Updated constraints on $Z'$ and $W'$ bosons decaying into bosonic and leptonic final states using the Run 2 ATLAS data

The full ATLAS Run 2 data set with time-integrated luminosity of 139 fb$^{-1}$ in the diboson and dilepton channels is used to probe benchmark models with extended gauge sectors: the $E_6$-motivated Grand Unification models, the left-right symmetric $LR$ and the sequential standard model (EGM). These all predict neutral $Z'$ vector bosons, decaying into lepton pairs, $\ell\ell$, or into electroweak gauge boson pairs $WW$, where one $W$ in turn decays semileptonically. 95% C.L. exclusion limits on the $Z'$ resonance production cross section times branching ratio to electroweak gauge boson pairs and to lepton pairs in the mass range of $\sim$ 1 - 6 TeV are converted to constraints on the $Z-Z'$ mixing parameter and the heavy resonance mass. We present exclusion regions on the parameter space of the $Z'$ which are significantly extended compared to those obtained from the previous analyses performed with LHC data collected at 7 and 8 TeV in Run 1 as well as at 13 TeV in Run 2 at time-integrated luminosity of 36.1 fb$^{-1}$ and are the most stringent bounds to date. Also presented, from a similar analysis of electrically charged $W'$ bosons arising in the EGM, which can decay through $W'\to WZ$ and $W'\to \ell\nu$, are limits on the $W$-$W'$ mixing parameter and the charged $W'$ vector boson mass.


I. INTRODUCTION
One of the main goals of the physics programme at the Large Hadron Collider (LHC) is to search for new resonant or non-resonant phenomena that become visible in high-energy proton-proton collisions. A prominent possible signature of such phenomena would be the production of a heavy resonance with its subsequent decay into a pair of leptons or into electroweak vector bosons. Many scenarios beyond the Standard Model (SM) predict such signals. Possible candidates are neutral and charged heavy gauge bosons which are commonly referred to as Z and W bosons, respectively [1]. Strong constraints have already been set on the production of such new heavy particles.
At the LHC, heavy Z and W bosons could be observed through their production as s-channel resonances with subsequent leptonic decays and respectively, where in what follows, = e, µ unless otherwise stated. The production of Z and W bosons at hadron colliders is expected to be dominated by the Drell-Yan (DY) mechanism, qq/qq → Z /W . The Feynman diagrams for the Z (W ) boson production at the parton level and their dilepton and diboson decays are illustrated in Fig. 1. Leptonic final states provide a low-background and efficient experimental signature that results in excellent sensitivity to new phenomena at the LHC. Specifically, these processes (1) and (2) offer the simplest event topology for the discovery of Z and W with a large production rate and a clean experimental signature. These channels offer the most promising discoveries at the LHC [2][3][4][5][6][7]. There have also been many theoretical studies of Z and W searches at the high-energy hadron colliders (see, e.g. [1,[8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]).
In the simplest models such as the Sequential Standard Model (SSM) [8] new neutral Z SSM and charged W SSM bosons have couplings to fermions that are identical to those of the SM Z and W bosons, but for which the trilinear couplings Z W W and W W Z are absent. The SSM has been used as a reference for experimental Z and W searches for decades, the results can be re-interpreted in the context of other models, it is therefore useful for comparing the sensitivity of different experiments. Another class of models considered here are those inspired by Grand Unified Theories (GUT), which are motivated by gauge unification or a restoration of the left-right symmetry violated by the weak interaction. Examples considered in this paper include the Z bosons of the E 6 -motivated [14] theories containing Z ψ , Z η , Z χ ; and high-mass neutral bosons of the left-right (LR) symmetric extensions of the SM, based on the SU (2) L SU (2) R U (1) B−L gauge group, where B − L refers to the difference between baryon and lepton numbers.
The data we consider were collected with the ATLAS and CMS detectors during the 2015-2018 running period of the LHC, referred to as Run 2 and corresponding to a time-integrated luminosity of 139-140 fb −1 . The ATLAS experiment has presented the first search for dilepton resonances based on the full Run 2 data set [2,7] and set limits on the Z and W production cross sections times branching fraction in the processes (1) and (2), σ(pp → Z X) × BR(Z → + − ) and σ(pp → W X) × BR(W → ν), respectively, for M Z and M W in the 0.25 TeV -6 TeV and 0.15 TeV -7 TeV ranges, correspondingly. Recently, similar searches have also been presented by the CMS Collaboration using 140 fb −1 of data recorded at √ s = 13 TeV [4]. The ATLAS and CMS collaborations set a 95% confidence level (CL) lower limit on the Z mass of ∼ 4.6 TeV-5. 2 TeV depending on the model [2,4], and 6.0 TeV for the W SSM [7].
