Unravelling the anomalous gauge boson couplings in $ZW^\pm$ production at the LHC and the role of spin-$1$ polarizations

We study the anomalous couplings in $ZW^\pm$ production in $3l$+missing $E_T$ channel at the LHC for $\sqrt{s}=13$ TeV. We use cross section, azimuthal asymmetry, forward-backward asymmetry and polarization asymmetries of $Z$ and reconstructed $W$ to estimate simultaneous limits on the anomalous triple gauge boson couplings (aTGC) for both effective vertex formalism as well as effective operator approach using Markov-Chai--Monte-Carlo (MCMC) method for luminosities $35.9$ fb$^{-1}$, $100$ fb$^{-1}$, $300$ fb$^{-1}$ and $1000$ fb$^{-1}$. The trilepton invariant mass ($m_{3l}$) and the transverse momentum of $Z$ ($p_T(Z)$) are found to be sensitive to the aTGC for the cross section as well as for the asymmetries. We observed that the asymmetries significantly improve the measurement of anomalous couplings if a deviation from the Standard Model (SM) is observed.


Introduction
After the discovery of Higgs [1,2], the Large Hadron Collider (LHC) has been looking for new physics beyond the SM (BSM) needed to address many open questions such as neutrino oscillation, dark matter, baryogenesis, etc. with higher energies and higher luminosities. Unfortunately no new physics has been found [3] except few fluctuations (e.g., Refs. [4][5][6]). One could expect that the new physics scale is too heavy to be directly explored by the LHC and they may leave some footprints in the available energy range. They will modify the structure of the SM vertices or bring some new vertices, often through higher dimensional operator with the SM fields. These new vertices and/or the extra contribution to the SM vertices are termed as anomalous in the sense that they are not present in the SM at leading order. The electro-weak sector will get affected by the anomalous bosonic self couplings which alter the paradigm of electro-weak symmetry breaking (EWSB). To understand the EWSB mechanism, one needs precise measurements of the couplings in the bosonic sector of the SM. Here, we choose to focus on the charge sector by probing the WW Z anomalous couplings in the ZW ± production at the LHC. The WW Z anomalous triple gauge boson couplings (aTGC) may be obtained by higher dimension effective operators made out of SM fields suppressed by a new physics scale Λ. The effective Lagrangian including the higher dimension effective operators (O) to the SM Lagrangian (L SM ) is treated to be . The effective operators up to dimension-6 contributing to WW Z/γ couplings in general are [7,8] where W ± µν = ∂ µ W ± ν − ∂ ν W ± µ , Z µν = ∂ µ Z ν − ∂ ν Z µ , Z µν = 1/2ε µνρσ Z ρσ , and the overall coupling constants is given as g WW Z = −g cos θ W , θ W being the weak mixing angle. The couplings ∆g Z 1 , ∆κ Z and λ Z of Eq. i , respectively. These anomalous gauge boson self couplings may be obtained from some high scale new physics such as MSSM [11][12][13], extra dimension [14,15], Georgi-Machacek model [16], etc. by integrating out the heavy degrees of freedom. Some of these couplings can also be obtained at loop level within the SM [17,18]. There has been a lot of studies to probe the anomalous WW Z/γ couplings in the effective operators method as well as in the effective vertex factor approach subjected to SU(2) × U(1) invariance for various colliders: for e + -e − linear collider [9,[19][20][21][22][23][24][25][26][27][28][29][30], for Large Hadron electron collider (LHeC) [31][32][33], e-γ collider [34] and for LHC [26,27,[35][36][37][38][39][40][41][42][43][44][45][46]. Some CP-odd WWV couplings has been studied in Refs. [29,44]. Direct measurement of these charged aTGC has been performed at the LEP [47][48][49][50], Tevatron [51,52], LHC [53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68] and Tevatron-LHC [69]. The most stringent constraints on the operators (c O i ) are obtained in Ref.
[66] for CPeven ones and in Ref. [55] for CP-odd ones and they are listed in Table 1. These limit translated to the effective vertices (c L g i ) are also given in Table 1.  [49] In this article we intend to study the WW Z anomalous couplings in ZW ± production at the LHC at √ s = 13 TeV using the cross section, forward backward asymmetry and polarizations asymmetries [47,[70][71][72][73][74][75] of Z and W ± in the 3l + E T channel. The polarization of Z and W have been used recently for various BSM studies [76][77][78][79][80][81][82] along with studies with anomalous gauge boson couplings [47,74,83,84]. Recently the polarizations of W ± /Z has been estimated in ZW ± production [43] at NLO in QCD and also been measured at the LHC [85].
