It is usually hard to challenge ideas that serve as the foundation of physics. Causality, i.e., the notion that an effect is preceded by a cause, is a vivid example. State-of-the- art research machines, like the Large Hadron Collider (LHC) [1] at the CERN laboratory in Switzerland, offer a unique opportunity to test the notion of causality, not at the macroscopic level but at the microlevel in physics.

Known limitations of the Standard Model (SM) of particle physics [2] have encouraged physicists to formulate diverse extensions of this theory. One of those ideas, on which some of the least tested versions are based, is known as the Lee-Wick (LW) Standard Model (LWSM) [3]. It is founded on the finite theory of quantum electrodynamics model proposed by Lee and Wick [4]. Among other consequences, the model predicts an observable that could change the understanding of causality in the realm of fundamental interactions [5].

In the LWSM re-normalization process, each standard field is paired to an LW field. In the context of the by-products of high-energy hadronic collisions, those LW fields are associated to fast decaying, heavier particles, which could microscopically violate causality. In a causal resonant decay, after a particle is produced in the so-called primary vertex, it travels a certain distance, to a secondary vertex (SV), before decaying into other particles (Figure 1.A). On the other hand, in the case of LW resonances, their final decay products appear at the same time (or before) the resonance was generated, as if the LW field would have propagated backwards in time [6]. These decays have a wrong displaced vertex (WDV) that can be distinguished because the total momentum of the final particles at the SV seems to point towards the collision spot [7], as depicted in Figure 1.B. For a macroscopic observer, the time differences between these two types of events are impossible to measure, yet they could be distinguished by observing their WDV topology.

Theoretical results suggest that it would be possible to test this acausal behaviour usingcollisions data from the LHC experiment. To the best of our knowledge, direct tests for these processes and the large amounts of data needed. In this work, we used CMS open data to test the feasibility of carrying out these studies even if the volume of data is not yet sufficient to search for LW particles. These data were taken by the CMS experiment, one of the largest detectors at the LHC, during the so-called Run 1 period. Details about the CMS detector can be found elsewhere [8]. Among other characteristics, CMS had, at the time, a spatial resolution of about 0.02 mm for the determination of primary and secondary vertices. This feature im- posed the main restriction to this investigation.

The best approach to overcome the experimental limitations is to study processes of LW-electrons (ℓ^e±) emerging from the Neutral Current (NC) sector of the LWSM [7],

where the suffix R (L) indicates the right-handiness (left-handiness) and g_Z^eR,L are the coupling constants between Z bosons and SM electrons. These particles,created in pairs, would later decay into a pair of SM electrons and Z bosons, namely,

Similar to reference [7], we calculated the Feynman rules and the average flight distances λ for this process with the Mathematica [9] package named FeynRules [10]. A summary of the values of λ , which are close to the minimum resolution of the CMS detector, are shown in Table 1 for different values of LW electron mass. The cross sections (σ) for each mass value, which were calculated from simulated events in Madgraph 5 [11] for proton-proton collisions at 8 TeV, are shown in the same table. We used Pythia 8 [12] for the simulation of the showering and hadronization processes, after restricting the Z-boson decays to the hadronic channel ( Z→jj ) The final state is characterized by the presence of a pair of energetic electrons and 4 jets ( j ) coming from a displaced vertex, . Finally, the interaction of the decay products with the CMS subdetectors was simulated using CMS software (CMSSW)[13] built-intools [14]. Further descriptions are made with LW electron mass equal to 200GeV, because of being the best candidate considering cross section and average flight distance.

Causally decaying vertices (Figure 1.A) only differ from WDV events (Figure 1.B) in thedirection of the displacement vector from the PV to the SV PV SV. A parallelity () variable was defined, for the first time, as the scalar product between PV SV and the total momentum of the decaying products, P_total. This quantity was used as the discriminator between causal decays, with mostly () > 0, and LW-decays, which tend to have values of < 0. Because of conservation laws, we cannot simulate proper WDV decays, so we simulated our particles as causal decaying resonances. For this reason, weselected the opposite of when the vertex comes from our signal.

To assess the feasibility of distinguishing events with displaced vertices, we lookfor candidate events with two jets and one energetic electron in the final state andstudy their distribution. The two experimental datasets used ([15] and [16]) arecharacterized by the presence of events with at least two energetic photon- or electron-like (egamma) particles, corresponding to total integrated luminosity of4.412 and 7.055 fb⁻¹, respectively. To consider background processes that could give similar signatures, we use the following simulated datasets: Drell-Yang (DY) [17], topquark pair production (TTbar) [18], and W boson production in association with jets(WJets) [19, 20, 21], corresponding to cross-sections of 3503.7, 225.2 and 1.75 × 10⁻¹ pb, respectively. The background, which originated from quantum chromodynamic (QCD) processes, was estimated from a region orthogonal to our signal. We fitted QCD to better match background and experimental data when the most energeticegamma particles have the same charge. The fitting was then extrapolated to theregion that contains our signal.