Alternative Z and W search channels are the diboson reactions and pp → W X → W ZX.
The study of gauge boson pair production offers a powerful test of the spontaneously broken gauge symmetry of the SM and can be used as a probe for new phenomena beyond the SM. Specifically, in contrast to the DY processes (1) and (2) diboson reactions are not the primary discovery channels, but can help to understand the origin of new gauge bosons.
As mentioned above, heavy resonances that can decay to gauge boson pairs are predicted in many scenarios of new physics, including extended gauge models (EGM) [8,23], models of warped extra dimensions [24,25], technicolour models [26,27] associated with technirho and other technimesons, composite Higgs models [28,29], and the heavy vector-triplet (HVT) model [30], which generalises a large number of models that predict spin-1 neutral (Z ) and charged (W ) resonances. In the SSM, the coupling constants of the Z (W ) boson with SM fermions are the direct transcription of the corresponding SM couplings, while the Z (W ) coupling to W W (W Z) is strongly suppressed, g Z W W = 0 and g W W Z = 0. This suppression may arise naturally in an EGM: if the new gauge bosons and the SM ones belong to different gauge groups, a vertex such as Z W W (W W Z) is forbidden. They can only be induced after symmetry breaking due to mixing of the gauge eigenstates. Searches for exotic heavy particles that decay into W W or W Z pairs are complementary to searches in the leptonic channels + − and ν of the processes (1) and (2). Moreover, there are models in which new gauge boson couplings to SM fermions are suppressed, giving rise to a fermiophobic Z and W with an enhanced coupling to electroweak gauge bosons [1,31]. It is therefore important to search for Z and W bosons also in the W W and W Z final states.
The properties of possible Z and W bosons are also constrained by measurements of electroweak (EW) processes at low energies, i.e., at energies much below their masses. Such bounds on the Z-Z (W -W ) mixing are mostly due to the constraints on deviation in Z (W ) properties from the SM predictions. In particular, limits from direct hadron production with subsequent diboson decay at the Tevatron [32] and from virtual effects at LEP, through interference or mixing with the Z boson, imply that any new Z boson is rather heavy and mixes very little with the Z boson. At LEP and the SLC, the mixing angle is strongly constrained by very high-precision Z pole experiments [33]. These include measurements of the Z line shape, the leptonic branching ratios as well as leptonic forward-backward asymmetries. The measurements show that the mixing angles, referred to as ξ Z-Z and ξ W -W , between the gauge eigenstates must be smaller than about 10 −3 and 10 −2 , respectively [1,15].
Previous analyses of the Z-Z and W -W mixing [34][35][36] were carried out using the diboson and dilepton production data sets corresponding to the time-integrated luminosity of ∼ 36 fb −1 collected in 2015 and 2016 with the ATLAS and CMS collaborations at √ s = 13 TeV where, in the former case, electroweak Z and W gauge bosons decay into the semileptonic channel [37] or into the dijet final state [38]. The results of the present analysis benefit from the increased size of the data sample, now amounting to an integrated luminosity of 139 fb −1 recorded by the ATLAS detector in Run 2 [39][40][41][42], almost four times larger than what was available for the previous study. 1 In addition, further improvement in placing limits on the Z and W mass and Z-Z and W -W mixing parameters can be achieved in semileptonic W W/W Z final states in which one vector boson decays leptonically (Z → , νν, W → ν) while the other decays hadronically (Z/W → qq). 2 Also, here we extend our analysis presented in [43] where we utilized the full Run 2 ATLAS data set for EGM (SSM) to various Z models, including E 6 -based Z χ , Z ψ , Z η , and also Z LR boson appearing in models with left-right symmetry. Thus, our present analysis is complementary to the previous studies [43].
We present results as constraints on the relevant Z-Z (W -W ) mixing angle, ξ Z-Z (ξ W -W ), and on the mass M Z (M W ) and display the combined allowed parameter space for the benchmark Z (W ) models, showing also indirect constraints from electroweak precision data. Previous direct search constraints from the Tevatron and from the LHC with 7 and 8 TeV in Run 1 (where available) are compared to those obtained from the LHC at 13 TeV with the full ATLAS Run 2 data set of time-integrated luminosity of 139 fb −1 in the semileptonic [41,42] and fully hadronic (qqqq) [39] final states.