We will begin in Sect. 2 by providing the estimates of the cross sections for CMS fiducial phase-space by MATRIX [86], MADGRAPH5_aMC@NLO [87] and investigate the sensitivities of them to the anomalous couplings. Section 3 is devoted to polarization asymmetries of Z and W and the reconstruction of longitudinal momenta of neutrino. In Sect. 4, we do simultaneous analysis using MCMC to obtain limits on the anomalous couplings along with a toy measurement of non-zero aTGC and conclude in Sect. 5. Figure 1: Sample of Born level Feynman diagram for ZW + production in the e + e − µ + ν µ channel at the LHC. The diagram for ZW − can be obtained by charge conjugation. The shaded blob represents the presence of anomalous WWV couplings on top of SM.

Signal cross sections and their sensitivity to anomalous couplings
The process of interest is the ZW ± production in the 3l + E T channel at the LHC. The representative Feynman diagram at Born level are displayed in Fig. 1 containing doubly-resonant processes (upper-row) as well as singly-resonant processes (lower-row). The presence of anomalous WW Z couplings is shown by the shaded blob. While, this may contains the WW γ couplings due to the off-shell γ, this has been cut out by Z selection cut, described later. The leading order result (148.4 fb estimated by MATRIX in Ref. [88]) for the 3l + E T cross section at the LHC is way below the measured cross section at the LHC (258 fb measured by CMS [89]  The differential distribution of m 3l (top-row) and p T (Z) (bottom-row) in the ZW + (leftcolumn) and ZW − (right-column) production in the e + e − µ ± + E T channel at the LHC for √ s = 13 TeV at LO, NLO and NNLO obtained using MATRIX [86,88,[90][91][92][93][94][95] for CMS fiducial phase-space.
connected to the quarks (see, Fig. 1) with either QCD loops or QCD radiations from the quarks. The cross sections of ZW ± production in the SM in the e + e − µ ± channel obtained by MATRIX and MADGRAPH5_aMC@NLO v2.6.4 (mg5_aMC) for √ s = 13 TeV for the CMS fiducial phase-phase region are presented in the Table 2. The CMS fiducial phase-phase region [89] is given by The combined result for all leptonic channel given in Ref. [88] and the measured cross section by CMS [89] are also presented in the same table. The uncertainties in the theoretical estimates are due to scale variation. The result obtained by MATRIX and mg5_aMC matches quite well at both LO and NLO level. The NLO corrections has increased the LO cross section by up to 100 % and the NNLO cross section further increased by 10 % from the NLO value. It is thus necessary to include QCD corrections to leading order result. The higher order corrections to the cross section vary with kinematical variable like m 3l and p T (Z) as shown in Fig. 2 obtained by MATRIX [86,88,[90][91][92][93][94][95].
The lower panels display the respective bin-by-bin ratios to the NLO central predictions. The NLO to LO ratio does not appear to be constant over the range of m 3l and p T (Z). Thus a simple k-factor with LO events can not be used as proxy for NLO events. We use results from mg5_aMC including NLO QCD corrections for our analysis in the rest of the paper.   pp → e + e − μ + ν μ @NLO Figure 3: The differential distribution of m 3l and p T (Z) in the W + Z production in the e + e − µ + ν µ channel at the LHC at √ s = 13 TeV and L = 35.9 fb −1 at NLO for SM and five benchmark anomalous couplings.
The signal for the e + e − µ + and e + e − µ − are generated separately using mg5_aMC with pdf sets NNPDF30 at NLO in QCD for SM as well as for SM including aTGC. We use the FeynRules [96] to generate QCD NLO UFO model of the Lagrangian in Eq. (1.3) for mg5_aMC. These signal are then used as a proxy for the 3l + E T final state upto a factor of four for the four channels. For these, the p T cut for e ± and µ ± are kept at the same value, i.e., p T (l) > 10 GeV. We use a threshold for the trilepton invariant mass (m 3l ) of 100 GeV. The event selection cuts for this analysis are thus, (2.2) We use the values of the SM input parameters same as used in Ref. [88] (default in MATRIX). A fixed renormalization (µ R ) and factorization (µ F ) scale of µ R = µ F = µ 0 = 1 2 (m Z + m W ) is used and the uncertainties are estimated by varying the µ R and µ F in the range of 0.5µ 0 ≤ µ R , µ F ≤ 2µ 0 and shown in Table 2.