In a first stage for event selection, collisions were subject to pass a trigger that requiresthe presence of two highly isolated and mild shower-shaped egamma objectswith transverse momentum over 36 GeV for the leading particle, and 22 GeV for the subleading one. Additionally, we require the egamma particles to be electrons and the presence of 4 jets with transverse momentum greater that 15 GeV. This selection requirement was named Trigger Filtering (TF). Details of the selection can be found instudies with SV reconstruction from leptons [22].

After the initial filtering, an Object Selection (OS) requirement was implemented. There, the electrons’ minimum transverse momentum was tuned up to 40 GeV and 25 GeV. This guarantees a trigger efficiency of 99% [23].Tracks associated to electrons are also required to match the high quality selection described in [22]. Additionally, at least 4 selected jets in the event must be from hadronic origin and have two or more constituents and a transverse momentum greater than 20 GeV. Details for this selection can be found in a previous study with SV reconstruction from hadronic jets [24]. Additionally, we required an electromagnetic jet energy fraction less than 0.6 to filter out mix of jets and electrons, more common in our topology than in others.

We also apply a Signal (SG) filter. We require that events must contain selected electrons of opposite charge. It is also required that the electron-jet structure follows this geometrical property: that the difference between the angular distance from theelectron to the closest jet (∆R₁) and the distance from the electron to the second closestjet (∆R₂) be (ΔR₂) be |ΔR₁ -ΔR₂| <0.2. Here, ΔR= √ΔΦ² + Δη² is defined by the difference in the azimuthal angles (φ ) and the pseudorapidities (η) of the objects.

This spatial topology of electrons and jets was most commonly found in the acausal signal than in the background processes.

The SV were reconstructed using the combination of selected electrons and any pair of jets that happen to reconstruct and invariant mass between 80 and 160 GeV. The total amount of tracks resulting from this combination was then fed to an SV reconstruction algorithm (SVRA) described in [24]. The SVRA forces any possible pair of tracks to a vertex fitting algorithm. Only SV with an adequate fitting chi-squared normalized to the number of degrees of freedom (X²_norm < 5) are selected. If any pair of SV share one or more tracks, the combined set of tracks is refitted to a new SV′. This process is repeated until none of the selected SV share tracks or there are no left well-fitted vertices. Resulting SV′ are then selected to guaranty they are well fitted, composed of at least 5 tracks, and follow other geometrical considerations as described in reference [24].We also calculated the value for each SV that contains the selected electrons. The comparative distributions for values of are found in Figure 2. It is evident that, while SM model contributions (solid colors) show a rather symmetrical histogram, the signalhistogram (in green dotted line) leans towards negative values of .

For the final WDV selection criteria (WDVC), we require events to have at least one adequately-selected WDV: the WDV must be associated to one electron track; every SV associated to the same electron must have a separation transverse to the collision axis PV SV ^t> 0.02 mm; and the total transverse momentum P ^t_total > 20 GeV.

Systematic uncertainties were estimated from the object and event selection comparable to the literature and can be visualized in Figure 2 as hashed bands. The uncertainty in the trigger efficiency was extracted from reference [23]. Uncertainties in electron energy and tracking alignment were inferred from [22]. Jet energy resolution and vertex reconstruction are comparable to reference [24], yet methods are not exactly the same, so we conservatively consider uncertainties of 3% and 18% respectively. Underlying events and integrated luminosity uncertainties are comparable to those found in reference [23]. A summary of the values can be found in Table 2.

In conclusion, we have confirmed the feasibility of identifying WDV decays in events containing a pair of energetic electrons and 4 jets in CMS open data. Considering the upgrades that have taken place for the LHC experiments, including CMS, and the greater amounts of data that will be collected in upcoming years, we believe that the WDV method described in this study, together with a refined handling of systematic uncertainties, could be used to challenge microscopic causality and search for acausal signatures at the LHC.

ACKNOWLEDGMENT

We gratefully acknowledge Edgar Carrera PhD. for his unsurpassable participation, guidance, and aid during this whole investigation. In the same way, we thank Santiago Paredes PhD for his technical assistance. We thankfully recognize the support from the University San Francisco de Quito and PoliGrants - HUBI Project 17462, for the financial assistance in this masteral program.

CONFLICT OF INTERESTS

The authors declare that they have no conflicts of interest to declare.

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