The paper is structured as follows. In Sect. II we present the theoretical framework, then, in Sect. III we summarize the relevant cross sections for the diboson and dilepton production processes (3) and (1) in the narrow-width approximation (NWA). Next, we discuss the relevant Z widths and branching ratios within the considered benchmark models. Further, we present an analysis of bounds on Z-Z mixing from constraints on diboson and dilepton production in the context of the benchmark models with extended gauge sector, employing the most recent searches recorded by the ATLAS (139 fb −1 ) detector in the semileptonic [41,42] and fully hadronic (referred to as qqqq) [39,40] final states at the LHC. Then, we show the resulting constraints on the M Z − ξ Z-Z parameter space obtained from these processes. Further, we collect and compare the indirect constraints obtained from electroweak precision data, direct search constraints derived from the LHC in Run 1 and early Run 2 data. In Sect. IV we present the corresponding analysis of bounds on W -W mixing, performed in a similar fashion as for the Z , from constraints on diboson and dilepton production processes (4) and (2) in the context of the EGM. Sect. V presents some concluding remarks.

II. MIXING AND PARAMETERS
We consider Z-Z mixing within the framework of models with extended gauge sector such as the E 6 models, the LR model and the EGM (see, e.g. [8,[14][15][16][17]). The mass eigenstates Z and Z are admixtures of the weak eigenstates Z 0 of SU (2) × U (1) and Z 0 of the extra U (1) , respectively: For each type of Z boson, defined by its gauge couplings, there are three classes of models, which differ in the assumptions concerning the quantum numbers of the Higgs fields which generate the Z-boson mass matrix [14,15]. In each case there is a relation between the Z 0 -Z 0 mixing angle φ and the masses M Z and M Z [14]: where the downward shift ∆M = M Z 0 − M Z > 0, and M Z 0 is the mass of the Z boson in the absence of mixing, i.e., for φ = 0, given by The mixing angle φ will play an important role in our analysis. Such mixing effects reflect the underlying gauge symmetry and/or the Higgs sector of the model as the ρ 0 parameter depends on the ratios of Higgs vacuum expectation values and on the total and third components of weak isospin of the Higgs fields. We set ρ 0 = 1 here, this corresponds to a Higgs sector with only SU (2) doublets and singlets [14]. Once we assume the mass M Z to be determined experimentally, the mixing depends on two free parameters, which we identify as φ and M Z , a parametrization that we will adopt throughout the paper. This Z 0 -Z 0 mixing induces a change in the couplings of the two bosons to fermions. From Eq. (5), one obtains the vector and axial-vector couplings of the Z and Z bosons to fermions: with unprimed and primed couplings referring to Z 0 and Z 0 , respectively, and found, e.g. in [17]. An important property of the models under consideration is that the gauge eigenstate Z 0 does not couple to the W + W − pair since it is neutral under SU (2). Therefore the W -pair production is sensitive to a Z only in the case of a non-zero Z 0 -Z 0 mixing. From Eq. (5), one obtains: where g W W Z 0 = e cot θ W . Also, g W W γ = e. In many extended models, while the couplings to fermions do not differ much from those of the SM, the Z W W coupling is substantially suppressed with respect to that of the SM. In fact, in the extended gauge models the SM trilinear gauge boson coupling strength, ) is the mixing factor 3 . We will set cross section limits on such Z as functions of the mass M Z and ξ.
In addition, we study W -W mixing in the process (4) within the framework of the EGM model [8,23]. At the tree-level mass mixing may be induced between the electrically charged gauge bosons. The physical (mass) eigenstates of W and W are admixtures of the weak eigenstates denoted asŴ andŴ , respectively, and obtained by a rotation of those fields [1,13]: in analogy with Eq. (5). Upon diagonalization of their mass matrix, the couplings of the observed W boson are shifted from the SM values. The properties of possible Z and W bosons, apart from collider experiments, are also constrained by measurements of electroweak (EW) processes at low energies, i.e., at energies much below their masses. Such bounds on the Z-Z (W -W ) mixing are mostly due to the constraints on the deviation in Z (W ) properties compared to the SM predictions. These measurements show that the mixing angles ξ Z-Z and ξ W -W (≡ | sin θ|) between the gauge eigenstates must be smaller than about ∼ 10 −3 and 10 −2 [1], respectively.

III. Z PRODUCTION AND DECAY IN pp COLLISION
We shall first consider Z production in some detail, and subsequently turn to the W case. In some sense the Z sector is richer than the W sector, different models predict different ratios of the vector and axial-vector couplings. The W models on the other hand, will all be restricted in the choice of pure left-handed couplings to fermions. Among the Z models, we start out with a detailed discussion of the ψ model.

A. Z resonant production cross section
The Z production and subsequent decay into W W in proton-proton collisions occurs via quark-antiquark annihilation in the s-channel. The cross section of the process (3) can at the LHC be observed through resonant pair production of gauge bosons W W . Using the narrow width approximation (NWA), one can factorize the process (3) into the Z production and its subsequent decay, Here, σ(pp → Z X) is the total (theoretical) Z production cross section and BR(Z → W W ) = Γ W W Z /Γ Z with Γ Z the total width of the Z . "Narrow" refers to the assumption that the natural width of the resonance is smaller than the typical experimental resolution of 5% of its mass [44,45]. This is valid for a large fraction of the parameter space of the considered models.