We explore the effect of aTGC in the distributions of m 3l and p T (Z) in both ZW + and ZW − production and show In Fig. 3. The distribution of m 3l in the left-panel and p T (Z) in the right-panel in the e + e − µ + ν µ channel are shown for SM (filled/green) and five anomalous benchmark couplings of ∆g Z 1 = −0.02 (solid/black), λ Z = +0.01 (dashed/blue), ∆κ Z = +0.2 (dotted/red), λ Z = +0.01 (dash-dotted/orange) and κ Z = +0.2 (dashed-dotdotted/magenta) with events normalised to an integrated luminosity of L = 35.9 fb −1 . The higher m 3l and higher p T (Z) seems to have higher sensitivity to the anomalous couplings which is due to higher momentum transfer at higher energies, for example see Ref. [30]. We study the sensitivity of total cross section to the anomalous couplings by varying lower cut on m 3l and p T (Z) for the above mentioned five benchmark scenarios. The sensitivity of an observable O(c i ) to coupling c i is defined as where δ O is the estimated error in O. For cross section and asymmetries, the errors are where L is the integrated luminosity and ε σ and ε A are the systematic uncertainty for the cross section and the asymmetries, respectively. The sensitivity of the cross sections, ignoring the systematic uncertainty, for the five benchmark cases (as used in Fig. 3) are shown in Fig. 4 for ZW + in the upper-row and for ZW − in the lower-row as a function of lower cut of m 3l (left-column) and p T (Z) (right-column) for luminosity of L = 35.9 fb −1 . It is clear that the sensitivity increases as the cut increases for both m 3l and p T (Z) for couplings ∆g Z 1 , λ Z and λ Z , while it decreases for couplings ∆κ Z and κ Z . This can also be seen in Fig. 3 where ∆κ Z and κ Z contribute more than other three couplings for m 3l < 0.8 TeV and p T (Z) < 0.6 TeV. Taking hints from Fig. 4, we identify four bins in m 3l -p T (Z) plane to maximize the sensitivity of all the couplings. These four bins are given by, (2.5) The sensitivities of the cross sections to the benchmark anomalous couplings are calculated in the said four bins for luminosity of L = 35.9 fb −1 and they are shown in Table 3 in both ZW + and ZW − productions. As expected, we see that Bin 22 has the higher sensitivity to couplings ∆g Z 1 , λ Z and λ Z , while Bin 11 has higher, but comparable sensitivity to couplings ∆κ Z and κ Z . The simultaneous cut on both the variable has increased the sensitivity by a significant amount as compared to the individual cuts. For example, the Fig. 4 shows that cross section in ZW + has a maximum sensitivity of 15 and 22 on ∆g Z 1 = −0.02 for individual m 3l and p T (Z) lower cut, respectively. While imposing a simultaneous lower cut on both the variable, the same sensitivity increases to 44.5 (in Bin 22 ).
At the LHC, the other contributions to the 3l + E T channel come from the production of ZZ, Zγ, Z + j, tt, Wt, WW + j, tt +V , tZ, VVV as has been studied by CMS [68,89] and ATLAS [85,97]. The total non-W Z contributions listed above is about 40 % of the W Z contributions [89]. We include this extra contributions to the cross sections while estimating limits on the anomalous couplings in Sect. 4.

