Search for Lepton Flavor Violating Signals at the Future Electron-Proton Colliders

In order to study the LFV scenario in an electron-proton collider, we consider the process

The LFV couplings introduced in the Lagrangian in Eq. (2) would induce such a process. In any electron-proton collider, the production of a single muon without any source of missing energy is unlikely in the SM because of the conservation of the lepton number of each generation. Therefore, this provides an interesting channel to search for lepton flavor violations. If no signal for such violation is observed in the collider, one may put a constraint on the LFV couplings introduced in Eq. (1). We then can use this bound on LFV couplings to provide an upper limit on the branching ratios of the LFVZD. In this work, we will be considering one type of coupling out of the $g^{e\mu}_{v}$, $g^{e\mu}_{av}$, $g^{e\mu}_{t}$, or $g^{e\mu}_{at}$, that is to say, one of vector, axial-vector, tensor, or axial-tensor operators at a time in our analysis. We have differed the scope of the study containing more than one operator in a separate work.

We have considered the Large Hadron Electron collider (LHeC) and the Future Circular lepton-hadron Collider (FCC-eh) for the searches for the above signal. Each machine has been proposed to run in two phases. We label these phases as ‘LHeC1’, ‘LHeC2’, ‘FCC-eh1’, and ‘FCC-eh2’ for further reference. We also list the most important specifications of these machines in Table 2.

In $e$-$P$ collision, the signal process comes up due to the flavor changing vertex of $V_{Z}^{e\mu}$ via a $t$-channel exchange of $Z$ boson. This particular channel is not present in the SM. However, some other SM processes can give rise to the final state as the signal. The following SM processes are considered to be potential backgrounds for our signal.

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  Photoproduction: The primary background comes from photoproduction Bauer:1977iq ; Chwastowski:2003aw ; ZEUS:2005nsy ; H1:2011rlk , wherein direct or resolved photons from the electron beam interact with the proton beam. The hadronic final states can give rise to muons from the (semi)leptonic decays of heavy $D$ or $B$ mesons.

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  mu: Another important background comes from the production of a $W$ boson and a jet along with an invisible neutrino. The $W$ boson can then decay into a muon and a neutrino. This background contains a single muon and has the same visible final states as that of the signal. However, the large missing energy from the invisible neutrinos makes the background reduction easier.

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  e-mu: The third main background primarily comes from the production of an electron, a jet, and a $W$ boson. The subsequent decay of $W$ boson to muon channel gives a similar final state as the signal if the electron in the final state gets missed. The source of missing energy makes the background reduction easier.

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  mu-mu: The fourth background is similar to the first one with the exception of the production of $\gamma^{*}/Z$ boson along with the jet and MET. The decay of $\gamma^{*}/Z$ to two muons and one of them being missed makes its final state identical to the signal.

We further note that the same event topology in the backgrounds also appears when the vector bosons decay to $\tau$ and the $\tau$ then decays leptonically. We have taken care of this aspect in the background event generation.

For further signal-background analysis, we have implemented the new Lagrangian described by Eq. (1) in Feynrules Alloul:2013bka to obtain the Universal Feynman Object (UFO) Degrande:2011ua files. Then, the UFO model is used to generate the signal events for the scattering process. The signal and background events at the parton-level have been generated using MadGraph5@aMCNLO (v3.4.2) Alwall:2011uj at the lowest order. At the parton-level event generation, we have imposed $p_{T}^{\mu,e}>5$ GeV, $p_{T}^{j}>10$ GeV, $|\eta_{e,\mu}|<3.5$, $|\eta_{j}|<8.0$ cuts. These parton-level events have then been showered using Pythia8Sjostrand:2014zea and stored in HepMC2Buckley:2019xhk formatted files. The photoproduction background has been generated using Pythia8 and stored in HepMC2 format. These events have then been passed onto DelphesdeFavereau:2013fsa for detector simulation. For the efficiencies of different objects, we have used the default card for LHeC and FCC-eh provided in Delphes (version 3.5.0). We then used fastjet3Cacciari:2011ma to form jets from the Delphes generated Particle Flow (PF) candidates.

The key to any signal-background analysis is to put some valuable cuts on some variables which can give us a signal-favored region with more signals and fewer backgrounds. To know those good variables in the signal and backgrounds plane, we illustrate, in Figs. 1 and 2, the normalized distributions of some important variables. In Fig. 1, we show the ${p_{T}}_{\mu}$ and ${p_{T}}_{j_{1}}$ distribution for the three different backgrounds and the signal with non-zero vector and non-zero tensor coupling. The variable distribution for the signal with non-zero axial-vector and non-zero axial-vector couplings are similar very much similar to their non-axial counterparts and, therefore, are not shown. Figure 1(a) is for the distribution of ${p_{T}}_{\mu}$ for LHeC1 and Fig. 1(b) is for FCC-eh1 machine. In the bottom row of Fig. 1, we show the normalized distribution of transverse momentum of the leading jet ($p_{T_{j_{1}}}$) for the signal and backgrounds having the same machine energies. Because of the momentum dependence factors in tensor couplings, the distributions of ${p_{T}}_{\mu}$ and ${p_{T}}_{j_{1}}$ have long tails for higher $p_{T}$ compared to the vector coupling scenario. Similarly, in the top row of Fig. 2, we show the distribution of missing transverse energy $\not{E}_{T}$ for both LHeC1 (Fig. 2(a)) and FCC-eh1 (Fig. 2(b)). In the second row, we illustrate the distribution of $\Delta R$ between the muon and the leading jet $\Delta R_{\mu j_{1}}$ for LHeC1 (Figs. 2(c)) and for FCC-eh1 (Fig. 2(d)).