We can schematically present the cross section for the process pp → W + W − X, within the gauge models discussed here, to next to leading order (NLO) in α s as where the K factor accounts for higher-order QCD contributions. Integrating over phase space, the interference term drops out and we are left with It is clearly a crude approximation to take the K-factor to be the same for the signal (Z ) as for the SM background, but that is the approach taken in the literature [46][47][48].

B. The Z width
In the calculation of the total width Γ Z we consider the following channels: Z → ff , W + W − , and ZH [34,35,43,49], where H is the SM Higgs boson and f refers to the SM fermions (f = l, ν, q). Throughout the paper we shall ignore the couplings of the Z to any beyond-SM particles such as right-handed neutrinos, which we take to be heavier than M Z /2, as well as to SUSY partners and any other exotic fermions. Such additional states may all together increase the width of the Z by up to about a factor of five [50] and hence lower the branching ratio into a W + W − pair by the same factor.
The total width Γ Z of the Z boson can then be written as follows: The two last terms, which are often neglected in studies at low and moderate values of M Z , are due to Z-Z mixing. For the range of M Z values below ∼ 3 − 4 TeV, the dependence of Γ Z on the values of ξ Z-Z (within its allowed range) is unimportant. Therefore, in this mass range, one can approximate the total width as Γ Z ≈ f Γ f f Z , where the sum runs over SM fermions only. The ratios of Γ f f Z /M Z for the benchmark models are summarized in Table I. One can appreciate the narrowness of the Z Z pole from this Table I.   TABLE I. Ratio Γ f f Z /M Z for the χ, ψ, η, LR and EGM models.
However, for larger Z masses, M Z > 4 TeV, there is an enhancement in the coupling that cancels the suppression due to the tiny Z-Z mixing parameter ξ Z-Z [49]. We note that the "Equivalence theorem" [51] suggests a value for BR(Z → ZH) comparable to BR(Z → W + W − ), up to electroweak symmetry breaking effects and phase-space factors. Throughout this paper, for definiteness, we adopt a scenario where both partial widths are comparable, For all M Z values of interest for the LHC the width of the Z boson is considerably smaller than the experimental mass resolution ∆M for which we adopt the parametrization in reconstructing the diboson invariant mass of the W + W − system, ∆M/M ≈ 5%, as proposed, e.g., in [44,45]. 4 The partial width of the Z → W + W − decay channel can be written as [8]: For a fixed mixing factor ξ Z-Z and at large M Z where Γ W W Z dominates over f Γ f f Z the total width increases rapidly with the mass M Z because of the quintic dependence of the W + W − mode on the Z mass as shown in Eq. (15). In this case, the W + W − mode (together with Z → ZH) becomes dominant and BR(Z → W + W − ) → 0.5 (this value arises from the assumption Γ ZH Z = Γ W W Z ), while the fermionic decay channels (Γ f f Z ∝ M Z ) are increasingly suppressed. These features are illustrated in Fig. 2, where we plot BR(Z → W + W − ) and BR(Z → e + e − ) vs M Z for the Z ψ model. C. Hadron production and diboson decay of Z In Fig. 3, we consider the full ATLAS Run 2 data set of time integrated luminosity of 139 fb −1 and show the observed 95% C.L. upper limits on the production cross section times the branching fraction, σ 95% × BR(Z → W + W − ), as a function of the Z mass, obtained from the semileptonic [41,42] and fully hadronic (qqqq) [39,40] final states. This allows for a comparison of the sensitivities of the data to mixing parameters and new gauge boson mass. This comparison demonstrates the dominating sensitivity to Z of the semileptonic channel with respect to the fully hadronic one, over almost the whole Z mass range.
Then, for Z ψ we compute the LHC production cross section multiplied by the branching ratio into two W bosons,   [41,42] and fully hadronic (thick solid) [39] final states for 139 fb −1 . Theoretical production cross sections σ(pp → Z ψ + X) × BR(Z ψ → W + W − ) are shown for mixing factors ξ Z-Z ranging from 3 · 10 −3 down to 3 · 10 −4 . Also, the cross section solid line labeled ξ EW ψ corresponds to the mixing parameter ξ EW Z indicated in Table 2 for the Z ψ model. The area lying below the long-dashed curve labelled NWA corresponds to the region where the Z resonance width is predicted to be less than 5% of the resonance mass, in which the narrow-width assumption is satisfied. The lower boundary of the region excluded by the unitarity constraints is also indicated. evaluated by the experimental collaborations and accounted for in σ 95% × BR(Z → W + W − ). Therefore, in our analysis we simulate only the Z ψ signal.