Polarization observables of Z and W ± along with other angular asymmetries
Being a spin-1 particle the Z/W (V ) offers eight additional observables related to their eight degrees of polarizations apart from their production cross section. The angular distributions of the daughter particle reveal the polarizations of the mother particle V . The normalised decay angular distribution of the daughter fermion f (l Z /l W ) from the decay of V is given by [72] 1 σ Here θ f , φ f are the polar and the azimuthal orientation of the fermion f , in the rest frame of the particle (V ) with its would be momentum along the z-direction. For massless final state fermions, The quantity p x , p y , p z are the three vector polarizations and T xy , T xz , T yz , T xx − T yy , T zz are the five independent tensor polarizations of the particle V . These polarizations p i and T i j are calculable from asymmetries constructed from the decay angular information of lepton using Eq. (3.1). For example, the polarization parameters p z and T xz can be calculated from the asymmetries A z and A xz , respectively as Similarly one can construct asymmetries corresponding to each of the other polarizations p i and T i j , see Ref. [74] for details. The Z and the W ± boson produced in the ZW ± production are not forward backward symmetric owing to only a t-channel diagram and not having an u-channel diagram (see Fig. 1). These provide an extra observable, the forward-backward asymmetry defined as θ V is the production angle of the V w.r.t. the colliding quark-direction. One more angular variable sensitive to aTGC is the angular separation of the lepton l W from W ± and the Z in the transverse plane, i.e, One can construct an asymmetry as, The sensitivities of A ∆φ to the five benchmark aTGC are shown in Fig. 5 as a function of lower cut on p T (Z) in both ZW ± for luminosity of L = 35.9 fb −1 . A choice of p T (Z) low = 300 GeV appears to be an optimal choice for sensitivity for all the couplings. The m 3l cut, however, reduces the sensitivities to all the aTGC.
To construct the asymmetries, we need to set a reference frame and assign the leptons to the correct mother spin-1 particle. For the present process with missing neutrino we face a set of challenges in constructing the asymmetries. These are discussed below.
Selecting Z candidate leptons The Z boson momenta is required to be reconstructed to obtain all the asymmetries which require the right pairing of the Z boson leptons l + Z and l − Z . Although the opposite flavour channels e + e − µ ± /µ + µ − e ± are safe, the same flavour channels e + e − e ± /µ + µ − µ ± suffer ambiguity to select the right Z boson candidate leptons. The right paring of leptons for the Z boson in the same flavoured channel is possible with ≥ 96.5 % accuracy for m 3l > 100 GeV and ≥ 99.8 % accuracy for m 3l > 550 GeV in both SM and benchmark aTGC by requiring a smaller value of |m Z − m l + l − |. This small miss pairing is neglected to use the 2eµν µ channel as a proxy for a 3l + E T final state with good enough accuracy.
The reconstruction of neutrino momentum The other major issue is to obtain the asymmetries related to W ± boson, which require to reconstruct its momenta. As the neutrino from W ± goes missing, reconstruction of W ± boson momenta is possible with a two-fold ambiguity using the transverse missing energyp T / & & E T and the W mass (m W ) constrain. The solutions for the longitudinal momenta of the missing neutrino are given by Because the W is not produced on-shell all the time, among the two solutions of neutrino longitudinal momenta, one of them will be closer to the true value and another will be far from true. There are no suitable selector or discriminator to select the correct solution from the two solutions based on on-shell W . Even if we substitute the Monte-Carlo truth m W to solve for p z (ν) we don't have any discriminator to distinguish between the two solutions p z (ν) ± . The smaller value of |p z (ν)| corresponds to the correct solution only for ≈ 65% times on average in ZW + and little lower in ZW − production. One more discriminator which is ||β Z | − |β W ||, the smaller value of this can choose the correct solution a little over the boundary i.e., ≈ 55%. We have tried machine-learning approaches (artificial neural network) to select the correct solutions, but the accuracy was not better than 65%. In some cases, we have D < 0 with the on-shell W , for these cases either one can throw those events (which affects the distribution and statistics) or one can vary the m W from its central value to have D > 0. Here, we follow the later. So as best available option, we choose smaller value of |p z (ν)| to be the correct solution to reconstruct the W boson momenta. At this point, it becomes important to explore the effect of reconstruction on asymmetries and corresponding sensitivities.
To this end, we consider three scenarios: Abs. True First thing is to use the Monte-Carlo truth events and estimate the asymmetries in the lab frame. The observables in this scenario are directly related to the dynamics upto rotation of frame [70,98,99].
Reco. True Using the pole mass of W in Eq. (3.7) and choosing the solution closer to the Monte-Carlo true value is the best one can do in reconstruction. The goal of any reconstruction algorithm would be to become as close to this scenario as possible.
Small |p z (ν)| This choice is a best available realistic algorithm which we will be using for the analysis.
