

The EFT description may not valid when the collision energy is approaching the masses of new states. Here we consider an ultraviolet complete model, the 2HDM Aoki:2009ha ; Branco:2011iw which is one of the simplest extension to the Higgs sector of the SM. The scalar sector of the 2HDM consists of two $SU_{W}(2)$ doublets. A discrete $Z_{2}$ symmetry is imposed to avoid tree-level flavor changing neutral currents. Depending on how this symmetry is extended to the fermion sector, four types of the 2HDM can be realized. The type-II case will be considered in this work. The 2HDM predicts many remarkable signatures at the hadron collider. In particular, there are resonant signals due to the existence of extra CP-even scalar, CP-odd scalar and charged scalar. Instead of proposing dedicated search for each of those signals, we will show that our method is sensitive to changes of the polarization and kinematic properties of the EW $W^{+}W^{-}jj$ production in the 2HDM. Comparing the latent features of the $W^{+}W^{-}jj$ process in the 2HDM with those from measurement, constraints on the parameters of the 2HDM can be obtained.

There are six parameters in the type-II 2HDM: mass of scalars ($m_{H_{1}},m_{H_{2}}$, $m_{A}$ and $m_{H^{\pm}}$), the mixing angle between two CP-even scalars $\alpha$ and the ratio between two vacuum expectation values $\tan\beta$. The $m_{H_{1}}$ has been measured to be close to 125 GeV. The $m_{A}$ and $m_{H^{\pm}}$ are not relevant in the $W^{+}W^{-}jj$ production. Their mass is set to 3 TeV to forbid the decays of $H_{2}$ into those states. The couplings of CP-even scalars to the $W$ bosons are given by

So the combination $\sin(\alpha-\beta)$ is usually used to replace the $\alpha$ parameter. Even though the $\tan\beta$ alone is not related to the $HWW$ couplings, it can modify the scalar to fermions couplings, which means the total decay width of the $H_{2}$ thus the kinematics of $W^{+}W^{-}jj$ can be affected. We will chose $\tan\beta=5$ for simplicity ⁷⁷7The influence of the $\tan\beta$ to the $W^{+}W^{-}jj$ production is mild as long as the decay width of the $H_{2}$ is not too large.. So we are left with two free parameters: $m_{H_{2}}$ and $\sin(\alpha-\beta)$. The partial widths of the $H_{2}$ are given by

with $g^{\prime}=\cos(\theta_{w})g_{w}+\sin(\theta_{w})g_{1}$, and $y_{t}$/$y_{b}$ is the Yukawa coupling of the top/bottom quark.

The model is implemented in FeynRules Alloul:2013bka , which generates the UFO model files for the MG5 to calculate the leading order production cross section and simulate the events. In Tab. 3, we present the production cross sections of the EW $W^{+}W^{-}jj$ process for a few points in the 2HDM as illustration. In particular, the contribution of the heavy scalar $H_{2}$ is taken into account, which lead to an increased total production rate for the most of the time ⁸⁸8The cross section in 2HDM can be smaller than that in SM when the mass of the $H_{2}$ is heavy and decay width of the $H_{2}$ is large, because of the destructive interference between $H_{1}$ and $H_{2}$ in some phase space..

Due to the facts that the cancellation between the amplitudes with and without Higgs exchange are delayed to the scale of $m_{H_{2}}$ and the heavy scalar dominantly decays into longitudinally polarized vector boson, the fraction of $W^{+}_{L}W^{-}_{L}jj$ is considerably larger than that of the SM. For relatively light $H_{2}$ and small $\sin(\beta-\alpha)$ (which means the contribution of $H_{2}$ is significant), the fraction of $W^{+}_{L}W^{-}_{L}jj$ can reach $\sim$ 30% before applying the preselection cuts, while the number is 6% in the SM. The preselections can increase the fraction even further. This feature renders our network very sensitive to the signals in the 2HDM.

Moreover, the existence of the $H_{2}$ resonance in the $W^{+}W^{-}jj$ production also gives rise to discriminative features in the final state. In Fig. 10, we plot the normalized distributions of latent features for the $W^{+}W^{-}jj$ production from pure $H_{2}$ resonance in the dileptonic channel. Different masses of the $H_{2}$ have distinct distributions in the latent space. It means the network is not only capable of classifying the polarizations of the vector bosons, but also sensitive to their kinematic properties, even though those 2HDM events are not used for trainning.