M (TeV
In Fig. 3, the theoretical production cross section σ(pp → Z ψ ) × BR(Z ψ → W + W − ) for Z ψ boson is calculated from a dedicated modification of PYHTHIA 8.2 [55]. As mentioned above, higher-order QCD corrections to the signal were estimated using a K-factor, for which we adopt a mass-independent value of 1.9 [46,56,57]. These theoretical curves for the cross sections, in descending order, correspond to values of the Z-Z mixing factor ξ Z-Z ranging from 3 · 10 −3 and down to 3 · 10 −4 . The intersection points of the measured upper limits on the production cross section with this theoretical cross section for various values of ξ Z-Z give the corresponding lower bounds on (M Z , ξ Z-Z ), presented in Fig. 4.   FIG. 4. The Z ψ model: 95%C.L. exclusion regions in the two-dimensional (M Z , ξ Z-Z ) plane obtained after incorporating indirect constraints from electroweak precision data (horizontal dashed straight line labeled "EW"), and direct search constraints from the LHC search for pp → Z → W W in semileptonic final states using the full Run 2 ATLAS data set. Limits obtained from the hadronic channel qqqq are overlaid for comparison. The region above each curve for the W W channel is excluded. The steep curve labelled "excluded by Z ψ → " shows the exclusion based on the dilepton channel pp → Z ψ → + X. The unitarity limit is shown as the dot-dashed curve. The overall allowed region is shown as a yellow area.
Different bounds on the Z parameter space are collected in Fig. 4 for the Z ψ model, showing that at high masses, the limits on ξ Z-Z obtained from the full Run 2 data set collected at √ s = 13 TeV and recorded by the ATLAS detector are substantially stronger than that derived from the global analysis of the precision electroweak data [15], which is also displayed. Limits obtained separately from the individual semileptonic channel νqq and the fully hadronic cannel qqqq are shown for comparison. It turns out that the semileptonic channel dominates the sensitivity over almost the whole resonance mass range 0.5 TeV ≤ M Z ≤ 5 TeV, while in the rather narrow mass range 2.2 TeV ≤ M Z ≤ 2.5 TeV the all-hadronic channel is most sensitive.
D. Z-Z mixing effects in dilepton decay of Z → The above analysis was for the diboson process (3), employing one of the most recent ATLAS searches for semileptonic [41,42] and fully hadronic [39] final states. Next, we turn to the dilepton production process (1), this process gives valuable complementary information.
We compute the Z theoretical production cross section at the LHC, σ(pp → Z X), multiplied by the branching ratio into two leptons, ( = e, µ), i.e., σ(pp → Z X) × BR(Z → ), as a function of M Z , and compare it with the upper limits established by the experiment [2] for 139 fb −1 . We make use of the relevant set of tables and figures (including additional results for dielectron and dimuon channels) available at the Durham HepData repository [3].
Results for σ 95% × BR(Z → ) are shown in Fig. 5. To account for next-to-next-to-leading order (NNLO) effects in the QCD strong coupling constant, the leading order (LO) cross sections calculated with PYHTHIA 8.2 [55] are multiplied by a mass-independent K-factor. The value of the K-factor is estimated at a dilepton invariant mass of ∼ 3.0 − 4.5 TeV and found to be consistent with unity [58,59].
For illustrative purposes we show theoretical production cross sections σ(pp → Z X) × BR(Z → ) for the ψ model Z , given by the dashed curves in Fig. 5. These curves, in descending order, correspond to values of the mixing factor ξ Z-Z from 0 to 5 · 10 −3 . Qualitatively, the decrease of the theoretical cross section with increasing values of ξ Z-Z can be understood as follows: For increasing ξ Z-Z , the Z → W + W − mode will at high mass M Z become more dominant (as illustrated in Fig. 2), and BR(Z → ) will decrease correspondingly. Notice also, that applying a mass-dependent K-factor (which for this process is less than 1.04), the ψ model mass limit of the Z changes by only ∼ O(50 GeV), justifying the use of the simpler mass-independent K-factor [58,59]. Comparison of σ(pp → Z X) × BR(Z → ) vs σ 95% × BR(Z → ) displayed in Fig. 5 permits us to read off an allowed mixing for a given mass value, higher masses are allowed for smaller mixing, for the reason stated above. This analysis of Z-Z mixing, illustrated here for the ψ model, can also be performed for the other benchmark models. The results of the numerical analysis for these tested models are presented in Figs. 6-9. Mass limits are calculated as the intersection between the observed limits with the model prediction. Table II lists the mass limits for two representative cases, namely for vanishing mixing (ξ Z-Z = 0) and for the mixing ξ EW Z-Z derived from the electroweak precision data [15]. The former are consistent with those derived in Refs. [2,4] whereas the mass limits at ξ EW Z-Z are weaker by ∼ 10 − 30%.