The values of reconstructed asymmetries and hence polarizations get shifted from Abs. True case. In case of Reco. True, the shifts are roughly constant, while in case of Small |p z (ν)|, the shifts are not constant over varying lower cut on m 3l and p T (Z) due to the 35 % wrong choice. It is, Abs. True

Reco. True
Small |p z (ν)|  thus, expected that the reconstructed sensitivities to aTGC remain same in Reco. True and change in Small |p z (ν)| case when compared to Abs. True case. In the Small |p z (ν)| reconstruction case, sensitivities for some asymmetries are less than that of the Abs. True case, while they are higher for some other asymmetries. This is illustrated in Fig. 6 comparing the sensitivity of some polarization asymmetries, e.g., A y to κ Z = +0.2 in cross (×) points, A z to ∆g Z 1 = −0.02 in square ( ) points and A zz to ∆κ Z = +0.2 in circular ( ) points for the three scenarios of Abs. True (solid/blue line), Reco. True (dotted/red) and Small |p z (ν)| (dash-dotted/blue) for varying lower cut on p T (Z) and m 3l in ZW + production with a luminosity of L = 100 fb −1 in the top-row. The sensitivities are roughly same for Abs. True and Reco. True reconstruction in all asymmetries for both p T (Z) and m 3l cuts. In the Small |p z (ν)| reconstruction case, sensitivity is lower for A zz ; higher for A y ; and depends on cut for A z when compared to the Abs. True case. When all the W asymmetries are combined, the total χ 2 is higher in the Small |p z (ν)| case compared to the Reco. True case for about 100 chosen benchmark point, see Fig. 6 (bottom-panel). Here a total χ 2 of all the asymmetries of W (A W i ) for a benchmark point ({c i }) is given by with a luminosity of L = 100 fb −1 . The said increment of χ 2 is observed in both W + Z ( /blue) and W − Z ( /red) production processes. So even if we are not able to reconstruct the W and hence its polarization observables correctly, realistic effects end up enhancing the overall sensitivity of the observables to the aTGC.
Reference z-axis for polarizations The other challenge to obtain the polarization of V is that one needs a reference axis (z-axis) to get the momentum direction of V which is not possible at the LHC as it is a symmetric collider. Thus, for the asymmetries related to Z boson, we consider the direction of total visible longitudinal momenta as an unambiguous choice for positive z-axis. For the case of W , the direction of reconstructed boost is used as a proxy for the positive z-axis. The later choice is inspired by the fact that in q q fusion the quark is supposed to have larger momentum than the anti-quark at the LHC, thus the above proxy could stand statistically for the direction of the quark direction.
List of observables The set of observables used in this analysis are, which make a total of N(O) = (4 + 8 + 1 + 1 + 8 + 1) × 2 = 46 observables including both processes. All the asymmetry from Z side and all the asymmetries from W side are termed as A Z i and A W i , respectively for latter uses. The total χ 2 for all observables would be the quadratic sum of sensitivities (Eq. (2.3)) given by We use these set of observables in chosen kinematical region to obtain limits on aTGC in the next section.

Measurement of the anomalous couplings
We study the sensitivity of all the (N(O) = 46) observables for varying lower cut on m 3l and p T (Z) separately as well as simultaneously (grid scan in step of 50 GeV in each direction) for the chosen benchmark anomalous couplings. The maximum sensitivities are observed for simultaneous lower cuts on m 3l and p T (Z) given in Table 4 for all the asymmetries in both ZW ± processes. Some of these cuts can be realised from Fig. 5 & 6. The SM values of the asymmetries of Z and W and their corresponding polarizations for the selection cuts (sel.cut in Eq. (2.2)) and for the optimized cuts (opt.cut in Table 4) are listed in Table 6 in appendix B for completeness. We use the cross section in the four bins and all asymmetries with the optimized cuts (opt.cut) to obtain limits on the anomalous couplings for both effective vertices and effective operators. We use the semianalytical expressions for the observables fitted with the simulated data from mg5_aMC. The details of the fitting procedures are described in appendix A. The uncertainty on the cross sections and asymmetries are taken as ε σ = 20 % and ε A = 2 %, respectively consistent with the analysis by CMS [89] and ATLAS [85]. We note that these uncertainties are not considered in the previous sections for qualitative analysis and optimization of cuts.  A zz (550, 0) (300, 400) " The sensitivities of all the observables to the aTGC are studied by varying one-parameter, two-parameter and all-parameter at a time. We look at the χ 2 = 4 contours in ∆κ Zκ Z plane for a luminosity of L = 100 fb −1 for various combinations of asymmetries and cross sections and show them in Fig. 7. We observe that the Z-asymmetries (A Z i ) are weaker than the W -asymmetries (A W i ); A W i provide very symmetric limits, while A Z i has a sense of directionality. The A ∆φ is better than  both A Z i and A W i in most of the directions in ∆κ Zκ Z plane. Combing the A Z i , A W i with A ∆φ we get a tighter contours; but the shape is dictated by A ∆φ . We see (Fig. 7 right-panel) that the cross sections have higher sensitivities compared to the asymmetries to the aTGC. The cross sections dominate constraining the couplings, while the contribution from the asymmetries remain subdominant at best. Although the asymmetries are not better than the cross sections in constraining the couplings, they certainly provide directional constrain in the parameter space. This would be helpful to extract the couplings should a non zero aTGC be observed. This possibility is discussed in the subsection 4.2.