In electron-proton collider, another useful variable for the signal, in which no sources of missing energy are expected, is $E-P_{z}$ variableZEUS:2005nsy ; H1:2011rlk , defined as the sum of the differences of energies and $z$-component of the momenta of all final state visible particles. The $z$-axis is defined in the direction of the proton beam, and, in the absence of invisible particles in the final state, $E-P_{z}$ should be equal to twice the electron beam energy ($E_{e}$). Since the signal is a $t$-channel process, $(E-p_{z})_{\mu}$ is expected to be closer to $2E_{e}$, and $(E-p_{z})_{j}$ is expected to be close to zero. In Fig. 3, we show the distribution of $(E-p_{z})_{\mu}$ for both LHeC1 (Fig. 3(a)) and FCC-eh1 (Fig. 3(b)). From the distribution of $\not{E}_{T}$ and $(E-p_{z})_{\mu}$, one can easily guess that these are good variables for reducing the SM backgrounds over signal with both types of couplings and in both colliders. Further inspection of the other variable might reveal more interesting signal region. Instead of manually inspecting on all possible event variables, we do the signal and background separation with the help of multivariate analysis. To gain the machine learning advantage in the computation time, we provide a cut of $\not{E}_{T}<50\text{GeV}$ before sending the variables to the multivariate analysis tool. Although the $\Delta R_{\mu j_{1}}$ also looks to have significantly different distributions between signal and background, we did not put any cut on it before the multivariate analysis. It is because the preselection cuts on the missing transverse energy reduce a significant amount of backgrounds.

Keeping these in mind, we have imposed the following preselection cuts for the signal and the backgrounds.

After putting the preselection cuts, the cross sections for our signal and backgrounds for all types of coupling for different machine energies are listed in Table. 3.

After applying the aforementioned cuts on the observables for signal and background events, we move on to investigate potential improvements in separating signal from the background using some established machine learning methods, such as Gradient Boosted Decision Trees Roe:2004na . In comparison to rectangular cut-based analysis, these techniques have been widely employed in the literature recently and have been found to offer a superior separation of the signal from the background. The key objective here is to build a one-dimensional observable after appropriately combining the important observables that may effectively distinguish our signals from the background. We have used XGBoost Chen:2016btl toolkit for gradient boosting in this analysis.

For further analysis, a total of $15$ input variables, called feature variables, have been used for the training and validation of our data sample. The variable, along with their definition and description, is provided in Table 4. With a maximum depth of 3 and a learning rate $\sim\!0.001$, we have taken about 1000-3000 estimators for the gradient-boosted decision tree technique of separation. We have used 70% of the whole dataset for training purposes and 30% for validation in both XGBoost analyses. Overtraining of the data sample is one potential drawback of these strategies. In cases of overtraining, the test sample cannot match the training sample’s incredibly high accuracy. With our selection of parameters, we have specifically verified that the algorithm is not overtrained.

For the discussion in this subsection, we have chosen only $g_{v}=3.68\times 10^{-4}$ to be non-zero. We first show, in Fig. 4(a), the performance of the BDT network through the Receiver Operating Characteristic (ROC) curve. The good performance of the network is evident from the figure. The background acceptance is below 0.01 with 50% signal efficiency. This high performance is because most of the backgrounds contain neutrinos, a source of missing energy, in contrast to the signal, where there is no source of missing energy. The network returns the BDT classifier variable, which can then, in principle, be used to set a cut and calculate the signal significance. We have shown these signal significances as a function of signal efficiencies.

For given signal efficiency $\epsilon_{S}$ and background efficiency $\epsilon_{B}$, the signal efficiencies are calculated as

where $\sigma_{S}^{0}(\sigma_{B}^{0})$ are the cross section for signal (backgrounds), and $\mathcal{L}$ represents the integrated luminosity. The Variation of signal significance as a function of signal efficiency is shown in Fig. 4(b) for all the four machine energies as described in Table 2. A feature of the curves shows that the signal significance is the best in the region 30% to 50% signal efficiencies. The background efficiencies in this region are actually substantially small which varies between $10^{-3}$ to $10^{-2}$. The region below 30% of signal significance suffered from a very low number of backgrounds (sometimes zero). As a result, this area is not reliable as there is a mismatch between the train and test samples.