As described above, both the diboson mode and the dilepton process yield limits on the (M Z , ξ Z-Z ) parameter space. These are rather complementary, as shown in Fig. 4, where we collect these limits for the ψ model. The limits arising from the diboson channel are basically excluding large values of ξ Z-Z , strongest at intermediate masses ... ... ... ... 2 · 10 −2 2 · 10 −2 0.4-0.9 [32] electroweak (EW) data [1,15] 1.8 · 10 −3 4.7 · 10 −3 1.6 · 10 −3 1.3 · 10 −3 2.6 · 10 −3 ∼ 10 −2 ... For reference, we plot also a curve labelled "Unitarity limit" that corresponds to the unitarity bound [35,60]. In [60], it was shown that the saturation of unitarity in the elastic scattering W + W − → W + W − leads to the constraint (g Z W W ) max = g ZW W · (M Z / √ 3M Z ) that was adopted here. In Table III, we collect our limits on the Z parameters for the benchmark models. Also shown in Table III are the current limits on the Z-Z mixing parameters, ξ Z-Z , from the EW precision data. They are generally weaker than those from the LHC limits obtained in Run 2 at 13 TeV with time-integrated luminosity of L int = 139 fb −1 . In fact, the latter improve the EW limits by approximately one order of magnitude, depending on the resonance mass.

IV. W PRODUCTION AND DECAY IN pp COLLISION
In contrast to the rich spectrum of Z models considered above, with different vector and axial-vector couplings, for W we consider only V − A couplings to fermions.

A. W resonant production cross section
We consider the simplest EGM model which predicts charged heavy gauge bosons. The analysis of W -W mixing in diboson and dilepton pair production which will be performed below is quite analogous to that carried out in previous sections for Z-Z mixing. At lowest order in the EGM, W production and decay into W Z in proton-proton collisions occurs through quark-antiquark annihilation in the s-channel. Using the NWA, one can factorize the process (4) into the W production and the W decay, Here, σ(pp → W X) is the total (theoretical) W production cross section and BR(W → W Z) = Γ W Z W /Γ W with Γ W the total width of the W .

B. The W width
In the EGM the W bosons can decay into SM fermions, gauge bosons (W Z), or W H. In the calculation of the total width Γ W we consider the following channels: W → ff , W Z, and W H, where f is a SM fermion (f = , ν, q) 5 . Only left-handed neutrinos are considered, possible right-handed neutrinos are assumed to be kinematically unavailable as final states. Also, like for the Z case, we shall ignore the couplings to other beyond-SM particles such as SUSY partners and exotic fermions. As a result, the total decay width of the W boson is taken to be Like for the Z case, the presence of the last two decay channels, which are often neglected at low and moderate values of M W , is due to W -W mixing which is constrained to be tiny. In particular, for the range of M W values below ∼ 1.0 − 1.5 TeV, the dependence of Γ W on the values of ξ W -W (within its allowed range) induced by Γ W Z W and Γ W H W is unimportant because f Γ ff W dominates over the diboson partial widths. Therefore, in this mass range, one can approximate the total width as Γ W ≈ f Γ ff W = 3.5% × M W [36], where the sum runs over SM fermions only. For heavier W bosons, the diboson decay channels, W Z and W H, start to play an important role, and we are no longer able to ignore them [36,43]. To be specific, in analogy with the Z case, we assume that both partial widths are comparable, Γ W H W Γ W Z W for heavy M W , as required by the Equivalence theorem [51].
The expression for the partial width of the W → W Z decay channel in the EGM can be written as [8,36]: For a fixed mixing factor ξ W -W and at large M W , the total width increases rapidly with the W mass because of the quintic dependence of the W Z mode on the W mass Γ W Z , corresponding to the production of longitudinally polarized W and Z in the channel W → W L Z L [8,36]. In this case, the W Z mode (as well as W H) becomes dominant and BR(W → W Z) → 0.5, while the fermionic decay channels, f Γ ff W ∝ M W , are increasingly suppressed, as illustrated in Fig. 10 (left panel). Bottom-left panel: Also shown are exclusion regions obtained after incorporating direct search constraints from the CDF and D0 collaborations which are referred to as Tevatron (the dark shaded area) in pp → W + W − X as well as those derived from the LHC measurement of pp → W W X in Run 1 (the gray area) [43].