Finally, we pass the preselected events in dileptonic channel and semi-leptonic channel to the pre-trained network, to extract the latent features. The binned log-likelihood test is performed in the latent space to find out the discovery potential of models with different parameters in 2HDM. Similar as before, the null hypothesis is taken as the SM backgrounds plus the SM EW $W^{+}W^{-}jj$ and the test hypothesis is taken as the SM backgrounds (assuming those processes are kept intact in 2HDM) plus the EW $W^{+}W^{-}jj$ in 2HDM with different sets of parameters. The required integrated luminosity for achieving 95% C.L. probing on the $m_{H_{2}}$-$\sin(\beta-\alpha)$ plane are shown in Fig. 11, for dileptonic channel and semi-leptonic channel, respectively. Unlike the traditional heavy Higgs resonant searches Aaboud:2018bun ; Sirunyan:2019pqw , the sensitivities of which drop quickly at large $m_{H_{2}}$ due to the suppressed production rate. Our method probe both the resonant feature and the modification to Higgs couplings simultaneously. The parameter space with $H_{2}$ as heavy as 1.5 TeV can be probed with relatively low integrated luminosity provided the $\sin(\beta-\alpha)$ is not too close to one. However, as $\sin(\beta-\alpha)\to 1$ (the alignment limit), our method loss the sensitivity completely. Searches for the resonances in fermionic channels are still able to constrain the model CMS:2019hvr ; Aad:2020zxo ; Kling:2020hmi ; Chen:2015fca , since their productions are mainly controlled by the Yukawa couplings. The production cross sections of both channel before applying the preselection cuts are indicated by the color grades in the figure. We can find the sensitivity of the method is roughly determined by the cross section, even though a slightly better sensitivity can be achieved in the small $\sin(\beta-\alpha)$ region, e.g. comparing to the the point ($m_{H_{2}}=300~{}\text{GeV},\sin(\beta-\alpha)=0.9$), lower integrated luminosity is required to probe the point ($m_{H_{2}}=550~{}\text{GeV},\sin(\beta-\alpha)=0.7$), even though their production cross sections are similar. The improvement of the sensitivity attribute to the fact that point with smaller $\sin(\beta-\alpha)=0.7$ contains larger fraction of the longitudinal $W$ boson.

In this work, we construct a neural network consists of a classification network and an autoencoder. With the input of low level information (4-momenta and the identities of particles in our case), the network is capable of reducing the dimensionality of the feature space for $WWjj$ production, without losing much discriminating power (discriminating the EW $WWjj$ from other processes, as well as discriminating different polarization modes of the EW $WWjj$). We find the feature space of both dileptonic and semi-leptonic channels can be compacted into three dimensions. Performing the binned log-likelihood test on the distributions of latent features, we can draw the conclusion whether the data is consistent with the SM predict. We have shown that those latent features are very sensitive to various possible new physics contributing to the VBS. Even though the scores given by the classifier network contain a certain amount of the process information, they are not as complete as the latent features. In Fig. 12, we present the sensitivities of the latent features and the sensitivities of the score ⁹⁹9Among the scores, we find the summation of scores of all polarization components of EW $WWjj$ lead to the best result. So it is used for calculating the $p$-value in the plots. obtained by the classifier for two benchmark points in the EFT and the 2HDM. It is not surprised to find out that the latent features have better sensitivities. In particular, the remarkable kinematic feature of the 2HDM is not very useful in classifying SM processes, which means this sort of information can be lost in the scores given by the classifier. Comparing to the EFT case, the improvements of using latent features are much more significant in the 2HDM model.

Considering both the dileptonic and semi-leptonic channel of the $W^{+}W^{-}jj$ production, we show that our network is capable of classifying different polarization modes efficiently. Without considering the background, the LHC dataset with integrated luminosity $\lesssim 600$ fb^-1 will be sufficient to probe the 1% change in the longitudinal $W^{+}W^{-}jj$ fraction, using the semi-leptonic channel. The dileptonic channel is less sensitive due to its small production rate. Then, the network is applied to the EFT with non-zero $\mathcal{O}_{H}$ operator and the type-II 2HDM taking into account the background effects, to obtain more complete and realistic results. In the EFT, our method will be able to constrain the coefficient $\bar{c}_{H}$ to [-0.2,0.1] providing the systematic uncertainty is around 5%. The dileptonic channel outperforms the semi-leptonic channel if the systematic uncertainty is higher than 5%. In the 2HDM, since our method is sensitive to both the resonant decay $H_{2}\to W^{+}W^{-}$ and the modification to the SM Higgs couplings, the whole region with $\sin(\beta-\alpha)\lesssim 0.95$ and $m_{H_{2}}\lesssim 1.5$ TeV can be probed with integrated luminosity $\sim$ 300 fb^-1 at the LHC.

We note that modifications of the SM are unlikely to be confined to VBS processes. Assuming a new physics scenario of some kind, the model dependent searches can be very effective in discovering the signal. Our method may not as sensitive as those model dependent searches for specific signals. For example, in the 2HDM with $\tan\beta=5$, our method is insensitive to the parameter space where $\cos(\beta-\alpha)=0.05$ (corresponds to $\sin(\beta-\alpha)=0.9987$). On the other hand, searches for $H\to\tau\tau$ at the LHC have already excluded the parameter space with $m_{H}\sim[200,350]$ GeV CMS:2019hvr ; Aad:2020zxo ; Kling:2020hmi . The advantage of our method is that it is suitable for detecting a wide class of new physics which contributes the VBS, i.e. related to the SM electroweak symmetry breaking. This is especially useful when the forms of new physics are not known.

This work was supported in part by the Fundamental Research Funds for the Central Universities, by the NSFC under grant No. 11905149 and No. 11875306.