We then choose our working points to be in the range of $\epsilon_{S}$ from 0.3 to 0.5, depending on maximum achievable signal significance. The four different machine energies, therefore, have different working points. We also ensured that there was a good agreement in the distribution between the test and train samples at those working points. The variation of signal significance as a function of integrated luminosity has been shown in Fig. 5(a) for four different colliders. One can observe that the signal significance is distinctively better ($\sim\!2.0-3.0$ times better) in FCC-eh1 and FCC-eh2 compared to all LHeC energies. Furthermore, for a given luminosity, the signal significance varies as the square of the coupling constant $g_{v}$. Now, we can, in principle, put an upper limit on the coupling constant at a $2\sigma$ signal significance. That is, to say, the value of $g_{v}$ at which 2$\sigma$ signal significance is achieved. These limits on $g_{v}$ can then be translated into the limit on the LFV $Z$ boson BRs by using Eq. 2. We plot, in Fig. 5(b), the upper limit on BR($Z\to e\mu$) at 95% C.L. as a function of luminosity for four different machines. In Fig. 5(b), the horizontal line represents the current bound by the ATLAS collaborationATLAS:2022uhq , which is achievable below 50 fb^-1 in all the future electron-proton colliders.

Similar to the previous subsection, we then performed an analysis with the tensor LFV coupling. In this case, we have only the tensor coupling to be non-zero at $g_{t}=5.71\times 10^{-6}~{}\text{GeV}^{-1}$. We first show the ROC curves for four different machines in Fig. 6(a).

The ROC curves actually feature a very good separation between the signal and the backgrounds. In all the machine energies, the background acceptance rate is between $10^{-3}$ to $10^{-2}$ signal significances between 30% to 70%. Signal significance as a function of signal efficiency has been plotted in Fig. 6(b). We can see that, in the case of tensor-only coupling, FCC-eh runs can provide better (approximately a factor of 1.2 - 3.0) significance than LHeC runs. Although a quick comparison between Fig. 4(b) and Fig. 6(b) may indicate that the tensor-only coupling provides better signal significances, we note that they should not be compared as the coupling values, at this point, are arbitrary and they are for demonstration only.

We then choose our working points to be in the range of $\epsilon_{S}$, from 0.3 to 0.5, which yields the maximum signal significance that we can achieve. We show in Fig. 7(a) the variation of signal significance as a function of luminosity. To affirm no overtraining, we ensure good agreements in the distribution between the test and train samples in our chosen $\epsilon_{S}$. The signal significances are above 2$\sigma$ even with 50 fb^-1 in the case of LHeC runs and are well above 2$\sigma$ in the case of FCC-eh runs.

The projected upper limit on BR($Z\to e\mu$) as a function of integrated luminosity at the best $\epsilon_{S}$ value with highest significance, has been plotted in Fig. 7(b). In the case of tensor-only coupling, one can see that the current bound on BR($Z\to e\mu$) at 95% C.L. can easily be achieved with less than 50 fb^-1 integrated luminosity in all four machine energies.

We now put together the results from the last two subsections by providing projected bounds on BR($Z\to e\mu$) for four different machine energies. As we mentioned previously, we consider only one effective coupling at a time. These projected bounds on the branching ratio and the effective coupling are shown in Tables 5 and 6. We have considered a one-year runtime of each machine and the integrated luminosity as provided in Table 2.

As was hinted at by the discussions in the previous two subsections, we note that the future electron-proton collider has the potential to provide a stronger upper bound. As mentioned in Table 2, if we look in the LHeC case, for run 1, an integrated luminosity around $50$ fb^-1 can be achieved, and for run 2, it can be $\sim\!90$ fb^-1 after collecting one year of data. Considering that we can marginally improve the limit on BR($Z\rightarrow e\mu$) compared to the current bound provided by the ATLAS collaborationATLAS:2022uhq for the case of vector-only and axial-vector-only coupling. On the other hand, if the new physics LFV couplings are governed by tensor or axial-tensor coupling, the bound on BR($Z\to e\mu$) can be $\sim\!4$ times stronger than the current upper limit. Overall, there is a scope for improvement in the current bounds at both runs of the LHeC machine. These numbers are calculated assuming that the run is for one year only. With more than one year run, there is further scope for improvement.

Let us look at the FCC-eh case now. In that case, with the run-1, we can get an integrated luminosity of $80$ fb^-1 and for run-2, it can be $150$fb ^-1 after collecting all data for one year. So, if one only focuses on the luminosity after one year run for FCC-eh run 1, the improvement on the BR($Z\rightarrow e\mu$) can be $\sim\!3$ times stronger than the existing bound set by ATLAS for the case of vector-only and axial-vector-only coupling. Whereas, for tensor-only or axial-tensor-only coupling, the improvement can be more than a factor of $10$. On the other hand, a more remarkable improvement can be shown if we take a look at the FCC-eh run-2 case. In that case, after the collection of one-year data, the bound can be $\sim\!17$ times stronger compared to the existing bound. For a run of more than one year, these projected bounds are expected to become stronger.