C. Hadron production and diboson decay of W
Our analysis employs the recent searches for diboson processes in semileptonic final states provided by ATLAS [41] with the full Run 2 data set with time-integrated luminosity of 139 fb −1 as well as, for the sake of comparison, in the fully hadronic (qqqq) final states [39].
In Fig. 10 (right panel), we show the observed 95% C.L. upper limits on the production cross section times the branching fraction, σ 95% × BR(W → W Z), as a function of the W mass.
Then, for W we compute the LHC theoretical production cross section multiplied by the branching ratio into W Z bosons, σ(pp → W X)×BR(W → W Z), as a function of the two parameters (M W , ξ W -W ) [36], and compare it with the limits established by the ATLAS experiment, σ 95% × BR(W → W Z). The simulation of signals for the EGM W is based on a suitably adapted version of the leading order PYHTHIA 8.2 event generator [55]. A mass-dependent K factor is adopted to rescale the LO PYTHIA prediction to the the NNLO one, using the ZWPROD [61] software. The result is presented as solid curves in the right panel for a mixing factor ξ W -W ranging from 10 −2 and down to 3 · 10 −4 . The factorization and renormalization scales are both set to the W mass.
The area below the long-dashed curve labelled "NWA" corresponds to the region where the W resonance width is predicted to be less than 5% of its mass, corresponding to the best detector resolution of the searches, where the narrow-width assumption is satisfied. We also show a curve labelled "Unitarity limit" that corresponds to the unitarity bound (see, e.g. [60] and references therein). It was shown that the saturation of unitarity in the elastic scattering W ± Z → W ± Z leads to the constraint (g W W Z ) max = g W W Z · M 2 Z /( √ 3 M W M W ) that was adopted in plotting this bound. The constraint was obtained under the assumption that the couplings of the W to quarks and to gauge bosons have the same Lorentz structure as those of the SM but with rescaled strength.
The   [39,41]. The theoretical production cross sections σ(pp → W X) × BR(W → W Z) for the EGM are calculated from PYTHIA with a W mass-dependent K-factor, given by solid curves, for mixing factor ξ W -W ranging from 10 −2 and down to 3 · 10 −4 . The NWA and unitarity constraints are also shown [36,60].
the production cross section with these theoretical cross sections for various values of ξ W -W give the corresponding lower bounds on (M W , ξ W -W ), displayed in Fig. 11, left panel.
Comparison of sensitivities of the process (4) to W with different decay channels, e.g., V V → νqq and qqqq, can be performed by the matching of 95% C.L. upper limits on the production cross section times the branching fraction, σ 95% × BR(W → W Z), which includes the SM branching fractions of the electroweak bosons to the final states in the analysis channel, effects from detector acceptance, as well as reconstruction and selection efficiencies. ATLAS bounds were included acccording to the Durham HEPdata repository [40,42].
From a comparison of the upper limits on the production cross section times the branching fraction for semileptonic vs. fully hadronic decay channels, one can conclude that the sensitivity of the semileptonic channel dominates over the fully hadronic one within the whole range of the W mass, from 0.5 TeV to 5 TeV. These features are illustrated in Fig. 10 (right panel) and Fig. 11 (left panel).
For reference, we display limits on the W parameters from the Tevatron (CDF and D0) as well as from ATLAS and CMS obtained at 7 and 8 TeV of LHC data taking in Run 1 denoted "LHC Run 1" [36]. Fig. 11 (left panel) shows that the experiments CDF and D0 at the Tevatron exclude EGM W bosons with ξ W -W > ∼ 2 · 10 −2 in the resonance mass range 0.25 TeV < M W < 1 TeV at the 95% C.L., whereas LHC in Run 1 improved those constraints, excluding W boson parameters at ξ W -W > ∼ 2 · 10 −3 in the mass range 0.2 TeV < M W < 2 TeV.
As expected, the increase of the time-integrated luminosity up to 139 fb −1 leads to dominant sensitivity of the semileptonic channel over the whole resonance mass range of 0.5 TeV < M W < 5 TeV and it allows to set stronger constraints on the mixing angle ξ W -W , excluding ξ W -W > 2.3 · 10 −4 as shown in Fig. 11. Our results extend the sensitivity beyond the corresponding CDF Tevatron results [32] as well as the ATLAS and CMS sensitivity attained at 7 and 8 TeV. Also, for the first time, we set W limits as functions of the mass M W and mixing factor ξ W -W from the study of the diboson production and subsequent decay into semileptonic final states at the LHC at 13 TeV with the full ATLAS Run 2 data set. The exclusion region obtained in this way on the parameter space of the W naturally supersedes the corresponding exclusion area obtained for time-integrated luminosity of 36.1 fb −1 in the semileptonic channel as reported in [36]. The limits on the W parameters presented in this section obtained from the diboson W Z production in semileptonic final states, corresponding to a time-integrated luminosity of 139 fb −1 , are the best to date.