Limits on the couplings
We extract simultaneous limits on all the anomalous couplings using all the observables using MCMC method. We perform this analysis in two ways: (i) vary effective vertex factors couplings (c L i ) and (ii) vary effective operators couplings (c O i ) and translate them in to effective vertex factors couplings (c L g i ) using Eq. (1.4). The 95 % BCI (Bayesian confidence interval) obtained on aTGC are listed in Table 5 for four choices of integrated luminosities: L = 35.9 fb −1 , L = 100 fb −1 , L = 300 fb −1 and L = 1000 fb −1 . The correlation among the parameters are studied (using GetDist [100]) and they are shown in Fig. 8 along with 1D projections for effective vertex factors. The limits on the couplings get tighter as the luminosity is increased as it should be. The shape of the contours are very circular in all two-parameter projections as the cross sections dominate in constraining the aTGC. The same conclusions are drawn when effective operators are varied as independent parameters. The limits on c L g i are tighter compared to the limits on c L i (see Table 5); the comparison between them are shown in the two-parameter marginalised plane in Fig. 9 in ∆g Z 1 - κ Z , λ Zλ Z and κ Zκ Z planes as representative for luminosity L = 100 fb −1 (outer contours) and L = 1000 fb −1 (inner contours). The limits and the contours are roughly same in λ Zλ Z plane. The contours are more symmetric around the SM for c L g i compared to c L i , e.g., see ∆g Z 1 -κ Z plane. The limits obtained here for luminosity 35.9 fb −1 are better than the experimentally observed limits at the LHC given in Table 1 except on c B and hence on ∆κ Z . This is due to the fact that the LHC analysis [66] uses WW production on top of W Z production whereas we only use W Z production process. But our limits on the couplings are better when compared with the W Z production process alone at the LHC [68]. In Fig. 10 The contour in the plane c WWW /Λ 2 -c W /Λ 2 in our estimate (We expect) (solid/green line) is tighter compared to both CMS ZW + WW (dashed/black line) and CMS ZW analyses (dotted/blue line). This is because we use binned cross section in the analysis. The limit on the couplings c B /Λ 2 (right-panel) on the other hand is tighter, yet comparable, with CMS ZW and weaker than the CMS ZW +WW analysis because the ZW process itself is less sensitive to c W .

The role of asymmetries in parameter extraction
The asymmetries are subdominant in constraining the couplings much like seen in Ref. [84] for pp → ZZ case. But the asymmetries help significantly giving directional constraint in the parameter 1 Figure 9: The marginalised 2D projections at 95 % BCI from MCMC in the ∆g Z 1 -∆κ Z , λ Zλ Z , and ∆κ Zκ Z planes are shown in solid/red when the effective vertex factors (c L i ) are treated independent, while shown in dashed/green when the operators are treated independent (c L g i ) for luminosities L = 1000 fb −1 (two inner contours) and L = 100 fb −1 (two outer contours) at √ s = 13 TeV using all the observables. using MCMC method. In Fig. 11 we show the posterior marginalised 1D projections for the couplings ∆g Z 1 , λ Z , ∆κ Z , and κ Z in top-panel and 2D projections at 95 % BCI on ∆g Z 1 -κ Z , λ Zλ Z ,  i ) are added, which can be seen from both 1D projections and 2D contours. The cross sections are blind to the orientation of aTGC-Bench couplings and sensitive only to the magnitude of deviation from the SM. The asymmetries however give direction to the measurement, e.g., in ∆κ Zκ Z plane σ i + A ∆φ + A Z i give two patches (excluding the SM) and we get one single (true) region when using all the asymmetries along with the cross sections. In the λ Zλ Z plane the asymmetries could not provide a direction, however, they shrink the 95 % contours from simply connected patch to an annular region (excluding the SM). For the other couplings the asymmetries favour the regions of the correct solution of aTGC-Bench couplings. For higher luminosities (not shown here) the contours become tighter and the 1D curves become sharper centred around the aTGC-Bench couplings when using σ i + A i , while σ i alone remain blind to the aTGC-Bench. Thus the asymmetries help in the measurement of anomalous couplings provided and excess of events are observed.