D. W -W mixing effects in dilepton decay of W → ν The above analysis was for the diboson process (4), employing one of the most recent ATLAS searches [39,41]. Next, we turn to the dilepton production process (2), this process gives valuable complementary information. Unlike (dark shaded area) as well as from the LHC searches for pp → W ZX at 7 TeV and 8 TeV (Run 1) (gray area) and at 13 TeV from diboson W → W Z production in hadronic and semileptonic final states using the full Run 2 ATLAS data set. The region above each curve for the W Z channel is excluded. The steep curve labelled "excluded by W → ν" shows the exclusion based on the dilepton channel pp → νX. The overall allowed region for the EGM W boson is shown as the yellow area. Right panel: Solid: observed 95% C.L. upper bound on the W production cross section times branching ratio to two leptons, σ 95% × BR(W → ν), obtained in the combined channels (electron and muon) at the LHC with integrated luminosity Lint=139 fb −1 by the ATLAS collaboration [7]. Dashed lines: theoretical production cross section σ(pp → W ) × BR(W → ν) for the EGM W boson, calculated from PYHTHIA 8.2 with an NNLO K factor. These curves, in descending order, correspond to values of the W -W mixing factor ξ from 0 to 0.01. the SSM, where there is no W -W mixing, in the EGM we consider a non-zero mixing ξ W -W in the analysis of the W → ν process. As described in Sec. IV B, this results in a modification of BR(W → eν).
We compute the W production cross section at LO with PYTHIA 8.2 [55], σ(pp → W ), multiplied by the branching ratio into two leptons, ν (here = e), i.e., σ(pp → W ) × BR(W → eν), as a function of M W . A mass-dependent K factor is applied, based on NNLO QCD cross sections as calculated with FEWZ 3.1 [62,63]. The K factor varies approximately from 1.3 to 1.1 for the range of W masses studied in this analysis, namely from 0.5 to 6.0 TeV. The NNLO corrections decrease with increasing W mass up to around 4.5 TeV [64]. For higher W masses, the K factor increases again and becomes similar to the low-mass values.
The product of the NNLO W theoretical production cross section and branching fraction, σ(pp → W ) × BR(W → eν), for the W boson for EGM strongly depends on the W mass, and is given by dashed curves, in descending order, corresponding to values of the mixing factor ξ from 0.0 to 0.01, as displayed in Fig. 11 (right panel).
Comparison of σ(pp → W ) × BR(W → eν) vs σ 95% × BR(W → ν) displayed in Fig. 11 (right panel) allows us to read off an allowed mixing for a given mass value, higher masses are allowed for smaller mixing, for the reason stated above. That comparison can be translated into constraints on the two-dimensional M W -ξ W -W parameter plane, as shown in Fig. 11 (left panel).
The above results are based on data corresponding to an integrated luminosity of 139 fb −1 taken by the ATLAS collaboration at √ s = 13 TeV in Run 2 [7]. The corresponding lower limits on the W boson mass of 6 TeV (at ξ W -W = 0) was set at 95% C.L. from combination of the electron and muon channels. Notice that, similar to the case of Z bosons, at ξ EW W -W = 10 −2 these limits become weaker, reaching ∼ 4.4 TeV, as illustrated in Fig. 11 (left and right panels).

V. CONCLUDING REMARKS
Examination of the diboson, W W and W Z, and dilepton, and ν, production at the LHC with the 13 TeV data set allows to place stringent constraints on the Z-Z and W -W mixing parameters as well as on the Z and W masses for benchmark extended models, respectively. We derived such limits by using the full ATLAS Run 2 data set recorded at the CERN LHC, with integrated luminosity of 139 fb −1 . The constraints are summarized in Table III. We note that if we had adopted the same K-factor for the W W (W Z) channel as for the dilepton channel, the bounds on ξ would have been slightly weaker, by a factor K 1/4 1.17 for the Z case [35].
By comparing the experimental limits to the theoretical predictions for the total cross section of the Z and W resonant production and their subsequent decays into W W or W Z pairs, we show that the derived constraints on the mixing parameters, ξ Z-Z and ξ W -W , are substantially improved with respect to those obtained from the global analysis of low energy electroweak data, as well as compared to the diboson production study performed at the Tevatron, and to those published previously and based on the LHC Run 1 as well as at 13 TeV in Run 2 at timeintegrated luminosity of ∼ 36 fb −1 and are the most stringent bounds to date. Further constraining of this mixing can be achieved from the analysis of data to be collected in Run 3 as well as at the next options of hadron colliders such as HL-LHC and HE-LHC [65,66].