Conclusion
To conclude, we studied the WW Z anomalous couplings in the ZW ± production at the LHC and examined the role of polarization asymmetries together with ∆φ (l W , Z) asymmetry and forwardbackward asymmetry on the estimation of limits on the anomalous couplings. We reconstructed the missing neutrino momenta by choosing small |p z (ν)| from the two-fold solutions and estimate the W polarization asymmetries, while the Z polarization asymmetries are kept free from any reconstruction ambiguity. We generated NLO events at mg5_aMC for about 100 sets of anomalous couplings and use them for numerical fitting of semi-analytic expressions of all the observables as a function of couplings. We estimate simultaneous limits on the anomalous couplings using MCMC method for both effective vertex formalism and effective operator approach for luminosities 35.9 fb −1 , 100 fb −1 , 300 fb −1 and 1000 fb −1 . The limits obtained for L = 35.9 fb −1 are tighter than the limits available at the LHC (see Table 1 & 5) except on c W (and ∆κ Z ). The asymmetries are helpful in extracting the values of anomalous couplings if a deviation from the SM is observed at the LHC. We performed a toy analysis of parameter extraction with some benchmark aTGC couplings and observed that the inclusion of asymmetries to the cross sections improves the parameter extraction significantly.

Acknowledgements
in both processes. The values of all the observables are obtained for the set couplings in the optimized cuts (Table 4) and then those are used for numerical fitting to obtain the semi-analytical expression of all the observables as a function of the couplings. For the cross sections the following CP-even expression is used to fit the data: For asymmetries, the numerator and the denominator are fitted separately and then used as The numerator (∆σ A ) of CP-odd asymmetries are fitted with the CP-odd expression pp → e + e − µ + ν µ σ fitted (fb) σ data (fb)  The denominator (σ A j ) of all the asymmetries and the numerator (∆σ A ) of CP-even asymmetries are fitted with the CP-even expression given in Eq. (A.1) We use MCMC method to fit the coefficients of the cross sections with positivity demand, i.e., σ ({c i }) ≥ 0. We use 80 % data to fit the coefficients of the cross sections and then the fitted expressions are validated against the rest 20 % of the data and found to be matching within 2σ MC error. We generated 10 7 events to keep the MC error as small as possible even in the tightest optimized cuts. For example, the A zz in ZW + has the tightest cut on m 3l (see Table 4) and yet have very small (0.2 %) MC error (see Table 6). In Fig. 12 fitted values of observables are compared against the simulated data for the cross section in two diagonal bins (top-panel) and the polarization asymmetries A z and A xz (bottom-panel) in ZW + production in e + e − µ + ν µ channel as representative.
The fitted values seems to agree with the simulated data used within the MC error.

B Standard Model values of the asymmetries and polarizations
In Table 6, we show the SM estimates (with 1σ MC error) of the polarization asymmetries of Z and W and their corresponding polarizations along with the other asymmetries for our selection cuts (sel.cut) given in Eq. (2.2) and optimized cuts (opt.cut) given Table 4. A number of events of N 9.9 × 10 6 satisfy our selection cuts which give same error (δ A i = 1/ √ N) for all asymmetries and hence they are given in the top row. As the optimized cuts for W is same for all asymmetries, the error for them are also given top row. For the optimized cuts for Z, however, the number of events vary and hence the MC error are given to each asymmetries. The CP-odd polarizations p y , T xy , T yz and their corresponding asymmetries are consistent with zero in the SM within MC error.